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Multi-stage discrete optimization with data uncertainty and lot-sizing

Permanent Link: http://ufdc.ufl.edu/UFE0042108/00001

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Title: Multi-stage discrete optimization with data uncertainty and lot-sizing
Physical Description: 1 online resource (155 p.)
Language: english
Creator: Zhou, Zhili
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: lotsizing, mixedinteger, robustoptimization, stochasticprogramming
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Multi-stage robust optimization and stochastic programming are two approaches for multi-stage decision making under data uncertainty. In this dissertation, three problems on multi-stage robust optimization and stochastic programming are discussed. First, we consider a robust lot-sizing problem as an example to analyze multi-stage robust integer programming problems. In the robust lot-sizing problem setting, we consider the cases in which severe events may happen such that the normal process will be disrupted. Our objective is to provide a robust schedule such that the total cost is minimized under the worst case scenario. This problem can be formulated as a multi-stage robust integer programming problem. Several cases are studied and corresponding algorithms are developed. Our preliminary study verifies the effectiveness of our approaches. Second, we consider two-stage stochastic uncapacitated lot-sizing problems with deterministic demands and Wagner-Whitin costs. We develop extended formulations in the higher dimensional space that can provide integral solutions by showing that their constraint matrices are totally unimodular. For the case without backlogging, we provide the convex hull description of the problem in the original space by projecting the extended formulation to the original space. For the case with backlogging, we provide a tighter extended formulation by projecting the extended formulation to a lower dimensional space. Third, we study a general stochastic dynamic knapsack polytope. We apply the pairing, mixing, and lifting schemes to the stochastic dynamic knapsack set and obtain strong valid inequalities. We investigate the algorithmic and implementation issues for the effective and efficient generation of lifted valid inequalities of the stochastic dynamic knapsack polytope in a parallel computing environment. The speedup, communication overhead, load balance, and effectiveness in closing the integrality gap for stochastic dynamic knapsack polytope are studied. Computational experiments show the effectiveness of our proposed approaches.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Zhili Zhou.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Guan, Yongpei.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-02-28

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Permanent Link: http://ufdc.ufl.edu/UFE0042108/00001

Material Information

Title: Multi-stage discrete optimization with data uncertainty and lot-sizing
Physical Description: 1 online resource (155 p.)
Language: english
Creator: Zhou, Zhili
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: lotsizing, mixedinteger, robustoptimization, stochasticprogramming
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Multi-stage robust optimization and stochastic programming are two approaches for multi-stage decision making under data uncertainty. In this dissertation, three problems on multi-stage robust optimization and stochastic programming are discussed. First, we consider a robust lot-sizing problem as an example to analyze multi-stage robust integer programming problems. In the robust lot-sizing problem setting, we consider the cases in which severe events may happen such that the normal process will be disrupted. Our objective is to provide a robust schedule such that the total cost is minimized under the worst case scenario. This problem can be formulated as a multi-stage robust integer programming problem. Several cases are studied and corresponding algorithms are developed. Our preliminary study verifies the effectiveness of our approaches. Second, we consider two-stage stochastic uncapacitated lot-sizing problems with deterministic demands and Wagner-Whitin costs. We develop extended formulations in the higher dimensional space that can provide integral solutions by showing that their constraint matrices are totally unimodular. For the case without backlogging, we provide the convex hull description of the problem in the original space by projecting the extended formulation to the original space. For the case with backlogging, we provide a tighter extended formulation by projecting the extended formulation to a lower dimensional space. Third, we study a general stochastic dynamic knapsack polytope. We apply the pairing, mixing, and lifting schemes to the stochastic dynamic knapsack set and obtain strong valid inequalities. We investigate the algorithmic and implementation issues for the effective and efficient generation of lifted valid inequalities of the stochastic dynamic knapsack polytope in a parallel computing environment. The speedup, communication overhead, load balance, and effectiveness in closing the integrality gap for stochastic dynamic knapsack polytope are studied. Computational experiments show the effectiveness of our proposed approaches.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Zhili Zhou.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Guan, Yongpei.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-02-28

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042108:00001


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MULTI-STAGE DISCRETE OPTIMIZATION UNDER UNCERTAINTY
AND LOT-SIZING


















By

ZHILI ZHOU


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2010































2010 Zhili Zhou































To my parents Guifeng Wang and Yaozong Zhou

for their love, support, and understanding









ACKNOWLEDGMENTS

I have worked with a great number of people whose contribution in assorted ways to

the successful completion of this dissertation. It is a pleasure to convey my gratitude to

them all in my humble acknowledgment.

I would like to record my gratitude to my supervisor, Dr. Yongpei Guan, for his

supervision, advice, and guidance from the initial to the final stage of this research. He

provided me unflinching encouragement and support in various ways. Dr. Guan taught

me how to question thoughts and express ideas. His patience and support helped me

overcome many crisis situations and finish this dissertation.

I gratefully acknowledge Dr. Cole Smith for his thought-provoking discussion and

timely help during the writing of this dissertation. I wish to thank Dr. Joseph Hartman for

the insightful advice on research and valuable mentor time. I am thankful to Dr. William

Hager for broadening my view of mathematical programming. I am much indebted to Dr.

Theodore Trafalis for his valuable advice in scientific discussions.

Many thanks go to my friends. They have helped me stay sane through these

difficult years. Their support and care helped me to overcome setbacks and to stay

focused on my graduate study. I greatly value their friendship and I deeply appreciate

their belief in me. Especially, I am obliged to Natassia Brenkus, Belle Brenkus, and

Tachun Lin.

Finally, I would like to express my heart-felt gratitude to my parents, Guifeng Wang

and Yaozong Zhou. My family has been a constant source of love, concern, support, and

strength all these years.









TABLE OF CONTENTS
page

ACKNOWLEDGMENTS ................... ............... 4

LIST O FTABLES ..................... ................. 8

LIST OF FIGURES .................... ................. 9

ABSTRACT .................... ................... .. 10

CHAPTER

1 INTRODUCTION .................... ............... 12

1.1 Stochastic Program m ing ............................ 12
1.1.1 Two-Stage Stochastic Programming .... 12
1.1.2 Multi-Stage Stochastic Programming .... 14
1.2 Robust Optimization ... .............. .......... 16
1.3 Mixed-integer Linear Programming . 18
1.3.1 Polyhedral Theory ................ ........... 18
1.3.2 The Branch-and-Cut Algorithm . ... 20
1.4 The Lot-sizing Problem ... .............. ...... ... 22
1.5 Dissertation O utline . .. 23

2 LOT-SIZING WITH DISRUPTION ... ...... ...... ...... ..26

2.1 Introduction ..... ..... ...... ..... .... .. 26
2.2 Lot-sizing Problem with Disruption and Outsourcing .... 27
2.3 Lot-sizing Problem with Disruption and Backlogging .... 30
2.3.1 Non-setup Cost Case ................ ......... 32
2.3.2 Setup Cost Case ............................ 34
2.3.2.1 A Branch-and-Bound Algorithm .... 36
2.3.2.2 Branching and Searching Strategies .... 36
2.3.2.3 The Lower and Upper Bounds .... 37
2.3.2.4 The Optimality Test ..................... 38
2.4 Com putational Results ................... .......... 41
2.4.1 Instance Generation .......................... 41
2.4.2 Heuristic: Maximum Pick ..... .. .. .. 42
2.4.3 Lot-sizing with Disruption and Outsourcing .... 42
2.4.4 Lot-sizing with Disruption and Backlogging .... 43

3 MULTI-STAGE ROBUST LOT-SIZING WITH DISRUPTIONS ... 46

3.1 Introduction . .. 46
3.2 The General Formulation ......... .... ... .......... 47
3.3 The Robust Lot-sizing Problem with Outsourcing .... 49
3.4 The Robust Lot-sizing Problem with Backlogging: Single Disruption case 53









3.4.0.1 Reform ulation .. .. .. .. .. .. 55
3.4.0.2 Facet-defining Inequalities ... 59
3.5 The Robust Lot-sizing Problem with Backlogging: Multiple Disruption case 70
3.5.1 Without Setup Cost Case ... .... .. 70
3.5.2 Setup Cost Case ............................ 77
3.6 Computational Results ................... .......... 77
3.6.1 Instance Generation .......................... 78
3.6.2 Two-stage Robust Lot-sizing Problem ..... 78
3.6.2.1 Two-stage Robust Lot-sizing Problem with Outsourcing 78
3.6.2.2 Two-stage Robust Lot-sizing Problem with a Single Disruption
and Backlogging ....................... 79
3.6.3 Multi-stage Robust Lot-sizing Problem with Backlogging and without
Setup C ost . .. 80

4 STOCHASTIC LOT-SIZING PROBLEM WITH DETERMINISTIC DEMANDS
AND WAGNER-WHITIN COSTS ..... ......... 84

4.1 Introduction ..... ..... .. 84
4.2 An Extended Formulation ........................... 85
4.3 An Integral Polyhedron in the Original Space ..... 94
4.4 Extensions . .. 98

5 STOCHASTIC LOT-SIZING PROBLEM WITH DETERMINISTIC DEMANDS
AND BACKLOGGING ................... ............. 99

5.1 Introduction . .. 99
5.2 An Extended Formulation for Two Stage SULSB-WW ... 100

6 LIFTING SCHEME FOR THE STOCHASTIC DYNAMIC KNAPSACK POLYTOPE123

6.1 Introduction . . 123
6.2 The Path Inequality . 125
6.3 The Pairing and Mixing Schemes for the Stochastic Dynamic Knapsack
S e t . . 12 7
6.4 The Lifting Scheme for the Stochastic Dynamic Knapsack Set ...... .132
6.5 Cutting Plane Generation under Parallel Environment ... 138
6.5.1 Partitions of Stochastic Scenario Tree and Local Initialization .. 139
6.5.2 The Parallel Cuts Management Decision Control ... 139
6.5.2.1 Cut Generation ........................ 139
6.5.2.2 Cutting Pool Management. ... 140
6.5.3 The Parallel Cut-and-Branch Algorithm ... 140
6.5.4 The LP Based Cutting Plane Algorithm ... 141
6.6 Com putational Results ............................. 142
6.6.1 Instance Generation .......................... 142
6.6.2 The Cut-and-Branch Algorithm .... 142
6.6.3 The LP Based Heuristics ........................ 143









7 CO NC LUSIO N . . 146

REFERENC ES . . .... .. 150

BIOGRAPHICAL SKETCH ................... ............. 155









LIST OF TABLES


Table page

2-1 Param eter setting . .. 41

2-2 Lot-sizing with disruption: outsourcing .... 45

2-3 Lot-sizing with disruption: backlogging and non-setup cost .... 45

2-4 Lot-sizing with disruption: backlogging and branch-and-bound algorithm 45

3-1 Param eter setting . .. 78

3-2 Robust lot-sizing with outsourcing: multiple disruptions 82

3-3 Robust lot-sizing with backlogging: a single disruption ... 82

3-4 Robust lot-sizing with backlogging: branch-and-cut ... 82

3-5 Multi-stage robust lot-sizing problem with 2 disruptions ... 83

3-6 Multi-stage robust lot-sizing problem with 3 disruptions ... 83

4-1 The matrix of constraints (4-1) to (4-5) for the example in Figure 4-2 92

5-1 The matrix of constraints (5-4), (5-5), and (5-10) to (5-17) for the example in
Figure 5-1 .... . .. 122

6-1 Parameter setting .................. ............. 143

6-2 Heuristics for low setup cost case . 143









LIST OF FIGURES
Figure page

1-1 Multi-stage stochastic scenario tree ... 15

2-1 An example for the lot-sizing problem with disruption and backlogging 34

4-1 The scenario tree for two-stage SULS .. 86

4-2 An example of two-stage SULS ..... ......... 88

4-3 The subtree of node i ................... ............ 88

5-1 The scenario tree for a 3 period SULS with backlogging 102

5-2 The subtree of node i ................... ............. 102

6-1 Path inequalities for a scenario tree ..... .. 126

6-2 G aps for P2-16 . . 144

6-3 Cuts num ber for P2-16 ................................ 144

6-4 G aps for P3-13 . . 145

6-5 Cuts num ber for P3-13 ................................ 145









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

MULTI-STAGE DISCRETE OPTIMIZATION UNDER UNCERTAINTY
AND LOT-SIZING

By
Zhili Zhou

August 2010

Chair: Yongpei Guan
Major: Industrial and Systems Engineering

Multi-stage robust optimization and stochastic programming are two approaches for

multi-stage decision making under data uncertainty. In this dissertation, three problems

on multi-stage robust optimization and stochastic programming are discussed.

First, we consider a robust lot-sizing problem as an example to analyze multi-stage

robust integer programming problems. In the robust lot-sizing problem setting, we

consider the cases in which severe events may happen such that the normal process

will be disrupted. Our objective is to provide a robust schedule such that the total

cost is minimized under the worst case scenario. This problem can be formulated as

a multi-stage robust integer programming problem. Several cases are studied and

corresponding algorithms are developed. Our preliminary study verifies the effectiveness

of our approaches.

Second, we consider two-stage stochastic uncapacitated lot-sizing problems with

deterministic demands and Wagner-Whitin costs. We develop extended formulations

in the higher dimensional space that can provide integral solutions by showing that

their constraint matrices are totally unimodular. For the case without backlogging, we

provide the convex hull description of the problem in the original space by projecting

the extended formulation to the original space. For the case with backlogging, we

provide a tighter extended formulation by projecting the extended formulation to a lower

dimensional space.









Third, we study a general stochastic dynamic knapsack polytope. We apply the

pairing, mixing, and lifting schemes to the stochastic dynamic knapsack set and obtain

strong valid inequalities. We investigate the algorithmic and implementation issues

for the effective and efficient generation of lifted valid inequalities of the stochastic

dynamic knapsack polytope in a parallel computing environment. The speedup,

communication overhead, load balance, and effectiveness in closing the integrality

gap for stochastic dynamic knapsack polytope are studied. Computational experiments

show the effectiveness of our proposed approaches.









CHAPTER 1
INTRODUCTION

Lot-sizing problems have been studied extensively during the last four decades. The

deterministic single item lot-sizing problem is to decide a production plan for a product to

satisfy demands for a fixed time horizon (i.e., T periods) while minimizing total costs.

The deterministic lot-sizing problem cannot provide robust production plans in

the presence of uncertainty. As such, the deterministic decision making may yield

unsatisfactory decisions. Therefore, we investigate stochastic programming and robust

optimization and analyze their roles in decision making and their application in lot-sizing

problems.

The goal of this dissertation is to analyze uncertainties in multi-stage discrete

optimization programs. We investigate multi-stage robust optimization and adopt

multi-stage stochastic programming approaches for mixed-integer programming

problems through the study of lot-sizing problems.

Before we describe the outline of this dissertation, we introduce the following

technique backgrounds to be used in the following chapters.

1.1 Stochastic Programming

Stochastic programming is an approach to process problems where the data

incorporated into the objective or constraints are uncertain. Uncertainty is usually

characterized by a probability distribution on the parameters. We seek a solution that

is feasible and optimizes a given objective function. Two-stage stochastic programs are

widely applied and studied stochastic programming models. The decision maker takes

some actions on the first stage, after which a random event occurs affecting the outcome

of the first stage decision.

1.1.1 Two-Stage Stochastic Programming

The study of stochastic programming was originated by Danzig (1955) and Beale

(1955). They investigated the classical two-stage stochastic linear programming which









can be formulated as


min z = c'x + E,(Q(x, w))

s.t. Ax = b, (1-1)

x> 0,

where


Q(x, ) = min f(w)Ty(w)

s.t. T(w)x + W(w)y(u) = h(w), (1-2)

y(w) > 0.

In this type of mathematical model, the data uncertainty can be represented as

random variables w. The particular values of random variables are known after the

realization of random experiments. The decision variables are divided into two types:

the first stage decision variable and the second stage decision variable. The first stage

variable, denoted as x, has to be decided before the random experiment, and the

period when this variable is taken is called the first stage. The second stage variable,

denoted as y, can be determined after the realization of the random parameters. The

corresponding period is called the second stage. The sequence of events and decisions

is as

Decision on x Observation on w Decision on y(x, U).

Formulations (1-1) and (1-2) are considered as the first stage and the second stage

problems, respectively. The matrix A and the vector b are deterministic parameters. The

recourse function Q(x, w) defines the expected second stage value function. The matrix

T(w), vectors h(w) and f(w), and the transition matrix W(w) can be random.

Several solution methods have been introduced for (1-1) and (1-2) with finite

distribution. Among them, decomposition methods are important for solving large scale









two-stage stochastic programming problems. They can be categorized according to
fundamental strategies as either outer linearization or inner linearization. Danzig and

Mandasky (1961) applied Danzig-Wolfe decomposition (Dantzig and Wolfe 1960)

to solve the dual of two-stage stochastic linear programming problem using inner
linearization. Van Slyke and Wets (1969) developed the two-stage L-shaped method

for solving the primal problem using outer linearization that is a form of Benders'

decomposition (Benders 1962). Kall (1976) and Strazicky (1962) presented another
dual method based on basis factorization. A method based on discrete distributions and

splines was proposed by Wets 1974, 1983. For the continuous sample space, Birge and

Louveaux (1997), Shapiro (2000), and Shapiro and de Mello (2001) proposed the Monte

Carlo sampling method with finitely many scenarios. In general, Ahmed and Shapiro

(2002) represented that the larger the number of scenarios, the more accurate is the
provided model. This, in turn, makes the resulting formulation very large.

1.1.2 Multi-Stage Stochastic Programming

The multi-stage stochastic programming problem is an extension of the two-stage

stochastic program in the multi-stage setting. Based on available information at each

time period, multi-stage programs model problems where decisions should be made

sequentially in certain time periods.

Birge and Louveaux (1997) and Louveaux and Schultz (2003) introduced the

special stochastic scenario tree structure for the deterministic equivalent multi-stage

stochastic linear programming. The structure can be interpreted as a scenario tree
with T stages (or levels) where node i in stage t of the tree constitutes the state of the

world that can be distinguished by information available up to time stage t. We use

T = T(0) = (V, S) = (V(0), S(0)) to represent the whole scenario tree. The set of leaf

nodes in V is denoted as L. Node i, i e V, i 7 0 (the root node indexed as i = 0), has a

unique parent a(i). Node i, i e V \ is the root node of a subtree T(i) = (V(i), S(i)),
which contains all descendants of node i, and has AN immediate children set C(i), i.e.,









C(i) = {j : a(j) = i}. (i) denotes the leaf nodes of the subtree T(i). The set of nodes

on the path from the root node to node i is denoted by P(i). If i e L, P(i) corresponds

to a scenario, and represents a joint realization of the problem parameters over this

scenario. We define 2P(i,j) = {k : k e P(j) n V(i)}, thus P(i) = P(0, i). t(i) denotes the

time stage(or level) of node i, i.e., t(i) = I|(i)|. The probability associated with the state

represented by node i is pi. Let V(i) be the subtree with node i as root node. Figure 1-1

shows a multi-stage stochastic scenario tree model.










(g ----------- ----- ----
I I












I -


stage 1 stage t(i) stage k stage T

Figure 1-1. Multi-stage stochastic scenario tree


The multi-stage stochastic linear program with recourse and tree structure can be

formulated as


minZ = pic;x;
iEV

s.t. A1x1 = bl,

T.xj + Aixi = bi, i V
jxP 0(a(i))

xi > 0, i E V.









Methods have been applied to the multi-stage stochastic programming with

block-separable recourse. Louveaux (1980) first performed the generalization of the

primal approach for the multi-stage problem. Birge (1985) and Pereira and Pinto (1985)

generalized the L-shaped method to the multi-stage problem as a nested decomposition

method. Birge (1988) also extended the basic-factorization techniques to the multi-stage

problem, but these techniques brought hieratical computational difficulties. Grinold

(1976) formulated the multi-stage stochastic program as a finite Markov chain and as

equivalent primal and dual optimization problems. Beale et al. (1986) performed the first

order approach for the multi-stage program.

1.2 Robust Optimization

The robust optimization is also an approach to address parameter uncertainty in the

optimization model. Unlike stochastic programming, it does not assume the uncertain

parameters are random variables with known distributions, rather, its parameters are

considered in uncertain sets or uncertain intervals. Robust optimization looks for the

feasible solution for all possible values of unknown parameters, normally the optimal

solution of the best worst case under data uncertainty. Currently, most studies are about

single stage and two-stage robust optimization.

The research on robust optimization has recently received renewed attention.

Kouvelis and Yu (1997) proposed a framework for robust discrete optimization, which

seeks to find a solution that minimizes the worst case performance under a set of

scenarios for the data. Their framework is a scenario based model.

They let the uncertainty data set be Q, which is a scenario uncertainty set. Let x be

the decision variable and D(_) be the instance of the feasible region that corresponds

to the uncertain parameter e Q. The function Q(x, ) evaluates the feasible solution

x. The robust optimization problem is designed to find the optimal solution x and the

corresponding such that

z = min max Q(x, ). (1-3)
x e









Ben-Tal and Nemirovski 1998, 2000 proposed the following framework on robust

optimization:


min max fo(x, Do)
x DoE9o

s.t. f(x, Di) > 0, i = 0, T, Di e Qi,


where Qi, i = 0, T, are the given uncertainty sets. Compared with Kouvelis and Yu

(1997), their uncertainty set is more general. They showed that under the assumption

that the set Qi is ellipsoids with the form


Q = {D y, D = DO + ADJyj, ||y| < Y}. (1-4)
jEN

The robust counterparts of convex optimization problems are either exact or approximated

tractable problems. But under the ellipsoid uncertainty, the robust counterpart of an LP

becomes a nonlinear program. Under the same framework, Ben-Tal et al. (2004)

studied the two-stage robust linear programming under the name adjustable robust

linear programming. They presented that two-stage robust linear programming is

computationally intractable and proposed a tractable alternative approach, referred to as

affinely-adjustable robust linear programming.

Soyster (1973) proposed a linear robust optimization model to construct a solution

that is feasible for all data that belong in a convex set. Bertsimas et al. (2004) and

Bertsimas and Sim (2004) considered LPs whose coefficients of the objective function

and constraints are assumed to be in uncertain intervals. Their approach retained

the advantage of the linear framework of Soyster (1973) and protested constraints

against violation. Bertsimas and Simchi-Levi (1996), Bertsimas and Sim (2003), and

Bertsimas and Thiele (2006) applied their robust approach to address data uncertainty

in discrete optimization. The perspective areas are vehicle routing, network, and

inventory problems. They showed their robust approach is tractable in the above areas.









Bienstock and Ozbay (2008) discussed the relaxation of the two-stage robust

lot-sizing problem with basestock, and described a bender decomposition based

algorithm for robust linear optimization. AtamtOrk and Zhang (2007) described a

two-stage robust optimization approach for solving network flow and design problems

with uncertain demand. Two-stage robust lot-sizing with uncertain demand is an

application for their approach. Most of these works are concentrated on the two-stage

robust optimization.

1.3 Mixed-integer Linear Programming

The general form of the mixed-integer linear programming (MIP) is

min cx + hy

s.t. Ax+ Gy < b, (1-5)

x > 0, y > 0 and integer,

where A e RmX", G e Rmp, c e IR" and h e RP, decision variable x e R", integer

variable y e RP. Let S = {(x, y) : Ax + Gy > b, x > 0, y > 0 integer} represent

the feasible region of MIP. When x is also integral, we have a special case of MIP,

an integer programming (IP). Nemhauser and Wolsey (1999) and Wolsey (1998) are

comprehensive references for MIP.

Some MIP problems can be solved in polynomial time, such as the uncapacitated

lot-sizing problem, the shortest path problem, the max flow problem, and the assignment

problem. To date, no one has found a polynomial algorithm for these MIP problems,

such as the 0-1 knapsack problem, the set covering problem, the traveling salesman

problem, and the uncapacitated facility location problem. In the following section, we

present useful results on polyhedral theory and algorithms for MIP.

1.3.1 Polyhedral Theory

First,we introduce the definitions of polyhedron and convex hull.









Definition 1. The convex hull of S, denoted by conv(S), is the set of all points that are

convex combination of points in S.

Definition 2. A polyhedron P c R" is the set of points that satisfy a finite number of

linear inequalities; that is, P = {x e RI : Ax < b}, where (A, b) is an m x (n + 1) matrix.

Definition 3. A bounded polyhedron is called a polytope.

Definition 4. F is a face of P, if (7r, 7o) is a valid inequality for P, and F = {x E P : -x =

ro}. A face Fof P is a facet of P if dim(F) = dim(P) 1.
In essence, every polyhedron P can be described either by listing its facet-defining

inequalities, or by its extreme points and extreme rays in the following theorems. Note

that every polytope can be described by only its extreme points.

Theorem 1.1. (Nemhauser and Wolsey 1999) A full-dimensional polyhedron P has a

unique minimal representation by a finite set of linear inequalities.

Theorem 1.2. (Minkowski's Theorem) If P 0, then

P = {x e Rn : x = kXk + j, Y Ak = 1, Ak > 0, for k e K, p > 0, forj c J.},
kEK jEJ kEK

where (Xk)keK is the set of extreme points of P and (rj)jej is the set of extreme rays of

P.

Theorem 1.3. (Nemhauser and Wolsey 1999) The projection of a polyhedron is a

polyhedron.

Now, we show that the integer program can be reduced to be a linear program by

the following theorems.

Theorem 1.4. (Nemhauser and Wolsey 1999) If S {x : Ax < b,x e R"}, where (A, b)

is an integer m x (n + 1) matrix, then conv(S) is a rational polyhedron.

Theorem 1.5. (Nemhauser and Wolsey 1999) IP is either infeasible or unbounded or

has an optimal solution.

The following definition and proposition show that there is a type of polyhedron

having only integral extreme points.









Definition 5. A nonempty polyhedron P c R" is integral if each of its nonempty faces

contains integral points.

Proposition 1.1. The following statements are equivalent:

1. P is integral.

2. LP has an integral optimal solution for all c e R" for which it has an optimal
solution.

3. LP has an integral optimal solution for all c e Zn for which it has an optimal
solution.

We describe a class of matrices for which the integrality of LP holds with integral b.

Definition 6. An m x n integral matrix A is totally unimodular (TU) if the determinant of

each square submatrix of A is equal to 0, 1, or -1.

Proposition 1.2. If A is TU, then P(b) = {x e R n : Ax < b} is integral for all b e Zn for

which it is not empty.

1.3.2 The Branch-and-Cut Algorithm

First, we introduce the branch-and-bound algorithm. Let S represent the feasible

region of MIP problem (1-5). The LP relaxation of MIP is min cx + hy, Ax + Gy < b, x >

0, y > 0}. Let P represent the feasible region of this LP problem.

Definition 7. The collection {S' :i = 1, k} is called a division of S ifuS' = S.

Let L be a collection of MIP' with MIP' = min {cx + hy, (x, y) e S'}, where

S' c S. The general branch-and-bound algorithm for mixed-integer programming can be

described as follows:

The branch-and-bound algorithm (Nemhauser and Wolsey 1999)

Step 1 [Initialization]: L = {MIP}, zo = +oo, and ZMIP = +00.

Step 2 [Termination test]: If L = 0, then the solution (xo, y) with objective value

ZMIP = cxO + hyo is optimal.









Step 3 [Problem selection and relaxation]: Select and delete problem MIP' from L.

Solve its relaxation RP'. Let zR be the optimal value of the relaxation and let (xx, yR) be

an optimal solution if one exists.

Step 4 [Pruning]:

a. If zi > ZMIP, go to Step 2.

b. If (xR, y) i S', go to Step 5.

c. If (x4, yi) e Si and cxR + hyR < ZM/Ip, let ZMIP = cxi + hyk. Delete from L all

problems with z' < ZM/p. If cxR + hyk = ZMIp, go to Step 2; Otherwise, go to Step 5.

Step 5 [Division]: Let {Suj = 1, 1 k} be a division of S'. Add problems

{MIPU,j = 1, .. k} to L, where z' = z- forj = 1,.. k. Go to Step 2.

The branch-and-cut algorithm is a generalization of the branch-and-bound

algorithm. In Step 4, instead of going to Step 5, then branch-and-cut algorithm finds

a valid inequality a'x + a2y < aO of MIP', such that alxR + a2yi > a. After adding this

inequality to LP relaxation RP', it is re-solved and and then goes back to Step 4.
In general, the number of nodes in the branch-and-bound enumeration tree is

exponential in terms of the problem size, even for the integer program with binary

variables. Almost all general MIP codes use a branch-and-bound algorithm with
LP relaxation. The LP relaxation in the branch-and-bound algorithm provides the

lower bound of the corresponding MIP. By adding cuts, the branch-and-cut algorithm

improves the lower bound and reduces the number of branching nodes. In practice, the

branch-and-cut algorithm normally performs better than the branch-and-bound algorithm

in terms of computational time.

In this dissertation, we generate the customized branch-and-bound algorithm

for lot-sizing with disruption in Chapter 2. We explore the polyhedral structures for

stochastic lot-sizing problems in Chapters 4 and 5.









1.4 The Lot-sizing Problem


The mixed-integer programming formulation for the single item, uncapacitated

lot-sizing problem (ULS) is (cf. Nemhauser and Wolsey (1999)):
T
min (a;xi + fiy, + hisi)
i-1
s.t. si_l + xi = di + si =0, ,T

xi < My, i= 0, ,T

xi si > yi {0,1} i =0, ,T,

where xi represents the production quantity in period i, si represents the inventory

amount at the end of period i, and yi indicates the setup decision at period i. Parameters

ai, fj, hi, and di represent the production cost, setup cost, inventory cost, and demand in

time period i, respectively. Without loss of generality, we can assume so = 0 and tighten

Mi = JTi di.

The study of lot-sizing problems can be traced back to Wagner and Whitin (1958)

in which a polynomial time dynamic programming algorithm was developed to get

an optimal solution of the problem. Later on, there is significant research progress

in solving this class of problems. To solve ULS to optimality, dynamic programming,

reformulation, and polyhedron study have been proposed. As a polynomially solvable

approach, dynamic programming applied in ULS was first introduced by Wagner and

Whitin (1958). They proposed an O(n2) dynamic algorithm to solve ULS. Federgrun and

Tsur (1991), Wagelmans et al. (1992), and Aggarwal and Park (1993) independently

obtained O(n log n) implementations of the Wagner and Whitin (1958) algorithm.

Wagelmans et al. (1992) introduced the definition of the Wagner-Whitin costs, i.e.,

ai + h' > ai 1, 1 < i < T 1, where a; and h' are the unit production and inventory costs

for time period i, and implemented an O(T) time dynamic programming algorithm.









The classic extended reformulation of the lot-sizing problem are the facility location,

the shortest path, and the multicommodity network flow formulations. Krarup and Bilde

(1977) first introduced the facility location reformulation of the lot-sizing problems.

Martin (1987) developed a reformulation technique to transform ULS to a shortest

path formulation. Radin and Choe (1979) provided the multicommodity network flow

reformulation.

The compact description of the convex hull of all feasible solutions for ULS has

been derived through different approaches. Barany et al. (1984b) introduced the

well-known (f, 5) inequalities


x, + diye > doe, (1-6)
iES iEz\s

together with xi > 0 and 0 < yi < 1 to describe the convex hull of ULS, where

Se I = {0, 1, T}, S C I, and dij = k, dk. In addition, for the uncapacitated

lot-sizing problem with start-up costs, van Hoesel et al. (1994) presented an extended

formulation and an O(T2) time separation algorithm. For the Wagner-Whitin costs case,

Pochet and Wolsey (1994) generated an extended formulation for ULS with O(T2)

constraints. Dynamic programming has been applied the uncapacitated lot-sizing

problem and capacitated lot-sizing problems. For the capacitated lot-sizing problem,

Baker et al. (1978) studied the dynamic lot-sizing prlbem with time-varying production

capacity constraints. Fisher et al. (2001) proposed a method to mitigate end-effects

in the dynamic lot-sizing by evaluating the end-of-horizon inventory level based on

the classic EOQ model. Hartman et al. (2010) derived a new set of valid inequalities

for the capacitated lot-sizing problem from the end-of-stage solutions of a dynamic

programming algorithm.

1.5 Dissertation Outline

In Chapter 2 and Chapter 3, we study the lot-sizing problem with disruptions in the

robust lot-sizing problem setting. In Chapter 2, we investigate cases in which a severe









event may happen such that the normal process will be disrupted. Our objective is to

provide a robust schedule such that the total cost is minimized under the worst case

scenario. In order to solve this problem more efficiently, we generate the customized

branch-and-bound algorithm with optimality testing.

In Chapter 3, we extend the previous study to multiple disruption cases. We

formulate the problem to be a multi-stage robust optimization problem. Our objective is

to provide a multi-stage robust schedule such that the total cost is minimized under the

worst case scenario. We explore the polyhedral structure of the lot-sizing problem with

single disruption by generating facet-defining inequalities. The general reformulation

scheme has been provided to transform the multi-stage robust optimization problem into

an equivalent single stage program.

In Chapter 4 and Chapter 5, we consider a two-stage stochastic uncapacitated

lot-sizing problem with deterministic demands and Wager-Whitin costs. In Chapter 4,

we consider cases without backlogging. The optimal form of the inventory level has

been explored. Based on the optimal form, we provide an extended formulation. Further,

we provide the integral polyhedron description in the original space by projecting the

extended formulation to the original space. In Chapter 5, we consider the case with

backlogging. We investigate the relationship among the inventory, the setup, and the

backlogging in the optimal solution and provide the optimal forms of inventory and

backlogging levels. An extended formulation has been proposed.

In Chapter 6, we study the stochastic dynamic knapsack set which is naturally

raised from the stochastic lot-sizing problem. First, we extend the results in the

deterministic dynamic knapsack set to the stochastic setting, and generate valid path

inequalities for the stochastic dynamic knapsack set. Second, the mixing and pairing

schemes are applied for the stochastic dynamic knapsack set to generate valid tree

inequalities. Third, the lifting schemes are adopted to the stochastic dynamic knapsack

set. More valid inequalities are generated. Fourth, the parallel computing is applied to









the stochastic dynamic knapsack set to test the efficiency of generated inequalities for

the stochastic capacitated lot-sizing problem.

Finally, in Chapter 7, we conclude this dissertation and propose future research

directions.









CHAPTER 2
LOT-SIZING WITH DISRUPTION

2.1 Introduction

In practice, manufacturers expend lots of resources and efforts generating the

long-term production plan. However, disruptions often occur and interrupt the regular

planned production processes and cause demands in some time periods cannot

be satisfied. Extra productions are required in order to cover the unfilled demands

and bring extra reparation costs. The efficiency of new production plans is normally

measured by the extra cost generated due to disruption. Previously, researchers have

realized that a lot-sizing schedule has to be updated after the uncertain events. In

previous research, Carlson et al. (1979) and Kazan et al. (2000), etc. have studied the

nervousness of lot-sizing problems in which future demands are gradually acquired

and the initial schedules have to be updated, which leads to extra production cost.

Yang et al. (2005) studied how to recover the lot-sizing problems after the realization of

disruptions. All these works are focused on the recovery planning after the occurrence

of uncertain events and no recourse information is involved in the model to obtain the

original schedule. As compared to the robust optimization approach, the above studies

focus on studying the efficiency for the second-stage problem.

In this paper, we study the lot-sizing problem with potential disruptions as recourse

via robust optimization to handle the uncertainty. We consider cases in which there is

potentially one disruption, and in which the exact time of the disruption is uncertain.

Once the disruption occurs, the corresponding recovery production follows to cover

the unfilled demand and the extra production cost is generated. Our objective is to

maintain the production planning with the consideration of a potential disruption during

the production process and achieve the minimum objective value that considers the

worst case scenario for the disruption.









The contribution of this paper lies in the fact that this paper proposes a robust

production planning to address disruptions. As compared to the case in which the

recovery production is performed after the occurrence of a disruption, our approach

provides a more robust planning. To the best of our knowledge, our approach is the first

to use the robust optimization approach to solve the lot-sizing problem with disruption. In

the remaining part of this paper, we study different lot-sizing problems with disruptions.

In Section 2.2, we study the lot-sizing problem with disruption and outsourcing. For

this case, the lot-sizing problem with disruption can be reformulated as a two-stage

min max problem. A corresponding primal-dual approach can be constructed to solve the

problem. In Section 3.4, we study the lot-sizing problem with disruption and backlogging.

In Section 2.3.1, we study the non-setup cost case and develop a pre-processing

algorithm to pre-calculate the parameters for the second stage problem. Then, we

can formulate the problem as a single-stage problem. In Section 2.3.2, we develop

a customized branch-and-bound algorithm to solve the case in which setup cost is

considered. Finally, in Section 2.4, we provide the computational results that show the

tractability and efficiency of our approaches.

2.2 Lot-sizing Problem with Disruption and Outsourcing

In this section, we consider outsourcing as the reparation approach for the lot-sizing

problem with disruption. We assume that if disruption happens in time period i, then

the production amount xi originally scheduled in this time period will be purchased from

other suppliers. That is, outsourcing happens in the same time period as the disruption

happens, and the outsourcing amount is equal to xi with unit outsourcing cost o;.

Since after the disruption, we consider that outsourcing is an option besides

production to satisfy demands, we also allow outsourcing as an option in he original

schedule. For the case in which disruption is considered, all first-stage variables, the

production levels, the original outsourcing levels, the inventory levels, and the setup

decision, are not influenced by the disruption; but the second-stage decision determines









the time period when the disruption happens. If we assume the disruption time period to

be t, then the corresponding formulation can be described as follows:


min pixi + h(is + oi wi + fy + max ((ot pt)xt ft)
x,s,y,w t


(PO) s. t. x; + wi + s_-1 = di + si, i = 1, T (2-1)

xi < Myi, i= 1, T (2-2)

x, w, s eR+, ye{0,1}7,


where wi represents the outsourcing amount in time period i and 1 < t < T.

Constraint (3-2) indicates the inventory flow balance and constraint (3-3) indicates

that production happens in the setup time period; no upper bound limit exists for the

production amount. By introducing an artificial binary decision variable ao to indicate

whether disruption happens in time period i, PO can also be described as follows:



x,s,y,w \a
in (i Px + hisi + oiw + fjy, + maxa, ((o p,)x
m Y 1 /i 1


(POi) s.t. xi + wi + si-_ = di + si, i = 1, T

xi < Myi, i = 1, T
T

i 1

x, w, s RT, Ea {,O1}


We can also relax ai for each i = 1, T to be fractional and the following

conclusion holds.










Proposition 2.1. For the lot-sizing problem with a single disruption and outsourcing, the

formulation can be simplified as the following two-stage min-max problem:

ST T

i 1i 1
min piXi + hisi + oiwi + fiyi + max ai ((oi pi)xi, fi)
x,s,y,w \a
i-1 i-l



(P02) s.t. xi + wi + si-1 = di + s, i = 1, T

xi < Myi, i = T

ai< 1, i = 1,..- ,T
T
S-ao < 1,
-i

x, s, w, a R, ye {0, 1}T


Proof. We only need to prove that there exists an optimal solution for the above problem

such that a* is integral. We prove the claim by the contradiction method. For a given

optimal solution (x*, s*, y*, w*, a*), we can first observe that there exists an optimal

solution in which ao = 0 if (oi pi)xi* f < 0. Now we prove that for the given optimal

solution (x*, s*, y*, w*), there exists a corresponding integral optimal solution a* for the

following subproblem

T
max 5a ((oi- pj)x f)
a
i= 1

(PO-SUB) s.t. a;i < 1, i = 1,... T
T

i 1

a c R T


If there exists a time period i in which ao* (0, 1) and T1, oa < 1, then we can

increase ao to be 1, which leads to a larger objective value for the above subproblem.

Contradiction.









If there exists a time period i in which a* e (0, 1) and T 1 a = 1, then there must

exist at least one more time period j in which oa e (0, 1). If (oi-pi)xi*-fi = (oj-p)x<-fj,

then we can increase ao and decrease oj such that either ao or oj becomes integral.

Thus, we obtain a solution with a fewer number of fractional solutions. Following this

same step, we can either obtain an integral solution with the same objective value or find

a case in a certain step in which (oi pi)xi* fi (oj pj)xj /. Under this scenario,

without loss of generality, we can assume (oi pi)xi* f > (oj pj)xj fj. Then we can

obtain a larger objective value for the subproblem if we increase ao and decrease aj by

a small value c > 0. Contradiction.

Therefore, the original conclusion holds. O

By introducing new extra variable 0, we can formulate P02 as follows:

Theorem 2.1. The lot-sizing problem with a single disruption and outsourcing can be

transformed as the following single stage mixed-integer program:
T
mmin (pixi + hisi + oiwi + fyi) + 0
x,s,y,w,q
i-1
(DO) s.t. (oi pi)xi fi < 8, i = 1, T

x; + w, + si_- = di + si, i = 1, T

xi < Myi, i = 1,... T

x, s, we RT, 0 R+, y {0,1}T


From above analysis, we can solve the lot-sizing problem with a single disruption

and outsourcing by a mixed-integer programming problem. We can also observe that the

following (, 5)-type inequalities (see, e.g., Barany et al. 1984a) are valid for DO.

2.3 Lot-sizing Problem with Disruption and Backlogging

In this section, we discuss the lot-sizing problem with disruption that utilizes

backlogging as the reparation approach. We assume that the unit outsourcing cost

is much larger than the unit backlogging cost and it is only utilized when no backlogging









can be obtained, e.g., in the pseudo time period T + 1. In the backlogging setting, the

inventory and backlogging amounts after the time period in which disruption happens

are changed. Therefore, we cannot adopt the same approach as the lot-sizing problem

with disruption and outsourcing to solve this problem.

We introduce parameters cji and auxiliary decision variables xi, as described

in Pochet and Wolsey (1988). That is, parameter cji represents the total cost of

producing an item in time period i (except setup cost) to satisfy the demand in time

period f. For instance, for each i < T, we have cj; = pi + hk if > i,
ci- = Pi + ~,bk if < i, and ci = Pi If i = T + 1, then cj = Oi + Ek Tbk.

Problem parameters hi and bi represent the unit inventory and backlogging costs in

time period i, respectively. Decision variable x1i represents the amount produced in

time period i to satisfy the demand in time period f. When a disruption happens in

time period i, decision variable x;i represents the quantity of the unsatisfied demand in

period f. Therefore, the makeup production amount for period f can be decided by the

disruption period i and the corresponding original production amount used for this period

x;i. In order to provide the smallest cost to cover the unfilled demand in time period f

due to the disruption in time period i, we choose a period q() that is setup and could

provide the smallest cost to cover the unfilled demand in time period .

Note here once the first stage decision variables are given, for a given disruption

period t, we can separately calculate the minimum makeup cost for each xtu. Since

there is no capacity, and the setup decision is provided in the first stage, the period

which provides the minimum unit cost for the demand in period t only depends on the

time period f. Then yT-, cq(xt, is the smallest makeup cost.

Based on this idea and the application of the facitliy location reformulation, the

lot-sizing problem with disruption and backlogging can be formulated as a two-stage









robust optimization problem as follows:
T+1 T+1 T T
min fcyx + c + max mi (cq()jXtj CtiXtj)
(xy) i1 i=1 j1 j1 t

T+1
(PR) s.t. xi = d, j = 1, T (2-3)
i=1
xij < Myi, j =1, -- ,T; i =1, -- ,T +1 (2-4)

x e R+ )x, y {0, 1}T+1


The objective function is a two-stage optimization formulation. In the objective

function, the second stage decision variable t is the index of time period. In the second

stage, the minimum total makeup cost for a disruption can be determined by the sum of

the minimum makeup cost for each xj, j = 1, -. T. Therefore, time periods q(j) are

the decision variables which provide the minimum makeup cost for xt, j = 1, -. T. The

parameter Cqj)j is determined by q(j), t < q(j) < T + 1. Constraint (2-3) indicates that

the demand in time period j should be satisfied. Constraint (2-4) indicates that for each

time period, production happens only in the time period in which production is set up.

In this formulation, in the second stage, we only need to decide the time period in

which the disruption happens and the time periods during which reparation productions

happen. The reparation production quantity for each period j is equal to the first stage

decision variable xtj, where t is the disruption period.

2.3.1 Non-setup Cost Case

In this subsection, we consider a case without setup costs. Suppose disruption

happens in time period t, with the second term of the objective function in PR, i.e.,

Z =1 mint first and second stage variables. Since there is no setup cost, we propose the following

Pre-processing algorithm to determine Cqo)j. We have mt = mini>t cy for each pair (tj).

By using backward induction and setting m(t-1)j = min{c(t_l)j, mtj}, we observe that mt










for all t,j = 1, T can be pre-calculated in O(T2) time with O(T2) storage space.

With the Pre-processing algorithm, Cq(j)j = mj when disruption happens in time period t.

Proposition 2.2. For the non-setup cost lot-sizing problem with disruption and back-

logging, suppose the disruption happens in time period t, the time period to cover the

unfilled demand in time period t is argmin{cie, i > t}, and the corresponding minimum

unit makeup cost is mte = min{cji, t < i < T + 1}.

With the pre-processing algorithm, we can pre-solve the second stage problem and

get the following formulation:

TT T T
min cx + mcax m+x c
mn Z Icuxijmax (fmtjxtj- CtjXtj
i=1 j=1 j=1 j=1
T
s.t. xij = dj, j = -. T
i= 1

x e RTxT


With the above formulation, we can reformulate PR to be a single stage optimization

problem which is presented in the following result.

Theorem 2.2. The non-setup cost lot-sizing problem with disruption and backlogging

has an equivalent linear programming formulation as follows:

T T
min Zcux&O
min j- cx + 0
i= j=1
T T
(PB1) s.t. Z mtjxtj- cjxj < 0, t = 1, T
j=1 j=1
T
x = dj, t= l, T
i-i 1

X I TxT


where mt is calculated from a pre-processing algorithm.










( ,1-- D2 ... X32 3

X41


Figure 2-1. An example for the lot-sizing problem with disruption and backlogging


2.3.2 Setup Cost Case

When the setup cost is considered as shown in PR, we cannot determine the

second stage parameters Cqo)j until the first stage decision variables yqo) are decided,

because the makeup production can only happen in the periods that are set up, i.e.,

y; = 1. In this section, we introduce a reformulation of PR such that the problem can be

formulated as a single stage problem and can be solved as a single stage mixed-integer

programming problem.

We first introduce the following additional decision variables:

qtj: The smallest total cost incurred in the second stage in order to satisfy the unfilled
demand in time period j, due to the disruption in time period t.

zj: An auxiliary binary decision variable to indicate if we produce in time period k to

satisfy the unfilled demand in time period j, where t < k < T + 1, when disruption

happens in time period t. If yes, then zj = 1, otherwise, zj = 0.

0: An auxiliary decision variable to represent the maximum increment cost after a

disruption.

The main idea here is to formulate and study a single-stage mixed-integer

programming problem, instead of solving a two-stage robust optimization problem.

Before we describe the mathematical reformulation, Figure 2-1 shows an example to

demonstrate the reformulation process. The figure shows a five period example. We

assume periods 1, 3, and 4 are set up for production. Production in period 1 satisfies

demands for itself and period 2, production in period 3 satisfies the demand for itself,

and production in period 4 satisfies demands for periods 4 and 5.









As shown in Figure 2-1, suppose a disruption happens in time period 1. Then qll

and q12 represent the smallest makeup costs for the unsatisfied demand in periods 1

and 2, respectively. The makeup production may be from periods 3 or 4. Since variable 0

represents the maximum increment cost after a disruption, we have qll + q12 c11l11 -

C12X12 < 0. Finally, time period T+1 serves as the pseudo-period to provide outsourcing.
Thus, we have YTr+ = 1 with zero setup cost.

Theorem 2.3. For the setup cost case, the lot-sizing problem with disruption and

backlogging can be formulated as the following single-stage mixed-integer programming

problem:
T+1 T+1 T
mm Y fY + y cu x + 0
x,y,O,q,z
i=1 i=1 j=1
T T
s.t. qt- ctxt < 0, t= 1, T (2-5)
j=1 j=1
ckjx, + M(z 1) < q, t,j = 1, T; k = t, T + 1 (2-6)
T
(PB2) zkj = 1, t,j = 1, T (2-7)
k=t+l
z~j < yk, t,j = 1, -. T; k = t, T + 1 (2-8)

(2-3), (2-4),

x E R(T+)XT, y {0, 1}T1

z {0, 1}TxTx(T1), qc RTxT, 0 R+.


In the above formulation, artificial variable 0 represents the total cost incurred in

the second stage. Constraint (2-5) represents that 0 should be the maximum increment

cost due to the disruption at each time period t = 1, T. Constraint (2-6) indicates

the backlogging cost to fulfill the unfilled demand in time period j due to disruption in

time period t. Constraint (2-7) represents that, due to the unlimited production capacity,

there is only one later period needed to produce more to fulfill the unfilled demand for

each time period j due to disruption in time period t. Constraint (2-8) represents that









the period provides reparation should be originally setup. Besides these, constraints

(2-3) and (2-4) described in PR are also necessary.

2.3.2.1 A Branch-and-Bound Algorithm

Due to a large amount of binary decision variables zj in the formulation, it is very

difficult to solve the problem into optimality. In order to efficiently solve PB2, we generate

a customized branch-and-bound algorithm. This algorithm guarantees to find an exact

optimal solution and terminate in a finite number of steps.

In PB2, there are two types of binary decision variables z and y. In our branch-and-bound

algorithm, except leaf nodes, we apply the branch-and-bound procedure only for setup

decision variable y and relax z to be fractional. At each leaf node in the resulting

branch-and-bound tree generated by our procedure, decision variables y are integral.

Then, we can solve a subproblem in which z is required to be integer and obtain a

corresponding feasible solution.

In the remaining part of this section, we describe in detail our branching and

searching strategies, lower and upper bounds in the branch-and-bound framework,

and Benders decomposition framework for the optimality test. Before we describe the

details, we first let L and V be the sets of leaf and total nodes in the enumeration tree,

respectively. For each particular node n c V, we let y(n) and d(n) be the solution of

variable y and the depth of this node; let a(n), C(n) and D(n) be the parent, children set

and descendant set of node n; let P(n) be the nodes on the path from the root node to

node n.

2.3.2.2 Branching and Searching Strategies

In our branch-and-bound framework, we relax z to be fractional in our search

process. Therefore, in our branch-and-bound process, we do not branch on z variables.

If a y decision variable, for instance, yi, becomes one at some node n e V, it may

become fractional when we solve the problem corresponding to a node which is a

child of node n, and then become zero when we solve the problem corresponding to









a node which is a descendant of node n. Therefore, in order to get the global optimal

solution, we need to consider all the combinations of integral y solutions. We will still

branch the y variable when y variable is integral. In our process, we first branch the

decision variable y, at the root node with depth 1. Then, in general, corresponding to

each node with depth d, we will branch the decision variable Yd. Finally, we apply the

depth first search, which allows us to find good feasible solutions at early stages during

the branch-and-bound process.

2.3.2.3 The Lower and Upper Bounds

We first can observe the following conclusion:

Proposition 2.3. In the branch-and-bound tree, corresponding to each node n, the

optimal objective value of the linear programming relaxation for this node provides the

lower bound for its descendant.

Proof. Our branch-and-bound policy is not the same as the traditional one. For instance,

we need to branch a yi decision variable even if we get an integral yi solution. However,

corresponding to each node in the tree, we solve the linear programming relaxation and

it still provides the lower bound for all its descendant. O

To obtain an upper bound, at each node in the branch-and-bound tree, if we have

all y decision variables integral, then we can solve a subproblem corresponding to this

integral y solution. As shown in PB2, once y decision variables are fixed, we can write

down a reformulation for the problem to decide the production quantity, the disruption

period, and the reparation production periods in the second stage. Corresponding to

each node n e V such that all y decision variables are integral in the linear programming

relaxation solution, let Z(n) = {t = 1, T : Y = 1}. Each period t e Z(n)

can be a candidate in which the disruption happens. Once a disruption happens in

time period t, only the periods that are set up after time period t can provide extra

production quantity to satisfy unfilled demands which are caused by the disruption

in time period t. Corresponding to each pair (t, ), where t represents the disruption









time period and f represents the period in which the demand is unfilled, we can define

mte = min {ci, i E Z(n), t < i < T + 1}.


If k = argmin{ci, I c 1(n), t < i < T + 1}, then zke = 1; otherwise z = 0. (2-9)


Then, we can write down the following subproblem to obtain a feasible solution for PB2,

which is an upper bound for the problem.
T+-1 fT ( T T
min ciexie + fi;i + max mtext T ctext
teZ(n)
i= l e e1 /
T+1
(PB-SUB) s.t. x~ = de, 1,--. ,T,
i-i
xie < Myi, = 1, T; i = 1, T + 1,

x E R(T+1)xT


The above formulation can be easily solved since it is a linear program. It is obvious

that for each leaf node, we have y decision variables integral and an upper bound

can be obtained by solving PB-SUB. Following a similar process as the traditional

branch-and-bound procedure, we initialize the upper bound UB = +oo and the lower

bound LB = -oo. At each node n c V, if not all y decision variables are integral by

solving the linear program corresponding to this node, then we only obtain a lower

bound LB(n) for the descendants of node n. If LB(n) > UB, then we prune node n and

its descendants. If all y decision variables are integral by solving the linear program

corresponding to this node, then we can obtain both an upper bound UB(n) and a lower

bound LB(n) for the problem. If UB(n) < UB, then we let UB = UB(n). If LB(n) > UB,

then we prune node n and its descendants. Otherwise, we continue branching and

performing the depth first search. Finally, we will terminate at an optimal solution.

2.3.2.4 The Optimality Test

We apply the Benders' decomposition approach at each leaf node to test if we have

obtained the optimal solution for the problem. In the Benders' decomposition framework,










we let (y, z) be the decision variables for the master problem and others be the decision

variables for each slave problem. The master problem can be described as follows:


min T
y,z
T T T T T T
s.t. I > ftyt + Mp (1 z) + M yt+ tdt, (2-10)
t=1 t=1 j=1 k=t+l t=1 t=1

(Master) zkj < yk, t,j = 1,- T; k = t,-** T+1,
T
Z 4=1, t,j= l, T,
k=t+l

ye {0, 1}T+ z {0, 1}TxTx(T 1)


where inequality (2-10) is generated based on the following subproblem and j, v, and w

are the dual values corresponding to constraints (2-11), (2-12), and (2-13).

T+1 T
Z(y, z) = m in cix
i l j i
i=1 j=1
T T
s.t. qt ctxt 0 < 0, t = 1, ** T,
j=l j=1
(Slave) CkXt qt < M(1 z), t,j = 1, T; k = t, T + 1(2-11)

xi < Myi, =1, T; i= 1, ,T+1, (2-12)
T+1
xi = e, = 1, T, (2-13)
i=1
xE e(T 1)xT qe ,TxT, E I .


In our optimality test procedure, for a given leaf node in which we have y decision

variables integral, we can obtain the corresponding integral z values. For instance,

suppose (y, z) is the solution corresponding to a leaf node. By solving PB-SUB, we

will obtain the optimal objective value Z(y, z) for the slave problem. Meanwhile, we

will obtain the dual values corresponding to constraints (2-11), (2-12), and (2-13).

Thus, we will obtain a corresponding r, in the master problem. Note here yT+ =









1 is predefined. Therefore, (Slave) is always feasible and bounded for any given y

solution and we do not need to put extreme ray constraints in the Master problem for the

optimality test. The optimality test for Benders' decomposition of PB2 is
T
Z(y, z) + ft < (2-14)
t= 1
If inequality (2-14) is satisfied, then (y, z, x) is an optimal solution of PB2. Since we

only need to test if the current integral solution y can lead to an optimal solution for

the original problem, we do not need to solve the master problem. Thus, we only need

to solve an extra linear programming problem to obtain an updated Tr each time. The

details are shown as follows:

Algorithm: The optimality test

Step 0: Initialize rT = +oo and U = 0, where U stores the dual solution of Slaves.

Step 1: At the kth leaf node in the branch-and-bound tree, denoted as node nk, we

perform the following operations:

Step 1.1: Based on y(nk), we obtain z(nk) as shown in (2-9).

Step 1.2: Solve the Slave problem to obtain the corresponding objective value Z,

and add the corresponding dual values to set U.
Step 1.3: Let T = max(,,, V)Eu{tT 1 ftyt(nk) + T Z T IL +i (l

zj (nk)) + T= Mvtyt(nk) +T=1 ~tdt}.
Step 2: Optimality test: If Z + Et=L fty(nk) < T, then (X(k (n k), y(n), q(nk), (k), (nk))

is the optimal solution for the original problem, stop. Otherwise, wait until the next

leaf node and go to Step 1.

Since the optimal y solution must exist at some leaf nodes, the optimality test will be

satisfied at a leaf node in the branch-and-bound framework. Thus, the algorithm will

terminate in finite steps.









Note here, the Bender's decomposition formulation can be used for both solving

the problem from scratch and serving as the optimality test. In the later computational

experiment, we apply the Bender's decomposition for both purposes.

2.4 Computational Results

In this section, we present the computational results to demonstrate the computational

tractability of solution approaches we studied for different cases of the robust lot-sizing

problems with disruption. All computational experiments were carried out on a Linux

workstation with a Pentium Dual 2.8G processor and 6G RAM. We used CPLEX 10.1

Callable Library to implement our algorithms, and run the reformulated models.

2.4.1 Instance Generation

We generate instances based on different ratios of setup cost f, to unit production

cost pi, and different time horizons T. For the instances, we set the time horizon

T = 10, 20, 30, 40 and 50, and the ratios of setup cost to unit production cost f/p = 10,

20, 30, and 40, respectively. There are 20 combinations in total.

For the cases with setup costs, corresponding to each of the combinations of T and

f/p, we generate random instances in which the unit production cost and the setup cost
are uniformly distributed in the intervals as shown in Table 2-1.

For the cases without setup costs, corresponding to each of the combinations of T

and f/p, we generate random instances in which the unit production cost is uniformly

distributed in the same interval as for the cases with setup costs shown in Table 3-1.

Table 2-1. Parameter setting
unit production cost pi setup cost fi
ratio 10 [50, 100] [500, 1000]
ratio 20 [50, 100] [1000, 2000]
ratio 30 [40, 60] [1000, 2000]
ratio 40 [40, 60] [500, 1000]









We also set demand di, unit inventory cost hi, and unit backlogging cost bi uniformly

distributed in the intervals [500, 1000], [5, 10], and [10, 20], respectively. Finally, we let

unit outsourcing cost be fixed at 200.

2.4.2 Heuristic: Maximum Pick

We first use maximum pick, a simple heuristic, to serve as a base for comparison

with other models. Based on the optimal solution of the single item lot-sizing problem

without disruption, the maximum pick heuristic picks a period, in which the outsourcing

brings the largest increment of the total cost, to do outsourcing. The detailed maximum

pick heuristic is listed as follows:

Heuristics: Maximum Pick (MPH)

Step 1: Solve the single item lot-sizing problem. Let x* be the optimal solution and z*

be the corresponding objective value.

Step 2: Sort (ot pt)x*, 1 < t < T from the largest to the smallest, and record them

as mpl, mp2, ..., mpT.

Step 3: Use z* + mp1 as the increment cost of the total cost due to the disruption.

2.4.3 Lot-sizing with Disruption and Outsourcing

We test the cases using outsourcing to recover unfilled demand. For this case,

as described in section 4.1, we test 20 combinations in which the time horizon

T = 10, 20, 30, 40, and 50, and the ratio f/p = 10, 20, 30, and 40, respectively. The

computational results are shown in Tables 2-2. For each of the 20 combinations, we

report the average values of 5 random instances. We report 1) the optimal objective

value of the lot-sizing problem without disruptions, denoted as "SLS", 2) the objective

value obtained by maximum pick heuristic, denoted as "MPH", 3) the objective value

for the robust optimization formulation obtained by using CPLEX to solve the dual

formulation (DO) for the outsourcing case and 4) the gap between (DO) and (MPH),

denoted as "GAP(M::R)=(ObjMPH-ObjDO)/(ObjDO)," and 5) the gap between (SLS) and

(DO), denoted as "GAP(R::S)= (ObjDo-ObjSLS)/(ObjSLs)." Compared with the maximum









pick heuristics, the average gap is 16.7%. That means the total cost can be reduced by

16.7% by applying robust optimization approach (DO). Compared with the uncapacitated

lot sizing problem without disruptions, the average gap between (SLS) and (DO) is

18.9%. That means total cost increases 18.9% by considering the disruption. We can

also observe that as T increases, the average gap between SLS and DO decreases.

2.4.4 Lot-sizing with Disruption and Backlogging

Finally, we test the cases using backlogging to recover unfilled demand. We perform

computational experiments for cases both with and without setup cost.

For the case without setup cost, we perform the computational experiments based

on the pre-processing algorithm and reformulation (PB1) described in Section 2.3.1. We

test 10 combinations such that the ratios between the setup cost and the production cost

equal to 10 and 30, and the planning horizons are 10, 20, 30, 40 and 50 respectively.

Corresponding to each combination, we report the values of "SLS," "MPH," "NoSetup,"

and "GAP(M::R)" as described in the outsourcing case.

The pre-processing algorithm needs O(T2) time. Formulation (PBi) is a linear

program with O(T2) variables and O(T) constraints. Therefore, the problems for this

case can be solved in short time and we do not show the computational time. Instead,

we only report the cost savings by applying robust optimization approach as compared

to the maximum pick heuristic, the case without considering the disruption during the

planning process. The comparison is shown on Table 2-3, and in average, the cost

saving is 27.8%.

For the case with setup cost, we compare different solution approaches that include

solving (PB2) described in Section 2.3.2 directly by default CPLEX, the proposed

branch-and-bound algorithm, and the Benders decomposition approach. We set the

time limit to be 2 hours and test the combinations that ratio=10, 20, 30 and 40, and the

time period T = 10, 20 and 30, respectively. The computational results are shown in

Table 2-4. We compare three different approaches that include the default CPLEX MIP









solver approach, our proposed branch-and-bound (BB) algorithm, and the Benders'

decomposition algorithm. If the optimal solution cannot be obtained within the time limit,

we report the final optimality gap. Otherwise, if the optimality gap is zero, we report the

computational time in the parenthesis, in terms of seconds. From Table 2-4, we can

observe that our proposed BB method performs much better than default CPLEX. For

T = 10 and 20 cases, no instances can be solved into optimality within the given time

limit by the default CPLEX. The final optimality gaps provided by the default CPLEX are

in the interval [1.13%, 5.39%]. Our proposed branch-and-bound algorithm can solve

all instances into optimality for T = 10 and T = 20 cases: for T = 10 cases, most

instances can be solved into optimality within 5 minutes by our BB approach; for T = 20

cases, most instances can be solved into optimality within 1 hour. For T = 30 cases,

neither the default CPLEX nor our proposed BB algorithm can solve the problems into

optimality within 2 hours. However, our proposed BB algorithm obtains smaller optimality

gaps as compared to the default CPLEX. For the Benders' decomposition approach to

solve (PB2) directly instead of serving as optimality test, we found that even for 10-period

instances, no instances can be solved into optimality within 24 hours. Therefore, the

computational results indicate that our proposed solution approach is the best among all

three approaches.














Table 2-2. Lot-sizing with disruption: outsourcing
T = 10 T = 20 T = 30 T = 40 T = 50
ratio=10 SLS 5.00 x 101 1.05 x 10" 1.54 x 106 2.06 x 106 2.50 x 10
MPH 8.44 x 10b 1.51x 10 2.04 x10 2.61x 10 2.97 x 10
DO 6.67 x 10 11.25 x 10 1.78 x 10 2.35 x 10 2.79 x 10
GAP(M::R) 26.5% 20.1% 14.6% 11.2% 6.6%
GAP(R::S) 33.3% 19.3% 15.9% 14.2% 11.5%
ratio=20 SLS 4.99 x 105 1.06 x 106 1.52 x 106 2.01 x 106 2.47 x 106
MPH 7.96 x 10b 1.48 x 10 2.05 x 10 2.60 x 10 3.09 x 10
DO 6.39 x 100 1.27 x 10 1.77 x 10 2.31 x 10 2.79 x 10
GAP(M::R) 24.5% 16.3% 15.8% 12.3% 10.7%
GAP(R::S) 28.0% 20.3% 16.3% 14.8% 12.6%
ratio=30 SLS 3.88 x 105 7.41 x 105 1.12 x 10 1.46 x 10b 1.88 x 10
MPH 6.85 x 10b 1.06 x 10b 1.42 x 10 1.81 x 10b 2.27x 10
DO 5.26 x 10b 8.98 x 105 1.29 x 10O 1.64 x 10b 2.08 x 10O
GAP(M::R) 30.2% 17.8% 9.9% 10.5% 9.1%
GAP(R::S) 35.7% 21.2% 14.8% 12.5% 10.4%
ratio=40 SLS 3.68 x 105 7.81 x 105 1.15 x 10 1.52 x 10b 1.92 x 10
MPH 7.28 x 10b 1.12 x 106 1.53 x 10 1.91 x 10b 2.29 x 10
DO 5.17 x 10b 9.45 x 105 1.31 x 10 1.69 x 10 2.10 x 10
GAP(M::R) 40.9% 18.8% 16.6% 12.9% 8.8%
GAP(R::S) 40.5% 20.9% 14.4% 11.7% 9.8%


Table 2-3. Lot-sizing with disruption: backlogging and non-setup cost
T = 10 T 20 T 30 T = 40 T = 50
ratio=10 SLS 5.03 x 105 9.86 x 105 1.56 x 106 2.08 x 106 2.44 x 106
MPH 8.23 x 105 1.35 x 10 2.01 x 10 2.54 x 10T 2.95 x 10
NoSetup 5.39 x 105 1.02 x 10 1.61 x 10 2.12 x 10 2.47 x 10
_ap(M::N) 34.6%o 24.2% 19.9% 16.5% 16.1%
ratio=30 SLS 3.49 x 105 7.04 x 105 1.08 x 10" 1.44 x 10b 1.85 x 10"
MPH 6.26 x 105 1.03 x 100 1.35 x 10 1.77 x 10 2.19 x 10
NoSetup 3.70 x 105 7.35 x 10 1.11 x 10 1 .47 x 10 1.88 x 10
Gap(M::N) 69.1% 40.3% 21.7% 20.5% 16.1%


Table 2-4. Lot-sizing with disruption: backlogging and branch-and-bound algorithm
T=10 T=20 T=30
ratio=10 Gap(Op) CPLEX 1.42% 4.39% 4.42%
BB 0(371) 0(3974) 2.75%
ratio=20 Gap(Op) CPLEX 2.76% 3.79% 4.71%
BB 0(322) 0(3400) 2.66%
ratio=30 Gap(Op) CPLEX 1.13% 1.59% 2.19%
BB 0(300) 0(2959) 1.68%
ratio=40 Gap(Op) CPLEX 1.21% 2.33% 2.43%
BB 0(365) 0(2918) 1.79%









CHAPTER 3
MULTI-STAGE ROBUST LOT-SIZING WITH DISRUPTIONS

3.1 Introduction

Uncertainty is a natural property of disruptions in production planning problems.

The exact times when disruptions happen are unpredictable. In this paper, we study the

multi-stage robust lot-sizing problem as an example to analyze solution approaches for

multi-stage robust integer programming problems. We consider the lot-sizing problem

in which the disruption occurrence is uncertain. Once a disruption occurs, subsequent

recovery productions follow to cover the unfilled demands with their corresponding

recovery production costs. Our objective is to maintain the production planning with

disruptions. Meanwhile, the production plan still can provide the minimal production

cost under the worst case scenario (the largest production cost growths). In general, a

multi-stage robust mixed-integer programming setting can be described as follows:


min [fi(xi,y) + max [ min f2(X1,1yi, w1,x2,y2) ...
(xl,yi)O Mi l R1 (X2,Y2)EM2(xi,y1,u1)
+ max [ min fT(x i, i, ..., X(T 1), y(T 1), (T 1), T,YT)]]], (3-1)
(7(T-1)( (T-1) (XT,YT)CM T(X1 l,...,(T-1))

where (xt, yt) represents the decision in stage t, and Qt represents the uncertainty set

in stage t. Decision variables in each stage t will satisfy constraints that describe the

feasible region in stage t, denoted as 4Mt(x,2,y, x, y2,... (t-)). For the multi-stage

robust lot-sizing problems, the above general multi-stage robust integer programming

formulation can be applied to formulate lot-sizing with uncertain disruptions. The

objective is to minimize the total cost under the worst case scenario.

The contributions of this chapter are threefold. First, this paper proposes a robust

production planning to address disruptions. As compared to the case in which the

recovery production is performed after the occurrence of disruptions, we provide more

robust initial production planning. Second, this paper proposes a robust production

planning to address multiple disruptions. As compared to the case in which there

is only one disruption, the multiple disruptions case is much more challenging to









solve. Third, this paper studies solution approaches to solve both two-stage and

multi-stage robust integer programming problems. In particular, this paper studies the

polyhedral structures of the corresponding models for the lot-sizing problem with a

single disruption and shows the computational effectiveness of our approaches. To

the best of our knowledge, all these three aspects have never been studied before.

In the remaining part of this paper, we study different multi-stage robust lot-sizing

problems which are tractable. In Section 3.2, we present the motivation, introduce

our robust lot-sizing problem, and develop a general formulation for the multi-stage

robust lot-sizing problem. In Section 3.3, we study the robust lot-sizing problem

with outsourcing. For this case, the multi-stage robust lot-sizing problem can be

reformulated as a two-stage min max problem. A corresponding primal-dual approach

can be constructed to solve the problem. In Section 3.4, the robust lot-sizing problem

with backlogging and a single disruption is studied. We formulate this problem as a

two-stage robust mixed-integer program. We propose a reformulation rule to transfer

this robust model to a mixed-integer program. For this problem, we study the polyhedral

structure and generate the corresponding facet-defining inequalities based on the

reformulation. In Section 3.5, we consider cases with and without setup cost of the

robust lot-sizing problem with backlogging and multiple disruptions. We develop a

deterministic equivalent formulation for the multi-stage robust optimization problem,

based on the enumeration of the periods in which disruptions happen. Finally, in Section

3.6, we provide the computational results that show the tractability and efficiency of our

approaches.

3.2 The General Formulation

For a single-item production planning process, when disruptions happen in given

time periods, the scheduled production is terminated in these time periods. We adopt

outsourcing and backlogging as approaches to accommodate the supply shortage.

That is, unfilled demands due to disruption in certain time period can be backlogged by









increasing the production in later time periods. This approach is referred to as a robust

lot-sizing problem with backlogging case. For this case, if there is only one disruption,

then the scheduled productions after the disruption time period can be processed,

and extra production amount can be introduced at a time period which is set up after

the disruption time period to accommodate the supply shortage. If there are multiple

disruptions, then there are multiple iterations in terms of disruptions and recoveries. For

instance, after the first disruption, the production process after the disruption time period

will be rescheduled to counteract the disruption. Similarly, when the second disruption

happens, reparation production will happen again. Consequently, disruptions and the

corresponding reparation productions will happen consecutively.

To formulate the robust lot-sizing problem with uncertain disruptions, we introduce a

hidden disruption parameter e {0, 1}T. If there is a disruption that happens in period

i e I = {1,..., T}, then _i = 1; otherwise, _i = 0. Accordingly, we denote the set of

disruption periods So = {i, = 1, i e I}. We assume that the total number of disruption

periods is 3. We also let x1, s1 and y1 represent the first stage production, stocking, and

set-up decision variables, and xk, sk and yk represent the corresponding variables after

the k 1th disruption. Then, the the objective function of the multi-stage robust lot-sizing

problem can be described as follows:



mm J>(p,x,1 + fy,l + h,s,) + max min t' J(tl,x, y) + E (ix,2 + hi,s2)
(xysl) t { tiSo (x2,s2y2) { T
i1 i ti+1
r T
+ max min -t202(t2, x,y, X y1), + ) ( px,3 + hs )
t2CSo,t2>tl (x3,s3,y3) _


t3+ So'tm >t2 (X4,S4m y 4-(
t s=1 t3+

te3 So,t3>t3-1 (x ,S ,y )
i= t3+41









where we generalize the cost parameter for the recovery production quantity to be the

unit recovery production cost if backlogging is applied in time period i for each i e I.

Besides this, in our objective setting, if thejth disruption happens in time period tj,

then the corresponding cost bj(tj, x, y, x1, yl, ... xi-1, y-1) is saved. For instance, if

there is a disruption that happens in time period t, then the scheduled production cost

cannot be processed, and the corresponding costs (including production, inventory,

and backlogging costs) are saved. The exact formulation of saving cost function ;i(...)

and the constraints set depend on the recover approaches. In the remaining part of this

paper, we study the robust lot-sizing problem with backlogging cases.
3.3 The Robust Lot-sizing Problem with Outsourcing

In this section, we study the multi-stage robust lot-sizing problem with outsourcing.

For this case, we consider outsourcing as the reparation approach. We assume

if disruption happens in a time period t, then the production amount xt originally

scheduled in this time period will be purchased from other suppliers. That is, outsourcing

happens in the same time period as the disruption happens, and outsourcing amount is

equal to xt with unit outsourcing cost ot.

Under this scenario, since we consider that outsourcing is also an option besides

production to satisfy demands after the disruption, for the deterministic lot-sizing

problem without disruptions, we will also have outsourcing option considered. For the

case in which disruption is considered, the production amount, the original outsourcing

amount, the inventory amount, and the setup decision are all first-stage decision

variables and not influenced by the disruption. The only second-stage decision variable

is the time period t when the disruption happens. If multiple disruptions happen, in

each disruption time period, the outsourcing amount is equal to the reduced production

quantity due to the disruption. Accordingly, we still only need to decide the periods

when the disruptions happen. Therefore, the multi-stage robust lot-sizing problem with

outsourcing can be transferred to be a two-stage robust optimization problem with









multiple disruptions happening in the second stage. In this case, if thejth disruption

happens in time period tj, the production costs in this time period can be saved. Thus,

the lost function is formulated as bj(tj, x, y) = ptjxt,. If we assume the set of disruption

periods to be So, a subset of I, and the number of disruptions to be 3, i.e., ISol = 3. The

corresponding formulation can be described as follows


min piXi + hisi + o;wi + fyi + max Y ((Ot t)xt)
x,s,y,w So
-i1 teSoC /_


s.t. xi + wi + si-_ = di + si, i e (3-2)

xi < Myi, I eI (3-3)

x, w, se RT, ye {0,1}T,


where wi represents the outsourcing amount in time period i. Constraint (3-2) indicates

the inventory flow balance and constraint (3-3) indicates that production happens in

the time period in which the production is set up and the production amount does not

have an upper bound limit since M is a very large number. Let artificial binary decision

variable a; indicate if the disruption happens in time period i, the above formulation can

also be described as follows:



minx pixi + hisi + owi + fIyi + max ai ((oi Pi)xi)
x's'y'w \a
i=1 i=l


(RFLS) s.t. x; + w; + si_ = d; + Si, i I

xi < My;, iEI
T
aj < 0,
i 1

x, w, s E y, ce {0, 1}T










We can also relax ai for each i e i to be fractional and the following conclusion

holds.

Proposition 3.1. For a multi-stage robust lot-sizing problem with outsourcing, the

formulation can be simplified as the following two-stage min-max problem:


min pixi + hisi + owi + fIyi + max ai ((oi P)xi)
x,s,y,w a
i=1 i=l



(RPLS) s.t. xi + wi + s_ 1 = di + Si, i I

xi < Myi, i EI

ai <1, iel
T

i 1

x, s, w, a {R T, ye {0,1}T.


Proof. We only need to prove that there exists an optimal solution for the above problem

such that a* is integral. We prove the claim by the contradiction method. For a given

optimal solution (x*, s*, y*, w*, a*), we can first observe that there exists an optimal

solution in which ao = 0 if (o, pi)x < 0. Now we prove that for the given optimal

solution (x*, s*, y*, w*), there exists a corresponding integral optimal solution a* for the

following subproblem:

T
max a, ((o, pi)x,)
a
i-1

(SUB) s.t. ai < 1, ie
T

i=1

a E <









If there exists a time period i in which ao e (0, 1) and y,T_ aO < 3, then we can

increase ao to be 1, which leads to a larger objective value for the above subproblem.

Contradiction.

If there exists a time period i in which ao e (0, 1) and ET, a = 3, then there must

exist at least one more time period j in which oj e (0, 1). If (oi pi)xi* = (oj pj)x*,

then we can increase ao and decrease ajo such that either ao or oj becomes integral.

Thus, we obtain a solution with a fewer number of fractional solutions. Following this

same step, we can either obtain an integral solution with the same objective value or find

a case in a certain step in which (oi pi)xi* (oj pj)x*. Under this scenario, without

loss of generality, we can assume (o, pi)xi > (oj pj)x*. Then, we can obtain a larger

objective value for the subproblem if we increase ao and decrease aj by a small value

e > 0. Contradiction.

Therefore, the original conclusion holds. O

We can further utilize a primal-dual approach to generate a dual formulation of

RPLS as follows:

Theorem 3.1. The multi-stage robust lot-sizing problem with outsourcing can be

transformed as the following single stage mixed-integer-programming problem.
T T
min (pix + hisi + oiwi + fiy) + 3qo + q
x,s,y,w,q
i= 1 i= 1
(RDLS) s.t. qo + qi > (oi pi)xi, i E

x, + wi + s-1 = di + si, i e

xi < Myi, E I

x, s, w R', q ER+ ye {0,1}T.

Proof. It is easy to observe that (x*, y*, s*, w*) are bounded since problem parameters

including demand are nonnegative and bounded. Then, from the formulation of SUB,

we can observe that the subproblem is bounded and feasible since a = 0 is an obvious









feasible solution. According to strong duality theorem, we can write down the dual
formulation of (SUB) and show that RDLS can provide the same optimal objective value

as the original problem RFLS. O

From above analysis, we can solve the multi-stage robust lot-sizing problem
with outsourcing by a mixed integer problem. We can also observe that the following

(f, S)-type inequalities (see, e.g., Barany et al. 1984a) are valid for RDLS.
Proposition 3.2. Corresponding to each time period 6 G and a given S c =

{1, 2,..., }, the following inequality

x, + diy, + w, > d>u,
iES iec\S iEc

where diu = Y=i dk, valid for RDLS.

Proof. If yi = 0 for each i e C \ S, then IEs xi + eir\s diayi + Ei wi = ELs xi +

E;. wi = KEL Xi+x1C+E wi > dle. Otherwise, let k = argmin {ie \S : y* = 1}. We have
iEs x, + EiEL\s dcyi + E, xWi >- Ees Xi + Ew + wWi + dk > i + w + dk die.
The conclusion holds. D

3.4 The Robust Lot-sizing Problem with Backlogging: Single Disruption case

In this section, we discuss the robust lot-sizing problem which utilizes backlogging
as the reparation approach and in which a single disruption happens. In the backlogging
setting, the inventory and backlogging levels in the time periods which are after the
disruption period are changed. We assume that every disruption in the lot-sizing
problem with backlogging should be recovered after the disruption without the
consideration of outsourcing. Under this assumption, we observe this problem and
obtain the following observation result:
Observation 1: Based on the problem setting, when the disruption happens
at period t, later periods should have the potential for recovery production. In the
production horizon, the last period is a special period, because if production processes









in the last period in the first stage and then, the disruption occurs in the last period, i.e.,

T, no later period can recover its production. Thus, we assume that the last period does

not produce in the first stage.

Note here, the purpose of the above assumption is to keep the problem self-contained.

In this case, all setup decisions are made before the disruption time period and obtain

the following observation result:

Observation 2: If period k is the last setup period in the first stage planning, then

no production happens in this period. If production and disruption happen in last setup

period, no later period can provide the recover production for its production.

Based on the general formulation for the lot-sizing problem with multiple disruptions

and the above observations, we formulate the problem as a two-stage robust problem:
T T T
rmm Zpixi1+ ((fyi'+hisi1+bi ,')+max mmin [ Yt(x1,s,)+ (j2+hs2+ .)
(x ',yl'sl) i t (x ,s ,y ) ( +
i= i=1 i=t+l1


(PLS) s.t. xi1 + sl, + f = di + sf + fi, i E IR (3-4)

xil +sl,++x2+s2 +_,+2= di+s+ + 2+s2+ 2 iel (3-5)

xi < Myi, i I (3-6)

x2 < My2, ieI (3-7)

x1, x2, s1, 2 RT, y1 BT. (3-8)

with xA- = 0.

In the objective function, if there is a disruption that happens at time period t, then

the scheduled production at t cannot be processed, the corresponding production cost

ptxt is saved. Meanwhile, the corresponding inventory and backlogging costs which
involve inventory and backlogging amounts from production amount xt are saved. The

total saved cost is evaluated by function ot(x1, s1, 1). The constraints (3-4) and (3-5)

guarantee the demand in every period can be satisfied before and after the disruption.









The constraints (3-6) and (3-7) keep that the production is according to the setup

decision before and after the disruption, respectively.

In PLS, the exact production amounts provided by the disruption period t for

demands in other time periods are hard to determine. Thus, the exact form of saving

function ot(...) cannot be determined based on the information in PLS. Therefore, in

the following section, we generate a reformulation to provide the exact mathematical

formulation for PLS.

3.4.0.1 Reformulation

Let parameter ci; represent the total cost involved for a single item produced in time

period i (except setup cost) to satisfy the demand in time period f. For instance, for each

i < T, we have ci = i + p1 i hk if I > i, cki = Pi + 'k- bk if < i, and ci = pi. If

S= T + 1, then c = oi + Ek bk. Problem parameters hi and bi represent the unit

inventory and backlogging costs in time period i, respectively. We introduce variable x~

to build the relationship between the stage and production amounts; xik is the production

amount in period i to satisfy the demand in period at stage k. If k = 1, variables xi

are the scheduled production amount in period i to satisfy the demand in period If

k 2, xk^ is the extra production amount in period i to satisfy the unsatisfied demand

in period f. We let binary variable yi and binary variable zi indicate the first stage set-up

decision and the second stage set-up requirement. If zi = 1, period i can do recovery

production after the disruption; otherwise, zi = 0. Because all set-up decisions are

made in the first stage, then, zi < yi, 2 < I < T. The index t indicates the period when

the disruption happens. With variable xi4, it is clear that the production amount for time
period i to satisfy demand in time period as inventory or backlogging is x. Hence, we

can reformulate the two-stage robust model PLS as follows:











T-1 T T
mm q ^c, + S y,
i 1 1 i 1
T T T
+ max mim n cYx, C Y ctxt1
t X2
i=t+1 1 =1
T-1
(LS-S) s.t. xi = de, f e (3-9)
-i 1
T
x, x14, eGI (3-10)
i=t+1

xi < diyi el (3-11)

x2f < dizi f e I (3-12)

zi < yi (3-13)

1 < t < T 1 and integer, (3-14)

yi, z, z xE { xi, Xi2> 0, i e I. (3-15)

In LS-S, constraints (3-9) and (3-10) guarantee that demand in each time period i

can be satisfied and that the last period does not produce before the disruption due to

Observation 1. Constraints (3-11) and (3-12) guarantee that production happens in the

setup time period in the first and second stages. Constraint (3-13) forces that all set-up

decisions are made before the disruption.

In order to solve this two-stage robust mixed-integer programming problem, we

enumerate periods when the disruption happens to reformulate LS-S as a single stage

linear programming problem. According to the problem setting, t represents the period

of the disruption. The decision of x 2 is based on the period when the first disruption

happens. Thus, we let the production quantity be a function of the disruption. For

instance, we let x2(t) represent the extra production quantities after the disruption.

We use these as decision variables in the formulation. Finally, for a given t, we use

0 to represent the total extra cost after the first disruption. In this way, we enumerate










all possible scenarios for the period when the disruption happens and rewrite the

formulation for the lot-sizing problem with single disruption and backlogging as follows:

T-1 T
x1
i= e=1
T T T
s.t. min c c ex2(t) Ct xt <0, 1< t T 1 (3-16)
x2
i=t+l 1 = 1
T
x (t) = 1, (3-17)
i=t+l
x(t) < dizi, 1 < t < T- 1, I (3-18)

Constraints (3-9), (3-11), (3-13), and (3-15).


We can claim that the above formulation is equivalent to the formulation without

"min" operation on the left side of constraints (3-16) as shown in the following:

T T T
SS c,.(t) 5- ct^ 4 < 0, 1< t < T 1. (3-19)
i=t+1 =1 =1

The reason lies in the following two facts:

In the optimal solution, for the formulation including (3-19), 01 should achieve the

maximum value for the left side corresponding to a certain time period t. That is,

there exists at least one tight inequality in (3-19) for 0.

In the optimal solution, for the formulation including (3-19), we have at least one

tight inequality in which the left side achieves the minimum. Otherwise, 0 can be

decreased and we have a contradiction. Note here, we do not need to consider

the "min" operation for the non-tight inequalities, since it will not affect the optimal

objective value.

With inequality (3-19), the single disruption case can be solved by using the

following mixed programming formulation (LS-S1)

T-1 T
min i 5 ciexi + 01
i, 011
-i f.-l









s.t. Constraints (3-9), (3-11), (3-13), (3-17), (3-18), (3-19).


For the lot-sizing problem with a single disruption and backlogging case, based on

the enumeration of the scenarios of t, we introduce an artificial variable 01 and there are

O(T) constraints of (3-9), (3-13) and (3-19). For the disruption, for a given scenario of

t, we also increase the dimension of the second stage decision variable x2, to be x2(t).

Therefore, there are O(T2) constraints of (3-17) and O(T3) constraints of (3-18).

To simplify the notation, we let xqi represent x2(i). With equation (3-17), we

substitute x4, by yT_ Xi. The detailed reformulation is as follows:

T T-1 T T
mmin fyi + c c x x+ 0
i-x i- 1 qi+
T-l T
s.t. x' = d, tE c (3-20)
i=1 q=i+1
T
,xi< deyi, 1 q=i+1
xq < dZq, 2 < q < T, c (3-22)

Zq < yq, 2< q < T (3-23)
T T T T
5 cqtx c1( 5 x,) 1 <, q=i+1 =-1 -=1 q=i+1
Yi, Zq E {0, 1}, Xq_ > 0, 1 < i < T, 2 < q < T, f e I. (3-25)

We divide de on both side of constraints (3-20), (3-21), and let (3-24). Let a' =

xq,/de. Then, we obtain the new formulation for robust lot-sizing with a single disruption

and backlogging as:
T T-1 T T
mim fy + y cjidi > a' +
i-1 i=1 =-1 q=i+l
T-l T
(LS-SR) s.t. a-= 1, C el (3-26)
i=l q=i+l









T
ia < yi, 1 < i< T-l, f eI (3-27)
q i 1

a'q < Zq, 2 < q < T, ef I (3-28)

Zq < q, 2< q < T (3-29)
T T T T
c dca', cid( d a',) < 0, 1 < < T 1 (3-30)
q=i+1 =l =1 q=i+l

SE {0, 1}, Zq E {0, 1},a' > 0, 1 < i < T- 1, 2 < q < T, je I. (3-31)

Next, we study the structure of feasible region of LS-SR. Let

TxTx(T-I) +2T
XD = {(x, y,z, ) R 2 : (x, y, z, 0) satisfies (3-26) to (3-31)}. (3-32)
TxTx(T-1) 2T
RD = {(x, y, z, 8) R 2 : (X, y, z, 0) satisfies (3-26) to (3-30))}. (3-33)


and PD is the polyhedron of XD. Note here, XD records the feasible region of LS-SR. RD

is the relaxation of XD.

Second, we derive the dimension of PD and show which above inequalities are

facet-defining inequalities.

Proposition 3.3. The dimension of the polyhedron PD is TxTx(T-1) + 2T.
2
Proposition 3.4. (a) For any 1 < i, f < T, i + 1 < q < T, a' > 0 defines a facet of PD.

(b) For any 1 < i < T, yi < 1 defines a facet of PD.

3.4.0.2 Facet-defining Inequalities

In this section, we investigate facet-defining inequalities which are not in LS-SR for

PD.

Proposition 3.5. The following inequalities

T ( q-1
j zq a + a'() > 2, where e(q) e and (ql) 7f(q2) if q q2 (3-34)
q= 2 i 1

are valid and facet-defining for LS-SR.

Proof. We prove Proposition 3.5 by two claims.









Claim 1. (3-34) is a valid inequality for LS-SR.

Claim 2. (3-34) is a facet-defining inequality for LS-SR.

Proof of Claim 1. From Observations 1 and 2, we know that in order to recover

the production, at least one period is set up only for the recovery production. Thus

Tq=2 Zq > 1. We discuss the following two cases.
Case 1. If there is more than one period which is set up for recovery production,

then q=2 Z > 2.

Case 2. If there is only one setup period for recovery production, then Y i2 Zi = 1.

With (3-29), we have Lq=2 i aq(q = 1. Because if Lq=2 aq(q) 1, demand in

each period cannot be satisfied.

Therefore, (3.5) is a valid inequality for LS-SR.

Proof of Claim 2. Let pz + 3a + 7y + 0K < 6 be a valid inequality for PD and assume

that
T q-1
1 = {(x, y, z, 0) PD, (Zq + aq(q)) = 2} (3-35)
q=2 i 1
T T-1 T T T
C {(x, y, z, 8) E PD, piz, + / + 7iYi + 0 = T} (3-36)
i=2 i 1 q=i+1 = 1 i 1

We prove that (3-34) represents a facet for PD by showing


S= 0, (3-37)

7 = 0, = 1,..., T (3-38)

i =p -i = 2, .* T (3-39)

3q p = -p + u, = (q) (3-40)

0,i = u- (q) (3-41)
T
= -2p + u. (3-42)
/ 1









First, we prove that K = 0. With a feasible and tight solution, (x, y, z, M) for

RI, we have another feasible and tight solution (z, x, y, 2M) for R1, where M =

max(;,c) ci 1,T di. Then, we have
T T-I T T T
:piz, i+ Y Y Oq/3ai + y 7i- + KM = -r]
i=2 i-1 q=i+l _1 i-1
T T-l T T T
Pzi f 3a6 + 7 yj y + 2nM =
i=2 i -1 q=i+1 1 i-1

Thus, K = 0. We assume 0 = M in the following discussion.

Second, we prove that 7i = 0 for all i = 1, T. We construct two tight feasible


solutions for R. as following:

Point 1: yi = Y2 = YT -= 1, Zq = Zq2 = 1, aql = 1, a',_,,,qi)

q, q2 < T- 1.
Point 2: yi = y2 = = T-1 = 1, Zql = Zq2 1, aqk = 1 a,,,,

qi, q2 < T -1.
Then, we have


= 1, k (qi),


= 1, k (q,),


T T-l T T T
Epiz + O E E + Y 'i = TI
i-=1 i=1 q=i+1 =-1 i-1
T T-l T T T-l
zPiZ, a + Y a + 7 = TI
i-1 i-1 q=i+1 =-1 i-1

Thus, 7T = 0. With different construction of tight feasible solutions, we have y7 = 0. We

assume y = 1 in the following discussion.

Now, we show the following coefficient relationship among 3, p and T. We construct

the following tight feasible solutions for Ri:

Point 3: Yi = Yi2 = yj- = y2 = z = z2 1, a 1, a= Zi 1, ak(1, a = 1,

where i1, 2
Point 4: yi = y2 = y1 = y2 = 1, ZI = Z2 = 1, a ) 1, a1 1, ,2 = 1, =
where i, < ,2, k, i 2) Jk k.
where 1',,1'2 < J1,J2, k 7$Cfji), f j2)fjCfji)j fj2), k.








Point 5: yi = y2 = yj = yj2 = 1, z = z, = 1 a = 1, a = 1, where

ii, i2 Point 6: yi = y = = 1, =2 1, = 1, a = 1, a = 1, ai = 1
where i, < J13,i2 I O), (j2).
Point 7: = y = Y~ = Y2 = 1, zj = 1, a = 1, where i, i2 <1 ,j2.
Point 8: y, = yi = y = y2 = 1, Zj = Zj2 = 1, = a = 1, = 1, where

il, /2 < j1,j2, k f(j2), f 7 -f(j2), k.
We show that 0/ = 0/ = Oq for f $ f(q) by putting Point 3, Point 4, and Point 8 into


(3-43)


T T-l T T
PiZi + 0i qaq = T.
i=2 i=1 q= i+1 = 1


Then, we have


PJi1 + Pj2 + 2 1'A)+ ) k+ k + i = (3-44)
S(ji), (j2), k
P^+ Pi+ 2 1+ k+ = TI, (3-45)
PJI + PJ2 + ,OJ'2) + Oj24i,) + OJ S/k + -j = (3-45)
U# Oi), (j2),k

PJ, + PJ2 + +k +j() + Y, Oj = (3-46)
-0(j2),k

Thus, with (3-44) and (3-45), we have = 3 = /32k, where k 7 f(ji), f(j2). With
(3-44) and (3-46), we have 3k = /,2 where k 7 f(jl). By the arbitrary construction of
Point 3, 4, and 8, we have

= =/2 = q for f 7 f(q). (3-47)

We show that P3i = P3j1 = 3. we put points 3 and 5 into (3-43), we have (3-44) and

pj, + p 2 + ,iJG2) + ) + k = -1. (3-48)
UIei),,e.2),k
With different constructions of feasible solutions, we have

/k = k -= /, where k f(ji), 2(j2). (3-49)









With (3-47) and (3-49), we have pj,'k= k = jk = 31. Thus, the following conclusion

holds.

3= u, 7j). (3-50)

Put Point 5 and Point 6 into (3-43), with (3-50), we have PJ2 = pj,.With the different

construction, we have

pj = -p.

Put points 3 and 7 into (3-43), we have j = -p u, where f = f(j). Hence,


!3C/) -P + u.

With any tight feasible solution for R1, we have Tr = -2p + tyT uI. D

Based on Proposition 3.5 and constraint (3-23), we generate the following

facet-defining inequality for PD as follows.

Proposition 3.6. The following inequalities
T ( q-1
y 4=i+ y+Yq a-ia >3, forall q(q) E Isuch that (qi) (q2) if q q2 (3-51)
q= 2 i= 1

are valid and facet-defining for LS-SR.

Proof. We proof Proposition 3.6 by two claims.

Claim 1. (3-51) is a valid inequality for LS-SR.

Claim 2. (3-51) is a facet-defining inequality for LS-SR.

Proof of Claim 1. From Observations 1 and 2, in order to satisfy demands and do

recovery production, ji yi > 2. We prove Claim 1 by two cases.

Case 1. If there are more than two production periods, then zT yi > 3.

Case 2. If there are only two setup periods, the recovery period covers the demand

in period (q) to satisfy the constraint (3-20). Thus, (3-49) holds.

Therefore, (3-51) is a valid inequality.









Proof of Claim 2. Let pz + 3a + 7y + K0 < 6 be a valid inequality for PD and assume

that

T q-1
2 = {(x, y, z, ) PD, YI + (q + a(q)) = 3} (3-52)
q=2 i=1
T T q-1 T T
C {(x, y,z, 0) E PD, PiZi + ,/3 qaqi+ 7;yi + 0 = T} (3-53)
i=2 q=2 i-1 = 1 i 1

We prove that (3-51) represents a facet by showing that


S= 0, (3-54)

Pi = 0, i = 2, T (3-55)

7 = -a, i .= 1,..., T (3-56)

,i = -a + u = f(q) (3-57)

,i = u$, f (q) (3-58)
T
TI = -3a + u (3-59)
=1

First, we prove that K = 0. With a feasible and tight solution (x, y, z, M) for R2, we have

another feasible and tight solution (x, y, z, 2M) for R2, where M = max(i ,) ci = 1 d;.

Then, we have
T T-l T T T
iizi ii + :M Y--q
i=2 i-1 q=i+1 =1 i=1
T T-l T T T
Pizi + ,ai i + +i r + 2iM = Y
i=2 i-1 q=i+1 i=1 i 1

Thus, K = 0. We assume 0 = M in the following discussion.

Second, we show that p, = 0 for 2 < i < T. We construct the following two feasible

points of 72.

Point 1: y = y2 = 1, z2 = 1, a = 1, 1 < ii < < T.

Point 2: y = yj = z= z = a, 1i <2 2< T.
Yi ~i2 1








Put points 1 and 2 into


T T-l T T T
S P1z, + f qa'. + y = 1,
i=2 i-1 q=i+1 =1 i=1
we have pi, = 0. For the arbitrary construction of points 1 and 2, we have pi = 0.
Now, we show the following relationship among 3, 7 and T1. We construct following
tight feasible solutions for 7R2.
Point 3: yi, = 1, yj, = yj = 1, x ) x1 2 = 1, x k = 1, x = 1, where j,J2 > 1,

k f(ji), f(j2), f f (jl), f(j2), k.
Point 4: y, = 1, y =yj = 1, 1 =, x,2)) = 1, j21k = 1, Xj = 1, wherejiJ > 1,

k 7 (ji), (j2), f (il), f(j2), k.
Point 5: y,i = 1, yi, = yi = 1, xXl j)= 1, x = 1, xk = 1, x = 1, where ji,2 > 1,

k f(ji), f(j2), 1 f (J,), f(j2), k.
Point 6: y;i = 1, y = y, = 1, x'U) = 1, xj'.) 1, xj = 1, whereji,j3 > 1,

f (jl), (j3).
Point 7: y. = yi, = 1, xy = 1.

First, we show that 0 = 3j =- /3k, k 7 f(j). Put points 3 and 4 into
T T-1 T T
7iyi + Ya = l. (3-6
i=1 i=1 q=i+1 e=1


0)


Then, we have


eU0i ) +(U2),k
7il+7+72+ OE, O+ )yj
eUA(ji) U(j2),k

We have = 0 = f2k, where k 7 (j2). Due to the arbitrary construction of points 3
and 4, then we have O' = OJ = /jk, where > 2, k $ f(j). Note here, under the problem
setting, 2'. is a T x 1 vector.








Second, we show that pjik = j2k = Uk, where k $ f(ji), (j2). Put points 3 and Point
5 into (3-60). Then,

7+7j+7j+ ii i ii+ + + =
iej )e(j2),k
7+7j, +7j, + + Oi+OiE) + O il i=i
e ) 02),(j2 k
We have -gk = Ok. Due to k = /3k, and jk = /ik. Thus, /3jk = pk = Uk, where
k $ f(ji), or f(j2). Due to the arbitrary choice of j1 and j2, we have pjk = Uk, where
k f(j#).
Third, we show that 7i = 7 = -a. We put points 5 and 6 into (3-60). Then,

n '7j+7 + E 3, + 12) =
4eUl)MCU2)
7j+j+7j+3 + +0:lA)+o ) = I
''jii),Mc3)

Because Ol, = u, where f # f(jl), With j'0) j i( = u i'), y7/ = y/ jWith the
arbitrary choice of j2 and j3, we have 7, = -a.
Forth, we show that Pf3) = -a + u j). Put points 3 and 7 to (3-60). We have


7+7+7+ E 0 +il+ 2)=
EUl),US2)
71+7j+ E hj)) + 12j2) = TI
-MOl)Ah2)


Then, -,3J = + 7 ,) = -a + u(jO). With the arbitrary choice of j1 and J2, we have

OeA) = -C + u U).
Finally, put any tight feasible solutions. Then, we get TI = -3a + Ti u1.
Therefore, (3-51) is a facet-defining inequality. D

The above two facet-defining inequalities are generated based on the relationship
between the recovery production and recovery setup requirement. Now, we construct









a facet-defining inequality for PD based on the relationship between the production and

the setup decision.

Proposition 3.7. The following inequalities

T q-1
a +< a' k=q+l i 1

are valid and facet-defining for LS-SR.

Proof. We prove that Proposition 3.7 holds by two claims.

Claim 1. (3-61) is a valid inequality for LS-SR.

Claim 2. (3-61) is a facet-defining inequality for LS-SR.

Proof of Claim 1. If yq = 0, we have Tk q+i a = 0 and z = 0. Thus, E=q+i ak

iTq 1 a = 0. If y, = 1, because

T T-1 T
a, < a', and a q, a= .
q=i1+ i=1 q=i+1

Then, we have kTq aq + 1 Tq+1 aq < 1. Therefore, (3-61) is a valid inequality for

LS-SR.

Proof of Claim 2. Let pz + 3a + 7y + K0 < 6 be a valid inequality for PD and assume

that
T q-1
3 = {(x, y, z, 0) PD, ak + a = Yq
k=q+l i=1
T T q-1 T T
C {(x, y, z, 0) e XD, Piz + q q3a + y + KO = 6}
i=2 q=2 i-1 1=1 i 1

We prove that (3-61) with a pair of (f, q) represents a facet-defining inequality by

showing


K= 0,
T
r]= 0










7q = 7q

i= 0, i= 2, ,T

i = 0, i q

3qt~= -7 +Pe, i 1,-- ,q-1

k= 7q-+~e, k=q+l,-- T

=jh = h, j q, or i q.

We prove that K = 0. With a feasible and tight solution, (x, y, z, M) for R3, we have

another feasible and tight solution (x, y, z, 2M) for R3, where M = max(i,) ci T1 d;.

Then, we have
T T-l T T T
pizi, + 0 ,a. + jiyf + KM = Tr
i=2 i 1 q=i+1 1 i 1
T T-l T T T
pizi + q q a + Z yli + 2KM = T
i=2 i= q=i+l =1 i 1

Thus, K = 0. We assume 0 = M in the following discussion.

We show that pi = 0 for 2 < i < T. We construct the following two feasible points of

R:

Point 1: y, = y = yj2=1, zj2 = 1, a 1< i < l,2 < T, 1 < k < T.
j2k ... .
Point 2: yi = Yy, = yj, = = 1, az = Z, 1< ii
Put points 1 and 2 into

T T-l T T T
C j0 'z + C C a'. + Tijy = 1,
i=2 i=1 q=i+1 =1 i=1

we have pj, = 0. For the arbitrary construction of points 1 and 2, we have pi = 0.

We show the relationship of 3, 7, and rT with equation

T-1 T T T
Yai + Y = I (3-62)
i=1 q= il 1 i 1

We construct the following feasible solutions for R3:









Point 3: yi, = y, = 1, a = 1, 1 < k < T.
qqk
Point 4: Yi, = y, = Yji = 1, aq = 1, 1 < k < T.



Point 6: yi, = yi, =y 1, a = 1, a = 1, where k .

Point 7: Yi, = Yq = j, = 1, a]i = a 1, where k 7 t.

Point 8: yi, = yj, = y, = 1, ajk = 1, 1 < k < T, iji,j2 7 q.

Point 9: y, = y, = yj, = 1, aq =1, a = 1,k .

Point 10: Yi, = Yj, = yj2 = 1, ajm = 1, ak = 1, il 2 j q, k 7 m.

Point 11: y = = yj, 1, ajk = 1, 1 < k < T, i, ji,j2 7 q.

Point 12: y,~ = Yj, = yj2 = 1, a, = a 2 1 q, k 2 m.

We show that 7i = 0, where i $ q. Put points 3 and 4 into (3-62). We get

T
7Yi + q +%+ =-'I
=1
T
7i, +7q +7j, = T1.
=1

Then, we have 7j, = 0, j1 7 q With the arbitrary construction of points 3 and 4, we have

7i = 0, where i $ q.

We show that / = /O = /q. With yi = 0, i 7 q, we put points 4 and 5 into (3-62)

and obtain


7(q + + Y 0 il = SI



k

Then, we have P' = / = .

We show that jk = .3, where i < q < k. We put points 6 and 7 into (3-62) and

obtain













q + S TI
7(q +J + 5 -
ki-

Then, we have /3 = 3, = 3qe.

We show that P' = e, where i,j $ q. Put points 8 and 10 into (3-61), we have

olk = fO3k = 3'. With the different construction of points 8 and 10, we have/3k = k ,
where i,j j q.
Put points 11 and 12 into (3-61), we have /3' = /3 = k, where ii,ji q. With the

different construction of points, we have /3k = -k, where i,j j q.

We show that /3q = e 7yq. Put points 8 and 9 into (3-61) and obtain

7 q + qe + Y/3Oj k = TI

T

k=l

Then, we have 7, + 0q = P3 = q+ P3q = .p Thus, /3q = (c q,.

Finally, we put any feasible solution of 7R3 into (3-61) and obtain rT = ,- e. E

3.5 The Robust Lot-sizing Problem with Backlogging: Multiple Disruption case

In this section, we present the formulation for the robust lot-sizing problem with

multiple disruptions, and with and without setup cost cases, respectively.

3.5.1 Without Setup Cost Case

Let the index ti indicate the period when the ith disruption happens, where i =
1, 3. We extend the three-stage robust model PLS to multi-stage, and formulate

the robust lot-sizing problem with multiple disruptions and without setup cost case as

follows:












T-I T
min ci x i
x1

T-I T 7
+ maxmin c ,xI -
tl X2
T-1l T 1
T-1 T 7
+max min c; x7 -
t2>tl X3
it2+l e 1 .e


T T
+ max min Vcix3i
t3>tp 1x1+1
i=-t+l =1


(MRLS) s.t.


-1


Ct2 t2
1 k=1


T /, 3 \

=1 k=l


T-l
xi = d, e
i-i
T-l
Z 2 I
C xi = xt 4, e 1,
i=-t-i
T-i 2
SX'3= x e e a,
i=t2+1 k=1


T 8
xS + = xtk4, e l,
i=tp+1 k=1

1 < ti < t2 < ... < t3 < T and integer.


We combine the enumeration of periods when the disruptions happen and the

pre-processing algorithm to reformulate MRLS as a single stage linear programming

problem. We extend the reformulation scheme in Section 3.4 for the single disruption

case to the multiple disruption case. First, we use a two-disruption case as an example

to explain our reformulation.

According to the problem setting, we have t2 > ti, where tj and t2 represent the

periods for the first and the second disruptions, respectively. The decision of x2, is based


(3-63)


(3-64)


(3-65)





(3-66)


(3-67)









on the period when the first disruption happens and xf3 is based on the period when

the first and the second disruptions happen. Thus, we let the production quantity be a

function of the disruption history. For instance, we let x2,(t) and xfj(t, t2) represent the

extra production quantities after the first and the second disruptions, respectively. We

use these as decision variables in the formulation. Finally, for a given ti, we use 02(tl)

to represent the total extra cost after the second disruption. In this way, we enumerate

all possible scenarios for the period when the second disruption happens and rewrite the

formulation for the two-disruption case as follows:
T T T T T
min cix+maxmin c x ~~c(ti) ctxt + 02(tl)
X1 ti X2(h) 1
i 1 1 i til 1 e 1
T T T
s.t. min cx,3(tl, t2)- ct2e(x (ti) + x12) < 82(t), 1 < t, < t2 (3- )
x3(tl,t2) i t2+ 1 6 1
T
Sx3(ti, t2)= x22(ti) +1, ,1 t1 < t < T, (3-69)
i=t2+l
Constraints (3- -64).


The left hand of the above constraint can be simplified to the case without "min"

operations. It results in a linear programming formulation. Because t2 is the last

disruption period, the extra production amount after t2 to satisfy the unsatisfied demand

for each period = 1, T, x3,(t, t2) is equal to x2 + x22,(ti). We can apply an

idea similar to that used to solve the robust lot-sizing problem with single disruption and

no setup cost case. We select a period after t2 that provides the minimal unit cost to

satisfy the unsatisfied demand in period f due to the disruption in t2. That is, we denote

mt2. = min{ce, > t2} to be the minimum unit production cost for the unsatisfied

demand in period Constraint (3-68) can be rewritten as:
T T T
Smt2(x2(t) + X)- c 2(x2(t) x1) < 2(t), 1 < t < t2 < T. (3-70)
i=t2+1 --1 i-1









Now, we enumerate all possible scenarios of the period when the first disruption

happens. We can reformulate the two-disruption cases as follows:

T T
min c x + 01
/-l ei
i 1 1


T T T
s.t. min E cx(t) ct1x + 02(t < 0, 1 < T, (3-71)
x2t' 112 1 1 1 < 13
i=t 1 (h)=1 t=1

Constraints (3-64), (3-69), (3-70).


We can claim that the above formulation is equivalent to the formulation without

"min" operation on the left side of constraints (3-71) as shown in the following.

T T T
S cex2(tl) C Ct 2(tl) < 1, 1 < < t T. (3-72)
i=+t1 1 -1 e=1

The reason lies in the following two facts:

In the optimal solution, for the formulation including (3-72), 01 should be the maximum

value of the left side corresponding to a certain time period t. That is, there exists

at least one tight inequality in (3-72) for 01.

In the optimal solution, for the formulation including (3-72), we have at least one tight

inequality in which the left side achieves the minimum. Otherwise, 01 can be

decreased and we have a contradiction. Note here, we do not need to consider

the "min" operation for the non-tight inequalities, since it will not affect the optimal

objective value.

Thus, we can substitute inequality (3-72) to (3-71). The two-disruption case can be

solved by using the following linear programming formulation (MB-2):

T T
min ciax; + 81
X1' 01
i-=1 -1

s.t. Constraints (3-64), (3-69), (3-70), (3-72).










Now, we can analyze the structure of (MB-2) and extend it to the multiple disruptions

case. For the two disruptions case, based on the enumeration of the scenarios of ti, we

introduce an artificial variable 01 and there are 0(T) constraints of (3-64) and (3-72).

For the second disruption, for a given scenario of ti, we introduce an artificial variable

02(t). We also increase the dimension of the second stage decision variable x, to

be x2,(ti). Therefore, there are O(T3) constraints of (3-69) and O(T2) constraints

of (3-70).

Based on the similar idea, we can introduce the notation xi(t,,..., tk-1) to be the

extra production quantity in period i to satisfy the demand in period f after the th-1

disruption. Then, we can observe that the following conclusion holds for multi-stage

robust lot-sizing problems.

Theorem 3.2. For the multi-stage robust lot-sizing problem with 3 disruptions and

without setup cost, the problem can be formulated as the following single stage linear

program.


T T
x1,m i =1 1
i= l = 1
T
s.t. xi = d,,
i--3

Xtl,





T -1
T
E x((tl) = x-
i= t 1.







i=t3 1 k=2


(3-73)


i e (3-74)


f e (3-75)


fi c (3-76)



fel (3-77)









T T T
cOx(tl) Ct ext ,+ 02() 1, (3-78)

T T T
S- cixL(, t2) t c(xj2 + (t)) + 3(tl, t2) < 02(tl), (3-79)
i=t2+1 e 1 e1

T T T 3-1
c ij ,(ti --- ,''- ) Ct (xt *Xte + X 1, .t ,..., k-1))
i=t_ 1+l 1 e=1 k=2
+0 (tl, t3_1) < 0-1(t, t _-2), (3-80)
T /3 T 13
m m, (xr + x (t,, tk-1)) C, X + (s ..., 4k-))
e=1 k=2 e 1 k=2
< 0 (tl,... tp_1), (3-81)

0k free; Xk > 0, V1 < k < 3, (3-82)


where mt,3 = min{cie, i > t,} and1 < tj < t2 < ... < t < T.

Proof. We have shown that the conclusion holds for the cases in which there are

one or two disruptions. For the multiple disruption case, we can explore the different

combinations of t, t2, ..., tp to a scenario tree with /3+ 1 depth. The root node indicates

that there are no disruptions at the very beginning. The children of root node describe

the possible first disruption time ti. That is, there are T 1 children of root node.

Corresponding to each particular tree node ti, there are T t1 1 children to represent

the possible second disruption time period. For the general case, corresponding to each

tree node tk, there are T tk 1 children to represent the possible k + 1 disruption

time periods, based on the previous disruption realizations of (t,, t2 ..., tk) along the

path from the root node to the tree node representing tk. Leaf nodes with depth 3 + 1

represent the last disruption time period.

Therefore, in our approach, we enumerate all possible disruption combinations. For

each tree node, corresponding to a particular realization of disruptions (t, ..., tk), we

need to have the extra production in stage k + 1 to cover the unsatisfied demands due to

disruption at tk. Thus, constraints (3-74) to (3-77) hold.









Also, similar to the study for the two disruptions case, corresponding to the root

node, we have
T T T
min cix2,(tl) ) < 0 1 < t1 < T. (3-83)
X2,02
i 1" e 1 = 1

For each ti, we can also consider inequality (3-83) corresponding to the link between

root node and its child t,.

For the general case, corresponding to each tree node (t1, t2 ... tk), except leaf

nodes, we have
T T T k+1
Xkn2 Cmi > X 2c1' 2 k1) CEk1(X + XT, ... t i-1))
min C JX tk+1 XtkJt,
xk+2,0k+ 2
i=tk+l+l e=1 e=1 r=2
+0k+2(tl, tk+) < Ok+ (t1i, tk), tk < tk+1 < T. (3-84)


Similarly, for a given tk+1, we can consider the above inequality corresponding to the

link between tree nodes (t, ..., tk) and (t,..., tk+1). Therefore, constraints (3-84)

correspond to all links in the tree.

As shown in the proof for the two disruptions case, we can remove the "min"

operations on the left side. That is, we can derive the corresponding constraints

(3-78) to (3-81), which also correspond to all links in the tree. Then, we can obtain

the following two conclusions.

(1) It can be observed that there exists at least one path from the root node to a leaf

node such that, corresponding to each link along the path, the corresponding

inequality in constraints (3-78) to (3-81) is tight. This path can be obtained by

breadth-first search starting from the root node to find descendants along the

links in which the constraints are tight (i.e., named tight links). If the breadth-first

search terminates without reaching any leaf nodes, then we can reduce 0 by a

small positive value c, which leads to a smaller objective value. Contradiction!

(2) It can also be observed that among all the candidate paths as shown in (1), there

exists at least one path such that, for the inequality corresponding to each









link along the path, the left side achieves the minimum. We can also use the

breadth-first search on the links among the paths that are tight. We check if the left

side is tight for each tight inequality corresponding to each link. If the breadth-first

search terminates without reaching any leaf node, then we can reduce the left side

value and the 0 value by a small positive value c, which leads to a smaller objective

value. Contradiction!

Based on (1) and (2), we will eventually find at least one path from the root node to

a leaf node such that each inequality is tight corresponding to each link, and the left side

for each tight inequality is minimized. The leaf node index gives the disruption periods in

the optimal solution. Therefore, the conclusion holds. O

3.5.2 Setup Cost Case

In the above section, we show that the multi-stage robust lot-sizing problem can be

reformulated as a single-stage deterministic equivalent linear programming problem with

polynomial size of constraints and decision variables. For the case with setup cost in the

first stage, we can similarly obtain the following formulation:
T T T
min yii fi, + E cix + 01 (3-85)
1 i=1 i=1
i 1 1 1
s.t. (3-74) to (3-81); (3-86)

xi, x(t ..., tk-i) < Myi, Vi, e 1 < k < /3. (3-87)


3.6 Computational Results

In this section, we present the computational results to demonstrate the computational

tractability of solution approaches we studied for the different cases of multi-stage

robust lot-sizing problems. All computational experiments were carried out on a Linux

workstation with a Pentium Dual 2.8G processor and 6G RAM. We use CPLEX 10.1

Callable Library to implement our algorithms and run the reformulated models.









3.6.1 Instance Generation

We generate instances based on different ratios of setup cost fi to unit production

cost pi and different time horizons T. For the instances, we set the time horizon T = 10,

20, 30, 40 and 50, and the ratios of setup cost to unit production cost f/p = 10, 20, 30,

and 40, respectively. There are 20 combinations in total.

For the cases with setup costs, corresponding to each of the combinations of T and

f/p, we generate the random instances in which the unit production cost and the setup
cost are uniformly distributed in the intervals as shown in Table 3-1.

For the cases without setup costs, corresponding to each of the combinations of T

and p, we generate the random instances in which the unit production cost is uniformly

distributed in the same interval as for the cases with setup costs shown in Table 3-1.

We also set the demand di, unit inventory cost hi, and unit backlogging cost bi

uniformly distributed in the intervals [500, 1000], [5, 10], and [10, 20] respectively.

3.6.2 Two-stage Robust Lot-sizing Problem

For two-stage robust lot-sizing problems, we test the case using outsourcing and

backlogging to recover unfilled demands. The computational results for the robust

lot-sizing problem with outsourcing and the robust lot-sizing problem with backlogging

are illustrated in Section 3.6.2.1 and Section 3.6.2.2, respectively.

3.6.2.1 Two-stage Robust Lot-sizing Problem with Outsourcing

For this case, we test twenty combinations in which the time horizon T =

10, 20, 30, 40, and 50 respectively and the ratio f/p = 10, 20, 30, and 40, respectively.

The computational results are shown in 3-2. Table 3-2 reports the computational results

Table 3-1. Parameter setting
unit production cost pi setup cost fi
ratio 10 [50, 100] [500, 1000]
ratio 20 [50, 100] [1000, 2000]
ratio 30 [40, 60] [1000, 2000]
ratio 40 [40, 60] [500, 1000]









for multiple disruption case in which we assume the number of disruption periods is 10%

of the time horizon T. For each of the 20 combinations, we report the average values of

5 random instances. We report 1) the optimal objective value of the lot-sizing problem
without disruptions, denoted as "SLS", 2) the objective value obtained by maximum
pick heuristic, denoted as "MPSLS", 3) the objective value for the robust optimization

formulation obtained by using CPLEX to solve the dual formulation (RDLS) for the

outsourcing cost cases, and 4) the gap between (RDLS) and (MPSLS), denoted as
"GAP(M::R)=(ObjMPSLS-ObjRDLS)/(ObjRDLS)," and 5) the gap between (SLS) and (RDLS),

denoted as "GAP(R::S)= (ObjRDLS-ObjsLs)/(ObjsLs)." Compared with the maximum pick
heuristics, the average gap is 55.8% for multiple disruption case. That means the

total cost can be reduced by 55.8% by applying robust optimization approach (RDLS).

Compared with the uncapacitated lot sizing problem without disruptions, the average

gap between (SLS) and (RDLS) is 35.0%. That means total costs increase 35.0% by

considering disruptions.
3.6.2.2 Two-stage Robust Lot-sizing Problem with a Single Disruption and
Backlogging

First, we evaluate the efficiency of our formulation for lot-sizing with a single

disruption and backlogging by comparing the total cost of our formulation with the no

disruption case and maximum pickup heuristics. We test 20 combinations in which the

time horizon T = 10, 20, 30, 40, and 50 respectively and the ratio f/p = 10, 20, 30,

and 40, respectively. We report the performance of our formulation for lot-sizing with

single disruption in Table 3.6.3. Corresponding to each combination, we report the

value of 1) the objective value obtained by single item lot-sizing problem without the

disruption, denoted as "SLS", 2)the objective value obtained by maximum pickup

heuristics, denoted as "MP", 3) the objective value obtained by the lot-sizing with

backlogging case, denoted as "SLS-B", 4) the gap between SLS and SLS-B, denoted

as "GAP(B::S)=(ObjsL_-B ObjSLS)/ObjSLS. 5) the gap between SLS-B and MP,









denoted as "GAP(B:M)=(ObjMp ObjSLS-B)/ObjSLS-B". GAP(B::S) shows the growth
of the total production with the consideration of setup cost. The average increasing rate

is 11.92% for the robust lot-sizing with backlogging. GAP(B::M) shows the saving of the

total production of our formulation for the lot-sizing problem with a single disruption and

backlogging compared with the maximum pickup heuristics. The average saving rate are

24.55%.

The performance of the generated facet defining inequalities for the robust lot-sizing

with backlogging is reported in Table 3.6.3. We test twelve combinations in which the

time horizon T = 100, 120, and 140, respectively and the ratio f/p = 10, 20, 30, and

40, respectively. For each combination, we run five instances and report the average

performance of these five instances. Let "Cut" and "NoCut" denote the branch-and-cut

algorithm with the facet-defining inequalities as cuts and the default CPLEX without

adding facet defining inequalities. The testing instances for "Cut" and "NoCut" are same.

We set the computational time limit as 1800 seconds for both cases. For the "NoCut"

case, no instance can be finished within time limits. For the "Cut" case, all instances for

T = 100 and T = 120 can be finished within 1800 seconds and achieve the optimal

solution. The exact computational time and optimality gap are listed in Table 3.6.3.

3.6.3 Multi-stage Robust Lot-sizing Problem with Backlogging and without Setup
Cost

Finally, we test multi-stage robust lot-sizing problems with backlogging, without

setup cost, as described in Section 3.5. From Theorem 3.2, we can observe that

the optimal solution for this type of problems can be obtained by solving a linear

programming formulation that contains O(T3) constraints and O(T3) variables, where

3 is the number of total disruptions. Therefore, this formulation is pseudo-polynomial,

in terms of time periods. We report the computational results in Tables 3-5 and 3-6 for

two-disruption and three-disruption cases, respectively. For 2 disruption case, we solve

a linear formulation that contains O(T2) variables and constraints to get the optimal









solution. For the three-disruption case, we solve a linear formulation that contains O( T3)

variables and constraints to get the optimal solution. Similarly, we use the objective

value obtained by the maximum pick heuristic as an upper bound to evaluate the cost

savings by applying our robust optimization approach. It can be observed in Table 3-5

that, on average, our robust optimization approach has possible cost savings around

80.8%, compared with max pick heuristics. When compared with the uncapacitated

lot-sizing problem without disruptions, our approach increases cost around 11.6%. It

can be observed in Table 3-6 that, on average, our robust optimization approach has

possible cost saving around 82.1%, compared with max pick heuristics. When compared

with the uncapacitated lot-sizing problem without disruptions, our approach increases

cost around 37.9%. Tables 3-5 and 3-6 also indicate that the computational time,

which is reported in seconds, increases as the number of time periods and disruptions

increases. But in general, the optimal solution can be obtained within one second for the

two disruption case and within a minute for the three disruption case.













Table 3-2. Robust lot-sizing with outsourcing: multiple disruptions
T= 10 T = 20 T = 30 T = 40 T = 50
Dis =1 Dis =2 Dis = 3 Dis =4 Dis =5
ratio=10 SLS 4.84 x 10 1.04 x 10b 1.56 x 10 2.05 x 100 2.52 x 10
MPSLS 8.37 x 10b 1.94 x 10 3.16 x 10 3.96 x 10 5.58 x 10
RDLS 6.53 x 10b 1.39 x 10U 2.06 x 10 2.72 x 10U 3.37 x 10
GAP(M::R) 28.2% 39.6% 53.4% 45.8% 65.6%
GAP(R::S) 35.1% 33.0% 32.0% 32.6% 33.5%
ratio=20 SLS 5.17 x 105 1.11 x 10b 1.51 x 10 2.07 x 10b 2.55 x 10
MPSLS 9.42 x 10b 2.07 x 106 2.96 x 10b 4.38 x 10 5.18 x 10
RDLS 6.82 x 10b 1.44 x 10 2.00 x 10b 2.70 x 106 3.40 x 10b
GAP(M::R) 38.1% 44.0% 47.8% 62.0% 52.3%
GAP(R::S) 31.9% 28.7% 32.4% 30.4% 33.5%
ratio=30 SLS 3.80 x 10 7.43 x 103 1.11 x 10 1.52 x 10b 1.87 x 10
MPSLS 6.40 x 10b 1.45 x 10b 2.11 x 10b 2.83 x 10b 3.81 x 10b
RDLS 5.15 x 10b 1.03 x 101 1.54 x 10 2.09 x 10 2.59 x 10
GAP(M::R) 24.2% 41.5% 37.3% 35.7% 47.5%
GAP(R::S) 35.5% 38.0% 38.5% 37.7% 38.4%
ratio=40 SLS 3.75 x 10 7.60 x 103 1.14 x 10 1.51 x 10 1.87 x 10
MPSLS 7.37 x 10b 1.48 x 10 2.12 x 10 3.06 x 10 3.95 x 10
RDLS 5.14 x 10b 1.04 x 10b 1.58 x 10b 2.09 x 10b 2.58 x10
GAP(M::R) 43.4% 42.7% 34.3% 46.6% 52.8%
GAP(R::S) 37.0% 36.8% 38.7% 37.7% 38.3%


Table 3-3. Robust lot-sizing with backlogging: a single disruption
T=10 T=20 T=30 T=40 T=50
ratio= 10 SLS 5.03 x 103 9.86 x 10 1.56 x 10b 2.08 x 10b 2.44 x 10
MP 8.44 x 105 1.51 x 10b 2.04 x 10b 2.61 x 10b 2.97 x 10
SLS-B 5.38 x 105 1.17 x 10b 1.69 x 10b 2.28 x 10b 2.73 x 10b
GAP(B::S) 7.05% 18.75% 8.35% 9.83% 11.88%
GAP(B::7M) 56.87%7 29.06% 20.71%7 14.47% 8.79%
ratio= 20 SLS 5.03 x 103 9.86 x 10 1.56 x 10b 2.08 x 10b 2.44 x 10
MP 7.96 x 105 1.48 x 10 2.05 x 10 2.60 x 106 3.09 x 10
SLS-B 5.89 x 105 1.20 x 10 1.72 x 10 2.32 x 10 2.83 x 10
GAP(B::S) 17.02% 21.50% 10.50% 11.63% 16.09%
GAP(B::M) 35.14% 23.33% 19.18% 12.07% 9.18%
ratio= 30 SLS 3.49 x 103 7.04 x 10 1.08 x 10b 1.44 x 10 1.85 x 10
MP 6.85 x 105 1.06 x 10 1.42 x 10 1.81 x 10 2.27 x 10
SLS-B 4.48 x 1051 8.28 x 10b 1.13 x 10 1.51 x 100 2.02 x 10
GAP(B::S) 22.09% 17.54% 5.20% 4.85% 9.19%
GAP(B::M) 52.49% 28.02% 25.66% 19.86% 12.37%
ratio= 40 SLS 3.49 x 103 7.04 x 10 1.08 x 10 1.44 x 10 1.85 x 10
MP 7.28 x 105 1.12 x 10b 1.53 x 10b 1.91 x 10b 2.29 x 1
SLS-B 4.70 x 105 8.81 x 10b 1.15 x 10b 1.56 x 10b 2.09 x 10
GAP(B::S) 25.74% 25.23% 6.05% 4.41% 12.97%
GAP(B::M) 35.40% 27.12% 33.04% 22.43% 8.79%


Table 3-4. Robust lot-sizing with backlogging: branch-and-cut
T=100 T=120 T=140
Cut NoCut Cut NoCut Cut NoCut
Time Gap Time Gap Time Gap Time Gap Time Gap Time Gap
ratio=10 472 0% 1800 0.26% 1151 0% 1800 0.56% 1800 0.02% 1800 0.60%
ratio=20 576 0% 1800 0.49% 1273 0% 1800 0.61% 1800 0.04% 1800 0.65%
ratio=30 514 0% 1800 0.67% 1286 0% 1800 1.04% 1800 0.08% 1800 1.35%
ratio=40 644 0% 1800 0.98% 1168 0% 1800 1.21% 1800 0.11% 1800 1.20%





















Table 3-5. Multi-stage robust lot-sizing problem with 2 disruptions
T= 10 T=20 T 30 T =40 T 50
Obj Time Obj Time Obj Time Obj Time Obj Time
ratio=10 SLS(x106) 0.50 0.034 0.99 0.034 1.56 0.034 2.08 0.038 2.44 0.039
MPSLS(x 10) 1.29 0.034 2.06 0.034 2.64 0.034 3.16 0.038 3.66 0.039
NoSetup(x 10) 0.62 0.044 1.12 0.076 1.65 0.191 2.15 0.478 2.64 0.952
Gap(M::N) 116.4% 89.9% 60.2% 47.1% 44.6%
Gap(N::S) 23.2% 13.6% 5.7% 3.5% 8.2%
ratio=30 SLS(x 10) 0.35 0.034 0.70 0.034 1.08 0.036 1.44 0.038 1.87 0.038
MPSLS(x 10) 1.20 0.034 1.63 0.034 1.93 0.036 2.42 0.038 3.03 0.038
NoSetup(x 10) 4.45 0.041 8.03 0.074 1.19 0.209 1.55 0.555 1.91 1.153
Gap(M::N) 169.2% 102.6%/ 62.9 56.1% 58.7%
Gap(N::S) 27.5% 14.1% 10.3% 8.0% 2.1%


Table 3-6. Multi-stage robust lot-sizing problem with 3 disruptions
T = 10 T = 20 T = 30 T =40 T 50
Obj Time Obj Time Obj Time Obj Time Obj Time
ratio=10 SLS(xl10) 0.50 0.034 0.99 0.034 1.56 0.034 2.07 0.038 2.44 0.039
MPSLS(x10) 1.53 0.034 2.44 0.034 3.09 0.034 3.69 0.038 4.24 0.039
NoSetup(x10) 0.73 0.065 1.35 0.557 2.01 3.066 2.62 10.165 3.26 32.688
Gap(M::N) 147.8% 81.4% 53.5% 40.5% 30.2%
Gap(N::S) 45.4% 36.6% 29.0% 26.3% 33.8%
ratio=30 SLS(x 10) 0.35 0.034 0.70 0.034 1.08 0.036 1.44 0.038 1.86 0.038
MPSLS(x10) 1.55 0.034 2.14 0.034 2.49 0.036 3.03 0.038 3.73 0.038
NoSetup(x10) 0.54 0.059 1.01 0.592 1.52 2.615 1.96 9.571 2.45 28.079
Gap(M::N) 185.0% 112.9% 63.4% 54.6% 52.1%
Gap(N::S) 55.7% 42.9% 41.5% 36.7% 31.1%









CHAPTER 4
STOCHASTIC LOT-SIZING PROBLEM WITH DETERMINISTIC DEMANDS AND
WAGNER-WHITIN COSTS

4.1 Introduction

In practice, the cost parameters may be uncertain. This research studies cost

parameter uncertainty and is motivated by tactical production decisions, not operational

production decisions, for chemical companies. As typical supply chain characteristics

of chemical industry, a chemical company produces mainly functional products, which

are defined as ones that have a long product lifecycle and stable demand (see, e.g., Lee

and Chen 2005). Under this situation, the demand is stable and easily forecasted

accurately. However, the cost parameter forecasts may need to be adjusted monthly or

quarterly. We can formulate this problem as a two-stage stochastic lot-sizing problem

(SULS), in which we let a specific time period, e.g., time period p, represent the time

for which the forecast needs to be adjusted. The cost parameters after the given time

period will be uncertain and follow a discrete probability distribution. The detailed

two-stage stochastic integer programming formulation can be described as follows

(cf. Birge and Louveaux 1997; Louveaux and Schultz 2003):
P
min (aixi + 3'zi + h'si) + E Q(x, z, s, (w))
i-i
s.t. xi + si-_ = di + si, 1< i < p

xi < Mzi, < < p

x",s, > 0, z, e {0, 1, 1 < i < p,

where



mai(w)x(W) xl(w) + s2_1(w) = di + s?(w), p + 1 < i < T
Q(x,z,s, (w)) = X2mi2 +w)z(w) xf (w) < Mz (w), p + 1 x2Z2w ) > ) ,
S +h'(w)s#(w) x2(w), s(w) > 0, z2(w) e {0,1}, p + 1 < i < T









Note, here s2 (w) = Sp if i- 1 = p. Decision variables (xi, zi, si) and (x2(w), z2(w), s2(w))
represent the setup decision, and production and inventory levels on the first and the

second stages, respectively. The corresponding cost parameters are (a, 3', h') and

(a(w), '(w), h'(w)).
For SULS, polynomial time algorithms (see, e.g., Guan and Miller 2008; Huang and

K090kyavuz 2008) and efficient cutting planes (see, e.g., Guan et al. 2006b; Summa

and Wolsey 2006) have been studied recently for its deterministic equivalent scenario-tree

based formulations. The reformulation for the problem was originally introduced

in Ahmed et al. (2003). Later on, in Guan et al. (2006a), this reformulation was proved

to be equivalent to adding the (f, S) inequalities in the original formulation, in terms

of providing LP lower bounds. However, both approaches could not provide integral

solutions for SULS. An extended formulation that provides integral solutions for SULS

up to now is only for two-period cases, which was developed in Guan et al. (2006a). To

the best of our knowledge, there is no previous research on developing an extended

formulation that provides integral solutions for multi-period SULS. This is another

motivation for our research, and this paper contributes to the literature on deriving an

extended formulation for multi-period SULS.

4.2 An Extended Formulation

After exploring possible realizations of the second stage random variables

(ai, f3, h'), we can generate a two-stage stochastic scenario tree for the problem as
shown in Figure 4-1. Nodes 1, p are first stage nodes and nodes q, fw are

second stage nodes, where branching node p connects the first and the second stages

and is unique on the scenario tree. Assuming there are W possible scenarios, we let

P(w), 1 < < W, represent the branch where scenario w occurs and accordingly
let p,, 1 < w < W, represent the probability that scenario w will occur. Since there

is no demand uncertainty, we have di = dt(i) > 0 for the second stage nodes, where

t(i) represents the time period of node i. We let V represent the set of nodes on the









scenario tree, and V(i) present the set of nodes which are descendants of node i

(including node i itself). For each node i e V \ {1}, there exists a unique parent,

denoted as node i-. Finally, for each non-leaf node i, we let C(i) represent the set of its

children. For our setting, C(i) contains a single element if the non-leaf node i $ p. The

deterministic equivalent formulation for two-stage SULS can be described as follows:

q1 ,,






Figure 4-1. The scenario tree for two-stage SULS



p w
min Z(aix, + 3'zi + h'si) + p (aixi + 'zi + h'si)
i=1 W=1 i6eP(W);t(i)>t(p)+i

s. t. xi + si- = di + s;, i e V,

xi < Mzi, ie V,

X si > 0, z E { 01, 1, ie V,

where xi, si, and z, present the production level, inventory level, and production set-up

indicator in the state defined by node i whose corresponding time period is t(i). Without

loss of generality, we can assume so = 0.

To define Wagner-Whitin costs for two-stage SULS, we substitute xi = di + s, si to

eliminate the production variable x;. Then, we obtain a reformulation of two-stage SULS

in the (s, z) space as follows:


(,a;x; + 13';z + h' s) + P Pw (a),x + 3z, + h/s)
-i1 W=1 i6P(W); t(i)>t(p)+ i

= [ac(d + ssi-s,) + zi + hsi] + = lw p, (CH(di + si si-) + 3zi + hsi)
-i1 w i6P(w);t(i)>t(p)+











p-1 W p p
= (ai+ h' ai 1)si + (ap + h'p- paqjr)sp + izi + ai di
i- i w i-i i-i
W W
+ >E E()p,((i + h' cc(i))si + E E( Ptp+ 3zi
W=1 iEP(w);t(i)>t(p) w= 1 iEP(w);t(i)>t(p) 1
W
+ E E Paidi
w=1 iP(w);t(i)>t(p)+l

= E(hisi + 3izi) + (a constant),
iEV
where



ai h -ai+, 1< i < P-l

p ,=1 P t(i) > t(p) +1,

Pw(ai + h ac(i)), t(i) > t(p) + 1,

where ac(i) = 0 if t(i) = T.

Definition 1. A two-stage SULS is said to have Wagner-Whitin costs if, for each i C V,

we have

hi > 0 and Ai > /3, with /, = 0 ifj c C(i) and t(i) = T.
jec(i)
Accordingly, we denote the problem satisfying the Wagner-Whitin cost setting

as two-stage SULS-WW. A problem satisfying the Wagner-Whitin cost setting is also

referred to as "without speculative motives" (see, Wagelmans et al. 1992, Pochet and

Wolsey 1994, among others). Under this setting, we will not set up production at node

i e V, if the inventory entering i is sufficient to satisfy the demand at i. The condition

hi > 0 is the same as the deterministic ULS Wagner-Whitin cost setting. Now, we

provide a 4-period example in Figure 4-2, to indicate that 3i > Ejec(i) 3 is necessary to

guarantee the Wagner-Whitin property.

As shown in Figure 4-2, for each node, the unit production cost is zero. The setup

and the unit inventory costs are (0, +oo + oc, 0, 0, +oo) and (100, 0, 0, 0, 0, 0),









3 = +00 ( 5 =0
h3=0 h=5
355
SO -0 ( +_o "
hl 100 'h, 0

h4 0 h6=0
4 6

Figure 4-2. An example of two-stage SULS


respectively. Problem parameters satisfy the condition hi > 0, but do not satisfy the

condition 3i > Yj:e(i) /i in Definition 1. We assume the demand in each node is a

non-zero finite number. In order to satisfy the demand at each time period and minimize

the total cost, the productions are set up in nodes 1, 4, and 5. The production in node

1 covers the demands for nodes 1, 2, 3, and 4. Meanwhile, node 4 is set up to produce

and satisfy the demand in node 6. Thus, there is inventory left from parent node and

production set up for node 4 simultaneously. The Wagner-Whitin property for the

deterministic case, xisi 0, does not hold here.


cp(i)





t(i) ...... t(p) -...... (i)

Figure 4-3. The subtree of node i


Now, we consider the optimal solution form of inventory level si for each i C V. Let

W(i) represent the set of scenarios that will occur after node i. For instance, if i < p,

W(i) = {1,..., W}; otherwise, if t(i) > t(p) + 1, W(i) contains a single element w such

that i P(w). Let b(i) = min{t : [1 Ec),'(_)\} z]+ = 0, w c W(i)} if there exists

a k e TP(wo)\{i}, w e W(i) such that Zk = 1, and b(i) = T + 1 o.w., as shown in

Figure 4-3, where Pi(w) represents the path from node i to its descendant node at time








period t on the branch corresponding to scenario w. Let p(i) = Uecw(i) arg min{t(k)
k e 'P-(w) \ {i} and Zk = 1}. From observation, we see that the optimal inventory
level si covers the demands at time periods after t(i) and before b(i). In the following
proposition, we provide the closed form of inventory level si, i e V:
Proposition 4.1. For two-stage SULS-WW, corresponding to each node i e V, there
exists an optimal solution in the form
,(i)-1
s, = dt, z, = 1 if t(j) = (i), andz = 0 if t(i) < t(j) < (i).
t=t(i)+1

Proof. Let A = {i e : zi = 1}. For each i e V, the conclusion is obvious if p(i) = 0.
Now, we assume p(i) 0 and let 1* = arg max{t(j) : je p(i)}. We prove this proposition
by two claims:
Claim 1: For each i e A, s, = EZt()I dt.
Claim 2: Corresponding to each i e A, zj = 1 ifj c V(i) and t(j) = t(i*); zj = 0 if
j e V(i) and t(i) < t(j) < t(i*).
Proof of Claim 1: First, we have s, > C t) dt in order to satisfy the demands
along each path P ;()() \ {i, i*}, we W(i). If s, > dt()+ dt, for instance, si =

ZPtm)- dt + F for a small positive number e > 0. Then, due to the fact that hi > 0,
we can reduce the production at i by E and increase the production at each j, j e p(i)
by E, which leads to a non-larger total cost. The problem is still feasible, which is a
contradiction; thus, Claim 1 holds.
Proof of Claim 2: Based on Claim 1, we have si = -C- (i), dt. Since /3I > E,,E() 0,
setting up at nodes in the set {j : e V(i) and t(j) = t(i*)} instead of at nodes in the set

{j : J V(i) and t(i) < t(j) < t(i*)} leads to a non-larger total cost. Therefore, there
exists an optimal solution such that zj = 1 if j e V(i) and t(j) = t(i*); zj = 0 if j e V(i)
and t(i) < t(j) < t(i*). Thus, Claim 2 holds.
Based on Claims 1 and 2, the proposition holds. O









Proposition 4.1 provides a stronger claim, as compared to the production path

property for SULS described in Guan and Miller (2008), developed for general cost

and demand settings. Under these general settings, we cannot guarantee that the

Wagner-Whitin cost setting holds. The example in Figure 4-2 can still happen, and

xisi- = 0 does not hold. That is, we can not guarantee zj = 1 if j V(i) and t(j) = -(i),

and zj = 0 ifj c V(i) and t(i) < t(j) < b(i), as described in Proposition 4.1.

Let binary decision variable 6J represent whether the inventory left at node i covers

the demand at time period t, t > t(i). If yes, then 6i = 1; otherwise, 6i = 0. Then,

based on Proposition 4.1, the following three types of inequalities are valid for two-stage

SULS-WW.

1. Path I inequalities: these inequalities are for the second stage nodes. For a given

node i, t(i) > t(p) + 1, u is determined and

6 > zj- t > t(i)+ 1. (4-1)
jE'Pf()\{i}

That is, node i covers demands along the branch it belongs to until the next

production cycle.

2. Path II inequalities: these inequalities are for the first stage nodes (except node p).

For a given node i, 1 < t(i) < t(p) 1, we have

6' > 1 zi +, t = t(i) + 1, (4-2)

S> 6i+1 zi+,, t > t(i) + 2. (4-3)

For the first stage node i (except node p), node i + 1 is its child. Then, (4-2) holds.

Inequality (4-3) indicates that if node i + 1 is not set up and the inventory left from

node i + 1 covers the demand at time period t, t > t(i) + 2, then the inventory from

node i also covers the demand at time period t.









3. Connection inequalities: these inequalities are for the branching node p. Corresponding

to p and each q, e C(p), u e W(p), we have

6t > 1 z,,, t = t(p) + 1, (4-4)

6p > 6q' zq., t > t(p) + 2. (4-5)

Constraints (4-4) and (4-5) are similar to (4-2) and (4-3). Along each scenario

path, if there is no setup in time period t(p) + 1, the inventory left at node p covers

demands up to the same time period as the inventory left at the node in time

period t(p) + 1 does.

Knowing 6i, the inventory level at each node in the scenario tree is as follows:
T
s,= dt6, ieV. (4-6)
t=t(i)+l

Inequalities (4-1)-(4-5) and equation (4-6), assuming binary 6i and zi, guarantee

the feasibility of the reformulation for two-stage SULS-WW because the demand at each

time period will be covered.

Theorem 4.1. The linear program (4-1)-(4-6) plus 0 < 6i < 1 and0 < zi < 1,

i e V, t(i) + 1 < t < T provides an extended formulation for two-stage SULS-WW.

Proof. Constraint (4-6) can be put directly in the objective function. We prove this

theorem by showing that the constraint matrix for Constraints (4-1) to (4-5) is totally

unimodular.

To prove that the constraint matrix for constraints (4-1) to (4-5) is totally unimodular,

we order variables z, and 6i with an outer loop i ranging from 1 to |VI, and an inner loop

t ranging from t(i) + 1 to T. Table 4-1 describes the constraint matrix corresponding to

the example in Figure 4-2.









Table 4-1. The matrix of constraints (4-1) to (4-5) for the example in Figure 4-2

Zl Z2 Z3 Z4 Z5 Z6 61 61 61 62 62 63 64
1 1 1
2 1 1
3 1 1
4 1 1 -1
5 1 1 -1
6 1 1
7 1 1
8 1 1 -1
9 1 1 -1



As the submatrix corresponding to variables 6i, t(i) > t(p) + 2, is an identity matrix,

we need only consider the constraint submatrix for the rest variables, denoted as matrix

A.

We show that for any column subset J of matrix A, there exists a partition J1 and J2

of J such that

Sa,- a, < 1, for each row i. (4-7)
jEI1 jEJ2
We partition 6 and z variables in J, starting from branching node p, and then extend it in

both directions for nodes after p and before p, respectively.

Step 1. Allocate 6P to J1, Zp to J1, and z,, to J2, where t > t(p) + 1, q, E C(p).

Step 2. Allocate -", t > t(p) + 2, to the same set as Zq. (if zq e J), or to the same set

as 6P (if zq, I J and 6P e J), or to Ji(if Zq, J and 6P J).

Step 3. Allocate z;, t(i) > t(p) + 2, to the opposite set of ZM(i), if there exists ZM(i)

in J, where M(i) is the closest ancestor of node i, at or after time period t(p).

Otherwise, allocate z; to the opposite set of 6P if 6b e J. If 6P J, then allocate z, to

J2.

Step 4. Allocate z;, 1 < i < q 1, to the opposite set of Zm(i), if there exists zm(i) in

J, where m(i) is the closest descendant of node i at or before time period t(p).

Otherwise, allocate z, to J1.









Step 5. Allocate 6i, 1 < i < q 1, to the opposite set as Zm(i), if there exists Zm(i) as
shown in Step 4 in J. Otherwise, allocate 6i to J1.

Now, we verify that (4-7) holds for constraints (4-1) to (4-5) under the above

partition. Corresponding to each row, if J contains at most one decision variable in A,

then it is clear that (4-7) holds. In the following, we consider the case where J contains

at least two decision variables in A:

1. For constraint (4-4), if {f6, zq } C J, 6P and zq, go to J1 and J2, respectively, based on

Step 1. Thus, (4-7) holds.

2. For constraint (4-5), we only need to consider the following four cases:
2-1. {6, zq } c J. The argument is the same as constraint (4-4).

2-2. {6t", Z } c J. 6- and z, go to the same set based on Step 2.
2-3. {f6, 6"} c J. Because Zq, J, 6P and 6q" go to the same set based on Step

2.
2-4. {6, zq, z } c J. 6P and zq, go to J1 and J2, respectively, based on Step 1.

Then, (4-7) holds no matter the destination of t6.

3. For constraint (4-1), we consider two types of constraints: where t(i) > t(p) + 2 or

t(i) = t(p) + 1.
3-1. t(i) > t(p) + 2. For this case, first of all, we need not consider 6i based on the

identity matrix argument at the beginning of the proof. Then, based on Step 3,

z; is assigned alternatively to J1 and J2. Thus, (4-7) holds.
3-2. t(i) t(p) + 1. For this case, we should consider 6^ since t(q,) = t(p) + 1.

We discuss the cases where 6Jq" J and 6q" e J.

3-2-1. 6Jq" J. The argument is the same as 3-1.
3-2-2. 6" e J. Based on Steps 2 and 3, we see that t6 is in the opposite

subset of zj., where j* is the closest descendant of q,, because 6 goes
to the same set as zq (if z, e J), or 6P (if zq, J and 6P e J), or J1

(if 6P J and zq, J). Accordingly, zj. is in the opposite set of zq, or









6P, or J1. We also observe that zj,j e ('P"(w) \ {q,}) n J, is assigned
alternatively to J1 and J2 based on Step 3. Therefore, (4-7) holds.
4. For Constraint (4-2), we only need to consider the case where both 6i and zi+l are
in J. Based on Step 5, 65 goes to the opposite set of zi+l, since zi, is the closest
descendant of node i. Thus, (4-7) holds.
5. For Constraint (4-3), we discuss the following four cases:
5-1. {6i, zi+} c J and 6+1 J. The argument is the same as for Constraint (4-2).
5-2. {J6+1, Zi1} C J and 6 J. 6'+1 and Zit1 go to the same set because both 65+1

and zi+, are in the opposite set of Zm(i+i) (if Zm(i+i) e J) or in J1 (if zm(i+l) A J),
based on Steps 4 and 5.
5-3. {1i, 56i1} C J and zi+1 J. Because z i+ J, both 6i and 6~ 1 are in the

opposite set of Zm(i+i) (if Zm(i+,) e J) or in J1 (if Zm(i+l) J), based on Step 5.
5-4. {(1, 6i+, zi+1} c J. Based on Step 5, 65 goes to the opposite set of zi,, since

zi,+ is the closest descendant of node i. Then, (4-7) holds no matter the
destination of 6+1.
Therefore, the desired property (4-7) holds for constraints (4-1) to (4-5), and the
matrix for constraints (4-1) to (4-5) is totally unimodular. D
4.3 An Integral Polyhedron in the Original Space
Now, we study the integral polyhedron in (s, z) space. First, we introduce Qo as a
polyhedron in (s, z) space described as follows:


S= (s, z) : si > dt 1 z < z < 1, Si, > 0, t(i) + 1 < T < T, Wt E W(i)
I tEtP)f(wt)V i )
(4-8)
We prove that Qo is an integral polyhedron for two-stage SULS-WW by showing that it
is a projection of Qr in (s, z) space, where Qr represents the polyhedron of two-stage
SULS-WW in (s, z, 6) space, i.e., Q, = {(s, z, 6) (s, z, 6) satisfies (4-1) to (4-6)}.
Theorem 4.2. Qo is an integral polyhedron for two-stage SULS-WW









Proof. We prove this theorem by showing that Q = Proj(s,z)Q, in the following two

claims:

Claim 1: Inequalities in Qo are valid for Qr. To prove Claim 1, we only need to show

that


T
s, > d t[1 zj]
t t(i)+ 1 jpT( )\{i}

because for given Wt, t(i) + 1 < t < T, we have


Wt e W(i),


Sdt[l z -> z dt(1- zy), t(i) + < < T.
t t(i)+i jE'tp(Wt)\(i} t t(i)-+l jEPf(wt)\(i}

We prove (4-9) by discussing two cases: (1) t(i) > t(p) + 1 and (2) t(i) < t(p).

For t(i) > t(p) + 1, based on (4-6), (4-1), and nonnegativity of 6', we have


T
si d6'^
t=t(i)-1
T
> 5 d ma;
t=-t(i)-+
T
= dt 1i
t t(i)+1

where Wt is the single element in W(i)

holds.

For t(i) < t(p), we have


x 0, 1 z



i +z
jE'Pf(wt)\{i}

for each t(i) + 1 < t < T. Then, (4-9)


s- = dt' = dt(i)+1b dtb6
t=t(i)+l t=--t(i)+2
T
> d,+1[l1- z,+I]+ + dt [6+1 zi+] if t(i) < t(p)
t=t(i)+2
T
or > dq, ,[1- zq, ]+ + > dt [6t zqj]+ if t(i) = t(p),
t=t(i)+2


(4-9)


(4-10)


(4-11)








where (4-10) is based on (4-2), (4-3), and nonnegativity of 6', while (4-11) is
based on (4-4), (4-5), and nonnegativity of 6'. We also notice that if t(i) = t(p),
as shown in (4-11), [+6 z,] > 1 EjEp(wt)\i) z] based on (4-1). Then,
(4-9) holds for the t(i) = t(p) case.
Then, we only need to show that if t(i) < t(p),


[65 zi ]> 1- z( (4-12)
je#t1()\{/}
It is easy to see that (4-12) holds based on (4-2) and (4-3) if t < t(p).
If t > t(p) + 1, because 56 > 65+' z+, i + 1 < j < p 1, holds based on (4-3),
we have

[6 +- ] zi,4 > [ j I +

If t = t(p) + 1, then


Y-z> >_ 1- z- = 1- t zj.
j =i+1 j=i[ je[ r (wO)\{i}
where the inequality follows from (4-4).
If t > t(p) + 1, then

r p 1+ -+ r
6- z -> 6- t Z > 1- E z
ji-+1 j'P(I)+l(t)\{i} jE (Wt)\{i}

where the first inequality follows from (4-5) and the second follows from (4-1);
therefore, (4-12) holds and thus, (4-9) and Claim 1 hold.
Claim 2: For any extreme point (s, z) E Qo, we can construct 6 such that (s, z, 6) e Qr
That is, (s, z, 5) satisfies constraints (4-1) to (4-6). Now, for a given extreme point
(s, z) C Qo, we let
r 1+
= max 1- z (4-13)
S EW (i)
yeP^(A\{i









Since (s, z) is an extreme point in Qo, we first observe that (s, z) satisfies equation

(4-6) based on (4-8) and (4-13). We also observe that (4-1), (4-2), and (4-4)

hold, which directly follows from (4-13).

For (4-3), let w* be the scenario where +i1 achieves the maximum value. Then,

it is easy to observe that w* is also the scenario where 61 achieves the maximum.

Then, according to (4-13),


jEPf(*){is


=- 1


zi+.


Thus, (4-3) holds.

For (4-5), according to (4-13),


Now, for each w E W(p),


S= max 1 -
wEW(p)


S1 for each w c W(p).
jEP (W)\{P}


-+


jE'P (w )\{p,q}

= >[_ z]


(4-14)









where (4-14) follows from the fact that PP(wu) \ {p, q,} = ((w) \ {q,} and
t"- = 1 i, z{q} j. Then, (4-5) holds. Thus, (s,, c) Q,, and Claim 2

holds.

Therefore, the conclusion holds. D

We can observe that with Wagner-Whitin costs, the explicit formulation of the

polyhedral description for two-stage SULS in the original space is described by O(IV|)
variables and O(TWIVI) constraints, where IV| is the cardinality of V.

4.4 Extensions

A part of our results can be applied to a more general multi-stage stochastic

programming setting, which can address further uncertainties in periods p + 1, p +
2,..., T. Under the multi-stage setting, it can be observed that Proposition 4.1 still

holds, based on the Wagner-Whitin cost setting defined in Definition 1. Accordingly,

we can obtain a reformulation similar to constraints (4-1) to (4-6). For instance, we
have constraints (4-1) to (4-3) for the last and the first stage nodes, and equation (4-6)
for the inventory level expression. For the nodes between the first and the last stages,

constraints similar to (4-4) and (4-5) are valid (e.g., 6i > 1 zk, k e C(i), t = t(i) + 1 and
6 > 6tk Zk, k C(i) t > t(i) 2). Therefore, we can obtain a similar reformation with

binary 61 and zi for the multi-stage case. However, it is unknown if the reformulation can
provide an extended formulation that provides integral solutions for multi-stage SULS;

this possibility is currently under investigation.









CHAPTER 5
STOCHASTIC LOT-SIZING PROBLEM WITH DETERMINISTIC DEMANDS AND
BACKLOGGING

5.1 Introduction

Pochet and Wolsey (1988) provided the first polyhedral study of the uncapacitated

lot-sizing problem with backlogging and the convex hull description for the problem was

recently studied by K090kyavuz and Pochet (2007). In addition, for the uncapacitated

lot-sizing problem with start-up cost, van Hoesel et al. (1994) presented an extended

formulation and an O(T2) time separation algorithm. For the Wagner-Whitin costs case,

Wagelmans et al. (1992) introduced the Wagner-Whitin costs, i.e., ai + h, > aji+, for

all time period 1 < i < T 1,, where a, and h' are the unit production and inventory

costs for time period i, and implemented an O(T) time dynamic programming algorithm

to solve ULS with Wagner-Whitin costs. For the Wagner-Whitin cost case, Pochet and

Wolsey (1994) generated an explicit formulation of convex hull for ULS with backlogging

with O(2T) constraints and an O(T2 log T) time separation algorithm.

The deterministic equivalent formulation for two-stage SULS with backlogging and

Wagner-Whitin costs can be described as follows:

p W
min Z(aixi + 'z h's;+ bi)+ p (ai xi + 3'zi + h's; + b'i)
i=1-i w=1 iew,t(i)>t(p)+

s.t. xi + si- + i = di + si + fi i e V

(SULSB-WW) xi < Mizi, i E V

xi, si, Ii > 0, zi e {0, 1, ie V,

where xi, si, i, and zi present the production level, inventory level, backlogging level,

and production set-up indicator in the state defined by node i whose corresponding

period is t(i). Node i- is the parent node of node i. Without loss of generality, we can

assume so = 0, ti = 0, and tighten Mi = Tt(i) dk, where i C









In the remaining part of this chapter, we study two-stage stochastic ULS with

backlogging, Wagner-Whitin costs and deterministic demands, denoted as two-stage

SULSB-WW. We examine the optimal solution property of the model and use it to

generate an extended formulation in the higher dimensional space. Then we prove that

the constraint matrix for the extended formulation is totally unimodular. We also project it

back to a lower dimension space such that we can find valid inequalities that can provide

tighter extended formulation of the problem.

5.2 An Extended Formulation for Two Stage SULSB-WW

In this section, we study the optimal solution forms of inventory and backlogging

levels for two-stage SUSLB-WW and generate a reformulation which can describe the

integral polyhedra of the problem in the higher dimension space.

First, we define Wanger-Whitin costs for two-stage SULSB. We substite xi =

di + si i Si- + to eliminate decision variable xi in two-stage SULSB. Then, we get

a reformulation of two-stage SULSB in (s, f, z) space as follows:





i=1 w 1 iEPw,t(i)>t(p)+l
P
= [aJi(di + s,- s,_1 + ~ i) + ,/3zi + h'si + blil]
i-1

+ ii pw t a,(di + s, s, + -) + ';zi + h;si + b bf)
w=1 iePw,t(i)>t(p)+1
p-1 p
= [(a + hi- a,)s, + (b'- a, + i,)f + 3;zi] + aid;
i-1 i-1


+ (ca + h'p


W

w=1ljPw,t()>_t(p)+1


W W
5waqw)Sp + (b' ap + Y Pw+aq,)p + j3,z
w=1 w=1


W
p,(hj j+ ac))s, + p
W= jiePw,t()>_t(p)+l


pw(bj aj + ac()tj)


100








W W
+Y pwOzj + pajdj,
w=ljePwt(j)>t(p)+1 w= lje tw,')t( t(p)+1

= (hisi + bi ~ + 3izi) + (a constant),
iEV
where



a, + h aC,1, 1 < t(i) < t(p) 1 b a i, 1 < t(i) < t(p) 1
hi = p + h' EW I PCq, i t(p) = b' ca + EY I p t(i) = t(p

pw(h + a c(i)), t(i) > t(p) + 1. p a + c(i)) t(i)> t(p)+ 1.

and pi = 3, if 1 < t(i) < t(p); otherwise, pi = p-P/. We let ac(i) = 0 if i E where C(i)

represents the set of children of node i and L represents the set of leaf nodes.

Definition 2. A two-stage stochastic lot-sizing problem with backlogging is said to have

Wagner-Whitin costs if

hi > 0, bi > 0, and p3 > Y 0 (5-1)
jEC(i)
for alli c V.

For the deterministic version, i.e., ULSB with Wagner-Whitin costs, the optimal

solution forms of inventory and backlooging levels are studied by Pochet and Wolsey

(1994). For two-stage SULSB, we consider the optimal solution forms of inventory level

si and backlogging level i for each i e V. In the optimal solution for a two-stage SULSB

with Wagner-Whitin costs, the demand for each node i will be satisfied (1) by setting up

the production at node i, (2) by inventory left from its parent node, or (3) by backlogging

from its children. Before we describe the proposition, we show a 3-period example to

demonstrate the optimal solution form as shown in Figure 5-1.

In this example, productions are set up at nodes 1, 3, and 4. Demand at node 1 is

satisfied by the production of itself. Demand at node 2 is satisfied by backlogging from

node 4. Node 3 covers demands in nodes 3 and 5. Thus the second stage nodes 2 and











0


Figure 5-1. The scenario tree for a 3 period SULS with backlogging

3 are backlogged and set up respectively. Meanwhile, the first stage node 1 does not
provide inventory for any second stage node.
.............. ........................................... L ........
|,-"-----------'---"-- -" A(J
I I.
i I I 10--- 0


S---------------
t(i) (i)

Figure 5-2. The subtree of node i


Now we consider the optimal solution forms of inventory level si and backlogging

level f for two-stage SULSB-WW. First, we introduce binary decision variables ft and

gi to indicate if node i is stocked or backlogged. If node i is stocked, f = 1; otherwise,
S= 0. If node i is backlogged, gi = 1; otherwise, gi = 0. Second, extra notation is

introduced. As shown in Figure 5-2, let b(i) = min{t(j) : zj = 1 or gj = 1,j c V(i) \ {i}}.
That is, b(i) represents the time period of the earliest descendant of node i which is

set up or backlogged. Accordingly, we define node set V(i) = r r c V(i), t(i) <

t(r) < b(i)}. Let A(i) = Ul<
0(i) = UjeA(i)P(i,j) \ {i,j}, where 'P(w;) represents the path from node i to node
which is at time period t and on the branch corresponding to scenario w. From the

observation, it is obvious that the optimal inventory level si covers the demands in

periods after period t(i) and before period b(i). Each descendant of node i in time

period b(i) sets up or gets backlogging. And each descendant of node i in set A(i) sets


102









up. In the following proposition, we provide the closed form of the inventory level s; and

the backlogging level i, i E V.

Proposition 5.1. For two-stage SULSB-WW,(1) there exists an optimal inventory level of

the form:
y(i)-i
si = d,, ieV, (5-2)
t=-t(i)-+
and an optimal backlogging level of the form:

tU) F
J = max 5dt cg,jt)- zr jE Q<(i) (5-3)
1 7 tO") rEP(0(", t),j)

where Tl(j, t) is the ancestor node of nodej at time t. That is, Tr(j, t) = {k e P(j)

t(k) = t}.

(2) The optimal solutions of two-stage SULSB-WW satisfy


fV + g + zi = 1, i V (5-4)

gi+zi>gi ieV\{1} (5-5)

fi- + z- >_ fi, V\ {} (5-6)

Proof. Let A = {i e V: z, = 1}. First, for each i e A, we prove that (5-2) and (5-3) hold

by showing (5-2), and

tO)
=- dt for each j e Q(i) \ W(i),j e Q(i) \ 'j(i) (5-7)
t-- (i)

and (5-3) hold.

Proof of (5-7). We prove (5-7) under 3 cases.

Case 1: (5-2) does not hold and = E tOi) d holds. In this case, s* is either larger or

less than E ti dt.

Case 1.1. If s* < ZEit() dt, then there exists at least one node j in which

t(j) = b(i) 1 whose demand is not satisfied. Thus, s, is not a feasible solution.


103








Case 1.2. If si > d, then let s = E dt- where 0 si* EZot(~ 1 d. It can be observed that (s, f*) is also a feasible solution and leads
to a non-larger total cost, which is a contradiction. Thus Claim 1 holds.
Case 2: = E(i) dt does not hold, but (5-2) holds. We can give the similar proof as
in Case 1 to find a contradiction.
Case 3: Neither (5-2) or nor j = E t(i) dt holds. In order to satisfy the demand
in each node in 0(i), we can construct two feasible solutions for two-stage
SULSB-WW.
Let sil = s* + E, where E is a small positive number and nodej in A(i) produces E
less. Then f = J E, j e (i) \ 1(i). The corresponding objective value is

F1 = (hisi + bii + iy,) + hjE bjF.
iev jCe(i) jCe (i)\V(i)
Let s2 = s, *-, and nodej in A(i) produces e more. Then 2 = ,+c, j e 0(i)\ (i).
The corresponding objective value is

F2 = (his, + bifi + iy) h + bj .
iEv jE'.(i) jeT(i)\1(i)

If Zje^(i) hj < Zjeq(i) \(i) bj, then F1 < F*; F* is the optimal objective value.
If je,'(i) hj > Yj:(i)\;(i) bj, then F2 < F*; This contradicts with the assumption
that F* is the optimal objective value.
Note here, if E~ ~(i) h = jC, ()\*(i) bj, we increase (or decrease) s* and decrease
(or increase) to match the optimal form. Therefore, (5-7) holds.
Proof of (5-3): According to (5-7), s, covers demands for nodes in v(i). In order
to minimize the objective function, nodes in '(i) do not obtain backlogging. Thus
gj = O,j e j(i). Therefore, j- = max{,:1 tZt I(i) dt
(T = max:<- Therefore, (5-2) and (5-3) hold.









Second, we prove that conditions (5-4), (5-5), and (5-6) hold.

According to the definition of fi and gi, i E V, it is obvious that Conditions (5-5) and

(5-6) hold.

Now we prove that Condition (5-4) holds by two cases.

Case 1. Ifj c (i), then the demand in nodej is satisfied by inventory and f; = 1. In

order to keep the smallest production cost, fj + gj + zj = 1.

Case 2. Ifj c 0(i) \ V(i), then f; = 0. and we need to prove gj + zj = 1. In order to
satisfy the demand, gj + zj > 1.

Case 2.1. Ifj c A(i), zj = 1. If gi = 1, then we can let nodej produce more
to cover the backlogging amount and reduce the objective value. Contradiction!

Therefore, gj = 0 and f; + gj + zj = 1.

Case 2.2. Ifj e 0(i) \ W(i), gj = 1. According to (5-3), the demand of nodej
can be covered by backlogging from its children. In order to minimize the objective

function, zj = 0.

Therefore, based on cases 1 and 2, f; + gj + zj = 1,j e 0(i). D

Under (5-4), at most one of g(i,k) and Z,(i,k) equals to 1. Then g,(i,k) Z(i,k) = 1 if

gl(i,k) = 1; g9(i,k) Zq(i,k) < 0 if g(i,k) = 0. Thus,


t( ) r-
=- max dt g,.,t) Zr
t(J rEZ P()(jt),j)\{j}




tt)) +

= maxC t g(jt)- zr 1 < 7< t(j). (5-8)
t= r P(q(j,t),j)\{j,7(j,t)}


105









From (5-4), we have


f + gi + zi = f- + gi + zi- = 1. (5-9)

Thus one of constraints (5-5) and (5-6) is redundant. That is, if constraint (5-5) holds,

then (5-6) must hold from (5-9).

Let binary variable mt represent whether the inventory left from node i covers the

demand at time period t, t > t(i) + 1. If yes, m' = 1; otherwise, mi = 0. Let binary

variable ni represent whether the backlogging at node i covers the demand at time

period t, t < t(i). If yes, ni = 1; otherwise ni = 0. Finally we let (i, t, w) represent the

node which is a descendant of node i at time period t and at the branch corresponding

to scenario w.

Now we introduce three types of inequalities corresponding to the optimal inventory

level for node i on the scenario tree.

1. Path I inequality: this type of inequality is for the second stage nodes. For a given

node i on the second stage, w is determined and

m[i > ,- zj, t(i) > t(p) + 1, t > t(i) + 1. i > p. (5-10)
j P (w)\{i,i(i,t,w)}

That is, node i covers demands along the branch it belongs to until next backlogging

or production happens.

2. Path II inequality: this type of inequality is for the first stage nodes (except node p).

For a given node i on the first stage, node i + 1 is its child node and

m' > fi+t, 1 < t(i) < t(p) 1, t = t(i) + 1, (5-11)

m' > mi1 z+,, 1 < t(i) < t(p) 1, t > t(i) + 2. (5-12)

Inequality (5-11) indicates that if the demand of child node i + 1 is satisfied by

inventory, then the inventory left from node i covers the demand of node i + 1.

Inequality (5-12) indicates that if node i + 1 is not set up and the inventory left from


106









node i + 1 covers the demand up to time t, t > t(i) + 2, then the inventory left from

node i also covers the demand up to time t.

3. Connection inequality: this type of inequality is for the branching node p,


m > fq, qw E C(p), t = t(p) + (5-13)

m m' z, q, C(p), t > t(p) + 2. (5-14)


Inequalities (5-13) and (5-14) are similar to (5-11) and (5-12). The inventory left

from node p will cover demands from period t(p) +1 to t- 1 unless there is a setup

or a backlogging, before or at time period t along each scenario path.

With the information of mi, the inventory level left from each node i in the scenario

tree is as follows:
T
s,= dtim', ieV (5-15)
t=t(i)+1
Because for a given node i, its ancestor at time period t, rl(i, t), is unique, we have

the following two inequalities hold for each node i e V based on (5-8).


n't > g(it) zj, i e V and t < t(i). (5-16)
jE'P(zr i,t),i)\{ri(i,t)}
t(i)
= dtn', i eV. (5-17)
t=-
Constraints (5-4), (5-5), (5-10) (5-17) guarantee the feasibilities of the reformulation

for two-stage SULSB-WW problem since the demand for each time period is covered.

Now, we show that constraints provide the extended formulation for two stage SULSB-WW.

Proposition 5.2. Constraints (5-4), (5-5), and (5-10) to (5-17) provide the extended

formulation for the two-stage SULSB-WW problem.

Proof. Because constraints (5-15) and (5-17) can be directly transferred to the

objective function. We prove this proposition by showing the constraint matrix for

constraints (5-4), (5-5), (5-10) to (5-14), and (5-16) is a totally unimodular.


107









To prove the constraint matrix for constraints (5-4), (5-5), (5-10) to (5-14), and

(5-16) is totally unimodular, we order variable f., gi, and zi with loop i ranging from 1 to

|VI. Variable mi is ordered with an outer loop i ranging from 1 to |VI and an inner loop t
ranging from t(i) + 1 to T. ni is ordered with an outer loop i ranging from 1 to |VI and an

inner loop t ranging from 1 to t(i). Table 5-1 shows the constraints matrix corresponding

to Figure 5-1.

As the the submatrix corresponding to variable ni is an identity matrix for i e V

and t < t(i), we do not need to consider ni in our construction. As the submatrix

corresponding to variable mi is an identity matrix for t(i) > t(p) + 1, i e V, and

t > t(i) + 2, we only need to consider mi, t(i) < t(p), t > t(i) + 2 in our construction.

That is, we only consider the associated constraint submatrix for the rest variables,

denoted as A.

We show that for any column subset J of matrix A, there exists partition J1 and J2 of

J such that

a a < 1 (5-18)
jEJ1 jEJ2
for all i. We do the partition of variables f, g, z, m in J starting from branching node p

and then extend it to both direction for nodes after p and before p respectively.
First, we define M(i) as the closest ancestor of node i such that ZM(i) e J and m(i)

as the closest descendant of node i such that zm(i) E J.

In the following steps 1 and 5, we allocate the decision variables m and z:

Step 1. Allocate mp to J1, z, to J1, and q,, to J2, where t > t(p) + 1, q, E C(p).

Step 2. Allocate m", t > t(p) + 2 to the same set as z,, (if zq, e J), or to the same set

as mP (if zq,, J and mP e J), or to Ji (if Zq, J and mP J).
Step 3. Allocate z1, t(i) > t(p) + 2, to the opposite set of ZM(i), if M(i) exists and

t(M(i)) > t(p). Otherwise, allocate zi to the opposite set of mP if m[ e J. If mP J,
then allocate z; to J2.


108









Step 4. Allocate zi, 1 < i < q 1, to the opposite set of ZD(i) if m(i) exists and

t(m(i)) < t(p). Otherwise, allocate zi to J1.
Step 5. Allocate mi, 1 < < q 1, to the opposite set as Zm(i), if there exists Zm(i) as

shown in Step 4 in J. Otherwise, allocate mi to J1.

In the following Steps 6 and 7, we allocate the decision variables f and g:

Step 6. Allocate f, to the same set of ZM(i), if ZM(i) e J; allocate f, to the opposite set of

zi, if ZM(i) J, zi e J; allocate f, to the opposite set of Zm(i), if Zm(i) e J, ZM(i), zi J ;

allocate f, to J1, if Zm(i), ZM(i), Zi J J.

Step 7. Allocate gi to the same set of Zm(i), if Zm(i) e J; allocate gi to the opposite set of

zi, if Zm(i) J, zi e J; allocate gi to the opposite set of ZM(i), if Zm(i), z; i J, ZM(1) e J;

allocate gi to J2, if Zm(i), z;, ZM(i) f J.

Following the above partition steps, we observe the following two properties.

Claim 1. If {z;, ZM(;)} c J, z; goes to the opposite set of ZM(i) for all i e V \ {1}.

Proof of Claim 1. If 1 < t(M(i)) < t(i) < t(p), because the closest descendant of

M(i) is i and t(i) < t(p), m(M(i)) = i, ZM(i) goes to the opposite set of zi based on Step

4.

If 1 < t(M(i)) < t(p) < t(i) < T, ZM(i) goes to J1 based on Step 4 and zi goes to J2

based on Step 3.

If t(p) = t(M(i)) < t(i) < T, ZM(i) goes to J1 based on Step 1 (i.e., M(i) = p), z;

goes to J2 based on Step 2 (if i = q,) or Step 3 (if t(i) > t(p) + 2). Thus, z; goes to the

opposite set of ZM(i).

If t(p) + 1 < t(M(i)) < t(i) < T, zi goes to the opposite set of ZM(i) based on Step

3.

Therefore, Claim 1 holds. o

Claim 2. If j(Wkl) and j(wk2) be the first second-stage nodes corresponding to

scenarios Wkl and Wk2 such that z,(wk), Z,(Wk) e J, then z,(wk) and zj(wk2) go to the same

set.


109









Proof of Claim 2. For any first second-stage node j(w), if j(w) = qw, zj(w) goes
to J2 based on Step 1, otherwise j(w) goes to J2 based on Step 3 due to the fact that

j(w) is the first second-stage node in J and on the branch corresponding to w and
t(p) < t(j(w)) < T. Thus, the claim holds. o
Now we verify that (5-18) holds for constraints (5-4), (5-5), (5-10) to (5-14), and

(5-16) under the above partition. At first, corresponding to each row, if J contains at

most one decision variable in A, then it is obvious that (5-18) holds. In the following, we

consider the case that J contains at least two decision variables in A.

1. For constraint (5-4), we discuss the following 4 cases.

1-1. {f,,gi} J and zi J

1-1-1. If Zm(i), ZM(i) e J both M(i) and m(i) exist, fi goes to the same set of

ZM(i) based on Step 6 ; gi goes to the same set of Zm(i) based on Step 7.
Because M(m(i)) = M(i) due to z; J, zm(i) goes to the opposite set

of ZM(i) based on Claim 1. Thus, fi goes to the opposite set of gi. Then,

(5-18) holds.

1-1-2. If ZM(i) e J, Zm(i) i J, M(i) exists, but m(i) does not exist, f, goes to

the same set of ZM(i) based on Step 6; g, goes to the opposite set of ZM(i)

based on Step 7. Thus, fi goes to the opposite set of gi.

1-1-3. If Zm(i) e J, ZM(i) J, m(i) exists, but M(i) does not exist, g, goes to

the same set of Zm(i) based on Step 7; fi goes to the opposite set of zm(i)

based on Step 6. Thus, fi goes to the opposite set of gi.

1-1-4. If Zm(i), ZM(i) f J, both m(i) and M(i) do not exist, fi and g, go to J1 and

J2 based on Steps 6 and 7 respectively.

1-2. {gi,z} CJ and f, J

1-2-1. If Zm(i) e J, m(i) exists, g, goes to the same set of Zm(i) based on Step

7. Because M(m(i)) = i, Zm(i) goes to the opposite set of z, based on

Claim 1. Thus, g, goes to the opposite set of z;. Then, (5-18) holds.


110









1-2-2. If Zm(i) J, m(i) does not exist, gi goes to the opposite set of z, based

on Step 7. Thus, (5-18) holds.

1-3. {f,, zi} e J and gi J

1-3-1. If ZM(i) e J, M(i) exists, f, goes to the same set of ZM(i) based on Step

6; zi goes to the opposite set of ZM(i) based on Claim 1. Then, f, goes to

the opposite set of z1. Thus, (5-18) holds.

1-3-2. If ZM(1) i J, M(i) does not exist, f, goes to the opposite set of zi based

on Step 6. Thus, (5-18) holds.

1-4. {f,, gi, zi} e J, this conclusion directly follows 1-3, because f, goes to the

opposite set of z1. Then (5-18) holds no matter where gi goes.

2. For constraint (5-5), we discuss the following 4 cases.

2-1. If {gi, zi} e J and g J this condition is the same as 1-2.

2-2. If {g-, zl} e J and gi ( J g- goes to the same set as z, due to z, = Zm(i-)

based on Step 7. Thus, (5-18) holds.

2-3. If {gi, gi-} e J and zi J, we first have m(i) = m(i-). If (a) m(i) = m(i-)

exists, then g, and g- go to the same set as Zm(i) based on Step 7. If (b)

m(i) = m(i-) does not exist and z;- e J, then g- goes to the opposite set

of z- based on Step 7. Also,, because z; e J, we have ZM(i) = z;-. Then

g, goes to the opposite set of zi based on Step 7. Thus gi and gi go to the
same set. If (c) m(i) = m(i-) does not exist, z;- J and M(i-) exists, then

ZM(i) = ZM(i-). Note that z, J, we have gi and gi go to the opposite set
of ZM(i) = ZM(- ) based on Step 7. Thus, g, and g- go to the same set. If (d)

m(i) = m(i-), M(i-) do not exist, zi- J, then both gi and g- go to J2 based

on Step 7.

2-4. If {gj, gi-, z;} e J, this conclusion directly follows 2-2, gi and z, go to the

same set. Then (5-18) holds no matter where g, goes.

3. For constraint (5-10), we consider 2 cases:









3-1. t(i) > t(p) + 2. For this case, first, we do not need to consider mi based

on the identity matrix argument at the beginning of the proof. Then, based

on Step 6, f(i,t,w) goes to the same set of Zm(v(i,t,w)), where ZM(i(i,t,w)) is the

largest-index node in J and path pi(w). Note here, here must exist at least

one such M((i, t, w)) based on the assumption that we have at least two

elements in J for each constraint. Based on Step 4, zj alternatively goes to J1

and J2 based on Step 4, where t(p) + 2 < t(j) < t. Thus, (5-18) holds.

3-2. t(i) = t(p) + 1. For this case, i = qw, for some 1 < w < W. We discuss the

following two cases.

3-2-1. If mw J, this argument is the same as 3-1.

3-2-2. If mqw c J, under this condition, we discuss ft(q,,,t,) i J and

fi(qw,t,w) c J respectively.
3-2-2-1. f(q,,,t,w) i J. Under this case, if zq,, J, mq" and zm(qw) go

to the same set and the opposite set of zq based on Steps 2 and

3 respectively; if zq,, J, mq" goes to the same set as mP (i.e.,

Ji) if mP e J or J1 if mf J based on Step 2 and similarly zm(qw)

goes to J2 based on Step 3. Thus, m" and Zm(qw) go to the opposite

sets. Besides these, zj, J alternatively goes to J1 and J2 where

t(p) + 2 < t(j) < t based on Step 3. Thus, (5-18) holds.
3-2-2-2. f(q,,t,w) E J. Under this case, we discuss two cases depending

on if there exists a nodej c P'(w) \ {i, (i, t, w)} such that zj e J.
3-2-2-2-1. If no such node j exists, then {f (q,,t,w), mqI} e J,

based on our assumption that at least two elements in each

constraints in matrix A. If zq e J, M((qw, t, w)) = q,. Based

on Steps 6 and 2, f(qw,t,w) and m1" go to the same set of z,.

If zqw J, based on Step 2, mqw goes to J1. In the following,

we prove that f,(qt,w) goes to J1 in this case. Based on Step 6,


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f (qw,t,w) goes to (a) the same set of ZM(q(qw,t,w)) (if ZM( t(q,,t,w)) e J if
M((q,, t, w)) exists). For this case, ZM(,(qw,t,w)) E J M(_(qw, t, w))
exists and z,, J. Based on Step 4, ZM( (q(,t,w)) goes to J1, because
1 < t(M(((q,, t, w))) < t(p). (b) the opposite set of z (q,,t,w)

(if ZM((qw,t,w)) i J, M((q,, t, w)) does not exist, z(qw,t,w) E J).
For this case, z (qw,t,w) e J and ZM(,(qw,t,w)), Zqw J, Zqw J and

M(q(q,, t, w)) does not exist. Then, Z (qw,t,w) goes to J2 based
on Step 3 and according f(qw,t,w) goes to J1. (c) the opposite

set of Zm((qw,t,w)) (if ZM((qw,t,w)), Z(qw,t,w) i J, Zm((qw,t,w)) E J,

m(q(qw, t, w)) exists, but M(((q,, t, w)) does not exist, z(q,t,w) J).
For this case, zm( ,(qwt,w)) e J, Z q(q,,t,w), ZM((q,,t,w)), Zqw, J. Then,

Zm((qw,t,w)) goes to J2 based on Step 3. Thus, accordingly f(q,,,t,w)
goes to J1. Then f(qw,t,w) and mt" go to the same set. (d) J1 (if

ZM(n(qw,t,w)), Z(qw,t,w), Zm((qw,t,w)) J M( w(q~ t, )) and m(((qw, t, w))
do not exist and z(qw,t,w) J).
3-2-2-2-2. If there exists a such node j, then f (q,,t,w) goes to the

same set of ZM((q,,t,w)) based on Step 6. If Zq, e J, m"qw goes to the
same set of zq based on Step 2, and zm(qw) goes to the opposite set
of Zq, based on Step 3. If zqw J, m" goes to J1 based on Step 2
and Zm(qw) goes to J2 based on Step 3. Thus, for both Zq, e J and

z,, f J, we have mq" goes to the opposite set of Zm(qw). Besides
these, zj alternatively goes to Ji and J2,j e P'(w) \ {i, ((i, t, w)}.
Thus, (5-18) holds.
4. For constraint (5-11), we only need to consider the case if {mi, fi,+} e J.
4-1. If M(i+ 1) exists, f + goes to the same set as ZM(i+) based on Step 6. If m(i)
exists and t(m(i)) < t(p), mi goes to the opposite set of zm(i) based on Step
5. Because m(i) = m(M(i + 1)), ZM(+I) and zm(i) go to the opposite set based


113









on Step 4. Thus f,+l and mi go to the same set. If m(i) does not exist, or m(i)
exists and t(m(i)) > t(p), then m(M(i + 1)) does not exists. Thus, ZM(/+I)
and m, go to J1 based on Steps 4 and 5. Thus mi and f,+l go to the same set.
Therefore, (5-18) holds.
4-2. If M(i + 1) does not exist and z;,+ e J, then m(i) = i + 1, Thus, mi and f+l go
to the opposite set of zi1 based on Steps 5 and 6. Thus, (5-18) holds.
4-3. If M(i + 1) does not exist, zi+ 1 J, and m(i + 1) exists, then, m(i) = m(i + 1).
If 1 < t(m(i + 1)) < t(p), m, and f+ go to the opposite set of Zm(i+l) based
on Steps 5 and 6; otherwise, if t(m(i + 1)) > t(p), mi goes to J1 and f +j
goes to the opposite set of Zm(i+1). Since Zm(i +) goes to J2 based on Step 1 (if
t(m(i + 1)) = t(p) + 1) or Step 3, under this case, M(m(i + 1)) does not exist.

f4+ and ml go to the same set. Then (5-18) holds.
4-4. If neither M(i + 1) nor m(i + 1) exists and z;+i J, mi and f,+ go J1 based on
Steps 5 and 6. Then, (5-18) holds.
5. For constraint (5-12), we discuss the following four cases:
5-1. {mi, zi } c J, and mi~1 J. Based on Step 5, mi goes to the opposite set of

z, + since z/+~ is the closest descendant of node i. Thus, (5-18) holds.
5-2. {m~1, zz+1} c J, and m, J. Under this case, mn1 and za+ will go to the
same set, due to both z,+i and mi1 are in the opposite set of Zm(i) (Zm(i+)), if

m(i + 1) exists and t(m(i + 1)) < t(p) or in J, otherwise (otherwise, in J1),
based on Steps 4 and 5.
5-3. {mi, m '1} c J, and z/+l J. Because z,+l m J, m[ and mi1 are in the same
set based on Step 5.
5-4. {mi, m +, zi+l} c J. The conclusion follows from the fact that based on Step
5. ml goes to the opposite set of zi,+, since zi~1 is the closest descendant of
node i. Then we have (5-18) holds no matter where mi1 goes.









6. For constraint (5-13), we only need to consider the case in which {fqw, mP} c J.

Based on Step 1, mP goes to J1. In the following, we show that f,, goes to Ji for

the following four cases.

6-1 If M(q,) exists, then ZM(q,) goes to J1 based on Step 1 if M(q,) = p, or Step

4 because (m(M(q,))) > t(p). Thus, based on Step 6, f,, goes to the same

set of ZM(q,), which is J1. f,, and mP go to the same set. Thus, (5-18) holds.

6-2 If zq, e J, M(q,) does not exist, then q,, goes to J2 based on Step 1. Based

on Step 6, f,, goes to the opposite set of zq, which is J1.

6-3 If m(q,) exists, z,, J and M(q,) does not exist, then t(m(q,)) > t(p) + 2

and Zm(qw) goes to the opposite set of mP based on Step 3. Based on Step 6,

we have f,, goes to the opposite set of Zm(qw).

6-4 If m(q,) and M(q,) do not exist, zq,, J, f,, goes Jibased on Step 6. Thus,

fq, and mP go to the same set.
7. For constraint (5-14), we need to consider the following four cases.

7-1. {mP, q},, c J, and mq" J. Based on Step 1, mP and z,, go to J1 and J2

respectively. Thus (5-18) holds.

7-2. {mq", Z,} c J, and m J. mq, and z,, go to the same set based on Step 2.

7-3. {mp, m"w} c J, z,, J. Under this case, mP and m"w go to the same set

based on step 2.

7-4. {mp, m q, z,} c J. Under this case, mP and zq go to Ji and J2 respectively

based on Step 1. Then (5-18) holds no matter where mq" goes.

8. For constraint (5-16), note here, we do not need to consider ni based on the identity

matrix argument at the beginning of the proof. Based on Step 7, g,(i,t) goes to the

same set of zm((i,t)). If t(m(r((i, t))) < t(p) or t(rT(i, t)) > t(p) + 1, then r(i, t)

and m(T(i, t)) are one to one corresponding. Otherwise, all Zm(,(i,t)) in different

scenarios go to the same set based on Claim 2 because t(m(Tl(i, t))) > t(p) + 1.


115









Besides these, zj alternatively goes to Ji and J2 based on Claim 1. Thus, (5-18)

holds.

Therefore, the desired property (5-18) holds for constraints (5-4), (5-5), (5-10) to

(5-14), and (5-16), and the corresponding constraint matrix is totally unimodular. Thus,

the extended formulation provides an integral solution for two-stage SULSB-WW. D

Now we study the integral polyhedra in the (f, g, z, s, f) space to generate a tighter

extended formulation for the two-stage SULSB-WW. First, we define


QM ={(f, g, z, s, ) (f, g, z, s, ) satisfies
T
si > dt(f(it,w)- z,), wt W(i), t(i) + 1 < < T, i V,
t=t(i)+l jePT(wt)\{iI,(it,Wt)}
(5-19)
t(i)
S> t(gi,- zj), 1 < 7 < t(i), i (5-20)
t T jep(r7(i,t),i)\{i7(i,t)}

(5-4) and (5-5),

0 < fi, gi, zi < 1, Si, i > 0, i e V}.

We prove that QM is an integral polyhedra for the two-stage SULSB-WW in the

(f, g, z, s, f) space, by showing that it is a projection of QH in the (f, g, z, s, ) space,
where QH records the polyhedra of the two-stage SULSB-WW in the (f, g, z, m, n, s, f)

space, i.e.


QH ={(f, g, z, m, n, s, ) : (f, g, z, m, n, s, f) satisfies

(5-4), (5-5), (5-10)to (5-17),

O < fi, gi, zi, m't, n, <, si, ,i > 0, t(i) + 1 < t < T, 1 < t' < t(i), i E V}.

Proposition 5.3. Proj(f,g,z,s,)QH = QM-

Proof. We prove this proposition by showing the following two claims.


116










Claim 1. All inequalities in QM are valid for QH.

Claim 2. For an arbitrary extreme point (f, g, z, s, f) e

(f,g,z,m,n,s, ) e QH.

Proof of Claim 1. We prove Claim 1 by showing that


T
S> d[ft(i)+l
t=t(i)+1


t(i)
i > dt[gl(i,t)
t 1


je'P(wt)\{ii(i,t,wt)}


zj] wt E W(i),


je'P(n(i,t),i) \{n(i,t)}


QM, there exists a


(5-21)



(5-22)


are valid for QH, because if (5-21) and (5-22) hold, then


T
Si >
t=t(i)+-1


> 5 dt(f,(i,t,wt)
t=t(i)+-1
t(i)
t= >e d[gi,t)
t- 1 jEDP

> 5 dt(gq#i,t) -
t=T jET


jeVPi(wt)\{i,(i,t,wt)}


jep'(wt)\{i, (i,t,w)}


* (it),i)\{ (i,t)}
S(.-.O '')


t(i) + 1 < 7 < T;


1 < 7 < t(i).


Based on (5-16), (5-17) and the nonnegativity of ni, we have


t(i
t-i


= > dt[g i,t)
t 1


t(i)
dtn't > dct max{0, g,(i,t)
t=1


jEP(lq(i,t),i)\{l0(i,t)}


jEP(l(i,t),i)\ {l(i,t)}


Thus, (5-22) holds.

Now we prove (5-21) by 2 conditions. (1) t(i) > t(p) + 1; (2) t(i) < t(p).


117


dt[fI(i.t,wt)









For Condition 1, t(i) > t(p) + 1, based on (5-10), (5-15), and the nonnegativity of
m, we have

T T
si = dtm't> dt max 0, (,,t,) -
t=t(i)+l t=t(i)+l jEPi(wt)\{i,((i,t,wt)}
T
wt)
t=t(i)+l jr' P(wt)\{i, (i,t,wt)}

where wt is the single element in W(i) for each t(p) + 1 < t < T. Thus, (5-21) holds.

For Condition 2, t(i) < t(p), we have
T T
Si = d dtmi = dt(i) I+m(i) 1 + d my
>i= d dz]if) tpt(i)+z + ( dtmit
t=t(i)+l t=t(i)+2
T
> dt(i)+l[fil]+ + dt [ml- zi ,]+, if t(i) < t(p) (5-23)
t=t(i)+2
T
> dt(i) i+fql1]+ + dt[mq" _- Zq~]+, if t(i) = t(p), (5-24)
t=t(i)+2

where (5-23) is based on (5-11), (5-12), and the nonnegativity of m' and (5-24) is

based on (5-13), (5-14), and the nonnegativity of mi.

If t(i) = t(p), based on (5-10)


[Mn"_ z ,] > [f(q,,,t,wt) zj- z ,]
jEPtwtt (wt)\{qw ,(qw ,t,wt)}

= [ (y7,(,)w- Z z.]+
jEPqtwt (wt)\f{(qwt,t, wt)}

= [fi(p.t.,w) z ,
jEP (wt)\{p,((p,t,wt)}

where the second equation holds due to (q,,, t, wt) = (p, t, wt), Pwt (wt) c pp(wt)

and Pwt (wt) \ (qwt, t, wt) = Pt(Wt) \ {p, (qw,, t, wt)}, because wt is the single element
in W(p) for each t(p) + 1 < t < T. Thus, (5-21) holds.


118









Now we only need to show that, if t(i) < t(p),


[m1 zi,1+ > [f>(i,t,w) z]+. (5-25)
jeP (wt)\{i'.(i,'t,wt)}
It is easy to observe that if t < t(p), (5-25) holds based on (5-11) and (5-12). Hence
we discuss t = t(p) + 1 and t > t(p) + 1 respectively.
(a) If t = t(p) + 1, then
P P
[m'1 z] > [ z] [ = [f, z .
j +1 j =+1 jEPT(wt)\{i,qwt}

where the first inequality follows (5-12) and the second inequality follows (5-13).
(b) If t > t(p) + 1, then
P
[m~1,- z, 1] > [m zj] > [mr"t S z,]
j[ 1 j T (p)+ (wt)\{i}


JeP(p)+l(wt)\i jEtpqwt (wt)\{qwt ,(i,t,wt)}

> [fi.t.,w)- Sr zj],
j>P[(Wt)\{i,(itWt)}

where the first inequality follows (5-12), the second one follows (5-14), and the third

one follows (5-10). Thus (5-21) holds.
Therefore, Claim 1 holds.
Now we prove Claim 2 holds.
For any give exteme point (f, g, z, s, f) e QM, we construct m and n such that

(f, g, z, m, n, s, ) E QH. That is, (f, g, z, m, n, s, ) satisfies the conditions (5-4), (5-5),
(5-10) to (5-17). Now for a given point (f, g, z, ) e QM, let


't =max f(itw) zj (5-26)
wEW(i)
JC P (w)\{i, (i, t, w)}


119










and = g,4(i,t)


jEP(rq(it),i)\{(i,t)}


- .
zj


(5-27)


Since (f, g, z, s, ) is an extreme point in QM, we first observe that (f, g, z, s, ) satisfies

equation (5-15) based on (5-26) and (5-19). Similarly, (f, g, Z, ,, ) satisfies equation

(5-17) based on (5-27) and (5-20). It is also obvious that (f, g, z, m, n, s, ) satisfies

(5-4) and (5-5). Based on (5-26), inequalities (5-10), (5-11), and (5-13) hold. Based

on (5-27) and the nonnegativity of in, inequality (5-16) holds.

Now we only need to show that (5-12) and (5-14) hold.

For (5-12), let w* be the scenario where m+1 achieves the maximum value. Then

we observe that ini achieves the maximum value in the same scenario w*. We prove

(5-12) holds based on zi+i = 1 and z+i = 0 respectively. If z2+1 = 1, then mi = 0 based

on (5-26). Then mi = 0 > i 1 = i z,. If z,1 = 0, then
m t _- z + j If z2 + j = O t h e n


S f (i,t,w*)





= [ (i,t,w*)-


j
je'P'(w*)\{ia,(it,w*)}







jEPt+l (w*)\ i{+ ,(i,t,w ,)}


= rn+1 = h -i+1 -2 .
- mt mt z .


Thus, (5-12) holds.

For (5-14), according to (5-26),

mh = max (pt,w)
WtEW(p)


W(p,t,w) -[w
jEPt(w


- +


jEPf(wt)\{p, (p,t,wt)}

24 for each wt c W(p).
t)\{p,U(i,t,wt)}


120









Now for each particular w, E W(p),


[f(Pt'wt)



(q,,twt W

- mwt Zqw

tmqw _zqt


-+




jEPIP(wt)\{pw, (p,t, wt)}
Z

jEPqw"t (wt)\{qwt,(p,t,wt)}

1+'


7- Z wt


Zj Z wt


where the second equation follows 'PiP(wt) \ {p, qw} -= P^wt(wt) \ {qw} and the third
equation follows mqwt = f(q,twt.) jprwt(w)\{qw,,(p.twt)} z for a given wt E W(p). Then

(5-14) holds. Thus (f, g, z, m, n, s, ) e QH and Claim 2 holds.

Therefore, the conclusion holds. D























































































N- *-


- Ncn It LO


0- MO SIL ( D O C 0 0 -
cD r- CC) 0C) -- -- -- -- ,- CM-


122


I I


I I


I I









CHAPTER 6
LIFTING SCHEME FOR THE STOCHASTIC DYNAMIC KNAPSACK POLYTOPE

6.1 Introduction

Deterministic dynamic knapsack set is naturally generated from the deterministic

uncapacitated lot-sizing problem. Loparic et al. (2003) first introduced the deterministic

dynamic knapsack set XDK as follows:


XDK = (, y) R+ x B : s + a y, > bt, t e T ,(6-1)

where a, b e RT. They studied the polyhedral structure of the deterministic dynamic

knapsack set. With the application of the sequence independent lifting scheme, a family

of facet-defining inequalities for XDK were introduced.

For the deterministic mixed-integer programming problems, different schemes have

been explored to generate more valid inequalities based on existing valid inequalities.

Guan et al. (2007) proposed the pairing scheme which generated a family of inequalities

with the properly ordered combination of two existing valid inequalities. Recently,

GOnlOk and Pochet (2001) developed the mixing procedure to generate valid inequalities

based on the mixed-integer rounding inequalities (MIR). They demonstrated that the

mixing inequalities are strong inequalities for some special polyhedral structures. Miller

and Wolsey (2003) showed that the mixing inequalities can provide the convex hull

description for a special single-item lot-sizing problem.

The lifting scheme was first introduced by Wolsy (1976, 1977), and Zemel (1978), et

al. The lifting scheme can be applied sequentially to generate strong valid inequalities.

Gu et al. (2000) extended the sequence independent lifting scheme (Wolsy 1977) to

the mixed 0-1 integer programming, and showed that the sequence independent lifting

property holds as long as the lifting function is superadditive. That is the generation of

lifting coefficients are independent of the lifting sequence. AtamtOrk (2004) generalized


123









the sequence independent lifting property to the general mixed-integer programming

problem.

In this chapter, we study the extension of the deterministic dynamic knapsack set:

the stochastic dynamic knapsack (SDK) set. We investigate the polytope of the SDK

set based on a multi-stage stochastic scenario tree model as described in Ruszczyriski

and Shapiro (2003). For the stochastic mixed-integer programming problems, Guan

et al. (2006b) studied the multi-stage stochastic uncapacitated lot-sizing problem and

developed a family of valid inequalities, named (Q, 5S) inequalities. They showed that

under certain conditions, (Q, Sg) inequalities are facet-defining inequalities. Guan et al.

(2006a) examined that (Q, Sg) inequalities are sufficient to describe the convex hull of

the two-period problem. The pairing scheme described in Guan et al. (2007) can also

provide valid inequalities for the stochastic lot-sizing problem and generalize all (Q, SQ)

inequalities. Further more, based on these, Guan et al. (2009) proposed a general

approach to generate valid inequalities for the multi-stage stochastic mixed-integer

programming problems. They combined valid deterministic inequalities corresponding to

each scenarios to generate valid inequalities for the whole scenario tree.

The remaining part of this chapter is organized as follows. In Section 6.3, we

study the pairing and mixing schemes for the SDK set and show that pairing and

mixing inequalities are facet-defining inequalities under certain conditions. Section

6.4 demonstrates the sequence independent and sequence dominant lifting schemes

for the SDK set. We generate families of valid inequalities for the SDK set through

the lifting schemes. Then, to solve large-scale problem, in Section 6.5, we apply

parallel computing technique to solve the SDK set. We develop parallel algorithms to

generate valid inequalities via pairing, mixing, and lifting schemes for the stochastic

capacitated lot-sizing problem as an example of the SDK set. Finally, we demonstrate

the computational efficiency by showing the improvement of the optimality and integrality

gaps in Section6.6.









6.2 The Path Inequality


With the notation of stochastic scenario tree in Chapter 1, we extend XDK to a

stochastic dynamic knapsack set with a single continuous variable defined by:


XSDK = (s, y) R+ x B" : s + ajyi > b;, ie V ,
jEp(i)

where n = IVI, a c Rn, b e R". In the following sections, without loss of generality, we

assume bj < bi, if j e P(i).

Similar as the deterministic case, XSDK is a relaxation set of the feasible region of

stochastic capacitated lot-sizing problem, XCSLS-


XCSLS = (s, x, y) e R+ x R xB" n: s + xi > dci, x,< aiy,, i e V ,
jEP(i)

where di and a; are the demand and capacity at time period i, respectively, and dci =

Ejep(i) dj. Replacing x, with ajy,, we get the relaxation set of XCSLS as


XRSLS = (s, y) R+ x Bn : s + ay > bi, i V ,
jEP(i)

where bi = maxjEp(/){dij}. It is obvious that XRSLS is a XSDK set.

XSDK evolves the information of the whole scenario tree. Now let X(i) represent the
path set of XSDK for a given node i e V, i.e.,


X(i) = {s + akYk > bj, j c '(i)}, (6-2)
keP(j)

We can observe that X(i) is a t(i)-period deterministic dynamic knapsack set

corresponding to the scenario with the information on path P(i). We name the inequality

in X(i) as path inequality. All path inequalities in X(i), i e V are valid for XSDK. Note

here, when Il = 1, XSDK = XDK.


125









Proposition 6.1. XSDK can be represented by the joined set of X(i) for all i C V, i.e.,


XSDK = nX(i).
iEV
Now we show an example of path inequalities, based on Figure 6-1. The following

path inequalities are generated corresponding to path P(1), P(2), P(3), P(4), and P(5)

separately:
s +40yl > 5

s +40yl +15y2 > 15

s +40yl +20y3 > 17

s +40yl +20y3 +20y4 > 20

s +40yl +20y3 20y5 > 20


2
4
1
3
5

Figure 6-1. Path inequalities for a scenario tree

Because X(i) is a t(i)-period deterministic dynamic knapsack set, we extend

the conclusion of Loparic et al. (2003) for XDK to the stochastic setting. We let b&k =

EteP(,k)cp(j) bt and bk = blk. With the information of b, we construct a modified
nonnegative parameter b. Let bj = maXkEP(j){bk, 0}. We order nodes in path P(f,j) as

i., I ik, ik+I, iK with K = t(j) t() + 1.
Proposition 6.2. The inequality

s + -b)ye> kb


is valid for conv(X (i)) forj e P(i), and facet-defining when = i, where R = P(j),

O(J) = min({a, Eiik c)(bi bik-)} with R(f) = R n V(f).


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In the following sections, we apply the pairing and lifting schemes to generate valid

and strong inequalities for XSDK-

6.3 The Pairing and Mixing Schemes for the Stochastic Dynamic Knapsack Set

In this section, we apply the pairing and mixing schemes to the path inequalities

for XSDK and generate corresponding tree inequalities. We show that these generated

tree inequalities are facet-defining inequalities for XSDK under certain conditions.

We introduce the notation of the vector operation for the following discussion. Given

two same dimension vectors a1 and a2, min(a1, a2) and max(al, a2) are carried out

component-wise. For brevity, given a vector a and a scalar c, we let min{a, c} =

min{a, cl}, where 1 is a vector of ones of the same dimension as a.

Guan et al. (2007) studied the pairing scheme for the mixed-integer sets X e

Zn x RP. Let the pair (a, g) e R"+1 x RP define a valid inequality for X, if
n p
ajy,; + gyxy > an+l, for all (y, x) e X. (6-3)
i=l j=

Let (al, gl) and (a2, g2) define two valid inequalities for X. The pairing scheme for these

two inequalities is as follows:

Theorem 6.1. (Guan et al. 2007) If (a1, gl) and (a2, g2) define two valid inequalities for

X with an < a_ the inequality
n p
SokYk X. >a 2 (6-4)
k =l = 1
is valid forX, where = min{a1 + (a a)1, max{al, a2}}, = max(g, g2).

Guan et al. (2009) applied the above pairing scheme to path inequalities of XSDK

and obtained the following tree inequalities:

Theorem 6.2 (Guan et al. 2009). Given a setR = {i/, i2, iK} C V indexed such that

bi1 < b12 < .. < bi,, the inequality

s + j(7)yj > bi, (6-5)
jeVR


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where VR = UikRP(ik) and cj(7) = min{aj,Y ik,7)(bik bik1)} with R(j) = n V(j)
and bo = 0, is valid for XSDK-
Tree inequalities (6-5) can be strengthened as follows:
Theorem 6.3 (Guan et al. 2009). Given a setR = {il, i2, iK} C V indexed such that

bi, < b2 < ... < bi,, leti = argmin{u : u e P(ik) and bu > bik,} for each ik E R,
Q = UikERP( k, ik), and Q(j) = Q n V(j). Then, the inequality (6-5) is dominated by the
inequality

s + yj(Q)yj > bi, (6-6)
jEVn
where Vn = nikeP(ik) and (Q) = min{a,, ,ikE.)(bik bik1)} with bo = 0.
With a stronger assumption of the coefficient, (6-6) is a facet-defining inequality for
conv(XSDK)-
Theorem 6.4 (Guan et al. 2009). Inequality (6-6) is facet-defining for conv(XSDK) if

(1) biK = max{bi : i E V}.
(2) for each c Vn, aj > max{bi, i E 2(j)}.
Guan et al. (2009) discussed the convex hull description of XSDK under the large
coefficient case (Condition (2) in Theorem 6.4).
Theorem 6.5 (Guan et al. 2009). If aj > max{bk, k e V(j)} for each E V, then the family
of inequalities (6-6) for all 2 c V, together with 0 < s < by and 0 < yj < 1 for each j e V,
describe the convex hull of XSDK, where by = max{bi, i E V}.
They also discussed the separation algorithm of tree inequalities for XSDK. Based
on the shortest path algorithm, the corresponding separation algorithm is polynomial.
Theorem 6.6 (Guan et al. 2009). If aj > max{bk, k e V(i)} for alli e V, there exists a
polynomial-time separation algorithm for the tree inequalities (6-6).
We show an example of generating the tree inequality (6-4) based on the scenario
tree in Figure 6-1. For set R = {1, 2}, we have VR = {1, 2}, R(1) = {1, 2}, and R(2) =
{2}. The corresponding two path inequalities are s + 40y, > 5 and s + 40y + 15y2 > 15.


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Then,


01(R) = min{max(a1l, a12, bl bo + b2 bi)}

= min{max(40,40),5 0 + 15 5} = 15, and

02(7R) = min{max(a12, a22, b2 bi}

= min{max(40, 40), 15 5} = 10.


The corresponding pairing inequality is


s + 15yi + 10y2 > 15.


Next, we apply the mixing scheme to XSDK and generate tree inequalities.

Gunl0k and Pochet (2001) proposed the mixing scheme for MIR inequalities. Given

a mixed-integer region S c Rml x Zm2 and a collection of m > 2 valid inequalities for S


f'(x) + Bg'(x) > 7T', i E = {1,..., m}, (6-7)


where B E Rt, 7i e R1, fi(x) > 0, and gi(x) e Z. For any ie Z, the simple MIR

inequality

f'(x) >_ '(r' g'(x))

is valid for S, where 7- = [-'/B1, 7i = 'i (i' 1)B, -' e Z1, and B > 7' > 0. Note

that f' and g' can be nonlinear, 7' and g' can be negative. Without loss of generality, we

assume that / = 1,..., n, and y' > y-1 for all n > i > 2.

Theorem 6.7 (Ginl0k and Pochet 2001). The following two inequalities
n
f (x) >_ (i_ -1) (T g(x)) (6-8)
i=1

and
n
f(x) > (7' '-1)( g9(x)) + (B 7n)(1 g1(x) 1) (6-9)
Si
with yy = 0, are valid for S, where f(x) > f'(x) > 0 for all x E S and i /. .


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Now we apply the mixing scheme to XSDK, and prove that the mixing inequalities

can provide the convex hull description of XSDK under certain conditions. Let zi =

:jeP(i) y and A = max{ai,j e P(i)}. Then,

XSDK = {(W, z) E R x ZI'l, s + Azi > bi, i e V} (6-10)

is a relaxation set of XSDK. We generate the mixing inequalities for XSDK as follows:

Proposition 6.3. The following two inequalities
n
s > (7i i-1) (Ti- zi) (6-11)
i-i 1

and
n
s > (i- 7-)(Ti zi) + (B 7n)( 1- zi 1) (6-12)
i-=
are valid forXSDK, where 70 = 0, -i = [bi/A] andy7 = bi (r' 1)A.

Replacing z, by jEP(i) y, we obtain the valid inequalities for XSDK by the mixing

scheme.

Proposition 6.4. The following two inequalities
n
s > (7i 'i-1)- yj) (6-13)
i-1 jEP(i)

and
n
s > (7i _- -1)(Ti- y) + (B 7)(T1 1) (6-14)
i-1 jeP(i)
are valid for XDK, where 70 = 0, -i = [bi/A] andy7 = bi (r' 1)A.

Note here, the mixing inequalities (6-11) and (6-12) can also be constructed for

subset XSDK(V(i)) based on the subtree V(i).

We provide the convex hull description for XSDK with mixing inequalities (6-13) and

(6-14) under the condition that a; = A for i e V.

Theorem 6.8. If a- = A for i e V, the family of inequalities (6-13) and (6-14) are

sufficient to describe the convex hull for XSDK-


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Proof. We prove this conclusion by two claims.

Claim 1. Inequalities (6-11), (6-12), and 0 < zi Za(i) < 1 define the convex hull

description for

X' = {(w, z) ERx ZLv, w + Azi > f, O < zi za(i) < 1, i V}. (6-15)

Claim 2. Let Xs denote XSDK with ai = A for i e V. It is one to one correspondence

between X' and X'.

Proof of Claim 1: Let Z = {(w, z) E R x ZI, w + Azi > f, i V} and

Z2 = {z E Z Z2 is the transpose of a network matrix (arc-node incidence matrix) and the right-hand

coefficient is integer. Following the result in Miller and Wolsey (2003), (6-11), (6-12),

and 0 < z, Za(i) < 1 are sufficient to provide the convex hull description for X'.
Proof of Claim 2: we show that for a given (s, y) e X', there exists a reversible

function G : X X'.

We show that there is a function G : X X', where

w = s, (6-16)

zi= yI, i V. (6-17)
jEp(i)

Because yi c {0, 1}, we have zi e Z+. Let f = b; for all i e V. With s + A Ejep(i) Yj > bi,

(6-16), and (6-17), we have w + Azi > f. Thus, (w, z) e X'.

The reverse function of G, G-1 : X' X', is that

s = w, (6-18)

yi zi Za(i), i V (6-19)

Because zi e Z+ and 0 < zi za(i) < 1, we have yi e {0, 1}. With w + Azi > f,, (6-18),

and (6-19), we have s + A Cjep(i) YJ > fi. Thus, (s, y) c Xs.

Thus, it is one to one correspondence between Xs and X'.









Finally, we show that (6-13) and (6-14) are sufficient to provide the convex hull
description for X'. Given any ci c R and c2 RIVI, the problem, P1, max{cis +

cTy, (S, y) E R x R1I} with (6-13) and (6-14) can be transferred to be the problem, P2,
max{ciw + bTz, (w, z) c R x R } with (6-11), (6-12), and 0 < zi za(i) < 1, where
bi = jep(i) cC. Suppose that (w*, z*) is the optimal solution for P2. With Claim 1, z* is
integral. With Claim 2 and (w*, z*), we obtain the corresponding optimal solution (s*, y*)
for P1. With the integrality of z* and 0 < y =- z Za() < 1, y* is integral. Thus, (6-13)
and (6-14) are sufficient to provide the convex hull description for Xs. D

6.4 The Lifting Scheme for the Stochastic Dynamic Knapsack Set

In this section, we derive more valid inequalities for XSDK by the lifting scheme.
First, we apply the lifting scheme to the path inequalities and show that the sequence
independent lifting property holds when we lift back variables based on path inequalities.
Second, we discuss the sequence dominant lifting procedure for the tree inequality in

XSDK. Third, we discuss how to combine pairing and lifting schemes and apply them to
the two-stage stochastic dynamic knapsack set and the multi-stage stochastic dynamic
knapsack set.
Similar as Loparic et al. (2003), we derive other valid inequalities for XSDK by
the lifting scheme. We set some variables to be 1 and modify the corresponding
parameters. Then, we generate the basic inequality and lift back variables that have
been fixed.
Let U(i) represent a subset of P(i). We set ye = 1 for e U(i). Then we modify
the parameter be as de = be ae, for e (i), and generate the basic path inequality as
follows:

s + (R)yj > bi (6-20)
jEP(i)\U(i)
where R7 = P(i), jP(RP) = min{aj, Eie(j)(bi-bibk )}, and bi = max{maxtEp(i)(EJEP(t)) d), 0}.


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We define the function


u(i)(h) = min{s + j(R7Z)yj- bi (s,y) E X(i)\u(i)(d + h)} (6-21)
jEP(i)\u(i)
With (6-20) and (6-21) as the basic inequality and the lifting function, respectively,
we generate the lifted valid inequality for XSDK as follows:
Proposition 6.5. The inequality

s+ () >yj + u(i) (aje y, > bi + u(i)(ajeS
jEP(i)\U(i) jEu(i) jEu(i)
is valid for XSDK with R = P (i) and facet-defining for X(i), where i C V.

Proof. We let K = IP(i)| and re-index nodes on P(i) from root node to node i as

{1,2,-.. ,K}. ThenX(i) = {(s,y) Rx {0, 1}K, s + ajYj > bt, t = 1,... ,K}. X(i)
is a deterministic dynamic knapsack set. We apply the lifting scheme to s+l_ 1 ajyj > bi
and re-index node index in (6-20) and (6-21). Directly following Loparic et al. (2003), we
obtain the above result. D

Note that based on Proposition 6.5, the sequence independent lifting property holds
for lifting back variables based on the path inequality in XSDK-
Now we derive valid inequalities for XSDK based on the tree inequality. Let XSDK(VR)
represent the feasible solution set defined by the path inequalities corresponding to
nodes in VR. We let (R) represent the set of leaf node in V and CR(i) = UjEp()C(j) \
VR with all i ( (R).
First, we apply the lifting scheme to XSDK(VR). Suppose that

s + /3y, i> 7 (6-22)
jEVR


133








is a facet-defining inequality. We set ye = 0 for a given e CR(i) and let (6-22) be the
basic inequality. We define the lifting function as

c,,)(0) = min{s + y /jy 7: (s, y) E XSDK(VR) U {}}

Then, we obtain the following result:
Lemma 1. If s + ~jEVR O yj > 7 is a facet-defining inequality for XSDK (V), then
corresponding to each node i c (R),

s + /y A + O3ye > 7 + / (6-23)

is facet-defining for XSDK( U { }) for each e c CR(i), where P3 = min{s + C r /3Yjy- :
(s, y) XSDK(VR) U { }.
Second, we consider two variables yk and yk 1. If they satisfy 4k, 4-1 c CR(i) for
a given i c (J), bek > b k 1, and a(fk) E 'P(a(kl)), we discuss the following two
conditions to show the relationship between the coefficients of y~ and yek1 in the lifted
valid inequality, and the sequence of lifting y4k and y~ -.
Condition 1: y~ is lifted before y~ 1. With Lemma 1, after lifting back of y,k' we obtain
a facet-defining inequality (6-23) for XSDK(VR U {4k}) and the coefficient of y4k is

/, = min{s + E ,v jy 7 : (s, y) E XSDK(VR U {(k})}. Then, we lift y~ 1
based on (6-23) and obtain a facet-defining inequality for XSDK(VR U {4k} U {4k-}),

s + EY:ev /3jy + /34 + /34-IY4- > 7 + Yk + a -1 with

a k- = min{s + r/3jyj + /kyk (7 + /3) : (s, y) e XSDK(VR U {4k} U {4k-})}.
jEVRj
In order to obtain the value of / k1, we show that for any (s, y) E XSDK((VR U {4k})
is feasible for XSDK(VR U {4k} U { k-1}), where XSDK(VR {U U{k} {U k-1}) =
XSDK(V U{4k})U{(s,y), S+ -etp(, ) aJyJ > bk-1}. With an unbounded variable s,









Yek = 0 is feasible for XSDK(VR U {4k}). Then, any (s, y) E XSDK(VR U {-k}) satisfies

s+ a yj > b k.
jEP(4)\{4}
Due to a(k) e P(a(4k-1)), we have P(k) \ {kk} C P(4k-) \ {4k-} and

s + Y ajy
JEP(k4-1)
=s + a a + a4-14-1


=s+ ajYj + ajyj + ak-1 y-1
jEP(4k)\{k} jE(P( -k 1)\{-k 1})\(P(4k)\{4k})
> b 4

With b k > b 1, we have s + EiEp(k -) ajy > bk-1 for any (s, y) E XSDK(VR U {k }).
Therefore, all feasible solutions for XSDK(VR U {k}) are feasible solutions for

XSDK(VR U {Ik} U {tk-1}). Because variable s does not have an upper bound, (s, y)
with y4k = 0 is a feasible solution for XSDK(VR U {4k} U {4k-i}). Then, we have

ak 1 = min{s + Yj;/y + /kY -k (7 + k) : (s, y) E XSDK(V U {4}k}) U {Ik-I}}

= min{s + 3y 7 + 0y : (s, y) E XSDK(VR U {jk}) U {k-1}}


=/e /4k = 0.

Therefore, if y4 is lifted before y the coefficient for yk- is 0.
Condition 2. yk- is lifted before y4. With Lemma 1, after lifting back yk 1, we obtain a
facet-defining inequality (6-23) for XSDK(VR U {4k-1})} and the coefficient of y -1

is /k-1 = min{s + YjEVR/3jyj 7 : (s, y) E XSDK(VR U {4k-1})}. Then, we lift yk 1
based on (6-23) and obtain a facet-defining inequality. The coefficient of y k is

a4k = min{s+ /3/,J +/34-lY4k 1- ('7 + ik 1) : (S, y) E XSDK(V U {4k-1} U {k})}.
jEVR


135









We have shown that all feasible solutions for XSDK(VR U ({k}) are feasible for

XSDK(VR U {4k-1} U {4}). And yk-1 = 0 is feasible for XSDK(VR U {4k-1} U {k}).
Thus,

a k = min{s + /3jYj +0k lY-k1 ( (57 +y (s) E XSDK(VR U {4k-1} U {k})}
jEVR
= min{s + Y /3YJ 7 + 0k,-1 k : (S, y) E XSDK(VR U {4})}
jEVR

-k .k *

Hence, if yk- is lifted before y4k, the coefficient for Y4k will be /k /k 1.
We conclude these two conditions and obtain the lifting dominant property as
follows.
Proposition 6.6. Lifting Dominant Property: For any pair of nodes (4k-1, k) C CR(i)
such that be, > be, and a(4k) E P(a(4k-1)),
1. If yek is lifted before y~ 1, then the coefficient for y~k is /k and the coefficient for yk 1
is O.
2. If yek1 is lifted before y~,, then the coefficient for yk-1 is / k 1 and the coefficient for y~

is Ok -0 _,1'.
Based on the lifting dominant property, we apply the lifting scheme to XSDKR(V) with

ye,, ti E CR(i) sequentially.
Theorem 6.9. If s + YjEV 3jyj > 7 is a facet-defining inequality for XSDK( R), then the
following inequality

s + y jyj + Y (0 )yV4 > 7 + 1K (6-24)
jEVR 4EA(i)

is facet-defining for XSDK(VR U CR(i)), where A(i) c CR(i) in which 0 = 0o < / < ... <

/4-1 < /4 < < /K', ~K = argmax{bk : k e CR(i)} and a(4k) c P(a(k-1)).

Proof. We prove this conclusion by induction, following the lifting sequence ~, 2 ,. K


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Step 1. We lift y, from the facet-defining inequality s + YjEVR Ojy > 7. According
to Lemma 1, we know that s + Y:v, /3jy, + ~1,y1, > 7 is a facet-defining inequality for

VR U {1} and 3~, = min{s + EY,R fjJ 7 : (s, y) E XSDK(VR U { 1})}.
Step 2. We lift y2 after the lifting of ye,. Following the sequence dominant property,

ye, is lifted before y2, where by, > bY2 and a(2) e 'P(a(t)). Thus, the coefficient of y2
is O2 f3, and the corresponding new lifted facet-defining inequality is

s + Y O+ y1' + (A2 l)Y + 2 1- fl
jEVR

> 7 + 2. (6-25)

With j3o = 0, we rewrite (6-25) as

s + YS /JYJ + (l, Jo)Y,1 + ( ei e/2 j3J)Y2 > 7 + 2 .
jEVR
Step j. After lifting yj, we generate a facet-defining inequality as

s + ( yJ + Y(k ( k- )Yk>) > 7 + (6-26)
JEVR 1k:j
Stepj + 1. Now based on (6-26), we lift y,, and the corresponding coefficient of

Y+1 is

+- l = min{s+- 3,jyj+ Y (i3- /k ,)yk -(y7+,) :(s,y) e XSDK(VRU{ }U. .U{.j+ })}.
jEVR i Now we prove that aj, = Pf, P3 Because a(i+l) e P(a( have a(il) e P(a(fi)), where 1 < i < j. Thus if (s, ) e XSDK(VR U {(+1}), then

(k, ) e XSDK( VR U {(+j} U {(I} U U {}). Hence,

j+- = min{s + y /y + (k /k- )y (7 + ) : (, y) e XSDK(VRU {1} U U {j+1})}
jEVR 1 = min{s + y /3Oyy 7 + ((k /~k ,)Y :j (s, y) e XSDK(VR U {I1} U U {j+1})}
JEVR l
= e+1 f-


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Thus, the above conclusion holds by induction.


6.5 Cutting Plane Generation under Parallel Environment

Our parallel implementation involves assigning slave processors to generate cuts

for the capacitated stochastic lot-sizing problem (CSLS). The purpose of our parallel

implementation is to test the effectiveness and efficiency of our cuts based on pairing,

lifting scheme. In order to evaluate the the performance of our cuts, we implement two

parallel schemes for CSLS: 1). the cut-and-branch algorithm, as well described further

in section 6.5.3, and 2). the LP-based cutting plane algorithm with heuristic, as well

described further in section 6.5.4. The major issues of parallel implementation is that of

load balancing among processors and management of generated cuts. The desirable

condition is that each processor handles approximately an equal number of LPs and the

cutting management can control the cutting pool size, recognize the strong cuts, and

purge the ineffective cuts.

Our parallel implementation was developed on a Pentium4 Xeon64 quad core Linux

cluster and distributed-memory multiprocessors. We apply the open source software

SYMPHONY in COIN-OR as our mixed-integer programming solver. We let K and w

represent the total number of processors and the slave processors working on the cut

generations.

In order to generate our cuts, we use multiprocessors to generate cuts according to

the stochastic scenario tree structure. This section is organized as follows. In Section

6.5.1, we discuss the static partition for the stochastic scenario tree structure. In Section

6.5.2, we discuss the cut management of our two algorithms in a parallel distributed

environment. In Sections 6.5.3 and 6.5.4, we provide the detailed parallel algorithm

to obtain the optimality and integrality gaps for the stochastic integer programming

problems.


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6.5.1 Partitions of Stochastic Scenario Tree and Local Initialization

We develop two algorithms for a stochastic integer programming problem with

stochastic scenario tree model. In order to handle the load balance issue, we discuss

static partitioning for the stochastic scenario tree structure. After the partition, each

processor handles a stochastic scenario subtree to generate cuts.

Static Partitioning: We partition the stochastic scenario tree into several subtrees.

The partition is based on the total number of leaf nodes and each partition is

recognized by the index of leaf nodes, i.e., the root node of the scenario tree is

the root node of each subtree. Its leaf nodes are indexed from (/w) (i 1) + 1

to(/w) i. Thus, the total partition number is [7/uw and each processor handles a

|V| x w/| I subtree. Note that the local initialization on the processor only needs to
load a physical copy of the corresponding subtree structure.

6.5.2 The Parallel Cuts Management Decision Control

6.5.2.1 Cut Generation

For an individual processor, the cut generation involves generating cuts based on

each subtree structure. In Section 6.3 and 6.4, there are three cut generation schemes,

the path lifting scheme, the pairing scheme, the tree lifting scheme. These three scheme

cuts are generated on each processor.

For the static partitioning, first, each processor obtains the information of the

subtree from the master machine. Second, the corresponding basic path inequalities

are generated based on the subtree structure. Third, the path lifting inequalities are

generated. Fourth, the pairing scheme is applied to the subtree structure to generate

valid inequalities. Finally, the lifting scheme is applied to the pairing inequalities.

Due to each processor should handle these three types of inequality, we assign

time limits for each scheme for cut generation. If a scheme of cut generation obtained

long time intervals, the processor spends longer time on generating cuts based on the

scheme.


139









6.5.2.2 Cutting Pool Management

All generated cuts are stored locally. Before, each processor sends all generated

cuts to the master processor, local cutting pool management function is called. The local

cutting pool purges the duplicated cuts from the cutting pool.

The master machine handles the master cutting pool which obtained all cuts from

the local cutting pool and purge the duplicated cuts before all cuts are applied to the

process in the master machine.

6.5.3 The Parallel Cut-and-Branch Algorithm

First, we discuss a straightforward algorithm for CSLS, the parallel cut-and-branch

algorithm. We use multiprocessors to generate as many cuts as possible for CSLS

within a certain time limit. Then SYMPHONY uses the branch-and-bound algorithm

to solve a big MIP problem which combines all generated cuts with the original CSLS

problem. The detailed algorithm is described as follows:

The parallel cut-and-branch algorithm

1. The cut generation: generate cuts based on the subtree of the stochastic scenario

tree according to the leaf nodes.

2. The cuts management: purge the duplicated cuts.

3. The branch-and-bound algorithm for CSLS with all generated cuts in the cutting pool.

Step 1 handles the partitions for a stochastic scenario tree. The cut generation is

from the shortest branch to the longest branch starting from the root nodes. First, we

generate the basic inequalities. Second, we generate the lifting inequalities based

on the basic inequalities. Third, we generate the pairing inequalities based on the

generated basic and lifted inequalities. In Step 2, we apply the basic cut management

for the generated cuts. The purpose of cut management is to purge the duplicated cuts

and store all efficient cuts. In Step 3, we combine all generated cuts with the original

CSLS problem as a big MIP problem and using the branch-and-bound scheme in


140









SYMPHONY to solve within the time limits. Finally, we obtain the optimality gap, as the

evaluation standard, from the parallel cut-and-branch algorithm.

6.5.4 The LP Based Cutting Plane Algorithm

The parallel cut-and-branch algorithm generates cuts at the beginning and sends

all cuts to the branch-and-bound structure. Now we discuss a LP based cutting plane

algorithm with an embedded heuristics to obtain an integer feasible solution.

The LP based cutting plane algorithm

Step 1. Set k = 1. Let the upper bound of the problem be UIp = +00 and the lower

bound be Lip = -oo. Let the initial integer feasible solution be X1p = 1. Let SR

represent the feasible region of LP relaxation and S -= SR.

Step 2. Solve the relation problem of S' and obtain the solution xt and objective ZLP

Termination tests: If xt is an integer solution, stop. xt is the optimal solution of

CSLS.

Step 3. Heuristics: call Heuristics for obtaining the integer feasible solution to obtain

the integer feasible solution, XH. If ZH < UIP, UIP = ZH and X1p = XH. Otherwise,

N = N + 1, where N represents the number of heuristics calls.

Termination test: If N > K, stop.

Step 4. Update Lip = ZLp and Sk + from Sk with more generated cuts.

Heuristics for obtaining the integer feasible solution

Step 1. Let x, ={xf, x> E,}

Step 2. For xt > E1, if x > ax 2 X = 1.

Before the implementation of this heuristics, the synchronization (share) of cuts is

called among processors and the corresponding LP problem (St) is processed. After

the implementation of this Heuristics, the modified LP with fixed integer variables

is resolved. Then this heuristics is called again until an integer feasible solution is

generated.









6.6 Computational Results


In this section, we report the computational results to demonstrate the computational

efficiency of the three types of cuts for stochastic dynamic knapsack sets.

6.6.1 Instance Generation

We consider the stochastic capacitated lot-sizing problem as an example for the

stochastic dynamic knapsack sets. The stochastic capacitated lot-sizing problem based

on the stochastic scenario tree setting can be formulated as follows:


min pi(aixi + hisi + fiyi)
iEV
s.t. xi + si- = di + si V

xi < cy, iEV

xi > 0, yi {0,1} e V,

where decision variables xi, si are the production and inventory amounts at node i.

Binary decision variable yi is the setup decision at node i. Parameters di and ci are

the demand and capacity of node i. Parameter pi is the probability associated with the

realization of each scenario.

In the following, we consider two stochastic scenario trees. The first is a binary

tree with sixteen periods (P2-16). The second one is a tree with three branches at each

non-leaf nodes and thirteen periods (P3-13). For each tree, we set the ratio f/a =

10, 20, 30 and 40. We generate two instances with ratio 10. There are ten combinations

in total. We also set demands di, unit production cost ai, and unit inventory cost hi

uniformly distributed in [50, 50], [50, 100], [5,10]. For each setting, we test five instances

and report the average value.

6.6.2 The Cut-and-Branch Algorithm

We show the performance of the cut-and-branch algorithms for the stochastic

capacitated lot-sizing problem by the following figures. We evaluate the performance


142









Table 6-1. Parameter setting
unit production cost pi setup cost fi
1 ratio 10 [50, 100] [500, 1000]
2 ratio 20 [50, 100] [1000, 2000]
3 ratio 30 [40, 60] [1000, 2000]
4 ratio 40 [40, 60] [500, 1000]
5 ratio 10 [50, 50] [500, 500]


of the cut-and-branch algorithm by the gap percentage OGap = UB 100%, where

"UB" is the objective value corresponding to the best integer solution obtained in the

given time limit. "LB" is the lower bound obtained from linear programming relaxation.

Figure 6-2 and Figure 6-4 show the optimality gaps for (P2-16) and (P3-13) with the

high setup costs with 5 ratios. For (P2-16), with more generated cuts, OGap is decreased

from 6.18% to 2.10% in average. For (P3-13), OGap is decreased from 7.99% to 3.45% in

average. Figure 6-3 and Figure 6-5 show the corresponding cuts numbers generated for

both cases.

6.6.3 The LP Based Heuristics

In this section, we show the performance of the LP based heuristics for the

cut generation. We use the integrality gap IGap to evaluate the performance of cut

generation, where IGap zHp-zP x 100%. In Table 6-2, we provide the average of all five
ZLP-ZIp
parameter settings for (P2-16) and (P3-13). We can see that with the cuts generation

and heuristics, we provide integer feasible solutions and improve the integrality gaps.










Table 6-2. Heuristics for low setup cost case
LP MIP HEUR IGAP %
2-16 1.182 x 106 1.788 x 106 2.062 x 106 45.27%
3-13 9.582 x 106 1.335 x 107 1.711 x 107 52.90%


143
















Figure 6-2. Gaps for P2-16


7%-

6%

5%-


4%

3%


2% h


Figure 6-3.
1400 -


1200

1000

800

600

400

200


Cuts number for P2-16


2 cores
4 cores
8 cores


-- --- ---- ---





1 2 3 4 5


1 cores
2 cores
4 cores
8 cores


~


-- -- -- -- -

------------ -

----------- -














Figure 6-4.
9% --


8%

7%

6%-


5%


Gaps for P3-13
---------------------------------------

-- -- ----- -- -- -- -


4%-

3%-

2%-


Figure 6-5.
2500

2000-


1500

1000-

500


Cuts number for P3-13
--------------------------------------------


2 cores
4 cores
8 cores


------------------------


2 cores
4 cores
8 cores


~









CHAPTER 7
CONCLUSION

In this dissertation, we discussed the multi-stage discrete optimization problem

under data uncertainty. We studied the multi-stage robust lot-sizing with disruptions,

the polyhedron of two-stage stochastic lot-sizing problems, and the generation of valid

inequalities for the stochastic dynamic knapsack set. In Chapter 2, we studied the

lot-sizing problem with a potential disruption as recourse to handle the uncertainty.

Our objective is to achieve the minimum objective value that considers the worst

case scenario for the disruption. We developed a customized branch-and-bound

algorithm to solve the lot-sizing problem with a disruption, which is a two-stage robust

optimization problem. A Benders' decomposition based optimality test is generated

for the branch-and-bound algorithm. This study provides more robust production

planning to address a disruption, as compared to the deterministic lot-sizing problem

and previous studies on recovery production with the information of disruption.

In Chapter 3, we provided a general model for the multi-stage robust lot-sizing

problem with uncertain disruptions. We adopted outsourcing and backlogging as the

reparation methods to satisfy unfilled demands due to disruptions. For the outsourcing

case, based on a proper assumption, a two-stage robust model and the corresponding

primal-dual algorithm are generated. For the backlogging case, we first considered a

two-stage robust optimization problem. We reformulated this problem as a mixed-integer

program and investigated the corresponding polyhedral structure. We analyzed

the trivial facet-defining inequalities, and generated three families of facet-defining

inequalities for the polyhedron of the robust lot-sizing problem with disruption and

backlogging. Applying these facet-defining inequalities as cuts, the computational results

demonstrate that these inequalities accelerate the computational speed as compared

with default CPLEX.


146









In our multi-stage robust optimization setting, we considered the multi-stage robust

lot-sizing problem with multiple disruptions. We studied the multi-stage robust lot-sizing

problem without and with setup cost, respectively. We proposed a reformulation scheme

to transfer the multi-stage robust optimization problem to be a mixed-integer linear

program, and provided the theoretical proof to show that it is one-to-one correspondence

between the reformulation and the original multi-stage robust optimization model. Thus,

the multi-stage robust lot-sizing problem with multiple disruptions is computationally

tractable. For the non-setup cost case, we generated a reformulated minimization linear

program with a pre-processing algorithm. We proved that the reformulation scheme also

works for the multi-stage robust mixed-integer program. As compared to the two-stage

robust optimization model, the multi-stage model is much more challenging to solve.

The reformulation scheme proposes a solution approach for the multi-stage robust

mixed-integer program.

Chapters 4 and 5 contribute to the literature on deriving integral polyhedral

descriptions and extended formulations for multi-period stochastic uncapacitated

lot-sizing problems. To the best of our knowledge, there is no previous research on

developing an extended formulation which provides integral solutions for the multi-period

stochastic uncapacitated lot-sizing problem. In Chapter 4, we introduce a deterministic

equivalent formulation for a two-stage stochastic uncapacitated lot-sizing problem with

Wagner-Whitin costs and deterministic demands. We examined the optimal solution

property and used it to generate an extended formulation in the higher dimensional

space. Then, we proved that the constraint matrix for the extended formulation is totally

unimodular. Finally, we projected the extended formulation back to the original space so

that we can find valid inequalities which describe the integral polyhedron of the problem

in the original space with O( V|) variables and O(TWIVI) constraints, where |V1 is the

cardinality of V.


147









In Chapter 5, we studied the two-stage stochastic lot-sizing problem with backlogging

and Wagner-Whitin costs. We investigated the relationship among the setup, inventory,

and backlogging decisions and obtained an extended formulation. We proved that

the extended formulation provides integral solutions for this problem. Further, we

generated another extended formulation in lower dimension space by projection. With

the polyhedral descriptions and the extended formulations for two-stage stochastic

lot-sizing problems, the practical tactical production decision problems can be solved by

linear programs with O(|V|) variables and O(TWIVI) constraints.

In Chapter 6, we studied the extension of the deterministic dynamic knapsack sets,

the stochastic dynamic knapsack set, and investigated the polytope of the stochastic

dynamic knapsack set with a scenario tree model. We studied the pairing scheme,

mixing scheme, lifting scheme for the stochastic dynamic knapsack set to generate valid

inequalities, which are facet-defining under certain conditions. The stochastic scenario

tree model involves many scenarios and causes computational difficulties. Therefore, we

applied parallel computing to solve the stochastic dynamic knapsack set. We developed

parallel computing algorithms to solve the stochastic capacitated lot-sizing problem as

an example of the stochastic dynamic knapsack set.

For the multi-stage discrete optimization under uncertainty problem, the multi-stage

robust optimization and stochastic programming are two main approaches. In the

future research, there are other interesting settings and models to study. For the

multi-stage robust optimization model, in Chapter 3, we studied the polyhedral structure

of a two-stage robust mixed-integer program. There is potential to extend current

results to the multi-stage robust mixed-integer model, and generate the corresponding

facet-defining inequalities to describe the polyhedron of the multi-stage robust

mixed-integer program. We can also consider a more general setting for the multi-stage

robust mixed-integer program, in which the uncertain parameters are in given intervals.


148









For the more general setting, whether the multi-stage robust optimization is still tractable

and how to generalize the reformulation scheme are interesting research topics.

For the multi-stage stochastic programming, a part of our results for the two-stage

stochastic lot-sizing can be applied to a more general multi-stage stochastic programming

setting, which can address further uncertainties. Under the multi-stage setting, it can

be observed that the optimality condition still holds, based on the Wagner-Whitin costs

setting for cases without and with backlogging. Accordingly, we can obtain similarly

constraints for the reformulation. Whether the reformulation can provide an extended

formulation that provides integral solutions for the multi-stage stochastic uncapacitated

lot-sizing problem is also of interest for future study.


149









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BIOGRAPHICAL SKETCH

Zhili Zhou was born in 1981, Nanjing, Jiangsu, China to parents Guifeng Wang and

Yaozong Zhou. She was raised in Nanjing and attended Nanjing No.3 Middle school for

her middle school and high school education. She went on to attend college at Nanjing

University in Nanjing, China, in 1981, majoring in Information and Computational

Mathematics. She received her bachelor's degree in science in June 2003. Zhili then

elected to continue her education in Department of Mathematics, Nanjing University,

majoring in Operations Research and completed her master degree in science in June

2006.

Zhili enrolled in the Ph.D. programming in the School of Industrial Engineering,

University of Oklahoma, in 2006 and transferred to Department of Industrial and

Systems Engineering, University of Florida, in 2009 with her advisor, Dr. Yongpei

Guan to obtain a Doctor of Philosophy degree.


155





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2

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fortheirlove,support,andunderstanding 3

PAGE 4

Ihaveworkedwithagreatnumberofpeoplewhosecontributioninassortedwaystothesuccessfulcompletionofthisdissertation.Itisapleasuretoconveymygratitudetothemallinmyhumbleacknowledgment.Iwouldliketorecordmygratitudetomysupervisor,Dr.YongpeiGuan,forhissupervision,advice,andguidancefromtheinitialtothenalstageofthisresearch.Heprovidedmeuninchingencouragementandsupportinvariousways.Dr.Guantaughtmehowtoquestionthoughtsandexpressideas.Hispatienceandsupporthelpedmeovercomemanycrisissituationsandnishthisdissertation.IgratefullyacknowledgeDr.ColeSmithforhisthought-provokingdiscussionandtimelyhelpduringthewritingofthisdissertation.IwishtothankDr.JosephHartmanfortheinsightfuladviceonresearchandvaluablementortime.IamthankfultoDr.WilliamHagerforbroadeningmyviewofmathematicalprogramming.IammuchindebtedtoDr.TheodoreTrafalisforhisvaluableadviceinscienticdiscussions.Manythanksgotomyfriends.Theyhavehelpedmestaysanethroughthesedifcultyears.Theirsupportandcarehelpedmetoovercomesetbacksandtostayfocusedonmygraduatestudy.IgreatlyvaluetheirfriendshipandIdeeplyappreciatetheirbeliefinme.Especially,IamobligedtoNatassiaBrenkus,BelleBrenkus,andTachunLin.Finally,Iwouldliketoexpressmyheart-feltgratitudetomyparents,GuifengWangandYaozongZhou.Myfamilyhasbeenaconstantsourceoflove,concern,support,andstrengthalltheseyears. 4

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page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTION ................................... 12 1.1StochasticProgramming ............................ 12 1.1.1Two-StageStochasticProgramming ................. 12 1.1.2Multi-StageStochasticProgramming ................. 14 1.2RobustOptimization .............................. 16 1.3Mixed-integerLinearProgramming ...................... 18 1.3.1PolyhedralTheory ........................... 18 1.3.2TheBranch-and-CutAlgorithm .................... 20 1.4TheLot-sizingProblem ............................ 22 1.5DissertationOutline .............................. 23 2LOT-SIZINGWITHDISRUPTION .......................... 26 2.1Introduction ................................... 26 2.2Lot-sizingProblemwithDisruptionandOutsourcing ............ 27 2.3Lot-sizingProblemwithDisruptionandBacklogging ............ 30 2.3.1Non-setupCostCase ......................... 32 2.3.2SetupCostCase ............................ 34 2.3.2.1ABranch-and-BoundAlgorithm ............... 36 2.3.2.2BranchingandSearchingStrategies ............ 36 2.3.2.3TheLowerandUpperBounds ............... 37 2.3.2.4TheOptimalityTest ..................... 38 2.4ComputationalResults ............................. 41 2.4.1InstanceGeneration .......................... 41 2.4.2Heuristic:MaximumPick ........................ 42 2.4.3Lot-sizingwithDisruptionandOutsourcing .............. 42 2.4.4Lot-sizingwithDisruptionandBacklogging .............. 43 3MULTI-STAGEROBUSTLOT-SIZINGWITHDISRUPTIONS .......... 46 3.1Introduction ................................... 46 3.2TheGeneralFormulation ........................... 47 3.3TheRobustLot-sizingProblemwithOutsourcing .............. 49 3.4TheRobustLot-sizingProblemwithBacklogging:SingleDisruptioncase 53 5

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........................ 55 3.4.0.2Facet deningInequalities ................. 59 3.5TheRobustLot-sizingProblemwithBacklogging:MultipleDisruptioncase 70 3.5.1WithoutSetupCostCase ....................... 70 3.5.2SetupCostCase ............................ 77 3.6ComputationalResults ............................. 77 3.6.1InstanceGeneration .......................... 78 3.6.2Two-stageRobustLot-sizingProblem ................. 78 3.6.2.1Two-stageRobustLot-sizingProblemwithOutsourcing 78 3.6.2.2Two-stageRobustLot-sizingProblemwithaSingleDisruptionandBacklogging ....................... 79 3.6.3Multi-stageRobustLot-sizingProblemwithBackloggingandwithoutSetupCost ............................... 80 4STOCHASTICLOT-SIZINGPROBLEMWITHDETERMINISTICDEMANDSANDWAGNER-WHITINCOSTS .......................... 84 4.1Introduction ................................... 84 4.2AnExtendedFormulation ........................... 85 4.3AnIntegralPolyhedronintheOriginalSpace ................ 94 4.4Extensions ................................... 98 5STOCHASTICLOT-SIZINGPROBLEMWITHDETERMINISTICDEMANDSANDBACKLOGGING ................................ 99 5.1Introduction ................................... 99 5.2AnExtendedFormulationforTwoStageSULSB-WW ............ 100 6LIFTINGSCHEMEFORTHESTOCHASTICDYNAMICKNAPSACKPOLYTOPE 123 6.1Introduction ................................... 123 6.2ThePathInequality ............................... 125 6.3ThePairingandMixingSchemesfortheStochasticDynamicKnapsackSet ........................................ 127 6.4TheLiftingSchemefortheStochasticDynamicKnapsackSet ....... 132 6.5CuttingPlaneGenerationunderParallelEnvironment ........... 138 6.5.1PartitionsofStochasticScenarioTreeandLocalInitialization ... 139 6.5.2TheParallelCutsManagementDecisionControl .......... 139 6.5.2.1CutGeneration ........................ 139 6.5.2.2CuttingPoolManagement .................. 140 6.5.3TheParallelCut-and-BranchAlgorithm ................ 140 6.5.4TheLPBasedCuttingPlaneAlgorithm ................ 141 6.6ComputationalResults ............................. 142 6.6.1InstanceGeneration .......................... 142 6.6.2TheCut-and-BranchAlgorithm .................... 142 6.6.3TheLPBasedHeuristics ........................ 143 6

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.................................... 146 REFERENCES ....................................... 150 BIOGRAPHICALSKETCH ................................ 155 7

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Table page 2-1Parametersetting .................................. 41 2-2Lot-sizingwithdisruption:outsourcing ....................... 45 2-3Lot-sizingwithdisruption:backloggingandnon-setupcost ............ 45 2-4Lot-sizingwithdisruption:backloggingandbranch-and-boundalgorithm .... 45 3-1Parametersetting .................................. 78 3-2Robustlot-sizingwithoutsourcing:multipledisruptions .............. 82 3-3Robustlot-sizingwithbacklogging:asingledisruption .............. 82 3-4Robustlot-sizingwithbacklogging:branch-and-cut ................ 82 3-5Multi-stagerobustlot-sizingproblemwith2disruptions .............. 83 3-6Multi-stagerobustlot-sizingproblemwith3disruptions .............. 83 4-1Thematrixofconstraints( 4 )to( 4 )fortheexampleinFigure 4-2 ...... 92 5-1Thematrixofconstraints( 5 ),( 5 ),and( 5 )to( 5 )fortheexampleinFigure 5-1 ....................................... 122 6-1Parametersetting .................................. 143 6-2Heuristicsforlowsetupcostcase .......................... 143 8

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Figure page 1-1Multi-stagestochasticscenariotree ........................ 15 2-1Anexampleforthelot-sizingproblemwithdisruptionandbacklogging ..... 34 4-1Thescenariotreefortwo-stageSULS ....................... 86 4-2Anexampleoftwo-stageSULS ........................... 88 4-3Thesubtreeofnodei 88 5-1Thescenariotreefora3periodSULSwithbacklogging ............. 102 5-2Thesubtreeofnodei 102 6-1Pathinequalitiesforascenariotree ........................ 126 6-2GapsforP2-16 .................................... 144 6-3CutsnumberforP2-16 ................................ 144 6-4GapsforP3-13 .................................... 145 6-5CutsnumberforP3-13 ................................ 145 9

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Multi-stagerobustoptimizationandstochasticprogrammingaretwoapproachesformulti-stagedecisionmakingunderdatauncertainty.Inthisdissertation,threeproblemsonmulti-stagerobustoptimizationandstochasticprogrammingarediscussed. First,weconsiderarobustlot-sizingproblemasanexampletoanalyzemulti-stagerobustintegerprogrammingproblems.Intherobustlot-sizingproblemsetting,weconsiderthecasesinwhichsevereeventsmayhappensuchthatthenormalprocesswillbedisrupted.Ourobjectiveistoprovidearobustschedulesuchthatthetotalcostisminimizedundertheworstcasescenario.Thisproblemcanbeformulatedasamulti-stagerobustintegerprogrammingproblem.Severalcasesarestudiedandcorrespondingalgorithmsaredeveloped.Ourpreliminarystudyveriestheeffectivenessofourapproaches. Second,weconsidertwo-stagestochasticuncapacitatedlot-sizingproblemswithdeterministicdemandsandWagner-Whitincosts.Wedevelopextendedformulationsinthehigherdimensionalspacethatcanprovideintegralsolutionsbyshowingthattheirconstraintmatricesaretotallyunimodular.Forthecasewithoutbacklogging,weprovidetheconvexhulldescriptionoftheproblemintheoriginalspacebyprojectingtheextendedformulationtotheoriginalspace.Forthecasewithbacklogging,weprovideatighterextendedformulationbyprojectingtheextendedformulationtoalowerdimensionalspace. 10

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11

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Lot-sizingproblemshavebeenstudiedextensivelyduringthelastfourdecades.Thedeterministicsingleitemlot-sizingproblemistodecideaproductionplanforaproducttosatisfydemandsforaxedtimehorizon(i.e.,Tperiods)whileminimizingtotalcosts. Thedeterministiclot-sizingproblemcannotproviderobustproductionplansinthepresenceofuncertainty.Assuch,thedeterministicdecisionmakingmayyieldunsatisfactorydecisions.Therefore,weinvestigatestochasticprogrammingandrobustoptimizationandanalyzetheirrolesindecisionmakingandtheirapplicationinlot-sizingproblems. Thegoalofthis dissertation istoanalyzeuncertaintiesinmulti-stagediscreteoptimizationprograms.Weinvestigatemulti-stagerobustoptimizationandadoptmulti-stagestochasticprogrammingapproachesformixed-integerprogrammingproblemsthroughthestudyoflot-sizingproblems. Beforewedescribetheoutlineofthisdissertation,weintroducethefollowingtechniquebackgroundstobeusedinthefollowingchapters. was originatedby Danzig ( 1955 )and Beale ( 1955 ).Theyinvestigatedtheclassicaltwo-stagestochasticlinearprogrammingwhich 12

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into twotypes : therststagedecisionvariableandthesecondstagedecisionvariable.Therststagevariable,denotedasx,hastobedecised before therandomexperiment,andtheperiodwhen thisvariableis takeniscalledtherststage.Thesecondstagevariable,denotedasy,canbedeterminedaftertherealizationoftherandomparameters.Thecorrespondingperiodiscalledthesecondstage.ThesequenceofeventsanddecisionsisasDecisiononx!Observationon!!Decisionony(x, 1 )and( 1 )areconsideredastherststageandthesecondstageproblems respectively.ThematrixAandthevectorbaredeterministicparameters.TherecoursefunctionQ(x,!)denestheexpectedsecondstagevaluefunction.ThematrixT(!),vectorsh(!) Severalsolutionmethodshavebeenintroducedfor( 1 )and( 1 )withnitedistribution. Amongthem ,decompositionmethodsareimportant for solvinglargescale 13

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They canbecategorizedaccordingto fundamental strategies aseither outerlinearizationorinnerlinearization. DanzigandMandasky ( 1961 )applied Danzig-Wolfe decomposition( DantzigandWolfe 1960 ) tosolve thedualoftwo-stagestochasticlinearprogrammingproblemusing inner linearization. VanSlykeandWets ( 1969 ) developed thetwo-stageL-shapedmethodforsolvingtheprimalproblemusingouter linearization thatisaformofBenders'decomposition( Benders 1962 ). Kall ( 1976 )and Strazicky ( 1962 )presentedanotherdualmethodbasedonbasisfactorization. A methodbasedondiscretedistributionsandsplines was proposedby Wets 1974 1983 .Forthecontinuoussamplespace, BirgeandLouveaux ( 1997 ), Shapiro ( 2000 ),and ShapiroanddeMello ( 2001 )proposed the MonteCarlosamplingmethodwithnitelymanyscenarios.Ingeneral, AhmedandShapiro ( 2002 )represented thatthelarger the numberofscenarios, themoreaccurateistheprovidedmodel .This,inturn,makestheresultingformulationverylarge. in certaintimeperiods. BirgeandLouveaux ( 1997 )and LouveauxandSchultz ( 2003 )introducedthespecialstochasticscenariotreestructureforthedeterministicequivalentmulti-stagestochasticlinearprogramming.ThestructurecanbeinterpretedasascenariotreewithTstages(orlevels)wherenodeiinstagetofthetreeconstitutesthestateoftheworldthatcanbedistinguishedbyinformationavailableuptotimestaget.WeuseT=T(0)=(V,E)=(V(0),E(0))to represent thewholescenariotree.ThesetofleafnodesinVisdenotedasL.Nodei,i2V,i6=0(therootnodeindexedasi=0),hasauniqueparenta(i).Nodei,i2VnListherootnodeofasubtreeT(i)=(V(i),E(i)),whichcontainsalldescendantsofnodei,andhasANimmediatechildrensetC(i),i.e., 14

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1-1 showsamulti-stagestochasticscenariotreemodel. . . . . 0 Multi-stagestochasticscenariotree Themulti-stagestochasticlinearprogramwithrecourseandtreestructurecanbeformulatedasminZ=Xi2Vpicixis.t.A1x1=b1,Xj2P(a(i))Tjxj+Aixi=bi,i2Vxi0,i2V.

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block -separablerecourse. Louveaux ( 1980 )rst performed thegeneralizationof theprimal approachfor the multi-stageproblem. Birge ( 1985 )and PereiraandPinto ( 1985 )generalizedtheL-shapedmethodto the multi-stageproblemasanesteddecompositionmethod. Birge ( 1988 )alsoextendedthebasic-factorizationtechniquestothemulti-stageproblem,but thesetechniques broughthieraticalcomputationaldifculties. Grinold ( 1976 ) formulated themulti-stagestochasticprogramasaniteMarkovchainandasequivalentprimal and dualoptimizationproblems. Bealeetal. ( 1986 )performedtherstorderapproachforthemulti-stageprogram. itsparametersareconsideredinuncertainsetsoruncertainintervals. Robust optimizationlooksforthefeasiblesolutionforallpossiblevaluesofunknownparameters,normallytheoptimalsolutionofthebestworstcaseunderdatauncertainty.Currently,moststudiesareaboutsinglestageandtwo-stagerobustoptimization. Theresearchonrobustoptimizationhasrecentlyreceivedrenewedattention. KouvelisandYu ( 1997 )proposedaframeworkforrobustdiscreteoptimization,whichseekstondasolutionthatminimizestheworstcaseperformanceunderasetofscenariosforthedata.Theirframeworkisascenariobasedmodel. Theyletthe uncertaintydataset be whichisascenariouncertaintyset.LetxbethedecisionvariableandD()betheinstanceofthefeasibleregionthatcorrespondstotheuncertainparameter2.ThefunctionQ(x,)evaluatesthefeasiblesolutionx.Therobustoptimizationproblemis designed tondtheoptimalsolutionxand the correspondingsuchthat 16

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1998 2000 proposedthefollowingframeworkonrobustoptimization:minxmaxD020f0(x,D0)s.t.f(x,Di)0,i=0,,T,Di2i, arethegivenuncertaintysets.Comparedwith KouvelisandYu ( 1997 ),their uncertainty setismoregeneral.They showed thatundertheassumptionthatthesetiisellipsoidswiththeform The robustcounterpartsofconvexoptimizationproblemsareeither exact orapproximatedtractableproblems.Butundertheellipsoiduncertainty,therobustcounterpartofanLPbecomesanonlinearprogram.Underthesameframework, Ben-Taletal. ( 2004 )studiedthetwo-stagerobustlinearprogrammingunderthenameadjustablerobustlinearprogramming.Theypresentedthattwo-stagerobustlinearprogrammingiscomputationallyintractableand proposed atractablealternativeapproach,referredtoasafnely-adjustablerobustlinearprogramming. Soyster ( 1973 )proposedalinearrobustoptimizationmodeltoconstructasolutionthatisfeasibleforalldatathatbelonginaconvexset. Bertsimasetal. ( 2004 )and BertsimasandSim ( 2004 )consideredLPswhosecoefcientsoftheobjectivefunctionandconstraintsareassumed tobe inuncertain intervals .Theirapproach retained theadvantageofthelinearframeworkof Soyster ( 1973 )andprotestedconstraintsagainstviolation. BertsimasandSimchi-Levi ( 1996 ), BertsimasandSim ( 2003 ), and BertsimasandThiele ( 2006 )appliedtheirrobustapproachtoaddressdatauncertaintyindiscreteoptimization.Theperspectiveareasarevehiclerouting,network andinventoryproblems.They showed theirrobustapproachistractableintheaboveareas. 17

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( 2008 )discussedtherelaxationof the two-stagerobustlot-sizingproblemwithbasestock,anddescribedabenderdecompositionbasedalgorithmforrobustlinearoptimization. AtamturkandZhang ( 2007 ) described atwo-stagerobustoptimizationapproachforsolvingnetworkowanddesignproblemswithuncertaindemand.Two-stagerobustlot-sizingwithuncertaindemandis an applicationfortheirapproach.Mostofthese works areconcentratedonthetwo-stagerobustoptimization. decision variablex2Rn, integer variabley2Rp.LetS=f(x,y):Ax+Gyb,x0,y0integergrepresentthefeasibleregionofMIP.Whenxisalso integral ,wehaveaspecialcaseofMIP,anintegerprogramming(IP). NemhauserandWolsey ( 1999 )and Wolsey ( 1998 )arecomprehensivereferencesforMIP. SomeMIPproblemscanbesolvedinpolynomialtime,suchastheuncapacitatedlot-sizingproblem,theshortestpathproblem,themaxowproblem,andtheassignmentproblem. Todate,noone hasfoundapolynomialalgorithmfortheseMIP problems ,suchasthe0-1knapsackproblem,thesetcoveringproblem,thetravelingsalesmanproblem, and the uncapacitated facilitylocationproblem. Inthefollowingsection, wepresentusefulresultsonpolyhedraltheoryand algorithms forMIP. of polyhedronandconvexhull. 18

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a nitenumberoflinearinequalities;thatis,P=fx2Rn:Axbg,where(A,b)isanm(n+1)matrix. in thefollowingtheorems.Notethateverypolytopecanbedescribedbyonlyitsextremepoints. NemhauserandWolsey 1999 )Afull-dimensionalpolyhedronPhasauniqueminimalrepresentationbyanitesetoflinearinequalities. NemhauserandWolsey 1999 )Theprojectionofapolyhedronisapolyhedron. weshowthattheintegerprogramcanbereducedtobealinearprogrambythefollowingtheorems. NemhauserandWolsey 1999 )IfS=fx:Axb,x2Rng,where(A,b)isanintegerm(n+1)matrix,thenconv(S)isarationalpolyhedron. NemhauserandWolsey 1999 ) IP iseitherinfeasibleorunboundedorhasanoptimalsolution. 19

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: 1.Pisintegral 2.LPhasanintegraloptimalsolutionforallc2Rnforwhichithasanoptimalsolution. 3.LPhasanintegraloptimalsolutionforallc2Znforwhichithasanoptimalsolution. 1 ).TheLPrelaxationofMIPisminfcx+hy,Ax+Gyb,x0,y0g.LetPrepresentthefeasibleregionofthisLPproblem. NemhauserandWolsey 1999 ) Step2[Terminationtest]:IfL=;,thenthesolution(x0,y0)withobjectivevalue 20

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Step4[Pruning]: a.IfziR b.If(xiR,yiR)=2Si,gotoStep5. c.If(xiR,yiR)2SiandcxiR+hyiR problemswithz Step5[Division]:LetfSij,j=1,,kgbeadivisionofSi.AddproblemsfMIPij,j=1,,kgtoL,wherez Thebranch-andcut algorithmisageneralizationofthebranch-andbound algorithm.InStep4,insteadofgoingtoStep5,thenbranch-and-cutalgorithmndsavalidinequalitya1x+a2ya0ofMIPi,suchthata1xiR+a2yiR>a0.AfteraddingthisinequalitytoLPrelaxationRPi,itisre-solvedandandthen goes backtoStep4. Ingeneral,thenumberofnodesinthebranch-and-boundenumerationtreeisexponentialin termsof theproblemsize,evenfortheintegerprogramwithbinaryvariables.AlmostallgeneralMIPcodesuseabranch-and-boundalgorithmwithLPrelaxation.TheLPrelaxationinthebranch-and-boundalgorithmprovidesthelowerboundofthecorrespondingMIP.Byaddingcuts,thebranch-and-cutalgorithm improves thelowerboundand reduces thenumberofbranchingnodes.Inpractice,thebranch-and-cutalgorithmnormallyperformsbetterthanthebranch-and-boundalgorithmin terms ofcomputationaltime. Inthisdissertation,wegeneratethecustomizedbranch-and-boundalgorithmforlot-sizingwithdisruptioninChapter2.Weexplorethepolyhedralstructuresforstochasticlot-sizingproblemsinChapters4and5. 21

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NemhauserandWolsey ( 1999 )):minTXi=1(ixi+fiyi+hisi)s.t.si1+xi=di+sii=0,,TxiMyii=0,,Txi,si0,yi2f0,1gi=0,,T, setup decisionatperiodi.Parametersi,fi,hi,anddirepresenttheproductioncost, setup cost,inventorycost anddemandin time periodi respectively.Withoutlossofgenerality,wecanassumes0=0andtightenMi=PTj=idi. Thestudyoflot-sizingproblemscan betraced backto WagnerandWhitin ( 1958 )inwhichapolynomialtimedynamicprogrammingalgorithmwasdevelopedtogetanoptimalsolutionoftheproblem.Lateron,thereissignicantresearchprogress insolving thisclassofproblems. Tosolve ULStooptimality,dynamicprogramming,reformulation,andpolyhedronstudyhavebeenproposed.As a polynomiallysolvableapproach,dynamicprogrammingappliedinULSwasrstintroducedby WagnerandWhitin ( 1958 ).TheyproposedanO(n2)dynamicalgorithmtosolveULS. FedergrunandTsur ( 1991 ), Wagelmansetal. ( 1992 ),and AggarwalandPark ( 1993 )independentlyobtainedO(nlogn)implementationsof the WagnerandWhitin ( 1958 ) algorithm Wagelmansetal. ( 1992 )introducedthedenitionoftheWagner-Whitincosts,i.e.,i+h0ii+1, wherei andinventory costs fortimeperiod 22

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the lot-sizingproblemare the facilitylocation, the shortestpath, andthe multicommoditynetworkowformulations. KrarupandBilde ( 1977 ) rst introducedthefacilitylocationreformulationofthelot-sizingproblem s Martin ( 1987 )developedareformulationtechniquetotransformULStoashortestpathformulation. RadinandChoe ( 1979 )providedthemulticommoditynetworkowreformulation. ThecompactdescriptionoftheconvexhullofallfeasiblesolutionsforULShasbeenderivedthroughdifferentapproaches. Baranyetal. ( 1984b )introducedthewell-known(`,S)inequalities togetherwithxi0and0yi1todescribetheconvexhullofULS,where`2I=f0,1,,Tg,SI,anddij=Pjk=idk.Inaddition,fortheuncapacitatedlot-sizingproblemwithstart-up costs vanHoeseletal. ( 1994 )presentedanextendedformulationandanO(T2)timeseparationalgorithm. FortheWagner-Whitin costs case, PochetandWolsey ( 1994 )generatedanextendedformulationforULSwithO( Bakeretal. ( 1978 )studiedthedynamiclot-sizingprlbemwithtime-varyingproductioncapacityconstraints. Fisheretal. ( 2001 )proposedamethodtomitigateend-effectsinthedynamiclot-sizingbyevaluatingtheend-of-horizoninventorylevelbasedontheclassicEOQmodel. Hartmanetal. ( 2010 )derivedanewsetofvalidinequalitiesforthecapacitatedlot-sizingproblemfromtheend-of-stagesolutionsofadynamicprogrammingalgorithm. 23

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InChapter3,weextendthepreviousstudytomultiple disruption cases.Weformulatetheproblemtobeamulti-stagerobustoptimizationproblem.Ourobjectiveistoprovideamulti-stagerobustschedulesuchthatthetotalcostisminimizedundertheworstcasescenario.Weexplorethepolyhedralstructureofthelot-sizing problem withsingledisruptionbygeneratingfacet-deninginequalities.Thegeneralreformulationschemehasbeenprovidedto transform themulti-stagerobustoptimizationproblem into anequivalentsinglestageprogram. InChapter4andChapter5,weconsideratwo-stagestochasticuncapacitatedlot-sizingproblemwithdeterministicdemandsandWager-Whitincosts.InChapter4,weconsidercaseswithoutbacklogging.Theoptimalformoftheinventorylevelhasbeenexplored.Basedontheoptimalform,weprovide an extendedformulation.Further,weprovidetheintegralpolyhedrondescriptionintheoriginalspacebyprojectingtheextendedformulationtotheoriginalspace.InChapter5,weconsiderthecasewithbacklogging.Weinvestigatetherelationshipamongtheinventory,the setup ,andthebackloggingintheoptimalsolutionandprovidetheoptimalformsofinventoryandbacklogginglevels. An extendedformulationhasbeenproposed. InChapter6,westudythestochasticdynamicknapsacksetwhichisnaturallyraisedfromthestochasticlot-sizingproblem.First,weextendtheresultsinthedeterministicdynamicknapsacksettothestochasticsetting,andgeneratevalidpathinequalitiesforthestochasticdynamicknapsackset.Second,themixingandpairingschemesareappliedforthestochasticdynamicknapsacksettogeneratevalidtreeinequalities.Third,theliftingschemesareadoptedtothestochasticdynamicknapsackset.Morevalidinequalitiesaregenerated.Fourth,theparallelcomputingisappliedto 24

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Finally,inChapter7,weconcludethisdissertationandproposefutureresearchdirections. 25

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expend lotsofresourcesand efforts generating thelong-termproduction plan .However,disruptionsoftenoccurandinterrupttheregularplannedproductionprocesses andcausedemandsinsometimeperiodscannotbesatised. Extraproductionsarerequiredinordertocovertheunlleddemands andbringextrareparationcosts .The efciency ofnewproductionplansisnormallymeasuredbythe extracostgeneratedduetodisruption .Previously,researchershaverealizedthat a lot-sizingschedule hastobeupdated aftertheuncertainevents Inpreviousresearch, Carlsonetal. ( 1979 ) and Kazanetal. ( 2000 ),etc.have studied thenervousnessoflot-sizingproblemsinwhichfuturedemandsaregraduallyacquiredandtheinitialscheduleshavetobe updated,whichleadstoextraproductioncost Yangetal. ( 2005 )studiedhowtorecoverthelot-sizingproblemsaftertherealizationofdisruptions.Allthese works arefocusedontherecoveryplanningaftertheoccurrenceofuncertainevents andnorecourseinformationisinvolvedinthemodeltoobtaintheoriginalschedule.Ascomparedtotherobustoptimizationapproach,theabovestudiesfocuson studyingthe efciencyforthesecond-stageproblem. Inthispaper,westudythelot-sizingproblemwith potential disruptions asrecourse viarobustoptimizationtohandletheuncertainty.Weconsidercasesinwhich thereispotentiallyonedisruption,and inwhich theexacttime of thedisruption isuncertain.Oncethedisruption occurs ,thecorrespondingrecoveryproductionfollowsto covertheunlleddemandandtheextraproductioncostisgenerated .Ourobjectiveistomaintaintheproductionplanningwith theconsiderationofapotentialdisruptionduringtheproductionprocessandachievetheminimumobjectivevaluethatconsiderstheworstcasescenarioforthedisruption 26

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thispaperproposesarobustproductionplanningtoaddressdisruptions.Ascomparedtothecaseinwhichtherecoveryproductionisperformedaftertheoccurrenceof a disruption, ourapproachprovidesa morerobustplanning.Tothebestofourknowledge, ourapproachisthersttousetherobustoptimizationapproachtosolvethelot-sizingproblemwithdisruption .Intheremainingpartofthispaper,westudydifferentlot-sizingproblemswithdisruptions.InSection 2.2 ,westudythelot-sizingproblemwith disruptionand outsourcing.Forthiscase,thelot-sizingproblem withdisruption canbereformulatedasatwo-stageminmaxproblem.Acorrespondingprimal-dualapproachcanbeconstructedtosolvetheproblem.InSection 3.4 ,westudythelot-sizingproblemwith disruptionandbacklogging .In Section 2.3.1 ,westudythe non-setup costcaseanddevelopapre-processingalgorithmtopre-calculatetheparametersforthe second stageproblem. Then,wecanformulatetheproblemasasingle-stageproblem InSection 2.3.2 we developacustomizedbranch-and-boundalgorithmtosolvethecaseinwhichsetupcostisconsidered.Finally inSection 2.4 ,weprovidethecomputationalresultsthatshowthetractabilityandefciencyofourapproaches. forthelot-sizingproblemwithdisruption .Weassume that ifdisruptionhappensintimeperiodi,thentheproductionamountxioriginallyscheduledinthistimeperiodwillbe purchasedfrom othersuppliers.Thatis,outsourcinghappensinthesametimeperiodasthedisruptionhappens,and the outsourcingamountisequaltoxiwithunitoutsourcing cost Since afterthedisruption,weconsiderthatoutsourcingisanoptionbesidesproductiontosatisfydemands,we alsoallow outsourcing asan option inheoriginalschedule .Forthecaseinwhichdisruptionisconsidered, allrst-stagevariables,theproductionlevels,theoriginaloutsourcinglevels,theinventorylevels,andthesetupdecision, arenotinuencedbythedisruption ;butthe second-stagedecisiondetermines 27

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timeperiod tobe : 3 )indicatestheinventory ow balanceandconstraint( 3 ) indicatesthatproductionhappensinthesetuptimeperiod;noupperboundlimitexistsfortheproductionamount .Byintroducinganarticialbinarydecisionvariableitoindicate whetherdisruptionhappens intimeperiodi, xiMyi, ,x,w,s2RT+,y,2f0,1gT. 28

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thelot-sizingproblemwithasingledisruptionandoutsourcing ,theformulationcanbesimpliedasthefollowingtwo-stagemin-maxproblem:minx,s,y,wTXi=1pixi+hisi+oiwi+fiyi+maxTXi=1i((oipi)xifi)! xiMyi, ,x,s,w,2RT+,y2f0,1gT. ,2RT+. 29

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1 Therefore,theoriginalconclusionholds. Byintroducingnewextravariable,wecanformulatePO2asfollows: lot-sizingproblemwithasingledisruptionandoutsourcing canbetransformedasthefollowingsinglestagemixed-integerprogram:minx,s,y,w,qTXi=1(pixi+hisi+oiwi+fiyi)+(DO)s.t.(oipi)xifi,i=1,,Txi+wi+si1=di+si,i=1,,TxiMyi,i=1,,Tx,s,w2RT+,2R+,y2f0,1gT. thelot-sizingproblemwithasingledisruptionandoutsourcing bya mixed-integerprogramming problem.Wecanalsoobservethatthefollowing(`,S)-typeinequalities(see,e.g., Baranyetal. 1984a )arevalidforDO. withdisruption thatutilizesbackloggingasthereparationapproach.Weassumethat the unitoutsourcingcostismuchlargerthantheunitbackloggingcostanditisonly utilizedwhennobacklogging 30

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Inthebackloggingsetting ,theinventoryandbackloggingamountsafterthe time periodinwhichdisruptionhappensarechanged.Therefore,wecannot adopt thesame approach asthelot-sizingproblemwith disruptionand outsourcingtosolvethisproblem. We introduceparametersci`andauxiliarydecisionvariablesxi`as described in PochetandWolsey ( 1988 ).Thatis,parameterci`representsthetotalcost ofproducinganitem intimeperiodi(exceptsetupcost)tosatisfythedemandintimeperiod`.Forinstance,foreachiT,wehaveci`=pi+P`k=ihkif`>i,ci`=pi+Pik=`bkif`
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two -stage optimization formulation.Intheobjectivefunction,the secondstage decision variabletistheindexoftimeperiod. In the second stage,the minimumtotal makeupcostforadisruptioncanbe determinedby thesumoftheminimummakeupcostforeachxtj, whichprovidetheminimummakeupcostforxtj, 2 )indicatesthatthedemandintimeperiodjshouldbesatised.Constraint( 2 )indicatesthatforeachtimeperiod,productionhappensonlyinthe time period inwhich productionis setup Inthisformulation, in the secondstage ,weonlyneedtodecidethetimeperiodinwhichthedisruptionhappensandthetimeperiods duringwhichreparationproductions happen. The reparationproduction quantityforeachperiodjis equaltotherststagedecisionvariablex j, wheretisthedisruptionperiod setup costs. Supposedisruption happensintimeperiodt, with the second termoftheobjectivefunctionin PR,i.e. ,PTj=1mint
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WiththePre-processingalgorithm,cq(j)j=mtjwhendisruptionhappensintimeperiodt. non-setupcost lot-sizingproblem withdisruptionandback-logging ,supposethedisruptionhappensintimeperiodt,the timeperiodtocovertheunlled demandintimeperiod`isargminfci`,i>tg andthecorrespondingminimumunitmakeupcostismt`=minfci`,t
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. 1 5 2 3 4 Anexampleforthelot-sizingproblemwithdisruptionandbacklogging setup costisconsideredasshownin PR ,wecannotdeterminethe second stageparameterscq(j)juntiltherststagedecisionvariablesyq(j) decided,becausethemakeupproductioncanonlyhappenintheperiodsthataresetup,i.e.,yi=1.Inthissection,weintroduceareformulationof PR suchthattheproblemcanbeformulatedasasinglestageproblemandcanbesolvedasasinglestagemixed integerprogrammingproblem. Werstintroducethefollowingadditionaldecisionvariables: second stageinordertosatisfytheunlleddemandintimeperiodj,duetothedisruptionintimeperiodt. produce intimeperiodktosatisfytheunlleddemandintimeperiodj,wheret
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2-1 ,supposeadisruptionhappensintimeperiod1.Thenq11andq12representthesmallestmakeupcostsfortheunsatiseddemandinperiods1and2 respectively.Themakeupproductionmaybefromperiods3or4.Sincevariablerepresentsthemaximumincrementcostafteradisruption,wehaveq11+q12c11x11c12x12.Finally,timeperiodT+1servesasthepseudo-periodtoprovideoutsourcing.Thus,wehaveyT+1=1withzerosetupcost. disruptionandbacklogging canbeformulatedasthefollowingsingle-stage mixed-integer programmingproblem: 2 ),( 2 ),x2R(T+1)T+,y2f0,1gT+1,z2f0,1gTT(T+1),q2RTT+,2R+. 2 )representsthatshouldbethemaximumincrementcostduetothedisruptionateachtimeperiod 2 )indicatesthebackloggingcosttofullltheunlleddemandintimeperiodjduetodisruption in timeperiodt.Constraint( 2 )representsthat,duetotheunlimitedproductioncapacity,thereisonlyonelaterperiodneededtoproducemoretofullltheunlleddemandforeachtimeperiodjduetodisruptionintimeperiodt.Constraint( 2 )representsthat 35

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2 )and( 2 )describedin PR arealsonecessary. PB2 a nitenumberofsteps. In PB2 requiredtobeintegerandobtainacorrespondingfeasiblesolution. Intheremainingpartofthissection,wedescribeindetailourbranchingandsearchingstrategies,lowerandupperboundsinthebranch-and-boundframework,andBendersdecompositionframeworkfortheoptimalitytest.Beforewedescribethedetails,werstletLandVbethe sets ofleafandtotalnodesintheenumerationtree respectively.Foreachparticularnoden2V,welety(n)andd(n)bethesolutionofvariableyandthedepthofthisnode;leta(n),C(n)andD(n)betheparent,childrensetanddescendantsetofnoden;letP(n)bethenodesonthepathfromtherootnodetonoden. 36

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descendant ofnoden.Therefore,inordertogettheglobaloptimalsolution,weneedtoconsiderallthecombinationsofintegralysolutions.Wewillstillbranchtheyvariablewhenyvariableisintegral.Inourprocess,werstbranchthedecisionvariabley1attherootnodewithdepth1.Then, ingeneral ,correspondingtoeachnodewithdepthd,wewillbranchthedecisionvariableyd.Finally,weapplythedepthrstsearch,whichallowsustondgoodfeasiblesolutionsatearlystagesduringthebranch-and-boundprocess. Proof. descendant Toobtainanupperbound,ateachnodeinthebranch-and-boundtree,ifwehaveallydecisionvariablesintegral,thenwecansolveasubproblemcorrespondingtothisintegralysolution.Asshownin PB2 the disruptionperiod,andthereparation production periodsinthesecondstage .Correspondingtoeachnoden2Vsuchthatallydecisionvariablesareintegralinthelinearprogrammingrelaxationsolution,letI(n)=f candidatein whichthedisruptionhappens.Onceadisruptionhappensintimeperiodt,onlytheperiodsthataresetupaftertimeperiodtcanprovide extraproduction quantityto satisfyunlleddemandswhicharecausedbythedisruption intimeperiodt.Correspondingtoeachpair(t,`),wheretrepresentsthedisruption 37

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Ifk=argminfci`,i2I(n),tUB,thenweprune node descendants .Ifallydecisionvariablesareintegralbysolvingthelinearprogramcorrespondingtothisnode,thenwecanobtainbothanupperboundUB(n)andalowerboundLB(n)fortheproblem.IfUB(n)UB,thenweprune node descendants .Otherwise,wecontinuebranchingandperformingthe depth rstsearch.Finally,wewillterminateatanoptimalsolution. approach ateachleafnodetotestifwehaveobtainedtheoptimalsolutionfortheproblem.IntheBenders'decompositionframework, 38

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: ,TXk=t+1ztkj=1, 2 )isgeneratedbasedonthefollowingsubproblemand,,and!arethedualvaluescorrespondingtoconstraints( 2 ),( 2 ),and( 2 ). Forinstance,suppose PB-SUB ,wewillobtaintheoptimalobjectivevalueZ( 2 ),( 2 ),and( 2 ).Thus,wewillobtainacorresponding

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PB2 Ifinequality( 2 )issatised,then( PB2 Thus, weonlyneedtosolveanextralinearprogrammingproblemtoobtainanupdatedeachtime.Thedetailsareshownasfollows: 2 ). andaddthecorrespondingdualvaluestosetU. Sincetheoptimalysolutionmustexistatsomeleafnodes,theoptimalitytestwillbesatisedataleafnodeinthebranch-and-boundframework.Thus,thealgorithmwillterminateinnitesteps. 40

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the robustlot-sizingproblems withdisruption .AllcomputationalexperimentswerecarriedoutonaLinuxworkstationwithaPentiumDual2.8Gprocessorand6GRAM.WeusedCPLEX10.1CallableLibrarytoimplementouralgorithms andrunthereformulatedmodels. anddifferenttimehorizonsT.Fortheinstances,wesetthetimehorizonT=10,20,30,40and50,andtheratiosofsetupcosttounitproductioncostf=p=10,20,30,and40 respectively.Thereare20combinationsintotal. Forthecaseswithsetupcosts,correspondingtoeachofthecombinationsofTandf=p,wegeneraterandominstancesinwhichtheunitproductioncostandthesetupcostareuniformlydistributedintheintervalsasshowninTable 2-1 Forthecaseswithoutsetupcosts,correspondingtoeachofthecombinationsofTandf=p,wegeneraterandominstancesinwhichtheunitproductioncostisuniformlydistributedinthesameintervalasforthecaseswithsetupcostsshowninTable 3-1 Table2-1. Parametersetting unitproductioncostpi [50,100] [500,1000] ratio20 [50,100] [1000,2000] ratio30 [40,60] [1000,2000] ratio40 [40,60] [500,1000] 41

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respectively.Finally,welet unit outsourcingcostbexedat200. asimpleheuristic, toserve asabaseforcomparisonwithothermodels.Basedontheoptimalsolutionofthesingleitemlot-sizingproblemwithoutdisruption,themaximumpickheuristicpicks aperiod ,inwhichtheoutsourcing brings thelargestincrementofthetotalcost,todooutsourcing.Thedetailedmaximumpickheuristicislistedasfollows : MPH ) Step1:Solvethesingleitemlot-sizingproblem.Letxbetheoptimalsolutionandzbethecorrespondingobjectivevalue. disruption Forthiscase ,asdescribedinsection4.1 ,wetest20combinationsinwhichthetimehorizonT=10,20,30,40,and50 andtheratiof=p=10,20,30,and40,respectively.ThecomputationalresultsareshowninTables 2-2 For eachofthe20combinations,wereporttheaveragevaluesof5randominstances.Wereport1)theoptimalobjectivevalueofthelot-sizingproblemwithoutdisruptions,denotedasSLS,2)theobjectivevalueobtainedbymaximumpickheuristic,denotedas MPH ,3)theobjectivevaluefortherobustoptimizationformulationobtainedbyusingCPLEXtosolvethedualformulation( DO )fortheoutsourcing case and4)thegapbetween( DO )and( MPH ),denotedasGAP(M::R)=(Obj DO ),denotedasGAP(R::S)=(Obj 42

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DO ).Comparedwiththeuncapacitatedlotsizingproblemwithoutdisruptions,theaverage gap between(SLS)and( DO ) is costincreases thedisruption WecanalsoobservethatasTincreases,theaveragegapbetweenSLSand DO decreases. Weperformcomputationalexperimentsforcasesbothwithandwithoutsetup cost Forthecasewithoutsetupcost,weperformthecomputationalexperimentsbasedonthepre-processingalgorithmandreformulation( PB1 2.3.1 .Wetest10combinationssuchthattheratiosbetweenthesetupcostandtheproductioncostequalto10and30,andtheplanning horizons are10,20,30,40and50respectively.Correspondingtoeachcombination,wereportthevaluesofSLS, MPH ,NoSetup,andGAP(M::R)asdescribedintheoutsourcingcase. Thepre-processingalgorithmneeds PB1 withO(T2) ascompared to the maximumpickheuristic,thecasewithoutconsidering thedisruption duringtheplanningprocess.ThecomparisonisshownonTable 2-3 andinaverage,thecostsavingis27.8%. Forthecasewithsetupcost,wecomparedifferentsolutionapproachesthatincludesolving( PB2 2.3.2 directlybydefaultCPLEX,theproposedbranch-and-boundalgorithm,and the Bendersdecomposition approach .Wesetthetimelimittobe2hoursandtestthecombinationsthatratio=10,20,30and40,andthetimeperiodT=10,20and30,respectively.ThecomputationalresultsareshowninTable 2-4 .WecomparethreedifferentapproachesthatincludethedefaultCPLEXMIP 43

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,intermsofseconds .FromTable 2-4 ,wecanobservethatourproposedBBmethodperformsmuchbetterthandefaultCPLEX.ForT=10and20cases,no instances canbesolvedintooptimalitywithinthegiventimelimitbythedefaultCPLEX.Thenaloptimalitygapsprovidedby thedefault CPLEXareintheinterval[1.13%,5.39%].Ourproposedbranch-and-boundalgorithmcansolveallinstancesintooptimalityforT=10andT=20cases :for ;for gapsascompared tothedefaultCPLEX.FortheBenders'decompositionapproachtosolve( PB2 instances,no instances canbesolvedintooptimalitywithin24hours. Therefore,the computationalresultsindicatethatourproposedsolutionapproachisthebestamongallthreeapproaches. 44

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Lot-sizingwithdisruption:outsourcing SLS 20.1% 14.6% 11.2% 6.6% 19.3% 15.9% 14.2% 11.5% SLS 16.3% 15.8% 12.3% 10.7% 20.3% 16.3% 14.8% 12.6% SLS 17.8% 9.9% 10.5% 9.1% 21.2% 14.8% 12.5% 10.4% SLS 18.8% 16.6% 12.9% 8.8% 20.9% 14.4% 11.7% 9.8% Lot-sizingwithdisruption:backloggingandnon-setupcost SLS 24.2% 19.9% 16.5% 16.1% SLS 40.3% 21.7% 20.5% 16.1% Lot-sizingwithdisruption:backloggingandbranch-and-boundalgorithm T=20 T=30 ratio=10 Gap(Op) CPLEX 4.39% 4.42% 0(3974) 2.75% Gap(Op) CPLEX 3.79% 4.71% 0(3400) 2.66% Gap(Op) CPLEX 1.59% 2.19% 0(2959) 1.68% Gap(Op) CPLEX 2.33% 2.43% 0(2918) 1.79%

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isa natural propertyofdisruptionsinproductionplanningproblems.Theexacttimeswhendisruptionshappenareunpredictable.Inthispaper,westudythemulti-stagerobustlot-sizingproblemasanexampletoanalyzesolutionapproachesformulti-stagerobustintegerprogrammingproblems.Weconsiderthelot-sizingprobleminwhichthedisruptionoccurrenceisuncertain.Onceadisruptionoccurs, subsequent recoveryproductions follow tocovertheunlleddemandswith theircorrespondingrecovery productioncosts.Ourobjectiveistomaintaintheproductionplanningwith disruptions .Meanwhile,theproduction plan stillcanprovidetheminimalproductioncostundertheworstcasescenario(thelargestproductioncostgrowths).Ingeneral,amulti-stagerobustmixed-integerprogrammingsettingcanbedescribedasfollows: andtrepresentstheuncertaintysetinstaget.Decisionvariables in eachstagetwillsatisfyconstraintsthatdescribethefeasibleregioninstaget,denotedasMt(x1,y1,x2,y2,...,! Thecontributionsofthischapterarethreefold.First,thispaperproposesarobustproductionplanningtoaddressdisruptions.Ascomparedtothecaseinwhichtherecoveryproductionisperformedaftertheoccurrenceofdisruptions,weprovidemorerobustinitialproductionplanning.Second,thispaperproposesarobustproductionplanningtoaddressmultipledisruptions.Ascomparedtothecaseinwhichthereisonlyonedisruption, the multiple disruptions caseismuchmorechallengingto 46

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structures ofthecorresponding models forthelot-sizingproblemwithasingledisruptionandshowsthecomputationaleffectivenessofourapproaches.Tothebestofourknowledge,allthesethreeaspectshaveneverbeenstudiedbefore.Intheremainingpartofthispaper,westudydifferentmulti-stagerobustlot-sizingproblemswhicharetractable.InSection 3.2 ,wepresentthemotivation,introduceourrobustlot-sizingproblem,anddevelopageneralformulationforthemulti-stagerobustlot-sizingproblem.InSection 3.3 ,westudytherobustlot-sizingproblemwithoutsourcing.Forthiscase,themulti-stagerobustlot-sizingproblemcanbereformulatedasatwo-stageminmaxproblem.Acorrespondingprimal-dualapproachcanbeconstructedtosolvetheproblem. InSection 3.4 ,therobustlot-sizingproblemwithbackloggingandasingledisruptionisstudied.Weformulatethisproblemasatwo-stagerobustmixed-integerprogram. Weproposeareformulationruletotransferthisrobustmodeltoamixed-integerprogram. Forthisproblem, westudythepolyhedralstructureandgeneratethecorrespondingfacet-deninginequalities basedonthereformulation .InSection 3.5 ,weconsider caseswithandwithoutsetupcost oftherobustlot-sizingproblemwithbackloggingandmultipledisruptions.Wedevelopadeterministicequivalentformulationforthemulti-stagerobustoptimizationproblem,based on theenumerationoftheperiodsinwhichdisruptionshappen.Finally,inSection 3.6 ,weprovidethecomputationalresultsthatshowthetractabilityandefciencyofourapproaches. disruptionshappeningiventimeperiods ,thescheduledproduction isterminated in these time periods .Weadoptoutsourcingandbackloggingasapproachestoaccommodatethesupplyshortage.Thatis,unlled demands duetodisruption in certaintimeperiodcanbebackloggedby 47

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in latertimeperiods.Thisapproachisreferredtoas a robustlot-sizingproblemwithbackloggingcase.Forthiscase,ifthereisonlyonedisruption,thenthescheduledproductionsafterthedisruptiontimeperiodcanbeprocessed,andextraproductionamountcanbeintroducedatatimeperiodwhichissetupafterthedisruptiontimeperiodtoaccommodatethesupplyshortage.Iftherearemultipledisruptions,thentherearemultipleiterationsintermsofdisruptionsandrecoveries.Forinstance,aftertherstdisruption,theproductionprocessafterthedisruptiontimeperiodwillberescheduledtocounteractthedisruption.Similarly,whentheseconddisruptionhappens,reparationproductionwillhappenagain.Consequently,disruptionsandthecorrespondingreparation productions willhappenconsecutively. Toformulatetherobustlot-sizingproblemwithuncertaindisruptions,weintroduceahiddendisruptionparameter2f0,1gT.Ifthereisadisruptionthathappensinperiodi2I=f1,...,Tg,theni=1;otherwise,i=0.Accordingly,wedenotethesetofdisruptionperiodsS0=fi,i=1,i2Ig.Weassumethatthetotalnumberofdisruptionperiodsis.Wealsoletx1,s1andy1representtherststageproduction, stocking, andset-updecisionvariables,andxk,skandykrepresentthecorrespondingvariablesafterthek1thdisruption.Then,the theobjectivefunctionofthe multi-stagerobustlot-sizingproblemcanbedescribedasfollows: +TXi=t1+1(ix2i+his2i)+maxt22S0,t2>t1min(x3,s3,y3)(t2 +TXi=t2+1(ix3i+his3i)+maxt32S0,t3>t2min(x4,s4,y4)(t3 +TXi=t3+1(ix4i+his4i)++maxt2S0,t>t1min(x,s,y)8<:t +TXi=t+1(ix+1i+his+1i)9=;9=;9=;9=;9=;,

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recovery productionquantitytobetheunit recovery productioncostifbackloggingisapplied in timeperiodiforeachi2I.Besidesthis,inourobjectivesetting,ifthejthdisruptionhappens in timeperiodt thecorrespondingcostj(tj,x,y,x1,y1,,xj1,yj1)issaved .Forinstance,ifthereisadisruptionthathappensintimeperiodt,thenthescheduledproductioncost cannotbeprocessed,andthecorrespondingcosts(includingproduction,inventory,andbackloggingcosts)aresaved.Theexactformulationofsavingcostfunctioni(...)andtheconstraintssetdependontherecoverapproaches .Intheremainingpartofthispaper,westudytherobustlot-sizingproblemwithbackloggingcases. purchasedfrom othersuppliers.Thatis,outsourcinghappensinthesametimeperiodasthedisruptionhappens,andoutsourcingamountisequaltoxtwithunitoutsourcing cost Underthisscenario,sinceweconsiderthatoutsourcingisalsoanoptionbesidesproductiontosatisfydemands afterthedisruption ,forthedeterministiclot-sizingproblemwithoutdisruptions,wewillalsohaveoutsourcingoptionconsidered.Forthecaseinwhichdisruptionisconsidered,theproductionamount,theoriginaloutsourcingamount,theinventoryamount,andthesetupdecisionareallrst-stagedecisionvariablesandnotinuencedbythedisruption.Theonlysecond-stagedecisionvariableisthetimeperiodtwhenthedisruptionhappens.Ifmultipledisruptionshappen, in eachdisruptiontimeperiod,theoutsourcingamountisequaltothereducedproductionquantitydueto thedisruption .Accordingly,westillonlyneedtodecidetheperiodswhenthedisruptionshappen.Therefore,themulti-stagerobustlot-sizingproblemwithoutsourcingcanbetransferredtobeatwo-stagerobustoptimizationproblemwith 49

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Inthiscase,ifthejthdisruptionhappensintimeperiodtj,theproductioncostsinthistimeperiodcanbesaved.Thus,thelostfunctionisformulatedasj(tj,x,y)=ptjxtj. IfweassumethesetofdisruptionperiodstobeS0,asubsetofI,andthenumberofdisruptionstobe,i.e.,jS0j=.Thecorrespondingformulationcanbedescribedasfollowsminx,s,y,wTXi=1pixi+hisi+oiwi+fiyi+maxS0Xt2S0I((otpt)xt)! 3 )indicatestheinventory ow balanceandconstraint( 3 )indicates thatproductionhappensinthetimeperiodinwhichtheproductionissetupandtheproductionamountdoesnothaveanupperboundlimitsinceMisaverylargenumber. Letarticialbinarydecisionvariableiindicateifthedisruptionhappensintimeperiodi,theaboveformulationcanalsobedescribedasfollows:

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Ifthereexistsatimeperiodiinwhichi2(0,1)andPTi=1i=,thentheremustexistatleastonemoretimeperiodjinwhichj2(0,1).If(oipi)xi=(ojpj)xj,thenwecanincreaseianddecreasejsuchthateitheriorjbecomesintegral.Thus,weobtainasolutionwithafewernumberoffractionalsolutions.Followingthissamestep,wecaneitherobtainanintegralsolutionwiththesameobjectivevalueorndacaseinacertainstepinwhich(oipi)xi6=(ojpj)xj.Underthisscenario,withoutlossofgenerality,wecanassume(oipi)xi>(ojpj)xj.Then,wecanobtainalargerobjectivevalueforthesubproblemifweincreaseianddecreasejbyasmallvalue>0.Contradiction. Therefore,theoriginalconclusionholds. Wecanfurtherutilizeaprimal-dualapproachtogenerateadualformulationofRPLSasfollows: 52

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Fromaboveanalysis,wecansolvethemulti-stagerobustlot-sizingproblemwithoutsourcingbyamixedintegerproblem.Wecanalsoobservethatthefollowing(`,S)-typeinequalities(see,e.g., Baranyetal. 1984a )arevalidforRDLS. Proof. which utilizesbackloggingasthereparationapproach andinwhichasingledisruptionhappens.Inthebackloggingsetting,theinventoryandbacklogginglevelsinthetimeperiodswhichareafterthedisruptionperiodarechanged.Weassumethateverydisruptioninthelot-sizingproblemwithbackloggingshouldberecoveredafterthedisruptionwithouttheconsiderationofoutsourcing. Underthisassumption,weobservethisproblemandobtainthefollowingobservationresult: the potential for recoveryproduction.Intheproductionhorizon,thelastperiodisaspecialperiod,becauseifproductionprocesses 53

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the disruptionoccursinthelastperiod,i.e.,T,nolaterperiodcanrecoveritsproduction.Thus,weassumethatthelastperioddoesnotproduceintherststage. Notehere,thepurpose ofthe aboveassumptionistokeeptheproblemself-contained. Inthiscase,allsetupdecisionsaremadebeforethedisruptiontimeperiodandobtainthefollowingobservationresult : noproductionhappensinthisperiod. Ifproductionanddisruptionhappeninlastsetupperiod,nolaterperiodcanprovidetherecoverproductionforitsproduction. Basedonthegeneralformulationforthelot-sizing problem withmultipledisruptionsandtheaboveobservations,weformulatetheproblemasatwo-stagerobustproblem: withx1T=0. Intheobjectivefunction,ifthereisadisruptionthathappensattimeperiodt,thenthescheduledproductionattcannotbeprocessed,thecorrespondingproductioncostptxtissaved.Meanwhile,thecorrespondinginventoryandbackloggingcostswhichinvolveinventoryandbackloggingamountsfromproductionamountxtaresaved.Thetotalsavedcostisevaluatedbyfunctiont(x1,s1,`1).Theconstraints( 3 )and( 3 )guaranteethedemandineveryperiodcanbesatisedbeforeandafterthedisruption. 54

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3 )and( 3 )keepthattheproductionisaccordingtothesetupdecisionbeforeandafterthedisruption,respectively. InPLS,theexactproductionamountsprovidedbythedisruptionperiodtfordemandsinothertimeperiodsarehardtodetermine.Thus,theexactformofsavingfunctiont(...)cannotbe determined basedontheinformationinPLS.Therefore, inthefollowingsection ,wegenerate a reformulationtoprovidetheexactmathematicalformulationforPLS. thetotalcostinvolvedforasingleitemproducedintimeperiodi(exceptsetupcost)tosatisfythedemandintimeperiod`.Forinstance,foreachiT,wehaveci`=pi+P`k=ihkif`>i,ci`=pi+Pik=`bkif`
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InLS-S, constraints( 3 )and( 3 )guaranteethatdemand in eachtimeperiodicanbesatisedand that thelastperioddoesnotproducebeforethedisruptionduetoObservation1.Constraints( 3 )and( 3 ) guaranteethatproductionhappensinthesetuptimeperiodintherstandsecondstages .Constraint( 3 )forcesthatallset-updecisionsaremadebeforethedisruption. Inordertosolvethistwo-stagerobustmixed-integerprogrammingproblem,weenumerateperiodswhenthedisruptionhappenstoreformulateLS-Sasasinglestagelinearprogrammingproblem.Accordingto the problemsetting,trepresentstheperiod of thedisruption.Thedecisionofx2i`isbasedontheperiodwhentherstdisruptionhappens.Thus,welettheproductionquantitybe a functionofthedisruption.Forinstance,weletx2i`(t)representtheextraproductionquantitiesafterthedisruption.Weusetheseasdecisionvariablesintheformulation.Finally,foragivent,weusetorepresentthetotalextracostaftertherstdisruption.Inthisway,weenumerate 56

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problem withsingledisruptionandbackloggingasfollows: Constraints( 3 ),( 3 ),( 3 ),and( 3 ). 3 )asshowninthefollowing: Thereasonliesinthefollowingtwofacts: Intheoptimalsolution,fortheformulationincluding( 3 ),1shouldachievethemaximumvaluefortheleftsidecorrespondingtoacertaintimeperiodt.Thatis,thereexistsatleastonetightinequalityin( 3 )for. Intheoptimalsolution,fortheformulationincluding( 3 ),wehaveatleastonetightinequalityinwhichtheleftsideachievestheminimum.Otherwise,canbedecreasedand wehaveacontradiction .Notehere,wedonotneedtoconsidertheminoperationforthenon-tightinequalities,sinceitwillnotaffecttheoptimalobjectivevalue. Withinequality( 3 ),thesingledisruptioncasecanbesolvedbyusingthefollowingmixedprogrammingformulation(LS-S1)minx1,1T1Xi=1TX`=1ci`x1i`+1

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3 ),( 3 ),( 3 ),( 3 ),( 3 ),( 3 ). problem withasingledisruptionandbackloggingcase,basedontheenumerationofthescenariosoft,weintroduceanarticialvariable1andthereareO(T)constraintsof( 3 ),( 3 )and( 3 ).Forthedisruption,foragivenscenariooft,wealsoincreasethedimensionofthesecondstagedecisionvariablex2i`tobex2i`(t).Therefore,thereareO(T2)constraintsof( 3 )andO(T3)constraintsof( 3 ). Tosimplify thenotation ,weletxiq`representx2q`(i).Withequation( 3 ),wesubstitutex1i`byPTq=i+1xiq`.Thedetailedreformulationisasfollows: Wedivided`onbothsideofconstraints( 3 ),( 3 ),andlet( 3 ).Letaiq`=xiq`=d`.Then weobtainthenewformulationforrobustlot-sizingwithasingledisruptionandbackloggingas:minx,y,TXi=1fiyi+T1Xi=1TX`=1ci`d`TXq=i+1aiq`+(LS-SR)s.t.T1Xi=1TXq=i+1aiq`=1,`2I 58

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Next,westudythestructureoffeasibleregionofLS-SR.LetXD=f(x,y,z,)2RTT(T1) 2+2T+:(x,y,z,)satises( 3 )to( 3 )g. 2+2T+:(x,y,z,)satises( 3 )to( 3 ))g. andPDisthepolyhedronofXD.Notehere,XDrecordsthefeasibleregionofLS-SR.RDistherelaxationofXD. Second,wederivethedimensionofPDandshowwhichaboveinequalitiesarefacet-deninginequalities. 2+2T. deningInequalities Proof. 3.5 by two claims. 59

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3 )isavalidinequalityforLS-SR. Claim2.( 3 )isafacet-deninginequalityforLS-SR. ProofofClaim1.FromObservations1and2,weknowthatinordertorecovertheproduction,atleast one periodissetuponlyfortherecoveryproduction.ThusPTq=2zq1.Wediscussthefollowing two cases. Case1.Ifthere is morethan one periodwhich is setupforrecoveryproduction,thenPTq=2zq2. Case2.Ifthereisonlyonesetupperiodforrecoveryproduction,thenPTi=2zi=1.With( 3 ),wehavePTq=2Pq1i=1aiq`(q)=1.BecauseifPTq=2Pq1i=1aiq`(q)6=1,demandineachperiodcannotbesatised. Therefore,( 3.5 )isavalidinequalityforLS-SR. ProofofClaim2.Letz+a+y+beavalidinequalityforPDandassumethatR1=f(x,y,z,)2PD,TXq=2(zq+q1Xi=1aiq`(q))=2g Weprovethat( 3 )representsafacetforPDbyshowing=0, 60

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Second,weprovethati=0foralli=1,,T.WeconstructtwotightfeasiblesolutionsforR1asfollowing: Point1:y1=y2==yT=1,zq1=zq2=1,aiq1k=1,aiq2`(q1)=1,k6=`(q1),q1,q2T1. Point2:y1=y2==yT1=1,zq1=zq2=1,aiq1k=1,aiq2`(q1)=1,k6=`(q1),q1,q2T1. Then,wehaveTXi=1izi+T1Xi=1TXq=i+1TX`=1iq`aiq`+TXi=1i=TXi=1izi+T1Xi=1TXq=i+1TX`=1iq`aiq`+T1Xi=1i= Now,weshowthefollowingcoefcientrelationship among construct thefollowingtightfeasiblesolutionsforR1 Point3:yi1=yi2=yj1=yj2=1,zj1=zj2=1,ai1j1`(j2)=1,ai1j2`(j1)=1,ai1j2k=1,ai1j2`=1,wherei1,i2j1,j2,k6=`(j1),`(j2),`6=`(j1),`(j2),k. Point4:yi1=yi2=yj1=yj2=1,zj1=zj2=1,ai1j1`(j2)=1,ai1j2`(j1)=1,ai2j2k=1,ai1j2`=1,wherei1,i2j1,j2,k6=`(j1),`(j2),`6=`(j1),`(j2),k. 61

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Point6:yi1=yi2=yj1=yj3=1,zj1=zj3=1,ai1j1`(j3)=1,ai1j3`(j1)=1,ai1j3k=1,ai1j3`=1,wherei1,i2j1,j3,`6=`(j1),`(j2). Point7:yi1=yi2=yj1=yj2=1,zj1=1,ai1j1`=1,wherei1,i2j1,j2. Point8:yi1=yi2=yj1=yj2=1,zj1=zj2=1,ai1j1`(j2)=1ai2j2k=1,ai1j2`=1,wherei1,i2j1,j2,k6=`(j2),`6=`(j2),k. Weshowthati1q`=i2q`=q`for`6=`(q)byputtingPoint3,Point4,andPoint8into Then,wehavej1+j2+i1j1`(j2)+i1j2`(j1)+i1j2k+X`6=`(j1),`(j2),ki1j1`=, Thus,with( 3 )and( 3 ),wehavei1j2k=i2j2k=j2k,wherek6=`(j1),`(j2).With( 3 )and( 3 ),wehavei2j2k=i1j2k,wherek6=`(j1).BythearbitraryconstructionofPoint3,4,and8,wehave Weshowthatij1`=ij2`=i`.weputpoints3and5into( 3 ),wehave( 3 )and Withdifferentconstructionsoffeasiblesolutions,wehave 62

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3 )and( 3 ),wehavej1k=i1j1k=i1j2k=i1k.Thus,thefollowingconclusionholds. PutPoint5andPoint6into( 3 ),with( 3 ),wehavej2=j3.Withthedifferentconstruction,wehavej=. 3 ),wehaveij`=+u`,where`=`(j).Hence,ij`(j)=+u`. BasedonProposition 3.5 andconstraint( 3 ),wegeneratethefollowingfacet-deninginequalityforPDasfollows. Proof. 3.6 by two claims. Claim1.( 3 )isavalidinequalityforLS-SR. Claim2.( 3 )isafacet-deninginequalityforLS-SR. ProofofClaim1.FromObservations1and2,inordertosatisfydemandsanddorecoveryproduction,PTi=1yi2.WeproveClaim1by two cases. Case1.Iftherearemorethan two productionperiods,thenPTi=1yi3. Case2.Ifthereareonlytwosetupperiods,therecoveryperiodcoversthedemandinperiod`(q)tosatisfytheconstraint( 3 ).Thus,( 3 )holds. Therefore,( 3 )isavalidinequality. 63

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Weprovethat( 3 )representsafacetbyshowingthat=0, First,weprovethat=0.With a feasibleandtightsolution(x,y,z,M)forR2,wehaveanotherfeasibleandtightsolution(x,y,z,2M)forR2,whereM=max(i,`)ci`PTi=1di.Then,wehaveTXi=2izi+T1Xi=1TXq=i+1TX`=1iq`aiq`+TXi=1iyi+M=TXi=2izi+T1Xi=1TXq=i+1TX`=1iq`aiq`+TXi=1iyi+2M=. Second,weshowthati=0for2iT.WeconstructthefollowingtwofeasiblepointsofR2. Point1:yi1=yi2=1,zi2=1,ai1i2`=1,1i1
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Now,weshowthefollowingrelationshipamong,and.WeconstructfollowingtightfeasiblesolutionsforR2. Point3:yi1=1,yj1=yj2=1,xi1j1`(j2)=1,xi1j2`(j1)=1,xi1j2k=1,xi1j1`=1,wherej1,j2>1,k6=`(j1),`(j2),`6=`(j1),`(j2),k. Point4:yi1=1,yj1=yj2=1,xi1j1`(j2)=1,xi1j2`(j1)=1,xj1j2k=1,xi1j1`=1,wherej1,j2>1,k6=`(j1),`(j2),`6=`(j1),`(j2),k. Point5:yi1=1,yj1=yj2=1,xi1j1`(j2)=1,xi1j2`(j1)=1,xi1j1k=1,xi1j1`=1,wherej1,j2>1,k6=`(j1),`(j2),`6=`(j1),`(j2),k. Point6:yi1=1,yj1=yj3=1,xi1j1`(j3)=1,xi1j3`(j1)=1,xi1j1`=1,wherej1,j3>1,`6=`(j1),`(j3). Point7:yi1=yj1=1,xi1j1`=1. First,weshowthati1jk=i2jk=jk,k6=`(j).Putpoints3and4into Then,wehavei1+j1+j2+X`6=`(j1),`(j2),ki1j1`+i1j2`(j1)+i1j1`(j2)+i1j2k=i1+j1+j2+X`6=`(j1),`(j2),ki1j1`+i1j2`(j1)+j1j1`(j2)+j1j2k= 65

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3 ).Then,i1+j1+j2+X`6=`(j1),`(j2),ki1j1`+i1j2`(j1)+i1j1`(j2)+i1j2k=i1+j1+j2+X`6=`(j1),`(j2),ki1j1`+i1j2`(j1)+i1j1`(j2)+i1j1k= Third,weshowthati=j=.Weputpoints5and6into( 3 ).Then,i1+j1+j2+X`6=`(j1),`(j2)i1j1`+i1j2`(j1)+i1j1`(j2)=i1+j1+j3+X`6=`(j1),`(j3)i1j1`+i1j3`(j1)+i1j1`(j3)= Forth,weshowthatij`(j)=+u`(j).Putpoints3and7to( 3 ).Wehavei1+j1+j2+X`6=`(j1),`(j2)i1j1`+i1j2`(j1)+i1j1`(j2)=i1+j1+X`6=`(j1),`(j2)i1j1`+i1j1`(j1)+i1j1`(j2)= Finally,putanytightfeasiblesolutions.Then,weget=3+PT`=1u`. Therefore,( 3 )isafacet-deninginequality. Theabovetwofacet-deninginequalitiesaregeneratedbasedontherelationshipbetweentherecoveryproductionandrecoverysetuprequirement.Now,weconstruct 66

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Proof. 3.7 holdsby two claims. Claim1.( 3 )isavalidinequalityforLS-SR. Claim2.( 3 )isafacet-deninginequalityforLS-SR. ProofofClaim1.Ifyq=0,wehavePTk=q+1aqk`=0andzq=0.Thus,PTk=q+1aqk`+PTi=q+1aiq`=0.Ifyq=1,becauseaiq`TXq=i+1aiq`andT1Xi=1TXq=i+1aiq`=1. 3 )isavalidinequalityforLS-SR. ProofofClaim2.Letz+a+y+beavalidinequalityforPDandassumethatR3=f(x,y,z,)2PD,TXk=q+1aqk`+q1Xi=1aiq`=yqgf(x,y,z,)2XD,TXi=2izi+TXq=2q1Xi=1TX`=1iq`aiq`+TXi=1iyi+=g 3 )withapairof(`,q)representsafacet-deninginequalitybyshowing=0,=TX`=1'`

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Weshowthati=0for2iT.WeconstructthefollowingtwofeasiblepointsofR: Point1:yi1=yj1=yj2=1,zj2=1,aj1j2k=1,1i1
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Point4:yi1=yq=yj1=1,ai1qk=1,1kT. Point5:yi1=yi2=yq=1,ai2q`=1,ai2qk=1,wherek6=`. Point6:yi1=yi2=yq=1,ai1q`=1,ai2qk=1,wherek6=`. Point7:yi1=yq=yj1=1,aqj1`=1,ai2qk=1,wherek6=`. Point8:yi1=yj1=yj2=1,ai1j1k=1,1kT,i1,j1,j26=q. Point9:yi1=yq=yj1=1,ai1q`=1,ai1j1k=1,k6=`. Point10:yi1=yj1=yj2=1,ai1j2m=1,ai1j1k=1,i1,j1,j26=q,k6=m. Point11:yi1=yj1=yj2=1,ai1j2k=1,1kT,i1,j1,j26=q. Point12:yi1=yj1=yj2=1,aj1j2m=1,ai1j2k=1,i1,j1,j26=q,k6=m. Weshowthati=0,wherei6=q.Putpoints3and4into( 3 ).Wegeti1+q+TX`=1i1q`=,i1+q+j1+TX`=1i1q`=. Weshowthati1q`=i2q`=q`.Withyi=0,i6=q,weputpoints4and5into( 3 )andobtainq+i1q`+Xk6=`i1q`=q+i2q`+Xk6=`i2q`= Weshowthatqk`=iq`,whereiqk.Weputpoints6and7into( 3 )andobtain 69

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Weshowthatij`='`,wherei,j6=q.Putpoints8and10into( 3 ),wehavei1j2k=i1j1k=i1k.Withthedifferentconstructionofpoints8and10,wehaveijk=ik,wherei,j6=q. Putpoints11and12into( 3 ),wehavei1k=j1k='k,wherei1,j16=q.Withthedifferentconstructionofpoints,wehaveijk='k,wherei,j6=q. Weshowthatq`=`q.Putpoints8and9into( 3 )andobtainq+i1q`+Xk6=`i1j1k=TXk=1i1j1k= Finally,weputanyfeasiblesolutionofR3into( 3 )andobtain=PT`=1'`. Weextendthethree-stagerobustmodelPLStomulti-stage,andformulatetherobustlot-sizingproblemwithmultipledisruptionsandwithoutsetupcostcaseasfollows: 70

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Wecombinetheenumerationofperiodswhenthedisruptionshappenandthepre-processingalgorithmtoreformulateMRLSasasinglestagelinearprogrammingproblem. WeextendthereformulationschemeinSection 3.4 forthesingledisruptioncasetothemultipledisruptioncase. First,weuseatwo-disruptioncaseasanexampletoexplainourreformulation. Accordingto the problemsetting,wehavet2>t1,wheret1andt2representtheperiodsfortherstandtheseconddisruptions respectively.Thedecisionofx2i`isbased 71

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Constraints( 364 ). anideasimilartothat usedtosolvetherobustlot-sizingproblemwithsingledisruptionandnosetupcostcase.Weselectaperiodaftert2thatprovidestheminimalunitcosttosatisfytheunsatiseddemandinperiod`duetothedisruptionint2.Thatis,wedenotemt2`=minfci`,i>t2gtobetheminimumunitproductioncostfortheunsatiseddemandinperiod`.Constraint( 3 )canberewrittenas: 72

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3 ),( 3 ),( 3 ). 3 )asshowninthefollowing. Thereasonliesinthefollowingtwofacts: Intheoptimalsolution,fortheformulationincluding( 3 ),1shouldbethemaximumvalueoftheleftsidecorrespondingtoacertaintimeperiodt.Thatis,thereexistsatleastonetightinequalityin( 3 )for1. Intheoptimalsolution,fortheformulationincluding( 3 ),wehaveatleastonetightinequalityinwhichtheleftsideachievestheminimum.Otherwise,1canbedecreasedandwehaveacontradiction.Notehere,wedonotneedtoconsidertheminoperationforthenon-tightinequalities,sinceitwillnotaffecttheoptimalobjectivevalue. Thus,wecansubstituteinequality( 3 )to( 3 ).Thetwo-disruptioncasecanbesolvedbyusingthefollowinglinearprogrammingformulation(MB-2):minx1,1TXi=1TX`=1ci`x1i`+1s.t.Constraints( 3 ),( 3 ),( 3 ),( 3 ).

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3 )and( 3 ).Fortheseconddisruption,foragivenscenariooft1,weintroduceanarticialvariable2(t1).Wealsoincreasethedimensionofthesecondstagedecisionvariablex2i`tobex2i`(t1).Therefore,thereareO(T3)constraintsof( 3 )andO(T2)constraintsof( 3 ). Basedonthesimilaridea,wecanintroducethenotationxki`(t1,...,tk1)tobetheextraproductionquantityinperioditosatisfythedemandinperiod`afterthetthk1disruption.Then,wecanobservethatthefollowingconclusionholdsformulti-stagerobustlot-sizingproblems. 74

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Therefore,inourapproach,weenumerateallpossibledisruptioncombinations.Foreachtreenode,correspondingtoaparticularrealizationofdisruptions(t1,...,tk),weneedtohavetheextraproductioninstagek+1tocovertheunsatiseddemandsduetodisruptionattk.Thus,constraints( 3 )to( 3 )hold. 75

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Foreacht1,wecanalsoconsiderinequality( 3 )correspondingtothelinkbetweenrootnodeanditschildt1. Forthegeneralcase,correspondingtoeachtreenode(t1,t2,...,tk),exceptleafnodes,wehave Similarly,foragiventk+1,wecanconsidertheaboveinequalitycorrespondingtothelinkbetweentreenodes(t1,...,tk)and(t1,...,tk+1).Therefore,constraints( 3 )correspondtoalllinksinthetree. Asshownintheproofforthetwodisruptionscase,wecanremovetheminoperationsontheleftside.Thatis,wecanderivethecorrespondingconstraints( 3 )to( 3 ),whichalsocorrespondtoalllinksinthetree.Then,wecanobtainthefollowingtwoconclusions. 3 )to( 3 )istight.Thispathcanbeobtainedby breadth-rst searchstartingfromtherootnodetond descendants alongthelinksinwhichtheconstraintsaretight(i.e.,namedtightlinks).Ifthe breadth-rst searchterminateswithoutreachinganyleafnodes,thenwecanreducebyasmallpositivevalue,whichleadstoasmallerobjectivevalue.Contradiction! 76

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breadth -rstsearchonthelinksamongthepathsthataretight.Wecheckiftheleftsideistightforeachtightinequalitycorrespondingtoeachlink.Ifthe breadth -rstsearchterminateswithoutreachinganyleafnode,thenwecanreducetheleftsidevalueandthevaluebyasmallpositivevalue,whichleadstoasmallerobjectivevalue.Contradiction! Basedon(1)and(2),wewilleventuallyndatleastonepathfromtherootnodetoaleafnodesuchthateachinequalityistightcorrespondingtoeachlink andtheleftsideforeachtightinequalityisminimized.Theleafnodeindexgivesthedisruptionperiodsintheoptimalsolution.Therefore,theconclusionholds. in therststage,wecansimilarlyobtainthefollowingformulation: s.t.( 3 )to( 3 ); 77

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Forthecaseswithsetupcosts,correspondingtoeachofthecombinationsofTandf=p,wegeneratetherandominstancesinwhichtheunitproductioncostandthesetupcostareuniformlydistributedintheintervalsasshowninTable 3-1 Forthecaseswithoutsetupcosts,correspondingtoeachofthecombinationsofTandp,wegeneratetherandominstancesinwhichtheunitproductioncostisuniformlydistributedinthesameintervalasforthecaseswithsetupcostsshowninTable 3-1 Wealsosetthedemanddi,unitinventorycosthi,andunitbackloggingcostbiuniformlydistributedintheintervals[500,1000],[5,10],and[10,20]respectively. 3.6.2.1 andSection 3.6.2.2 ,respectively. 3-2 .Table 3-2 reportsthecomputationalresults Table3-1. Parametersetting unitproductioncostpi [50,100] [500,1000] ratio20 [50,100] [1000,2000] ratio30 [40,60] [1000,2000] ratio40 [40,60] [500,1000] 78

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3.6.3 .Correspondingtoeachcombination,wereportthevalueof1)theobjectivevalueobtainedbysingleitemlot-sizingproblemwithoutthedisruption,denotedasSLS,2)theobjectivevalueobtainedbymaximumpickupheuristics,denotedasMP,3)theobjectivevalueobtainedbythelot-sizingwithbackloggingcase,denotedasSLS-B,4)thegapbetweenSLSandSLS-B,denotedasGAP(B::S)=(ObjSLS-BObjSLS)=ObjSLS.5)thegapbetweenSLS-BandMP, 79

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Theperformanceofthegeneratedfacetdeninginequalitiesfortherobustlot-sizingwithbackloggingisreportedinTable 3.6.3 .WetesttwelvecombinationsinwhichthetimehorizonT=100,120,and140,respectivelyandtheratiof=p=10,20,30,and40,respectively.Foreachcombination,werunveinstancesandreporttheaverageperformanceoftheseveinstances.LetCutandNoCutdenotethebranch-and-cutalgorithmwiththefacet-deninginequalitiesascutsandthedefaultCPLEXwithoutaddingfacetdeninginequalities.ThetestinginstancesforCutandNoCutaresame.Wesetthecomputationaltimelimitas1800secondsforbothcases.FortheNoCutcase,noinstancecanbenishedwithintimelimits.FortheCutcase,allinstancesforT=100andT=120canbenishedwithin1800secondsandachievetheoptimalsolution.TheexactcomputationaltimeandoptimalitygaparelistedinTable 3.6.3 3.5 .FromTheorem 3.2 ,wecanobservethattheoptimalsolutionforthistypeofproblemscanbeobtainedbysolvingalinearprogrammingformulationthatcontainsO(T)constraintsandO(T)variables,whereisthenumberoftotaldisruptions.Therefore,thisformulationispseudo-polynomial,intermsoftimeperiods.WereportthecomputationalresultsinTables 3-5 and 3-6 fortwo-disruptionandthree-disruptioncases,respectively.For2disruptioncase,wesolvealinearformulationthatcontainsO(T2)variablesandconstraintstogettheoptimal 80

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3-5 that,onaverage,ourrobustoptimizationapproachhaspossiblecostsavingsaround80.8%,comparedwithmaxpickheuristics.Whencomparedwiththeuncapacitatedlot-sizingproblemwithoutdisruptions,ourapproachincreasescostaround11.6%.ItcanbeobservedinTable 3-6 that,onaverage,ourrobustoptimizationapproachhaspossiblecostsavingaround82.1%,comparedwithmaxpickheuristics.Whencomparedwiththeuncapacitatedlot-sizingproblemwithoutdisruptions,ourapproachincreasescostaround37.9%.Tables 3-5 and 3-6 alsoindicatethatthecomputationaltime,whichisreportedinseconds,increasesasthenumberoftimeperiodsanddisruptionsincreases.Butingeneral,theoptimalsolutioncanbeobtainedwithinonesecondforthetwodisruptioncaseandwithinaminuteforthethreedisruptioncase. 81

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Robustlot-sizingwithoutsourcing:multipledisruptions Dis=2 Dis=3 Dis=4 Dis=5 ratio=10 SLS 39.6% 53.4% 45.8% 65.6% 33.0% 32.0% 32.6% 33.5% SLS 44.0% 47.8% 62.0% 52.3% 28.7% 32.4% 30.4% 33.5% SLS 41.5% 37.3% 35.7% 47.5% 38.0% 38.5% 37.7% 38.4% SLS 42.7% 34.3% 46.6% 52.8% 36.8% 38.7% 37.7% 38.3% Robustlot-sizingwithbacklogging:asingledisruption T=20 T=30 T=40 T=50 ratio=10 SLS 7.05% 18.75% 8.35% 9.83% 11.88% GAP(B::M) 56.87% 29.06% 20.71% 14.47% 8.79% ratio=20 SLS 17.02% 21.50% 10.50% 11.63% 16.09% GAP(B::M) 35.14% 23.33% 19.18% 12.07% 9.18% ratio=30 SLS 22.09% 17.54% 5.20% 4.85% 9.19% GAP(B::M) 52.49% 28.02% 25.66% 19.86% 12.37% ratio=40 SLS 25.74% 25.23% 6.05% 4.41% 12.97% GAP(B::M) 35.40% 27.12% 33.04% 22.43% 8.79% Robustlot-sizingwithbacklogging:branch-and-cut T=120 T=140 Cut NoCut Cut NoCut Cut NoCut Time Gap Time Gap Time Gap Time Gap Time Gap Time Gap ratio=10 472 0% 1800 0.26% 1151 0% 1800 0.56% 1800 0.02% 1800 0.60% ratio=20 576 0% 1800 0.49% 1273 0% 1800 0.61% 1800 0.04% 1800 0.65% ratio=30 514 0% 1800 0.67% 1286 0% 1800 1.04% 1800 0.08% 1800 1.35% ratio=40 644 0% 1800 0.98% 1168 0% 1800 1.21% 1800 0.11% 1800 1.20%

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Multi-stagerobustlot-sizingproblemwith2disruptions Time Obj Time Obj Time Obj Time Obj Time ratio=10 SLS(106) MPSLS(106) NoSetup(106) Gap(M::N) 116.4% 89.9% 60.2% 47.1% 44.6% Gap(N::S) 23.2% 13.6% 5.7% 3.5% 8.2% ratio=30 SLS(106) MPSLS(106) NoSetup(106) Gap(M::N) 169.2% 102.6% 62.9% 56.1% 58.7% Gap(N::S) 27.5% 14.1% 10.3% 8.0% 2.1% Multi-stagerobustlot-sizingproblemwith3disruptions Time Obj Time Obj Time Obj Time Obj Time ratio=10 SLS(106) MPSLS(106) NoSetup(106) Gap(M::N) 147.8% 81.4% 53.5% 40.5% 30.2% Gap(N::S) 45.4% 36.6% 29.0% 26.3% 33.8% ratio=30 SLS(106) MPSLS(106) NoSetup(106) Gap(M::N) 185.0% 112.9% 63.4% 54.6% 52.1% Gap(N::S) 55.7% 42.9% 41.5% 36.7% 31.1%

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Thisresearchstudiescostparameteruncertaintyandismotivatedbytacticalproductiondecisions,notoperationalproductiondecisions,forchemicalcompanies.Astypicalsupplychaincharacteristicsofchemicalindustry,achemicalcompanyproducesmainlyfunctionalproducts,whicharedenedasonesthathavealongproductlifecycleandstabledemand(see,e.g., LeeandChen 2005 ).Underthissituation,thedemandisstableandeasilyforecastedaccurately. However,thecostparameterforecastsmayneedtobeadjustedmonthlyorquarterly. Wecanformulatethisproblemasatwo-stagestochasticlot-sizingproblem(SULS),inwhichweletaspecictimeperiod,e.g.,timeperiodp,representthetimeforwhichtheforecastneedstobeadjusted.Thecostparametersafterthegiventimeperiodwillbeuncertainandfollowadiscreteprobabilitydistribution.Thedetailedtwo-stagestochasticintegerprogrammingformulationcanbedescribedasfollows(cf. BirgeandLouveaux 1997 ; LouveauxandSchultz 2003 ):

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here thesetupdecision, andproductionandinventorylevels ontherstand the secondstages,respectively.Thecorrespondingcostparametersare ForSULS,polynomialtimealgorithms(see,e.g., GuanandMiller 2008 ; HuangandKucukyavuz 2008 )andefcientcuttingplanes(see,e.g., Guanetal. 2006b ; SummaandWolsey 2006 )havebeenstudiedrecentlyforitsdeterministicequivalentscenario-treebasedformulations. Thereformulationfortheproblemwasoriginallyintroducedin Ahmedetal. ( 2003 ).Lateron,in Guanetal. ( 2006a ),thisreformulationwasprovedtobeequivalenttoaddingthe(`,S)inequalitiesintheoriginalformulation,intermsofprovidingLPlowerbounds.However,bothapproachescouldnotprovideintegralsolutionsforSULS.AnextendedformulationthatprovidesintegralsolutionsforSULSuptonowisonlyfortwo-periodcases,whichwasdevelopedin Guanetal. ( 2006a ). Tothebestofourknowledge, thereisnopreviousresearchondeveloping anextendedformulationthatprovidesintegralsolutionsfor multi-period SULS. Thisisanothermotivationforourresearch,andthispapercontributestotheliteratureonderivinganextendedformulationformulti-periodSULS. possiblerealizationsofthesecond stage randomvariables wecangenerate atwo-stagestochasticscenario tree fortheproblem asshowninFigure 4-1 .Nodes ,,parerststage nodes andnodesq1,, wherebranchingnodep thesecondstages andisuniqueonthescenariotree. AssumingthereareWpossiblescenarios,weletP(!),1!W,representthebranchwherescenario!occursandaccordinglylet!,1!W,representtheprobabilitythatscenario!willoccur.Sincethereisnodemanduncertainty,wehavedi=dt(i)0forthesecondstagenodes,wheret(i)representsthetimeperiodofnodei.WeletVrepresenttheset of nodesonthe 85

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descendants ofnodei(includingnodeiitself).Foreachnodei2Vnf1g,thereexistsauniqueparent,denotedasnodei. Finally,foreachnon-leafnodei,weletC(i)representthesetofits children .Foroursetting,C(i)containsasingleelementifthenon-leafnodei6=p. Thedeterministicequivalentformulationfortwo-stageSULScanbedescribedasfollows: . . . Thescenariotreefortwo-stageSULS 1 TodeneWagner-Whitincostsfor two-stage SULS wesubstitute theproductionvariablexi. Then,weobtain areformulationof two-stage SULS inthe(s,z)space asfollows:

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,andi=8><>:0i,1ip costsetting astwo-stageSULS-WW.A problem satisfyingtheWagner-Whitin costsetting isalsoreferredtoaswithout speculative motives(see, Wagelmansetal. 1992 PochetandWolsey 1994 ,amongothers). Underthissetting ,wewillnotsetupproductionatnodei2V,iftheinventoryenteringiissufcienttosatisfythedemandati.Theconditionhi0isthesameasthedeterministicULSWagner-Whitin cost setting.Now,we provide a 4-periodexample inFigure 4-2 ,to indicatethat Wagner-Whitin property. AsshowninFigure 4-2 foreachnode ,theunitproductioncostiszero.Thesetupand the unitinventorycostsare(0, +1 0, 0,0,0,0), 87

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. Anexampleoftwo-stageSULS respectively Problemparameterssatisfytheconditionhi0,butdonotsatisfytheconditioniPj2C(i)jinDenition 1 .Weassumethedemand in eachnodeisanon-zeronitenumber.Inordertosatisfythedemandateach time periodandminimizethetotalcost,theproductionsaresetupinnodes1, 4 ,and 5 .Theproductioninnode 1 covers thedemands fornodes 1,2,3,and4 .Meanwhile,node 4 issetuptoproduceandsatisfy thedemand innode 6 Thus ,thereisinventoryleftfromparentnodeandproductionsetupfornode 4 simultaneously. The Wagner-Whitin propertyforthedeterministiccase,xisi=0, doesnothold here. . . Thesubtreeofnodei t ,asshowninFigure 4-3 wherePit(!)representsthepathfromnodeitoitsdescendantnodeattime 88

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two-stage SULS-WW,correspondingtoeachnodei2V,thereexistsanoptimalsolutionintheform Foreachi2V, theconclusionisobviousif'(i)=;.Now,weassume'(i)6=;and leti=argmaxft(j):j2'(i)g Claim1:Foreachi2,si=P Claim2:Correspondingtoeachi2,zj=1ifj2V(i)andt(j)=t(i);zj=0ifj2V(i)andt(i)
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4.1 providesastrongerclaim,ascomparedtotheproductionpathpropertyforSULSdescribedin GuanandMiller ( 2008 ), developedforgeneralcostanddemandsettings.Underthesegeneralsettings,wecannotguaranteethat theWagner-Whitincostsettingholds.TheexampleinFigure 4-2 canstillhappen,andxisi=0doesnothold.Thatis,wecannotguaranteezj=1ifj2V(i)andt(j)=(i),andzj=0ifj2V(i)andt(i)
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are forthe branching node t t Constraints( 4 )and( 4 )aresimilarto( 4 )and( 4 ).Alongeachscenariopath,ifthereisnosetupintimeperiodt(p)+1,theinventoryleftatnodepcoversdemandsuptothesametimeperiodastheinventoryleftatthenodeintimeperiodt(p)+1does. Knowingi t t Inequalities( 4 )-( 4 )andequation( 4 ), assumingbinaryitandzi, guaranteethefeasibilityofthereformulationfortwo-stageSULS-WWbecausethe demandateachtimeperiodwillbecovered. 4 )( 4 )plus0it1and0zi1,i2V,t(i)+1tTprovides an extendedformulationfortwo-stageSULS-WW. Proof. 4 )canbe putdirectlyin theobjectivefunction.WeprovethistheorembyshowingthattheconstraintmatrixforConstraints( 4 )to( 4 )istotallyunimodular. Toprovethattheconstraintmatrixfor constraints ( 4 )to( 4 )istotallyunimodular,weorder variables 4-1 describes the constraint matrixcorrespondingto theexamplein Figure 4-2 91

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Thematrixofconstraints( 4 )to( 4 )fortheexampleinFigure 4-2 t theconstraintsubmatrix fortherestvariables, denotedasmatrixA. WeshowthatforanycolumnsubsetJofmatrixA, thereexists apartitionJ1andJ2ofJsuchthat Wepartition Step1. Allocatep t Step2. Allocateq t Step3. Allocatezi, totheoppositesetofzM(i), ifthereexistszM(i)inJ, where t t allocatezitoJ2 Step4. Allocatezi 92

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Allocateit t Now,weverifythat( 4 )holdsforconstraints( 4 )to( 4 ) under theabove partition Correspondingtoeachrow, if atmostonedecisionvariableinA,thenitisclearthat( 4 )holds. In thefollowing,weconsiderthecasewhereJcontainsatleasttwodecisionvariablesinA: 4 ),iffp t t Thus,( 4 )holds. constraint ( 4 ), weonlyneedtoconsiderthefollowingfourcases: t Theargumentis the sameasconstraint( 4 ). t t basedonStep 2 t basedonStep1.Then, ( 4 )holdsnomatterthedestinationofq 4 ),weconsidertwotypesofconstraints:wheret(i)t(p)+2ort(i)=t(p)+1. t 4 )holds. thecaseswhere ,orp t t orJ1(ifp t Accordingly, 93

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t orJ1 Wealsoobservethat is assignedalternativelytoJ1andJ2 4 )holds. 4 ), weonlyneedtoconsiderthecasewhereboth t t closest descendantofnodei. Thus,( 4 )holds. 4 ), wediscussthefollowingfourcases: t argument isthesameasforConstraint( 4 ). t tothesamesetbecauseboth ifzm(i+1)2J)or in basedonSteps4and5. t bothi t theoppositesetofzm(i+1) orinJ1 ,basedonStep5. t 4 )holdsnomatterthedestinationofi+1t. Therefore,thedesiredproperty( 4 )holdsforconstraints( 4 )to( 4 ), andthe matrixforconstraints( 4 )to( 4 )istotallyunimodular. t WeprovethatQoisanintegralpolyhedronfortwo-stageSULS-WWbyshowingthatitisaprojectionofQrin(s,z)space,whereQr thepolyhedronoftwo-stageSULS-WWin(s,z,)space,i.e.,Qr=f(s,z,)j(s,z,)satises( 4 )to( 4 )g.

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thefollowingtwo claims: Claim1:InequalitiesinQoarevalidforQr.ToproveClaim1,weonlyneedtoshowthatsiTXt=t(i)+1dt[1Xj2Pit( becauseforgiven!t,t(i)+1tT,wehaveTXt=t(i)+1dt[1Xj2Pit( 4 )bydiscussingtwocases:(1)t(i)t(p)+1and(2)t(i)t(p). Fort(i)t(p)+1,basedon( 4 ),( 4 ),andnonnegativityofit,wehavesi=TXt=t(i)+1dtitTXt=t(i)+1dtmax8<:0,1Xj2Pi t t 4 )holds. Fort(i)t(p),wehavesi=TXt=t(i)+1dtit=dt(i)+1it(i)+1+TXt=t(i)+2dtitdi+1[1zi+1]++TXt=t(i)+2dti+1tzi+1+ift(i)
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4 )isbasedon ( 4 ), ( 4 ),andnonnegativityofit,while( 4 )isbasedon( 4 ),( 4 ),andnonnegativityofit.Wealsonoticethatift(i)=t(p),asshownin( 4 ),q 4 ).Then,( 4 )holdsforthet(i)=t(p)case. Then,weonlyneedtoshowthatift(i)t(p)+1,then"ptpXj=i+1zj#+264q t(p)+1 4 )andthesecondfollowsfrom( 4 );therefore,( 4 )holdsandthus, ( 4 )andClaim1hold Claim2:Forany extremepoint wecanconstructsuchthat(s,z,)2Qr.Thatis,(s,z,)satisesconstraints( 4 )to( 4 ).Now,foragivenextreme point welet t t 96

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4 )basedon( 4 )and( 4 ).Wealsoobservethat( 4 ),( 4 ),and( 4 )hold,whichdirectlyfollowsfrom( 4 ). For( 4 ), let!bethescenariowherebi+1tachievesthemaximumvalue.Then,itiseasytoobservethat!isalsothescenariowherebitachievesthemaximum.Then,accordingto( 4 ) ,bi t t t 4 )holds. For( 4 ),accordingto( 4 ),b t t t t t

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4 )followsfromthefactthatPpt(!)nfp,q!g=Pq!t(!)nfq!gandbq!t=1Pj2Pq!t(!)nfq!gbzj.Then,( 4 )holds.Thus,(bs,bz,b)2Qr,andClaim2holds. Therefore,theconclusionholds. Wecan observethatwithWagner-Whitincosts,theexplicitformulationofthepolyhedraldescriptionfortwo-stageSULSintheoriginalspaceisdescribedby 4.1 stillholds,basedontheWagner-WhitincostsettingdenedinDenition 1 .Accordingly,wecanobtainareformulationsimilartoconstraints( 4 )to( 4 ).Forinstance,wehaveconstraints( 4 )to( 4 )forthelastandtherststagenodes,andequation( 4 )fortheinventorylevelexpression.Forthenodesbetweentherstandthelaststages,constraintssimilarto( 4 )and( 4 )arevalid(e.g.,it1zk,k2C(i),t=t(i)+1anditktzk,k2C(i),tt(i)+2).Therefore,wecanobtainasimilarreformationwithbinaryitandziforthemulti-stagecase.However,itisunknownifthereformulationcanprovideanextendedformulationthatprovidesintegralsolutionsformulti-stageSULS;thispossibilityiscurrentlyunderinvestigation. 98

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( 1988 )providedtherstpolyhedralstudyoftheuncapacitatedlot-sizing problem withbackloggingandtheconvexhulldescriptionfortheproblemwas recently studiedby KucukyavuzandPochet ( 2007 ).Inaddition,fortheuncapacitatedlot-sizingproblemwithstart-upcost, vanHoeseletal. ( 1994 )presentedanextendedformulationandanO(T2)timeseparationalgorithm. FortheWagner-Whitincostscase, Wagelmansetal. ( 1992 )introducedtheWagner-Whitincosts,i.e.,i+h0ii+1forall timeperiod1iT1, ,wherei andinventory costs fortimeperiod tosolve ULSwithWagner-Whitincosts.For theWagner-Whitincost case, PochetandWolsey ( 1994 )generatedanexplicitformulationofconvexhullforULSwithbackloggingwithO(2T)constraintsandanO(T2logT)timeseparationalgorithm. Thedeterministicequivalentformulationfortwo-stageSULSwithbackloggingandWagner-Whitincostscanbedescribedasfollows:minpXi= 1 inventorylevel,backlogginglevel,andproductionset-upindicator inthestatedenedbynodeiwhosecorrespondingperiodist(i). Nodeiistheparentnodeofnodei. Withoutlossofgenerality ,wecanassumes0=0,`i=0,andtightenMi=PTk=t(i)dk,wherei2L

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theremainingpartofthis chapter,westudytwo-stage stochasticULS withbacklogging,Wagner-Whitincostsanddeterministicdemands,denotedas two-stageSULSB-WW .Weexaminetheoptimalsolution property of the modelanduseittogenerate an extended formulation inthehigherdimensionalspace .Thenweprovethattheconstraintmatrixfor the extendedformulationistotallyunimodular. Wealso projectitbacktoalowerdimensionspacesuchthatwecanndvalidinequalitiesthatcanprovidetighterextendedformulationoftheproblem. First,wedeneWanger-Whitincostsfortwo-stageSULSB.Wesubstitexi=di+si`isi+`itoeliminatedecisionvariablexiintwo-stageSULSB.Then,wegetareformulationoftwo-stageSULSBin(s,`,z)spaceasfollows:

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PochetandWolsey ( 1994 ).Fortwo-stageSULSB,weconsidertheoptimalsolutionformsofinventorylevelsiandbacklogginglevel`iforeachi2V.Intheoptimalsolutionforatwo-stageSULSBwithWagner-Whitincosts,thedemandforeachnodeiwillbesatised(1)bysettinguptheproductionatnodei,(2)byinventoryleftfromitsparentnode,or(3)bybackloggingfromitschildren.Beforewedescribetheproposition,weshowa3-periodexampletodemonstratetheoptimalsolutionformasshowninFigure 5-1 Inthisexample,productionsaresetupatnodes1,3,and4.Demandatnode1issatisedbytheproductionofitself.Demandatnode2issatisedbybackloggingfromnode4.Node3coversdemandsinnodes3and5.Thusthesecondstagenodes2and 101

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. Thescenariotreefora3periodSULSwithbacklogging 3arebackloggedandsetuprespectively.Meanwhile,therststagenode1doesnotprovideinventoryforany second stage node . . . Thesubtreeofnodei notationis introduced. AsshowninFigure 5-2 ,let(i)=minft(j):zj=1orgj=1,j2V(i)nfigg. Thatis ,(i)representsthetimeperiodofthe earliest descendantofnodeiwhichissetuporbacklogged.Accordingly,wedenenodeset(i)=fr:r2V(i),t(i)
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(2)Theoptimalsolutionsoftwo-stageSULSB-WWsatisfyfi+gi+zi=1,i2V 5 )and( 5 )holdby showing ( 5 ), and and( 5 )hold. Proofof ( 5 ) .Weprove ( 5 ) under3cases. Case1 : ( 5 )doesnotholdand`j=Pt(j)t=(i)dtholds Inthiscase,siiseitherlargerorlessthanP(i)1t=t(i)+1dt. Case1.1.Ifsi
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"siP(i)1t=t(i)+1dt.Itcanbeobservedthat( also afeasiblesolutionandleadstoanon-largertotalcost,whichisacontradiction.ThusClaim1holds. Case2 : 5 )holds.WecangivethesimilarproofasinCase1tond a contradiction. Case3 : Neither( 5 )or nor .Inordertosatisfythedemandineachnodein(i),we canconstruct twofeasiblesolutionsfortwo-stageSULSB-WW. Lets1i=si+" andnodejin Thecorresponding objectivevalue is Thecorresponding objectivevalue is IfPj2(i)hj>Pj2(i)n ThiscontradictswiththeassumptionthatFistheoptimalobjectivevalue Notehere,if Therefore,( 5 )holds Proofof ( 5 ) :Accordingto ( 5 ) ,sicoversdemandsfornodesin(i).Inordertominimizetheobjectivefunction,nodesin(i)donotobtainbacklogging.Thusgj=0,j2(i).Therefore,`j=maxf:1t(j)gPt(j)t=dthg(j,t)Pr2P((j,t),j) ( ) Therefore,( 5 )and( 5 )hold. 104

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5 ),( 5 ),and( 5 )hold. Accordingtothedenitionoffiandgi,i2V,itisobviousthatConditions( 5 )and( 5 )hold. NowweprovethatCondition( 5 )holdsbytwocases. Case1.Ifj2(i),thenthedemandinnodejissatisedbyinventoryandfj=1.Inordertokeepthesmallestproductioncost,fj+gj+zj=1. Case2.Ifj2(i)n(i), then and weneedtoprovegj+zj=1.Inordertosatisfythedemand,gj+zj1. Case2.1.Ifj2 Ifgi=1,thenwecanlet nodejproducemoretocoverthebackloggingamountandreducetheobjectivevalue. Contradiction!Therefore,gj=0and Case2.2.Ifj2(i)n According to( 5 ),thedemandofnodejcanbecoveredbybackloggingfromitschildren.Inordertominimizetheobjectivefunction, Therefore, basedoncases1and2, Under ( 5 ),atmostoneofg(i,k)andz(i,k)equals to 1.Theng(i,k)z(i,k)=1 105

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5 ),wehave Thusoneofconstraints( 5 )and( 5 )isredundant.Thatis,ifconstraint( 5 )holds,then( 5 )mustholdfrom( 5 ) Letbinaryvariable whetherthebackloggingatnodeicoversthedemandattimeperiodt,tt(i).Ifyes, Finally welet(i,t,w)representthenodewhichisadescendantofnodeiattimeperiodtandatthebranchcorrespondingtoscenariow. Nowweintroducethreetypesofinequalities correspondingto theoptimalinventorylevel for nodeionthescenariotree. Thatis,nodeicoversdemandsalongthebranchitbelongstountilnextbackloggingorproduction happens Inequality( 5 )indicatesthatifthedemandofchildnodei+1issatisedbyinventory,thentheinventoryleftfromnodeicoversthedemandofnodei+1.Inequality ( 5 ) indicatesthatifnodei+1isnotsetupandtheinventoryleftfrom 106

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uptotimet uptotimet branching nodep, Inequalities ( 5 )and( 5 )aresimilarto( 5 )and( 5 ).Theinventoryleftfromnodepwillcoverdemandsfromperiodt(p)+1tot1unlessthereisasetuporabacklogging beforeorattimeperiodtalongeachscenariopath. Withtheinformationof Because foragivennodei,itsancestor attimeperiodt,(i,t), isunique, wehavethefollowingtwoinequalitiesholdforeachnodei2Vbasedon( 5 ). Constraints( 5 ),( 5 ),( 5 )-( 5 ) guaranteethefeasibilitiesofthereformulationfortwo-stageSULSB-WWproblem since thedemandforeachtimeperiod is covered. Now,weshowthatconstraintsprovidetheextendedformulationfortwostageSULSB-WW. 5 ),( 5 ),and( 5 )to( 5 )providetheextendedformulationforthetwo-stageSULSB-WW problem Proof. 5 )and( 5 )canbedirectlytransferredtotheobjectivefunction.Weprovethispropositionbyshowingtheconstraintmatrixforconstraints( 5 ),( 5 ),( 5 )to( 5 ),and( 5 )isatotallyunimodular. 107

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5 ),( 5 ),( 5 )to( 5 ),and( 5 )istotallyunimodular,weordervariablefi,gi,andziwithloopirangingfrom1tojVj.Variablemitisorderedwithanouterloopirangingfrom1tojVjandaninnerlooptrangingfromt(i)+1toT.nitisorderedwithanouterloopirangingfrom1tojVjandaninnerlooptrangingfrom1tot(i).Table 5-1 showstheconstraintsmatrixcorrespondingtoFigure 5-1 Asthethesubmatrixcorrespondingtovariablenitisanidentitymatrixfori2Vandtt(i),wedonotneedtoconsidernitinourconstruction. Asthesubmatrixcorrespondingtovariablemitisanidentitymatrixfort(i)t(p)+1,i2V,andtt(i)+2, weonlyneedtoconsidermit,t(i)t(p),tt(i)+2inourconstruction.Thatis, weonlyconsidertheassociatedconstraintsubmatrixfortherestvariables,denotedasA. WeshowthatforanycolumnsubsetJofmatrixA,thereexistspartitionJ1andJ2ofJsuchthat foralli.Wedothepartitionofvariables branching nodepandthenextendittobothdirectionfornodesafterpandbeforeprespectively. First,wedeneM(i)astheclosestancestorofnodeisuchthatzM(i)2Jandm(i)astheclosestdescendantofnodeisuchthatzm(i)2J. Inthefollowingsteps1and5,weallocatethedecisionvariablesmandz: Step1. Allocatemp t Step2. Allocatemq t Step3. Allocatezi, ifM(i)existsandt(M(i))t(p). Otherwise,allocatezitotheoppositesetofmp t t allocatezitoJ2 108

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Allocatezi ifm(i)existsandt(m(i))t(p). Otherwise,allocate Step5. Allocatemit t InthefollowingSteps6and7,weallocatethedecisionvariablesfandg: Step6.AllocatefitothesamesetofzM(i),if Step7.Allocategitothesamesetofzm(i),if Followingtheabovepartitionsteps,weobservethefollowingtwoproperties. Claim1.Iffzi,zM(i)gJ,zigoestotheoppositesetofzM(i)foralli2Vnf1g. ProofofClaim1.If1t(M(i))t(i)t(p),becausethe closest descendantofM(i)isiandt(i)t(p),m(M(i))=i,zM(i)goestotheoppositesetofzibasedonStep 4 If1t(M(i))
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Foranyrstsecond-stagenodej(w),if second-stage nodeinJandonthebranchcorrespondingtowandt(p)t( )T.Thus, theclaimholds .2 5 )holdsforconstraints( 5 ),( 5 ),( 5 )to( 5 ),and( 5 ) under theabove partition .Atrst, correspondingtoeachrow, if atmostonedecisionvariableinA,thenitisobviousthat( 5 )holds. In thefollowing,weconsiderthecasethatJcontainsatleasttwodecisionvariablesinA. 5 ),wediscussthefollowing4cases. ; Because 5 )holds. ,figoestothesamesetofzM(i)basedonStep6 ; ; bothm(i)andM(i)donotexist, respectively. m(i)exists, Because 5 )holds. 110

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m(i)doesnotexist, 5 )holds. M(i)exists, 5 )holds. M(i)doesnotexist, 5 )holds. because set ofzi.Then( 5 )holdsnomatterwheregigoes. 5 ),wediscussthefollowing4cases. the sameas1-2. 5 )holds. wersthavem(i)=m(i).If(a) andzi2J,thengigoestotheoppositesetofzibasedonStep7.Also,,becausezi2J,wehavezM(i)=zi.ThengigoestotheoppositesetofzibasedonStep7.Thusgiandgigotothesameset.If(c) ,zi=2Jand ,thenzM(i)=zM(i).Notethatzi=2J,wehavegiandgigototheoppositesetofzM(i)=zM(i)basedonStep7.Thus,giandgigotothesameset.If(d) ,zi=2J,thenbothgiandgigotoJ2basedonStep7. 5 )holdsnomatterwheregigoes. 5 ),weconsider2cases: 111

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For thiscase,rst,wedonot needto considermitbasedontheidentitymatrixargumentatthebeginningof the proof.Then,basedonStep6,f(i,t,w)goestothesamesetofzM((i,t,w)),wherezM((i,t,w))isthe largest-index nodeinJandpath .BasedonStep4,zjalternativelygoestoJ1andJ2basedonStep4,wheret(p)+2t(j) 5 )holds. thiscase, Wediscussthefollowingtwocases. the sameas3-1. thesamesetasmpt(i.e.,J1)ifmpt2JorJ1ifmpt=2J totheopposite sets Besidesthese, .Thus,( 5 )holds. thiscase,wediscuss twocasesdependingonifthereexistsanodej2Pit(w)nfi,(i,t,w)gsuchthatzj2J. 3-2-2-2-1.If nosuchnodejexists ,thenff(qw,t,w),mqwtg2J, basedonourassumptionthatatleasttwoelementsineachconstraintsinmatrixA. Ifzqw2J,M((qw,t,w))=qw. BasedonSteps6and2, basedonStep2, Inthefollowing,weprovethatf(qw,t,w)goestoJ1inthiscase. BasedonStep6, 112

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). Forthiscase,zM((qw,t,w))2J M((qw,t,w))exists andzqw=2J.BasedonStep4,zM((qw,t,w))goestoJ1,because1t(M((qw,t,w)))t(p) Forthiscase,z(qw,t,w)2JandzM((qw,t,w)),zqw=2J .Then,z(qw,t,w)goestoJ2basedonStep3andaccordingf(qw,t,w)goestoJ1. (c)theoppositesetofzm((qw,t,w))(ifzM((qw,t,w)),z(qw,t,w)=2J,zm((qw,t,w))2J Forthiscase, Then, accordingly toJ1.Thenf(qw,t,w)andmqwtgotothesameset.(d)J1(ifzM((qw,t,w)),z(qw,t,w),zm((qw,t,w))=2J M((qw,t,w))andm((qw,t,w))donotexistandz(qw,t,w)=2J 3-2-2-2-2.If thereexistsasuchnodej,then ,and Ifzqw=2J,mqwtgoestoJ1 andzm(qw)goestoJ2 .Thus, forbothzqw2Jandzqw=2J,wehave Besidesthese, 5 )holds. 5 ), weonlyneedtoconsiderthecase iffmit,fi+1g2J. ,fi+1goestothesamesetaszM(i+1)basedonStep6.If 113

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5 )holds. then Thus,mitand totheoppositesetofzi+1basedonSteps5and6.Thus,( 5 )holds. ,m(i)=m(i+1) If1t(m(i+1))t(p),mitandfi+1gototheoppositesetofzm(i+1)basedon Steps 5and6;otherwise, ift(m(i+1))>t(p), the oppositesetofzm(i+1). Since ,underthiscase,M(m(i+1))doesnotexist .fi+1andmitgotothesameset.Then( 5 )holds. neitherM(i+1)norm(i+1)existsandzi+1=2J Steps 5and6. Then,( 5 )holds 5 ), wediscussthefollowingfourcases: t t closest descendantofnodei. Thus,( 5 )holds. andmi t thiscase, tothesameset,duetoboth ,ifm(i+1)existsandt(m(i+1))t(p) (otherwise,inJ1) ,basedon Steps 4and5. t and t t Step 5.mitgoestotheoppositesetofzi+1,sincezi+1istheclosestdescendantofnodei.Thenwehave( 5 )holdsnomatterwheremi+1tgoes. 114

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5 ), weonlyneedtoconsiderthecaseinwhich onStep1,mptgoestoJ1. Inthefollowing,weshowthatfqwgoestoJ1 thefollowing four cases. because(m(M(qw)))>t(p) Thus,basedonStep6, tothe samesetofzM(qw), whichisJ1 5 )holds. BasedonStep6, BasedonStep6,wehave constraint ( 5 ), weneedtoconsiderthefollowingfourcases. t andmqwt=2J. BasedonStep1,mp t Thus( 5 )holds. andmpt=2J t Underthiscase t tothesameset basedonstep 2 t Underthiscase, basedonStep1.Then ( 5 ) holds nomatterwheremq 5 ), notehere, wedonot needto considernitbasedontheidentitymatrixargumentatthebeginningoftheproof.BasedonStep7,g(i,t)goestothesamesetofzm((i,t)).Ift(m((i,t)))t(p)ort((i,t))t(p)+1, then becauset(m((i,t)))t(p)+1 115

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5 )holds. Therefore,thedesiredproperty( 5 )holdsforconstraints( 5 ),( 5 ),( 5 )to( 5 ) ,and( 5 ), andthe correspondingconstraintmatrixistotallyunimodular. Thus ,theextendedformulationprovidesanintegralsolutionfortwo-stageSULSB-WW Nowwestudytheintegralpolyhedrain the a tighterextendedformulationfor the two-stageSULSB-WW.First,we dene (f,g,z,s, 1t(i),i2V 5 )and( 5 ), 0fi,gi,zi1,si,`i0,i2V g. the two-stageSULSB-WW inthe(f,g,z,s,`)space, byshowingthatitisaprojectionofQHin the the two-stageSULSB-WWin the ( 5 ),( 5 ),( 5 )to( 5 ), 0fi,gi,zi, ,i2V g. Proof. 116

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Claim2.Foranarbitraryextremepoint(f,g,z,s,`)2QM,thereexistsa(f,g,z,m,n,s,`)2QH. ProofofClaim1.WeproveClaim1byshowingthatsiTXt=t(i)+1 ,i)nf(i,t)gzj]+ arevalidforQH 5 )and( 5 )hold,thensiTXt=t(i)+1 ,i)nf(i,t)gzj]+ 5 ),( 5 )andthenonnegativityofnit,wehave`i=t(i)Xt=1dtnit t(i)Xt=1maxf0,g(i,t)Xj2P((i,t), 5 )holds. Nowweprove( 5 )by2conditions.(1)t(i)t(p)+1;(2)t(i)t(p). 117

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5 ),( 5 ),andthenonnegativityofmit,wehavesi=TXt=t(i)+1dtmit 5 )holds. ForCondition2,t(i)t(p),wehavesi=TXt=t(i)+1dtmit=dt(i)+1mit(i)+1+TXt=t(i)+2dtmitdt(i)+1[fi+1]++TXt=t(i)+2dt[mi+1tzi+1]+,ift(i)
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ift(i)t(p)+1respectively. (a)Ift=t(p)+1,then[mi+1tzi+1]+[mptpXj=i+1zj]+[fqwtpXj=i+1zj]+=[fqwtXj2Pit(wt)nfi,qwtgzj]+. 5 ) and thesecondinequalityfollows( 5 ). (b)Ift>t(p)+1,then[mi+1tzi+1]+[mptpXj=i+1zj]+[mqwttXj2Pit(p)+1(wt)nfigzj]+[f(i,t,wt)Xj2Pit(p)+1(wt)nfigzjXj2Pqwtt(wt)nfqwt,(i,t,wt)gzj]+[f(i,t,wt)Xj2Pit(wt)nfi,(i,t,wt)gzj]+ 5 ) thesecondonefollows( 5 ) andthethirdonefollows( 5 ).Thus( 5 )holds. Therefore,Claim1holds. NowweproveClaim2holds. Foranygiveextemepoint(f,g,z,s,`)2QM,weconstructmandnsuchthat(f,g,z,m,n,s,`)2QH.Thatis,(f,g,z,m,n,s,`) the conditions ( 5 ),( 5 ),( 5 )to( 5 ).Nowforagivenpoint(bf,bg,bz,bs,b`)2QM,letbmit=maxw2W(i)24bf(i,t,w)Xj2Pit(w)nfi, 119

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Since(bf,bg,bz,bs,b`)isanextremepointinQM,werstobservethat(bf,bg,bz,bs,b`)satisesequation( 5 )basedon( 5 )and( 5 ).Similarly,(bf,bg,bz,bs,b`)satisesequation( 5 )basedon( 5 )and( 5 ). Itis also obviousthat(bf,bg,bz,bm,bn,bs,b`) ( 5 )and( 5 ). Basedon ( 5 ), inequalities ( 5 ),( 5 ),and( 5 ) hold Basedon ( 5 )andthenonnegativityofbnit, inequality ( 5 )holds. Nowwe onlyneedto showthat( 5 )and( 5 )hold. For( 5 ),letwbethescenariowherebmi+1tachievesthemaximumvalue.Thenweobservethatbmitachievesthemaximumvalueinthesamescenariow. Weprove( 5 )holdsbasedonbzi+1=1andbzi+1=0respectively.Ifbzi+1=1,thenbmit=0basedon( 5 ).Thenbmit=0bmi+1t1=bmi+1tbzi+1.Ifbzi+1=0,then 5 )holds. For( 5 ),accordingto( 5 ),bm t t t

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t t wt t thethirdequationfollows wt t( ( 5 ) holds.Thus(bf,bg,bz,bm,bn,bs,b`)2QHandClaim2holds. Therefore,theconclusion holds 121

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Thematrixofconstraints( 5 ),( 5 ),and( 5 )to( 5 )fortheexampleinFigure 5-1

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dynamicknapsacksetisnaturallygeneratedfromthedeterministicuncapacitatedlot-sizingproblem. Loparicetal. ( 2003 )rstintroducedthedeterministicdynamicknapsacksetXDKas follows : wherea,b2RT+.Theystudiedthepolyhedralstructure of thedeterministicdynamicknapsackset.Withtheapplicationofthesequenceindependentliftingscheme,afamilyoffacet-deninginequalitiesforXDKwereintroduced. Forthedeterministicmixed-integerprogrammingproblems,differentschemeshavebeenexploredtogeneratemorevalidinequalitiesbasedonexistingvalidinequalities. Guanetal. ( 2007 )proposedthepairingschemewhichgeneratedafamilyofinequalitieswiththeproperlyorderedcombinationoftwoexistingvalidinequalities. Recently GunlukandPochet ( 2001 )developedthemixingproceduretogeneratevalidinequalitiesbasedonthemixed-integerroundinginequalities(MIR).Theydemonstratedthatthemixinginequalitiesarestronginequalitiesforsomespecialpolyhedralstructures. MillerandWolsey ( 2003 )showedthatthemixinginequalitiescanprovidetheconvexhulldescriptionfor aspecial single-itemlot-sizingproblem. Theliftingschemewasrstintroducedby Wolsy ( 1976 1977 ),and Zemel ( 1978 ),etal.Theliftingscheme canbe appliedsequentially togeneratestrongvalidinequalities Guetal. ( 2000 )extendedthe sequence independentliftingscheme( Wolsy 1977 )tothemixed0-1integerprogramming,and showedthat the sequence independentlifting propertyholdsaslongastheliftingfunctionissuperadditive Thatis thegenerationofliftingcoefcientsareindependentoftheliftingsequence. Atamturk ( 2004 )generalized 123

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sequence independentlifting property tothegeneralmixed-integerprogrammingproblem. Inthischapter,westudytheextensionofthedeterministicdynamicknapsackset:thestochasticdynamicknapsack(SDK)set.WeinvestigatethepolytopeoftheSDKsetbasedona multi-stagestochastic scenariotreemodel asdescribedin RuszczynskiandShapiro ( 2003 ).Forthestochasticmixed-integerprogrammingproblems, Guanetal. ( 2006b )studiedthemulti-stagestochasticuncapacitatedlot-sizingproblemanddevelopedafamilyofvalidinequalities,named(Q,SQ)inequalities.Theyshowedthatundercertainconditions,(Q,SQ)inequalitiesarefacet-deninginequalities. Guanetal. ( 2006a )examinedthat(Q,SQ)inequalitiesaresufcienttodescribethe convexhullofthe two-periodproblem.Thepairingscheme described in Guanetal. ( 2007 )canalsoprovidevalidinequalitiesforthestochasticlot-sizingproblemand generalize all(Q,SQ)inequalities. Furthermore,basedonthese Guanetal. ( 2009 )proposedageneralapproachtogeneratevalidinequalitiesforthe multi-stage stochasticmixed-integerprogrammingproblems.Theycombinedvaliddeterministicinequalities correspondingtoeachscenariostogenerate validinequalitiesforthewholescenariotree. The remainingpart ofthischapteris organized asfollows.InSection 6.3 ,westudythepairingandmixingschemesfortheSDKsetandshow that pairingandmixinginequalitiesarefacet-deninginequalitiesundercertainconditions.Section 6.4 demonstratesthesequenceindependentandsequencedominantliftingschemesfortheSDKset.WegeneratefamiliesofvalidinequalitiesfortheSDKsetthroughtheliftingschemes.Then,tosolvelarge-scaleproblem,inSection 6.5 ,weapplyparallelcomputingtechniquetosolvetheSDKset.Wedevelopparallelalgorithmstogeneratevalidinequalitiesviapairing,mixing,andliftingschemesforthestochasticcapacitatedlot-sizingproblemasanexampleoftheSDKset.Finally,wedemonstratethecomputationalefciencybyshowingtheimprovementoftheoptimalityandintegralitygapsinSection 6.6 124

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Similarasthedeterministiccase,XSDKisarelaxationsetofthefeasibleregionofstochasticcapacitatedlot-sizingproblem,XCSLS.XCSLS=8<:(s,x,y)2R+Rn+Bn:s+Xj2P(i)xjd1i,xiaiyi,i2V9=;, Wecanobservethat correspondingto thescenariowiththeinformationonpathP(i).WenametheinequalityinX(i)aspathinequality.AllpathinequalitiesinX(i),i2VarevalidforXSDK.Notehere,whenjLj=1,XSDK=XDK. 125

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6-1 .ThefollowingpathinequalitiesaregeneratedcorrespondingtopathP(1),P(2),P(3),P(4),andP(5)separately:s+40y15s+40y1+15y215s+40y1+20y317s+40y1+20y3+20y420s+40y1+20y320y520 Pathinequalitiesforascenariotree BecauseX(i)isat(i)-perioddeterministicdynamicknapsackset,weextendtheconclusionof Loparicetal. ( 2003 )forXDKtothestochasticsetting.Weletb`k=Pt2P(`,k)P(j)btandebk=b1k.Withtheinformationofb,weconstructamodiednonnegativeparameter

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Guanetal. ( 2007 )studiedthepairingschemeforthemixed-integersetsX2Zn+Rp+.Letthepair(a,g)2Rn+1RpdeneavalidinequalityforX,if Let(a1,g1)and(a2,g2)denetwovalidinequalitiesforX.Thepairingschemeforthesetwoinequalitiesisasfollows: Guanetal. 2007 )If(a1,g1)and(a2,g2)denetwovalidinequalitiesforXwitha1n+1a2n+1,theinequality ( 2009 )appliedtheabovepairingschemetopathinequalitiesofXSDKandobtainedthefollowingtreeinequalities: Guanetal. 2009 ). 127

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6 )canbestrengthenedasfollows: Guanetal. 2009 ). 6 )isdominatedbytheinequality 6 )isafacet-deninginequalityforconv(XSDK). Guanetal. 2009 ). 6 )isfacet-deningforconv(XSDK)if ( 2009 )discussedtheconvexhulldescriptionofXSDKunderthelargecoefcientcase(Condition(2)inTheorem 6.4 ). Guanetal. 2009 ). 6 )forallV,togetherwith0sbVand0yj1foreachj2V,describetheconvexhullofXSDK,wherebV=maxfbi,i2Vg. Guanetal. 2009 ). 6 ). 6 )basedonthescenariotreeinFigure 6-1 .ForsetR=f1,2g,wehaveVR=f1,2g,R(1)=f1,2g,andR(2)=f2g.Thecorrespondingtwopathinequalitiesares+40y15ands+40y1+15y215. 128

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GunlukandPochet ( 2001 )proposedthemixingschemeforMIRinequalities.Givenamixed-integerregionSRm1Zm2andacollectionofm2validinequalitiesforSfi(x)+Bgi(x)i,i2I=f1,...,mg, whereB2R1+,i2R1,fi(x)0,andgi(x)2Z.Foranyi2I,thesimpleMIRinequalityfi(x)i(igi(x))isvalidforS,wherei=di=Be,i=i(i1)B,i2Z1,andBi>0.Notethatfiandgicanbenonlinear,iandgicanbenegative.Withoutlossofgenerality,weassumethatI=1,...,n,andii1forallni2. GunlukandPochet 2001 ).

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isarelaxationsetofXSDK.WegeneratethemixinginequalitiesforX0SDKasfollows: 6 )and( 6 )canalsobeconstructedforsubsetXSDK(V(i))basedonthesubtreeV(i). WeprovidetheconvexhulldescriptionforXSDKwithmixinginequalities( 6 )and( 6 )undertheconditionthatai=Afori2V. 6 )and( 6 )aresufcienttodescribetheconvexhullforXSDK.

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Claim1.Inequalities( 6 ),( 6 ),and0ziza(i)1denetheconvexhulldescriptionfor Claim2.LetX0SdenoteXSDKwithai=Afori2V.ItisonetoonecorrespondencebetweenX0SandX0. ProofofClaim1:LetZ1=f(w,z)2RZjVj+,w+Azifi,i2VgandZ2=fz2ZjVj+,0ziza(i)1,i2Vg.WehaveX0=Z1\Z2.TheconstraintmatrixforZ2isthetransposeofanetworkmatrix(arc-nodeincidencematrix)andtheright-handcoefcientisinteger.Followingtheresultin MillerandWolsey ( 2003 ),( 6 ),( 6 ),and0ziza(i)1aresufcienttoprovidetheconvexhulldescriptionforX0. ProofofClaim2:weshowthatforagiven(s,y)2X0S,thereexistsareversiblefunctionG:X0S!X0. WeshowthatthereisafunctionG:X0S!X0,wherew=s, Becauseyi2f0,1g,wehavezi2Z+.Letfi=biforalli2V.Withs+APj2P(i)yjbi,( 6 ),and( 6 ),wehavew+Azifi.Thus,(w,z)2X0. ThereversefunctionofG,G1:X0!X0S,isthats=w, Becausezi2Z+and0ziza(i)1,wehaveyi2f0,1g.Withw+Azifi,( 6 ),and( 6 ),wehaves+APj2P(i)yjfi.Thus,(s,y)2X0S. Thus,itisonetoonecorrespondencebetweenX0SandX0. 131

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6 )and( 6 )aresufcienttoprovidetheconvexhulldescriptionforX0S.Givenanyc12Randc22RjVj,theproblem,P1,maxfc1s+cT2y,(s,y)2RRjVj+gwith( 6 )and( 6 )canbetransferredtobetheproblem,P2,maxfc1w+bTz,(w,z)2RRjVj+gwith( 6 ),( 6 ),and0ziza(i)1,wherebi=Pj2P(i)cj2.Supposethat(w,z)istheoptimalsolutionforP2.WithClaim1,zisintegral.WithClaim2and(w,z),weobtainthecorrespondingoptimalsolution(s,y)forP1.Withtheintegralityofzand0yi=ziza(i)1,yisintegral.Thus,( 6 )and( 6 )aresufcienttoprovidetheconvexhulldescriptionforX0S. Similaras Loparicetal. ( 2003 ),wederiveothervalidinequalitiesforXSDKbytheliftingscheme.Wesetsomevariablestobe1andmodifythecorrespondingparameters.Then,wegeneratethebasicinequalityandliftbackvariablesthathavebeenxed. LetU(i)representasubsetofP(i).Wesety`=1for`2U(i).Thenwemodifytheparameterb`ased`=b`a`,for`2U(i),andgeneratethebasicpathinequalityasfollows: whereR=P(i), 132

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With( 6 )and( 6 )asthebasicinequalityandtheliftingfunction,respectively,wegeneratetheliftedvalidinequalityforXSDKasfollows: Proof. 6 )and( 6 ).Directlyfollowing Loparicetal. ( 2003 ),weobtaintheaboveresult. NotethatbasedonProposition 6.5 ,thesequenceindependentliftingpropertyholdsforliftingbackvariablesbasedonthepathinequalityinXSDK. NowwederivevalidinequalitiesforXSDKbasedonthetreeinequality.LetXSDK(VR)representthefeasiblesolutionsetdenedbythepathinequalitiescorrespondingtonodesinVR.WeletL(R)representthesetofleafnodeinVRandCR(i)=[j2P(i)C(j)nVRwithalli2L(R). First,weapplytheliftingschemetoXSDK(VR).Supposethat 133

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6 )bethebasicinequality.WedenetheliftingfunctionasCR(i)(0)=minfs+Xj2VRjyj:(s,y)2XSDK(VR)[f`gg Condition1:y`kisliftedbeforey`k1.WithLemma 1 ,afterliftingbackofy`k,weobtainafacet-deninginequality( 6 )forXSDK(VR[f`kg)andthecoefcientofy`kis`k=minfs+Pj2VRjyj:(s,y)2XSDK(VR[f`kg)g.Then,welifty`k1basedon( 6 )andobtainafacet-deninginequalityforXSDK(VR[f`kg[f`k1g),s+Pj2VRjyj+`ky`k+`k1y`k1+`k+`k1with`k1=minfs+Xj2VRjyj+`ky`k(+`k):(s,y)2XSDK(VR[f`kg[f`k1g)g. 134

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Therefore,allfeasiblesolutionsforXSDK(VR[f`kg)arefeasiblesolutionsforXSDK(VR[f`kg[f`k1g).Becausevariablesdoesnothaveanupperbound,(s,y)withy`k=0isafeasiblesolutionforXSDK(VR[f`kg[f`k1g).Then,wehave`k1=minfs+Xj2VRjyj+`ky`k(+`k):(s,y)2XSDK(VR[f`kg)[f`k1gg=minfs+Xj2VRjyj+0`k`k:(s,y)2XSDK(VR[f`kg)[f`k1gg=`k`k=0. Condition2.y`k1isliftedbeforey`k.WithLemma 1 ,afterliftingbacky`k1,weobtainafacet-deninginequality( 6 )forXSDK(VR[f`k1g)gandthecoefcientofy`k1is`k1=minfs+Pj2VRjyj:(s,y)2XSDK(VR[f`k1g)g.Then,welifty`k1basedon( 6 )andobtainafacet-deninginequality.Thecoefcientofy`kis`k=minfs+Xj2VRjyj+`k1y`k1(+`k1):(s,y)2XSDK(VR[f`k1g[f`kg)g.

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Weconcludethesetwoconditionsandobtaintheliftingdominantpropertyasfollows. Proof. 136

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Step2.Welifty`2aftertheliftingofy`1.Followingthesequencedominantproperty,y`1isliftedbeforey`2,whereby`1by`2anda(`2)2P(a(`1)).Thus,thecoefcientofy`2is`2`1andthecorrespondingnewliftedfacet-deninginequalityiss+Xj2VRjyj+`1y`1+(`2`1)y`2+`1+`2`1+`2. With`0=0,werewrite( 6 )ass+Xj2VRjyj+(`1`0)y`1+(ell2`1)y`2+`2. Stepj+1.Nowbasedon( 6 ),welifty`j+1andthecorrespondingcoefcientofy`j+1isj+1=minfs+Xj2VRjyj+X1kj(`k`k1)y`k(+`j):(s,y)2XSDK(VR[f`1g[[f`j+1g)g.

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6.5.3 ,and2).theLP-basedcuttingplanealgorithmwithheuristic,aswelldescribedfurtherinsection 6.5.4 .Themajorissuesofparallelimplementationisthatofloadbalancingamongprocessorsandmanagementofgeneratedcuts.ThedesirableconditionisthateachprocessorhandlesapproximatelyanequalnumberofLPsandthecuttingmanagementcancontrolthecuttingpoolsize,recognizethestrongcuts,andpurgetheineffectivecuts. OurparallelimplementationwasdevelopedonaPentium4Xeon64quadcoreLinuxclusteranddistributed-memorymultiprocessors.WeapplytheopensourcesoftwareSYMPHONYinCOIN-ORasourmixed-integerprogrammingsolver.Weletand!representthetotalnumberofprocessorsandtheslaveprocessorsworkingonthecutgenerations. Inordertogenerateourcuts,weusemultiprocessorstogeneratecutsaccordingtothestochasticscenariotreestructure.Thissectionisorganizedasfollows.InSection 6.5.1 ,wediscussthestaticpartitionforthestochasticscenariotreestructure.InSection 6.5.2 ,wediscussthecutmanagementofourtwoalgorithmsinaparalleldistributedenvironment.InSections 6.5.3 and 6.5.4 ,weprovidethedetailedparallelalgorithmtoobtaintheoptimalityandintegralitygapsforthestochasticintegerprogrammingproblems. 138

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6.5.2.1CutGeneration 6.3 and 6.4 ,therearethreecutgenerationschemes,thepathliftingscheme,thepairingscheme,thetreeliftingscheme.Thesethreeschemecutsaregeneratedoneachprocessor. Forthestaticpartitioning,rst,eachprocessorobtainstheinformationofthesubtreefromthemastermachine.Second,thecorrespondingbasicpathinequalitiesaregeneratedbasedonthesubtreestructure.Third,thepathliftinginequalitiesaregenerated.Fourth,thepairingschemeisappliedtothesubtreestructuretogeneratevalidinequalities.Finally,theliftingschemeisappliedtothepairinginequalities. Duetoeachprocessorshouldhandlethesethreetypesofinequality,weassigntimelimitsforeachschemeforcutgeneration.Ifaschemeofcutgenerationobtainedlongtimeintervals,theprocessorspendslongertimeongeneratingcutsbasedonthescheme. 139

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Themastermachinehandlesthemastercuttingpoolwhichobtainedallcutsfromthelocalcuttingpoolandpurgetheduplicatedcutsbeforeallcutsareappliedtotheprocessinthemastermachine. 1. Step1handlesthepartitionsforastochasticscenariotree.Thecutgenerationisfromtheshortestbranchtothelongestbranchstartingfromtherootnodes.First,wegeneratethebasicinequalities.Second,wegeneratetheliftinginequalitiesbasedonthebasicinequalities.Third,wegeneratethepairinginequalitiesbasedonthegeneratedbasicandliftedinequalities.InStep2,weapplythebasiccutmanagementforthegeneratedcuts.Thepurposeofcutmanagementistopurgetheduplicatedcutsandstoreallefcientcuts.InStep3,wecombineallgeneratedcutswiththeoriginalCSLSproblemasabigMIPproblemandusingthebranch-and-boundschemein 140

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Step2.SolvetherelationproblemofStRandobtainthesolutionxtandobjectivezLP Step3.Heuristics:callHeuristicsforobtainingtheintegerfeasiblesolutiontoobtaintheintegerfeasiblesolution,xH.IfzHUIP,UIP=zHandxIP=xH.Otherwise,N=N+1,whereNrepresentsthenumberofheuristicscalls. Terminationtest:IfNK,stop. Step4.UpdateLIP=zLPandSk+1fromSkwithmoregeneratedcuts. Step2.Forxtj"1,ifxtjxtmax"2,xtj=1. Beforetheimplementationofthisheuristics,thesynchronization(share)ofcutsiscalledamongprocessorsandthecorrespondingLPproblem(StR)isprocessed.AftertheimplementationofthisHeuristics,themodiedLPwithxedintegervariablesisresolved.Thenthisheuristicsiscalledagainuntilanintegerfeasiblesolutionisgenerated. 141

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Inthefollowing,weconsidertwostochasticscenariotrees.Therstisabinarytreewithsixteenperiods(P2-16).Thesecondoneisatreewiththreebranchesateachnon-leafnodesandthirteenperiods(P3-13).Foreachtree,wesettheratiof==10,20,30and40.Wegeneratetwoinstanceswithratio10.Therearetencombinationsintotal.Wealsosetdemandsdi,unitproductioncosti,andunitinventorycosthiuniformlydistributedin[50,50],[50,100],[5,10].Foreachsetting,wetestveinstancesandreporttheaveragevalue. 142

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Parametersetting unitproductioncostpi ratio10 [50,100] [500,1000] 2 ratio20 [50,100] [1000,2000] 3 ratio30 [40,60] [1000,2000] 4 ratio40 [40,60] [500,1000] 5 ratio10 [50,50] [500,500] ofthecut-and-branchalgorithmbythegappercentageOGap=UBLB UB100%,whereUBistheobjectivevaluecorrespondingtothebestintegersolutionobtainedinthegiventimelimit.LBisthelowerboundobtainedfromlinearprogrammingrelaxation.Figure 6-2 andFigure 6-4 showtheoptimalitygapsfor(P2-16)and(P3-13)withthehighsetupcostswith5ratios.For(P2-16),withmoregeneratedcuts,OGapisdecreasedfrom6.18%to2.10%inaverage.For(P3-13),OGapisdecreasedfrom7.99%to3.45%inaverage.Figure 6-3 andFigure 6-5 showthecorrespondingcutsnumbersgeneratedforbothcases. 6-2 ,weprovidetheaverageofallveparametersettingsfor(P2-16)and(P3-13).Wecanseethatwiththecutsgenerationandheuristics,weprovideintegerfeasiblesolutionsandimprovetheintegralitygaps. Table6-2. Heuristicsforlowsetupcostcase LP MIP HEUR IGAP% 2-16 3-13 143

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GapsforP2-16 1% 2% 3% 4% 5% 6% 7% 8cores 4cores 2cores 1cores Figure6-3. CutsnumberforP2-16 200 400 600 800 1000 1200 1400 8cores 4cores 2cores 144

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GapsforP3-13 1% 2% 3% 4% 5% 6% 7% 8% 9% 8cores 4cores 2cores Figure6-5. CutsnumberforP3-13 500 1000 1500 2000 2500 8cores 4cores 2cores 145

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Inthisdissertation,wediscussedthemulti-stagediscreteoptimizationproblemunderdatauncertainty.Westudiedthemulti-stagerobustlot-sizingwithdisruptions,thepolyhedronoftwo-stagestochasticlot-sizingproblems,andthegenerationofvalidinequalitiesforthestochasticdynamicknapsackset.InChapter2,westudiedthelot-sizingproblemwitha potential disruptionasrecoursetohandletheuncertainty.Ourobjectiveistoachievetheminimumobjectivevaluethatconsiderstheworstcasescenarioforthedisruption.Wedevelopedacustomizedbranch-and-boundalgorithmtosolvethelot-sizingproblemwithadisruption,whichisatwo-stagerobustoptimizationproblem.ABenders'decompositionbasedoptimalitytestisgeneratedforthebranch-and-boundalgorithm.Thisstudyprovidesmorerobustproductionplanningtoaddressadisruption,ascomparedtothedeterministiclot-sizingproblemandpreviousstudiesonrecoveryproductionwiththeinformationofdisruption. InChapter3, we provided a generalmodelforthemulti-stagerobustlot-sizingproblemwithuncertaindisruptions.Weadoptedoutsourcingandbackloggingasthereparationmethodstosatisfyunlleddemands dueto disruptions.Fortheoutsourcingcase,basedonaproperassumption,atwo-stagerobustmodelandthecorrespondingprimal-dualalgorithmaregenerated.Forthebackloggingcase,we rst consideredatwo-stagerobustoptimizationproblem.Wereformulatedthisproblem as amixed-integerprogramandinvestigatedthecorrespondingpolyhedralstructure.Weanalyzedthetrivialfacet-deninginequalities,andgenerated three familiesoffacet-deninginequalitiesforthepolyhedronoftherobustlot-sizing problem withdisruptionandbacklogging.Applyingthesefacet-deninginequalitiesascuts,thecomputationalresultsdemonstratethattheseinequalitiesacceleratethecomputationalspeedascomparedwithdefaultCPLEX. 146

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thatitisone-to-onecorrespondencebetweenthereformulationandtheoriginalmulti-stagerobustoptimizationmodel .Thus,themulti-stagerobustlot-sizingproblemwithmultipledisruptions is computationallytractable.Forthenon-setupcostcase,wegeneratedareformulatedminimizationlinearprogramwith a pre-processingalgorithm.Weprovedthatthereformulationschemealsoworksforthemulti-stagerobustmixed-integerprogram.Ascomparedtothetwo-stagerobustoptimizationmodel,themulti-stagemodelismuchmorechallengingtosolve.Thereformulationscheme proposes asolutionapproach for themulti-stagerobustmixed-integerprogram. Chapters4and5contributetotheliteratureonderivingintegralpolyhedraldescriptionsandextendedformulationsformulti-periodstochasticuncapacitatedlot-sizing problems .Tothebestofourknowledge,thereisnopreviousresearchondevelopinganextendedformulationwhichprovidesintegralsolutionsforthemulti-periodstochasticuncapacitatedlot-sizingproblem.InChapter4,weintroduceadeterministicequivalentformulationforatwo-stagestochasticuncapacitatedlot-sizingproblemwithWagner-Whitincostsanddeterministicdemands.Weexaminedtheoptimalsolutionpropertyandusedittogenerate an extended formulation inthehigherdimensionalspace .Then,weprovedthattheconstraintmatrixfor the extendedformulationistotallyunimodular.Finally, we projectedtheextendedformulationbacktotheoriginalspace sothatwecanndvalidinequalitieswhichdescribethe integralpolyhedron oftheproblem intheoriginalspacewith 147

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InChapter6,westudiedtheextensionofthedeterministicdynamicknapsacksets,thestochasticdynamicknapsackset,andinvestigatedthepolytopeofthestochasticdynamicknapsacksetwithascenariotreemodel.Westudiedthepairingscheme,mixingscheme,liftingschemeforthestochasticdynamicknapsacksettogeneratevalidinequalities,whicharefacet-deningundercertainconditions.Thestochasticscenariotreemodelinvolvesmanyscenariosandcausescomputationaldifculties.Therefore,weappliedparallelcomputingtosolvethestochasticdynamicknapsackset.Wedevelopedparallelcomputingalgorithmstosolvethestochasticcapacitatedlot-sizingproblemasanexampleofthestochasticdynamicknapsackset. Forthemulti-stagediscreteoptimizationunderuncertaintyproblem,themulti-stagerobustoptimizationandstochasticprogrammingaretwomainapproaches.Inthefutureresearch,thereareotherinterestingsettingsandmodelstostudy.Forthemulti-stagerobustoptimizationmodel,inChapter3,westudiedthepolyhedralstructureofatwo-stagerobustmixed-integerprogram.Thereispotentialtoextendcurrentresultstothemulti-stagerobustmixed-integermodel,andgeneratethecorrespondingfacet-deninginequalitiestodescribethepolyhedronofthemulti-stagerobustmixed-integerprogram.Wecanalsoconsideramoregeneralsettingforthemulti-stagerobustmixed-integerprogram,inwhichtheuncertainparametersareingivenintervals. 148

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Forthemulti-stagestochasticprogramming,apartofourresultsforthetwo-stagestochasticlot-sizingcanbeappliedtoamoregeneralmulti-stagestochasticprogrammingsetting,whichcanaddressfurtheruncertainties.Underthemulti-stagesetting,itcanbeobservedthattheoptimalityconditionstillholds,basedontheWagner-Whitincostssettingforcaseswithoutandwithbacklogging.Accordingly,wecanobtainsimilarlyconstraintsforthereformulation.Whetherthereformulationcanprovideanextendedformulationthatprovidesintegralsolutionsforthemulti-stagestochasticuncapacitatedlot-sizingproblemisalsoofinterestforfuturestudy. 149

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ZhiliZhouwasbornin1981,Nanjing,Jiangsu,ChinatoparentsGuifengWangandYaozongZhou.ShewasraisedinNanjingandattendedNanjingNo.3Middleschoolforhermiddleschoolandhighschooleducation.ShewentontoattendcollegeatNanjingUniversityinNanjing,China,in1981,majoringinInformationandComputationalMathematics.Shereceivedherbachelor'sdegreeinscienceinJune2003.ZhilithenelectedtocontinuehereducationinDepartmentofMathematics,NanjingUniversity,majoringinOperationsResearchandcompletedhermasterdegreeinscienceinJune2006.ZhilienrolledinthePh.D.programmingintheSchoolofIndustrialEngineering,UniversityofOklahoma,in2006andtransferredtoDepartmentofIndustrialandSystemsEngineering,UniversityofFlorida,in2009withheradvisor,Dr.YongpeiGuantoobtainaDoctorofPhilosophydegree. 155