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Models and Algorithms for Reliable Facility Location Problems and System Reliability Optimization

Permanent Link: http://ufdc.ufl.edu/UFE0021262/00001

Material Information

Title: Models and Algorithms for Reliable Facility Location Problems and System Reliability Optimization
Physical Description: 1 online resource (120 p.)
Language: english
Creator: Zhan, Lezhou
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: algorithms, approximation, facility, fortify, global, heuristics, integer, location, models, monotonic, nonlinear, optimization, programming, reliability, reliable, stochastic, system, uncertainty, unreliable
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: Uncertainty is one of the elements that make this world so fascinating and dynamic. However, the existence of uncertainty also poses a great challenge to reliable system design. Our study uses various models and algorithms to address reliability issues in the context of (1) the uncapacitated facility location problem where facilities are vulnerable, and (2) the system reliability problem where components are subject to fail. We first study the uncapacitated reliable facility location problem in which the failure probabilities are site-specific. The problem is formulated as a two-stage stochastic program and then a nonlinear integer program. Several heuristics that can produce near-optimal solutions are proposed for this computationally difficult problem. The effectiveness of the heuristics is tested through extensive computational studies. The computational results also lead to some managerial insights. For the special case where the probability that a facility fails is a constant (independent of the facility), we provide an approximation algorithm with a worst-case bound of 2.674. Another part of our research is related to the application of a monotonic branch-reduce-bound algorithm, a powerful tool to obtain globally optimal solution to problems in which both the objective function and constraints possess monotonicity. We tailor the algorithm to solve a mixed integer nonlinear programming problem. Its convergence analysis and acceleration techniques are also discussed. The algorithm is then successfully applied to solve system reliability optimization problems in complex systems, including the redundancy allocation optimization problem and the reliability-redundancy allocation optimization problem. Compared to the existing techniques, the monotonic branch-reduce-bound algorithm is not only versatile but also very efficient in dealing with different types of problems in system reliability. We also develop several models that can be used to fortify the reliability of the existing facilities. They are the extensions to the models in the first part of the dissertation and offer insights on which facility to choose and to what extent it should be fortified. The properties and solution methodologies of the models are discussed. In particular, a monotonic branch-reduce-bound algorithm is used to solve one of these models. The efficiency of the algorithm is demonstrated in the computational results.
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 Lezhou Zhan.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Shen, Zuo-Jun.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021262:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021262/00001

Material Information

Title: Models and Algorithms for Reliable Facility Location Problems and System Reliability Optimization
Physical Description: 1 online resource (120 p.)
Language: english
Creator: Zhan, Lezhou
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: algorithms, approximation, facility, fortify, global, heuristics, integer, location, models, monotonic, nonlinear, optimization, programming, reliability, reliable, stochastic, system, uncertainty, unreliable
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: Uncertainty is one of the elements that make this world so fascinating and dynamic. However, the existence of uncertainty also poses a great challenge to reliable system design. Our study uses various models and algorithms to address reliability issues in the context of (1) the uncapacitated facility location problem where facilities are vulnerable, and (2) the system reliability problem where components are subject to fail. We first study the uncapacitated reliable facility location problem in which the failure probabilities are site-specific. The problem is formulated as a two-stage stochastic program and then a nonlinear integer program. Several heuristics that can produce near-optimal solutions are proposed for this computationally difficult problem. The effectiveness of the heuristics is tested through extensive computational studies. The computational results also lead to some managerial insights. For the special case where the probability that a facility fails is a constant (independent of the facility), we provide an approximation algorithm with a worst-case bound of 2.674. Another part of our research is related to the application of a monotonic branch-reduce-bound algorithm, a powerful tool to obtain globally optimal solution to problems in which both the objective function and constraints possess monotonicity. We tailor the algorithm to solve a mixed integer nonlinear programming problem. Its convergence analysis and acceleration techniques are also discussed. The algorithm is then successfully applied to solve system reliability optimization problems in complex systems, including the redundancy allocation optimization problem and the reliability-redundancy allocation optimization problem. Compared to the existing techniques, the monotonic branch-reduce-bound algorithm is not only versatile but also very efficient in dealing with different types of problems in system reliability. We also develop several models that can be used to fortify the reliability of the existing facilities. They are the extensions to the models in the first part of the dissertation and offer insights on which facility to choose and to what extent it should be fortified. The properties and solution methodologies of the models are discussed. In particular, a monotonic branch-reduce-bound algorithm is used to solve one of these models. The efficiency of the algorithm is demonstrated in the computational results.
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 Lezhou Zhan.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Shen, Zuo-Jun.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021262:00001


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9b43b9f85f4ce40c852a1ad3ce2bcabaf1e37478







MODELS AND ALGORITHMS FOR RELIABLE FACILITY LOCATION PROBLEMS
AND SYSTEM RELIABILITY OPTIMIZATION


















By
ROGER LEZHOU ZHAN


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

2007



































S2007 Roger Lezhou Zhan

































To my parents,



and my brother, Lep~ing,

for their love and support










ACKENOWLED GMENTS

I would never he able to adequately thank Dr. Zuo-.Jun Max Shen, my supervisor and

mentor, for helping me in developing my research topics. He allowed me to work freely,

but he was ah-- .1-4 there when I needed advice or guidance. I want to thank hint not only

for his tremendous guidance and encouragement throughout my study, but also his endless

trust and understanding.

Dr. Panos Pardalos deserves many sincere thanks. He recruited me to the University

of Florida and was ahr-l- .- very supportive through my years there.

I thank Dr. .Joseph P. Geunes and Dr. Edwin Romeijn for their many good

-II_0-r~!--- 0.. in this research as well as their wonderful courses I have taken with them.

I also thank another nienter of my coninittee, Dr. .Juan F in- for her time and guidance

in the last three years.

I am grateful to both of my co-authors Dr. .Jiawei Z1! Iu-, and Dr. Mark Daskin

for their incisive insights. I also thank Prof. H. Tuy for introducing his nionotonic

optimization methodology to me while he visited Gainesville in August, 2003.

I acknowledge all the help and financial support front the Department of Industrial

and Systems Engineering. In particular, I thank Dr. Hearn for assigning me to teach

various courses at the University of Florida.

My deepest appreciation goes to my entire family, including my wife, Ngana, my

brother, Leping, my parents and in-laws. I could not have finished this research without

their support, love, and encouragement. Most specially, I want to thank my wife for being

with me in numerous late nights in Gainesville, Miami, and Greenwich, and helping to

proofread nly manuscript.

Last but not the least, I thank my friends, Leon, Lian, Shu, Gang, Bin, .Jean,

Chunhua, .Jun, Altannar, .Jie, .Jonathon, Cindy, Silu, .Junnmin, and many others who

made my experience at the University of Florida nienorable.










TABLE OF CONTENTS


page


ACKNOWLEDGMENTS

LIST OF TABLES.

LIST OF FIGURES

ABSTRACT

CHAPTER


1 INTRODUCTION

2 RELIABLE FACILITY LOCATION PROBLEM: MODELS AND HEURISTICS

2.1 Introduction.
2.2 Literature Review.
2.3 Notations and Acronyms .. ........ ....
2.4 Uncapacitated Reliable Facility Location Problem: a Scenario-Based Model
2.5 Uncapacitated Reliable Facility Location Problem with a Single-level Failure
Probability
2.5.1 Nonlinear Integer Programming Model
2.5.2 Model Properties
2.5.3 A Special Case: Uniform Failure Probabilities


2.6 Uncapacitated Reliable Facility Location Problem
Probabilities.
2.7 Solution Methodologies.
2.7.1 Sample Average Approximation Heuristic
2.7.2 Greedy Methods.
2.7.3 Genetic Algorithm Based Heuristic
2.8 Computational Results.
2.8.1 Sample Average Approximation Heuristic
2.8.2 Greedy Methods: GAD-H and GADS-H .
2.8.3 Genetic Algorithm Based Heuristic
2.8.4 Applying Heuristics to Solve ITRFLP-SFP
2.8.5 ITRFLP-MFP: GADS-H vs. GA.
2.9 Conclusions


with Multi-level Failure


:3 ITNIFORM INCAPACITATED RELIABLE FACILITY LOCATION PROBLEM:
A 2.674-APPROXIMATION ALGORITHM

:3.1 Introduction.
:3.2 Formulations
:3.3 Approximation Algforithms.
:3.4 Conclusions










:3.5 Proofs ......... ... . 56

4 SYSTEM RELIABILITY OPTIMIZATION AND MONOTONIC OPTIMIZATION 61


4.1 Introduction ....... ... ......
4.2 A Monotonic Branch-Reduce-Bound Algorithm ..........
4.2.1 Select and Branch ...........
4.2.2 Reduce and Bound .........
4.2.3 Convergence Analysis ..........
4.2.4 Acceleration Techniques ...... ... ....
4.3 Using Monotonic Branch-Reduce-Bound Algorithm to Solve System Reliabi
Optimization Problems ....... .....
4.3.1 Redundancy Allocation Optimization .........
4.3.2 Reliability-Redundancy Allocation Optimization ..........
4.4 Conclusions .........


61
67
6;9
70
71
72
lity
7:3
7:3
76
79


5 FORTIFYING THE RELIABILITY OF EXISTING FACILITIES AND MONOTONIC
OPTIMIZATION ......... . .. .. 81

5.1 Introduction ......... . .. .. 81
5.2 Continuous Facility Fortification Model ... .. . .. 8:3
5.2.1 Properties of the Continuous Facility Fortification Model .. .. 84
5.2.2 An Example of the Continuous Facility Fortification Model .. .. 87
5.3 Discrete Facility Fortification Model ..... ... .. 92
5.3.1 Properties and Algorithms ...... ... . 9:3
5.3.2 Computational Experiments ...... ... .. 96
5.4 Conclusions ......... .. .. 99

6 CONCLUDING REMARK(S ......... .. .. 102


APPENDIX


A SYSTEM RELIABILITY COMPUTATION IN CHAPTER 4 .


104

107


B DATASET ITSED IN CHAPTER 2 .

C DATASET ITSED IN CHAPTER 5.

REFERENCES ........

BIOGRAPHICAL SKETCH ....











LIST OF TABLES


Table page

2-1 Acronyms ......... .... . 19

2-2 Sample chromosome for model URFLP-MFP ..... .... 32

2-3 GA-H parameters ......... . . 32

2-4 Selection probabilities for a population with NVp solutions .. .. .. 33

2-5 Objective values from SAA-H for the 50-node dataset ... .. .. .. 36

2-6 Runs from SAA-H for the 50-node dataset ...... .. 39

2-7 Fifty-node uniform case: greedy adding and exact solution .. .. .. .. 40

2-8 Fifty-node uniform case: GADS-H and exact solution .. .. . .. 40

2-9 Values of the GA-H parameters ......... ... .. 41

2-10 Fifty-node uniform case: GA and exact solution ... . .. 41

2-11 Objective values obtained from SAA-H on URFLP-SFP with different sample
sizes ........... ........ .... 44

2-12 Computational performance on URFLP-SFP: GAD-H, GADS-H, GA and the
enumeration method ......... .. .. 44

2-13 Computational results for URFLP-MFP using GADS-H and GA .. .. .. .. 46

4-1 Coefficients in Example 1 ......... . 74

4-2 Coefficients in Example 3 ......... . 77

4-3 Performance comparison of Example 3 . ..... 78

4-4 Performance comparison of Example 4 . ..... 79

5-1 Solution of a CFFM model ........ .. .. 89

5-2 Input data (Ve(yi) and Pi(yi)) for the 3-level model ... ... .. 96

5-3 Solutions of the 3-level model ........ ... .. 97

5-4 Solutions of the 2-level model ........ ... .. 98

B-1 Dataset of URFLP-SFP ........ . .. 107

B-2 Dataset of URFLP-MFP: 3-level ....... ... .. 109

C-1 Dataset of DFFM ........ . .. 113










LIST OF FIGURES


Research structure ...

Example of crossover operation at position :3 .....

Objective ratio across different sample sizes for the 50-node dataset ...

CPIT time across different sample sizes for the 50-node dataset ......

Evolution of the solutions from GA .......

Comparison of objective values from GAD-H, GADS-H, GA, and SAA-H


Figu:

1-1

2-1

2-2

2-3

2-4

2-5

4-1

4-2

4-3

4-4

4-5

5-1

5-2

5-3

5-4

5-5


page

1:3

:34

:37

:37

42

45

61l

6i2

64

70

78

88

91

92

98


Series system .....

Parallel-series system ......

Five-conmponent bridge network .....

Reduce process .....

Seven-link ARPA network .....

Total cost at different system reliability level ......

Failure probability at individual facility vs system reliab

Frequency of completely open facility in Table 5-1 ...

Tradeoff between objective and resource used .....

Fortification level at individual facility by resource used


,ility Level ....


A-1 Configurations based on state of component 5 in Figure 4-:3: A) Component 5
works; B) Component 5 fails ..........

A-2 Configurations based on state of subsystem 4 in Figure 4-5: A) Subsystem 4
works; B) Subsystem 4 fails .........


104


106









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

MODELS AND ALGORITHMS FOR RELIABLE FACILITY LOCATION PROBLEMS
AND SYSTEM RELIABILITY OPTIMIZATION

By

Roger Lezhou Zhan

August 2007

Cl.! ny~: Zuo-Jun Alax Shen
Major: Industrial and Systems Engineering

Uncertainty is one of the elements that make this world so fascinating and dynamic.

However, the existence of uncertainty also poses a great challenge to reliable system

design. Our study uses various models and algorithms to address reliability issues in the

context of (1) the uncapacitated facility location problem where facilities are vulnerable,

and (2) the system reliability problem where components are subject to fail.

We first study the uncapacitated reliable facility location problem in which the failure

probabilities are site-specific. The problem is formulated as a two-stage stochastic program

and then a nonlinear integer program. Several heuristics that can produce near-optimal

solutions are proposed for this computationally difficult problem. The effectiveness of the

heuristics is tested through extensive computational studies. The computational results

also lead to some managerial insights. For the special case where the failure probability

at each facility is a constant (independent of the facility), we provide an approximation

algorithm with a worst-case bound of 2.674.

Another part of our research is related to the development and application of a

monotonic branch-reduce-bound algorithm, a powerful tool to obtain globally optimal

solution to problems in which both the objective function and constraints possess

monotonicity. We tailor the algorithm to solve a mixed integer nonlinear programming

problem. Its convergence analysis and acceleration techniques are also discussed. The

algorithm is then successfully applied to solve system reliability optimization problems










in complex systems, including the redundancy allocation optimization problem and

the reliability-redundancy allocation optimization problem. Compared to the existing

techniques, the nionotonic branch-reduce-bound algorithm is not only versatile but also

very efficient in dealing with different types of problems in system reliability.

We also develop several models that can he used to fortify the reliability of the

existing facilities. They are the extensions to the models in the first part of the dissertation

and offer insights on which facility to choose and to what extent it should be fortified. The

properties and solution methodologies of the models are discussed. In particular, a

nionotonic branch-reduce-bound algorithm is used to solve one of these models. The

efficiency of the algorithm is demonstrated in the computational results.









CHAPTER 1
INTRODUCTION

Our study focuses on reliability issues arising in facility location design problems and

complex systems. In the former case, we consider mathematical models that minimize

the sum of facility opening costs and expected service and penalty costs when facilities

are subject to fail from time to time. These failures may come from disruptive events

(e.g. labor strikes, supplier business failures, terrorist attacks), or natural disasters (e.g.

hurricanes, earthquake). Facili T--p~ecific failure probabilities are explicitly considered in

our models. To the best of our knowledge, these appear to be the first such models in

the literature. These models help to make decisions in the system design phase. Several

heuristics and an approximation algorithm are proposed for solving these models.

If facilities have been built but are still subject to fail, we consider models to fortify

the reliability of the existing system given limited fortification resources. These models

can be reduced to a special class of global optimization problems, called monotonic

optimization, in which both the objective function and constraints possess monotonicity.

A specialized monotonic branch-reduce-bound algorithm is developed to efficiently solve

these problems.

We also examine reliability issues in complex industrial and military systems.

The reliability of such a system is measured by the probability of successful operation.

We address the issue of allocating unreliable components in the system to achieve

the maximum probability of successful operation, a different objective from that

used in the facility location model. The problem is generally categorized as a system

reliability optimization problem, including the classes of redundancy allocation and

reliability-redundancy allocation optimization problems. In redundancy allocation

optimization, one is given the option to allocate the appropriate levels of redundancy

to maximize reliability or minimize the cost of a system given the design constraints.

For example, if a component of reliability level at 0.9 is assigned in parallel to backup










another component of the same reliability level, then the overall reliability level is at 0.99,

because the probability that both fail is 0.01 assuming both components are independently

operational. The redundancv allocation problem can he formulated as a nonlinear integer

programming problem. In the reliability-redundancy allocation optimization problem, one

is given not only the option to choose different levels of redundancy, but also the reliability

levels of each component in a system. For example, in the previously discussed setting,

a redundancy level of 2 provides a reliability level of 0.99; one may choose a component

whose reliability level is 0.99 at a different cost. The problem can he formulated as

a nonlinear mixed integer programming problem. In our study, we show that system

reliability optimization is a particular type of monotonic optimization problem. Therefore,

it can he efficiently solved by the monotonic branch-reduce-bound algorithm as shown in

our computational study for a number of representative problem instances.

The contributions of our study encompass theoretical developments, computational

algorithms and practical applications. In particular, we make the following contributions:

* We develop several models to optimally choose facility locations and assign demands
to facilities in order to minimize the sum of fixed costs, the expected service, and the
penalty cost by taking into consideration that a facility may fail. These models are
suitable in the system design phase when no facility has been built yet.

* Several heuristics are proposed to effectively solve the complicated models presented
in our work along with the extensive computational tests.

* We present an approximation algorithm for a special case of the reliable facility
location problem where the failure probability is not facili'i- specific. We show
that the algorithm admits a worst-case bound of 2.674. This is the first such
approximation algorithm for the reliable facility location problem.

* We effectively develop and apply the branch-reduce-bound algorithm for system
reliability optimization. The algorithm can he easily adapted to solve various types of
problems arising from system reliability optimization and provides some benchmark
results for the existing examples in the literature.

* We develop two models to help decision makers to choose which facility to fortify
and to what extent it should be fortified when they face the issue of enhancing the










existing system. Such models are among the earliest fortification models in the
facility location literature. Several efficient algorithms are also provided.

The structure of our research is depicted in Figure 1-1. In C'!s Ilter 2, we present

several models for the uncapacitated reliable facility location problem in which some

facilities are subject to failure from time to time. These models are the foundation of

our research. Besides the general scenario-based model, they include the case in which

each facility has a site-specific failure probability, and the case in which each facility has

multi-level failure probabilities. The properties and different formulations of the models

are thoroughly discussed. Several heuristics are presented along with the computational

results.

Models Algorithms


Uncapacitated Reliable Facility Location Model, ~ISample Average Approximation Heuristic
Chapter 2


Single Failure Probability, Greedy Add Heuristic
Multi-level Failure Probabilities,
Chapter 2 Genetic Algorithm


Uniform Failure Probability, Chapter 3 I 2.674-Approximation Algorithm


Discrete/Continuous Facility Fortification Model,
Chapter 5
Monotonic Branch-Reduce-Bound
Algorithm

System Reliability Model, Chapter 4 -


Figure 1-1. Research structure



In C'!s Ilter 3, we present a tighter approximation algorithm with a worst-case bound

of 2.674 for a special case of the uncapacitated reliable facility location problem, where all

failure probabilities are identical.

In C'!s Ilter 4, we present a monotonic branch-reduce-bound algorithm for a

special case of the nonlinear mixed/pure integer programming problem where both










the objective function and constraints possess monotonicity. Its convergence analysis and

acceleration techniques are also discussed. The algorithm then is applied to solve system

reliability optimization problems in complex systems, including the redundancy allocation

optimization problem and the reliability-redundancy allocation optimization problem. The

efficiency of the algorithm is demonstrated via the computational results.

Based on the models in C'!s Ilter 2, we develop two models that are used to fortify

the reliability of the existing facilities. The properties and solution methodologies of the

models are discussed. In particular, the monotonic branch-reduce-bound algorithm

presented in Chapter 4 is used to solve one of these models. The efficiency of the

algorithm is demonstrated through the computational results.

This dissertation is concluded in OsI Ilpter 6 with a discussion on future research

directions.










CHAPTER 2
RELIABLE FACILITY LOCATION PROBLEM: MODELS AND HEURISTICS

2.1 Introduction

Facility location models have been extensively studied in the literature. Different

kinds of facilities have been modeled, such as routers or servers in a communication

network, warehouses or distribution centers in a supply chain, hospitals or airports in a

public service system. Facility location models typically try to determine where to locate

the facilities among a set of candidate sites, and how to assign 'customers' to the facilities,

so that the total cost can he minimized or the total profit can he maximized ([34], [8], and

[45]). Most models in the literature have treated facilities as if they would never fail; in

other words, they were completely reliable. In this chapter, we relax this assumption to

model a more realistic case.

The reliability issue we consider is under the framework of the so-called uncapacitated

facility location problem (ITFLP). In ITFLP, we are given a set of demand points, a set of

candidate sites, the cost of opening a facility at each location, and the cost of connecting

each demand point to any facility. The objective is to open a set of facilities from the

candidate sites and assign each demand point to an open facility so as to minimize the

total facility opening and connection costs.

ITFLP and its generalizations are NP-hard, i.e., unless P = NVP they do not admit

polynomial-time algorithms to find an optimal solution. There is a vast literature on these

NP-hard facility location problems and many solution approaches have been developed in

the last four decades, including integer programming, meta-heuristics, and approximation

algorithms. One common assumption in this literature is that the input parameters of

the problems (costs, demands, facility capacities, etc.) are deterministic. However, such

assumptions may not he valid in many realistic situations since many input parameters in

the model are uncertain during the decision-making process.










The uncertainties can he generally classified into three categories: provider-side

uncertainty, receiver-side uncertainty, and in-hetween uncertainty. The provider-side

uncertainty may capture the randomness in facility capacity and the reliability of facilities,

etc.; the receiver-side uncertainty can he the randomness in demands; and the in-hetween

uncertainty may be represented by the random travel time, transportation cost, etc. 1\ost

stochastic facility location models focus on the receiver-side and in-hetween uncertainties

([47]). The common feature of the receiver-side and in-hetween uncertainties is that

the uncertainty does not change the topology of the provider-receiver network once the

facilities have been built. However, this is not the case if the built facilities are subject to

fail (provider-side uncertainty). If a facility fails, customers originally assigned to it have

to be reassigned to other (operational) facilities, and thus the connection cost changes

(usually increases).

We focus on the reliability issue of provider-side uncertainty in this chapter. The

uncertainty is modeled using two different approaches: 1) by a set of scenarios that

specify which subset of the facilities will become non-operational; or 2) by an individual

and independent failure probability inherent in each facility. Although each demand

point needs to be served by one operational facility only, it should be assigned to a

group of facilities that are ordered by levels: in the event of the lowest level facility

becoming non-operational, the demand can then he served hv the next level facility that is

operational; and so on. If all operational facilities are too far away from a demand point,

one may choose not to serve this demand point by p wiing a penalty cost. The objective is

thus to minimize the facility opening cost plus the expected connection and penalty costs.

This problem will be referred to as the uncapacitated reliable facility location problem

(ITRFLP).

In particular, two variants of ITRFLP are considered in this chapter in terms

of the characteristics of the failure probability at each facility. In the first one, we

assume that there is only one site-specific failure probability at each facility. We










call it the uncapacitated reliable facility location problem with a single-level failure

probability(URFLP-SFP). In the other variant, we assume that there are multiple levels

of failure probabilities that can he chosen at each facility. We call it the uncapacitated

reliable facility location problem with multi-level failure probabilities(URFLP-1\FP).

Both of them can he modeled hv a scenario-based stochastic programming approach and a

nonlinear integer programming approach.

ITRFLP is clearly NP-hard as it generalizes ITFLP. We propose several heuristics

to solve ITRFLP. They include the sample average approximation heuristic for the

scenario-based model, the greedy adding heuristics, the greedy adding and substitution

heuristics, and the genetic algorithm for the nonlinear integer programming model.

The rest of this chapter is organized as follows. In Section 2.2, we review the related

literature and provide some basic background for our models. The notation and acronyms

are introduced in Section 2.:3. In Section 2.4, a scenario-based model is proposed, which

is followed by the nonlinear integer model for ITRFLP-SPF in Section 2.5. Section 2.6

contains the nonlinear integer model for ITRFLP-SPF. The three heuristics are presented

in Section 2.7. In Section 2.8, we conduct computational studies on the performance of the

heuristics. In Section 2.9, we conclude the chapter by -II---- -r; h.-; several future research

directions.

2.2 Literature Review

The importance of uncertainty in decision making has promoted a number of

researchers to address stochastic facility location models (e.g., [:38, 47]). However, as

we pointed out in the Introduction, a 1 in ~dl~y of the current literature mainly deals with

the receiver-side and/or in-hetween uncertainties. This includes [63], [10], [9], [7] and [42]

among others.

The following two papers, [48] and [5], are closely related to this chapter. In [48],

the authors assume that some facilities are perfectly reliable while others are subject to

failure with the same probability. On the contrary, we assume that the failure probability










is site-specific, a much more general case. They formulate their problem as a linear integer

program and propose a Lagrangian relaxation solution method. Another related model

is proposed in [5], which is based on the p-niedian problem rather than the framework of

ITFLP.

There is also a small strand of literature devoted to addressing the fortification of

reliability for existing facilities, which includes [4:3], [44], and [49]. These models typically

focus on the interdiction-fortification framework hased upon the p-niedian facility location

problem. They are generally formulated as bilevel progranining models. Their main focus

is to identify the existing critical facilities to protect under the events of disruption.

In our model, the failure probabilities are site-specific, which significantly complicates

the problem when formulating it as a niathentatical program. The model is further

extended to ITRFLP-1\FP, where each facility is allowed to have multiple levels of the

failure probabilities. We propose two different modeling approaches: a scenario-based

stochastic progranining approach and a nonlinear integer progranining approach. The

scenario-based model is attractive due to its structural simplicity and its ability to

model dependence among random parameters. But the model becomes computationally

expensive as the number of scenarios increases. If the number of scenarios is too large,

the nonlinear integer progranining approach provides an alternative way to tackle the

problem.

The sample average approximation method is widely used for solving complicated

stochastic discrete optimization problems, e.g., [24], [42], and [61]. The basic idea of this

method is to use a sample average function to estimate the expected value function. Thus

the original problem is transformed to the one that can he efficiently solved.

Other types of the coninonly used heuristics in optimization are the local search and

iterative intprovenient algorithms ([1:3], [16], [4] and [:3]). These heuristics start with an

empty set and repeatedly consider adding a potential facility into the solution. Or they

start with an non-enipty set and repeatedly consider deleting or substituting a facility in









the original set. Typically such algorithms are simple and easy to be incorporated in more

sophisticated algorithms, such as Tabu search in [60].

We also apply a genetic algorithm (GA) based heuristic to solve the models we

develop. GAs imitate the natural selection in biological evolution. As solution techniques,

they maintain a large number of solutions, called the population, and allow each member

of the population (called a chromosome) to evolve iteratively into good ones. Some good

descriptions of GAs are provided in [11, 14, 39].

GAs have been used to solve many combinatorial optimization problems with success,

including various facility location problems, e.g. [22] and [2]. In this chapter, we design a

specialized GA for the models in which we are interested. A computational study on all

three heuristics is also provided.

2.3 Notations and Acronyms

We first introduce some common notations that will be used throughout the chapter.

Let D denote the set of clients or demand points and F denote the set of facilities. |F| is

the number of the facilities. Let fi be the facility cost to open facility i, dj be the demand

of client j, and cij be the service cost if j is serviced by facility i. The service costs, cij,

are assumed to form a metric, i.e., they satisfy triangle inequalities. For each client j E D,

if it is not served by any open and operational facility, then a penalty cost rj will be

incurred.

The acronyms listed in Table 2-1 are frequently used in this chapter.

Table 2-1. Acronyms
Acronym Meaning
UJFLP U~ncapacitated facility location problem
URFLP-SFP Uncapacitated reliable facility location problem with a single-level
failure probability
UJRFLP-MFP Uncapacitated reliable facility location problem with multi-level
failure probabilities
GA Genetic algorithm
SAA-H Sample average approximation heuristic
GAD-H Greedy adding heuristic
GADS-H Greedy adding and substitution heuristic









2.4 Uncapacitated Reliable Facility Location Problem: a Scenario-Based
Model

We first discuss a scenario-based approach to model URFLP. Given a finite set of

scenarios, where each scenario specifies the set of operational facilities, we can formulate

UJRFLP as a two-stage stochastic program with recourse. The first stage decision is to

determine which facilities to open before knowing which facilities will be operational.

When the uncertainty is resolved, the clients (demand points) will be assigned to the

operational facilities. These are the second stage decisions. In this model, we are not

allowed to build new facilities in the second stage. In other words, no remedy can be made

to the first stage decision, except for optimally assigning the clients to the operational

facilities. The objective is to minimize the total expected cost which includes the first

stage cost and the expected second stage cost. The expected cost is the sum of the cost of

all scenarios times their specific probabilities.

Let S be the set of scenarios. For any A E S, let pA be the probability that scenario

A happens. Then URFLP can be formulated as the following two-stage stochastic

program.


minimize fg~yi + plA9A y) Subhjct to ye, { 0, 1}, (2-1)
iLEF AES
where gA 9/) = mi Cdjli djz (2-2)
j6D iLEF j6D
s.t. xs + zf = 1, Vj EDn (2-3)
i6EF
x y, i t F,i jeD (2 4)

x ~ 1 5 AiV F, jeD (2 5)

x ,~ zf c{, 1}. (2 -6)


In the above formulation, the binary variable yi indicates if facility i is opened in the first

stage. Parameter IA,i indicates if facility i is operational under scenario A, which is an

input regardless of th~e value of ys. Viariabhle x is th~e assignment variabhle which? indicates










whether client j is assigned to facility i in scenario A or not. Finally, the vrariable zf

indicates whether client j receives service at all or is subject to a penalty. The objective

in the formulation, i.e. 2-1, is to minimize the sum of the fixed cost and the expected

second stage cost. The objective of the second stage, i.e. 2-2, is to minimize the service

and penalty cost. Constraints (2-3) ensure that client j is either assigned to a facility or

subject to a penalty at each level k in Scenario A. Constraints (2-4) and (2-5) make sure

that no client is assigned to an unopen facility or a nonfunctional facility respectively.

It is straightforward to show that the formulation (2-1) is equivalent to the following

mathematical program.


(URFLP-SP)


iLEF ACS \jeD iEF j6 / J-;
s.t.i x + Z=1,V ,A ACS
iEF
x 5 elAi, i t F, je D A CS

yei, X ,3 zf 6 {, 1}.


One advantage of the scenario-based formulation is that it can easily capture the

dependence of the failure probabilities of different facilities by properly defining the

scenarios. If the number of scenarios is not too large, it is possible to solve URFLP-SP

efficiently and effectively.

However, when the failure probabilities are independent, the possible number of

scenarios can be extremely large. Therefore, the number of variables and constraints

in URFLP-SP is exponentially large accordingly, which makes it extremely difficult to

solve. Under this situation, we propose several alternative nonlinear integer programming

formulations and efficient solution algorithms. We discuss these alternative formulations in

the next two sections.









2.5 Uncapacitated Reliable Facility Location Problem with a Single-level
Failure Probability

2.5.1 Nonlinear Integer Programming Model

In this section, we consider URFLP-SFP, assuming that the failure probabilities of the

facilities are independent and each facility has only one failure probability. Let pi denote

the probability that facility i fails. Without loss of generality, we assume that 0 < pi < 1.

The 1!! ri ~ difficulty here is to compute the expected service cost for each client. In order

to overcome this difficulty, we extend Snyder and Daskin's formulation [48] to a more

general setting. Comparing to [48], URFLP-SFP can be interpreted slightly differently as
follows. Each client should be assigned to a set of facilities initially. The facilities assigned

to any client can be differentiated by the levels: in case a lower level facility fails, the next

level facility, if operational, will provide service instead.

Mathematically, we define two types of new binary variables xf zf to capture

different level of facilities for a client j. In particular, x@ = 1 if facility i is the k-th level

ba~ckup? facility of clien~t j anld otherTwise, xi = 0. zy = 1 if j h~as (k )-th backup facility,

but has no k-th backup facility so that j incurs a penalty cost at level k.

Given? the variables xfy, zf on~e canl compute th~e expected total service cost a~s

follows. Consider a client j and its expected service cost at its level-k facility. Client

j is served by its level-k facility only if all its assigned facilities at lower levels become

non-operational. On the other hand, for any facility 1, if it is on the lower level (i.e, less

th~anl k) for demanld node j, thenl P xt) = 1, otherwise CE xt = 0. It follows that for

client j, the probability that all its lower level facilities fail is lier p, If j is severed

by facility i, as j's level-k backup facility, then facility i has to be operational which occurs

with probability (1 pi). Therefore, the expected service cost of client j at level k is

CiE ic1JxyT(1 ps) nlEPI. Sim~ila~rly, we canl calculate th~e expected penaRlty cost
of client j at level k, which is ter pI iiz









The above discussion leads to a nonlinear integer programming formulation for

UJRFLP-SFP.



i6EF j6D k=1 iEF 16F
|F|+1

j6D k=1 16F

subject to x2 z = 1c Vj e D, k = 1..., |F| + 1 (2 8)
i6EF t=1
x$ y /, Vi e F% je D, k = 1,... | F | (2-9)

x@ :5 1, Vi le F (2(10)

xfyzf ,ye {0,1}.(2-11)

The decision variables xfy, zf are defined earlier. The indicator variable yi = 1 if facility i

is open in the first stage, otherwise yi = 0. The objective function (2-7) is the summation

of the facility cost, the expected service cost, and the expected penalty cost. Constraints

(2-8) ensure that client j is either assigned to a facility or subject to a penalty at each

level k. Constraints (2-9) make sure that no client is assigned to an unopen facility.

Constraints (2-10) prohibit a client from being assigned to a specific facility at more than

one level. Note that constraints (2-9) and (2-10) can be tightened as


x@2 < ye Vi e F, je D. (2 12)


2.5.2 Model Properties

In formulation (URFLP-SFP), we do not explicitly require that a closer open facility

be assigned as a lower level facility to a particular demand point. However, according to

the following proposition, it is true that the level assignments among the open facilities are

based on the relative distances between the demand point and the facilities regardless of

the failure probabilities.










Proposition 2.1. In r,.;, optimal solution to URFLP-SFP, for r,.;, client j, ifx~

x' = 1, then cap < c,4.

Proof. We prove the proposition by contradiction. Suppose cej > cej, we will show that by

I1l1!1! the assignment of a and v, the objective function will strictly decrease.

In particular, if we set x =l 1 and Zk = 1 With the values of other variables

unchanged, we can compute the new objective value. The difference between the new

objective value and the original one is






< 0,

where


16F
The last inequality holds because pl > 0 and it is assumed that p, < 1, p, < 1, and

cmj < cy. This is clearly a contradiction to the optimality of the original solution.

Therefore, cej < ce. O

An implication of Proposition 2.1 is that if the set of open facilities is determined,

then it is trivial to solve the level assignment problem for each client: assigning levels

according to the relative distances of different facilities to the client. If at some level the

distance is beyond the penalty cost, then no facility will be assigned at this level (and

higher ones) and the demand node simply takes the (cheaper) penalty.

We would like to point out the relationship between formulation (URFLP-SP) and

formulation (URFLP-SFP). Since these two formulations are just two wwsi~ of modeling

the same problem, they should have the same minimum cost as long as the inputs to the

two models are consistent. In formulation (URFLP-SFP), each facility i has independent

failure probability pi. This implies that there are 2 *I scenarios and the probability that









each of the scenarios occurs can be easily calculated. The corresponding values serve as

inputs to formulation (URFLP-SP). We also notice that, in formulation (URFLP-SP), it is

straightforward to obtain an optimal second stage solution for a given first stage solution.

In particular, at the second stage, every client will be assigned to and be served by an

open and operational facility that is closest to the client at the lowest possible level, if the

service cost is higher than the penalty cost, then it takes the penalty.

2.5.3 A Special Case: Uniform Failure Probabilities

Now we consider a special case of URFLP where all facilities have the same failure

probability, i.e., pi = p, Vi E F. This assumption simplifies formulation (URFLP-SFP)

considerably based on the following observation. Because pi = p, Vi E F, it is

straightiforward that Ipli pk_-1,Whch is independent of the values of xie.
16F
This property is implicitly used in a multi-objective formulation proposed in [48].

Based on the above observation, we are able to reduce formulation (URFLP-SFP) to

a linear integer program as follows.

( URFLP-IP)

|FI IFI+1
minimize feyI/ CCdirceyx\ (1 Cp-1k-djrjzf (2-13)
i6EF j6D k=1 iEF j6D k=1

subject to x z)=1I~ Vj eD, k=1,...,|IF|+1
i6EF t=1
|F|

xf~y,, zfy l E {, 1}



This special case can be solved to optimality by using commercial solvers such as

CPLEX. We use this case as a bench mark in the computational tests conducted in

Section 2.8.









2.6 Uncapacitated Reliable Facility Location Problem with Multi-level
Failure Probabilities

In this section, we extend URFLP-SFP to URFLP-MFP so that each facility has

multi-level failure probabilities. In this model, the decision makers can make some key

facilities more sustainable than others by investing more if necessary. In URFLP-MFP, we

model the failure probabilities as functions of the initial fixed investment. To do so, we

introduce t to devote different investment levels. In addition, fee is denoted as the fixed

cost at the facility i at the level t; yti, the decision binary variables for the facility i at the

level t. That is, yti = 1, if a level t investment is put at facility i; otherwise, ymi = 0. We

assume 0 < t < U.

Given different investment levels at facility i, the output of the failure probability at

facility i, ~Pf(yti), is determined by yes:


I~I(I1L) C~P(.fL.i)~lZ)


(214)


with foi = 0, P(0) = 1, and P(-) is a decreasing function. foi = 0 and P(0) = 1 imply that

if there is no investment at facility i, then it is completely nonfunctional.

After the failure probability at facility i is determined, URFLP-MFP is essentially no

different from URFLP-SFP. In the following formulation, pi in the objective function 2-7

of URFLP-SFP is replaced by ~Pf(yti).










( URFLP-M~FP-1)


|F|
minimize CC s 1Jf,+Cc 2) +i~ EE ge,,1_ ivs) Jre))
i6EF t j6D k=1 iEF 16F
|F|+1

j6D k=1 16F

sub~jct to x z~r = ~ 1 Vk=1 .,||+1
i6EF t=1

xi 5 yr, Vi E Fj jE D, k = 1..., |F|

yti < Vi =1,..., |F|

|F|
xi $3-~ 1 Vi EF,j ED


(2-16)

(2-17)

(2-18)


where constraints (2-18) ensure at most one investment level is allowed at each facility

and all the other constraints are similar to the ones in URFLP-SFP. Note that Proposition

2.1 still holds in this model.

The above formulation, URFLP-MFP-1, is a binary model. URFLP-MFP can also be

modeled as a regular integer model by reinterpreting the definition of yi as the investment

level at facility i. Thus, yi is not binary any more; 0 < yi < Ui, where Ui is the highest

level at which facility i can be possibly built. The corresponding failure probability

and the fixed cost at facility i are denoted by functions Pi(yi) and Fi(yi) respectively.

Pi(yi), iE F, are nonincreasing functions of yi and Pi(0) = 1, whereas Fi(yi), iE F,

are nondecreasing functions of yi and Fi(0) = 0. Then URFLP-MFP can be modeled as

follows.










( URFLP-M~FP-2)


minimize F ,(yi) +CC dy~c,,x@(1 y)) (IU
i6EF j6D k=1 iEF 16F
|F|+1

j6D k=1 16F

sub~jct to xrk z~ = j ,.,||+1
i6EF t=1
x -EImin~yi,1} Vi EF, j ED,k~= 1,...,|IF|


k=1



xi, zfE ( {, 1}, y,: integer,


(2-22)

(2-23)

(2-24)

(225)

(2-26)


where constraints (2-23) ensure no client is assigned to an unopen facility. That is, when

yI = 0, xi = 0, Vi, j, k = 1,..., |F|. Because of th~e binary con~straints (2-26) on? xi, th~e

right hand side of constraints (2-23) can be replaced by yi without affecting the feasible
domain.

2.7 Solution Methodologies

2.7.1 Sample Average Approximation Heuristic

The Sample Average Approximation (SAA) method is widely used for solving

complicated stochastic discrete optimization problems ([24], [42], and [61]). The basic idea

of this method is to randomly generate samples, then use a sample average function to

estimate the true expected value function. By doing so, the original problem is reduced

to a relatively small problem that can be repeatedly solved. Such an approach has been

used by various authors over the years. We apply the following procedures to solve model

UJRFLP-SP.










The SAA Heuristic


* Step 1: Randomly generate a sample of NV scenarios {A AN} and solve the
followingf SAA problem:


minimize feS 1J+C dyCicqx@ +C drlyzf~
iLEF s=1 j6D iLEF jED

s.t. xf +zyg=1, VjtD,s=1,...N
i6EF
xy I yilAs~i Vi E F, j e D, a = 1, .. NV
yi, xy zy E [0, 1]

Repeat this step M~ times. For each m = 1, 2, M~, let y7m and vm be the
corresponding optimal solution and its optimal objective value respectively. In
view of Proposition 2.1, the level assignment decision (the second stage decision)
can be solely determined by ym. Compute the true objective value im using the
formulation of URFLP-SFP for each ym, m = 1, 2, M~.

* Step 2: Among the M~ solutions obtained in the first step, output the one with the
minimum objective value, i.e., imm min im
m=1,...,M
Two remarks are in order. First, in a standard SAA approach ([42] and [61]), one

additional independent sample is needed to estimate the true expected value im. But in

our case, an analytical formula is ready for estimating the true expected value. Second,

the average of the vm" values, i.e. vM~ mM=1 Vm, does not provide a statistical lower

bound for the optimal value. This is different from the result in [42] where the uncertainty

only comes from demand-side. In the current model, different samples may lead to

different solution spaces of the problem, due to the provider-side uncertainty. Therefore

the average value vM iS no longer unbiased to the true expected value. Nonetheless, vM

may still serve as a good indication of the quality of the solution from the SAA approach,

as we will illustrate in the computational tests.

2.7.2 Greedy Methods

In this section, we propose two related heuristics to solve the general URFLP: Greedy

Adding Heuristic (GAD-H), and Greedy Adding and Substitution Heuristic (GADS-H).

These two heuristics based on greedy local search and iterative improvement algorithms.










In formulation (URFLP-SFP), Pro~position 2.1 ensures that the level assignments can he

easily derived for a given set of open facilities. Therefore, one can concentrate on selecting

a set of open facilities without worrying too much on the decisions of level assignment. Let

t'(T) denote the objective function value given by the set of open facilities, T. Let Tt be

the set of open facilities at step t, and # he the empty set.

The Greedy Adding Heuristic

* Step 1: Initially the set of open facilities is empty. Set t = 0 and T' = 0.

* Step 2: Ch....~--- a facility front the remaining candidates to open such that it can
reduce the total cost the most. Add this facility to the facility set. That is,

t= t+1,


T' = Tt-l U {.t)}.


* Step 3: Repeat Step 2 until the current solution cannot he improved further.

In general, as we can see front the computational tests later, the greedy adding

heuristic is able to find a high quality solution very efficiently. The complexity of this

heuristic is O(n4 log n), where n = |F|. Given T'-l, it takes O(n log n) to do the level

assignments for each node, mainly because it involves a sort process that is in complexity

of O(n log n). There is n such nodes, so it takes O(n2 log n) to evaluate the value of

t'(Tt-l). In the worst case, it takes n such evaluations to get the most cost effective

facility, .), at step t. The greedy adding process iterates at most n times, which leads to

the complexity of O(n4 log n) for GAD-H.

After the greedy adding heuristic, we perform the following greedy substitution

heuristic to further improve the solution: at each iteration, a substitute facility is chosen

to replace the existing open facility if doing so reduces the total cost the most. This

procedure is repeated until no substitute facility can he found to further reduce the

total cost. The substitution can he a null facility. Replacing an open facility with a null










facility means that we close the facility. After the substitution process, another greedy

adding procedure is performed to further improve the solution. The whole process (a

greedy adding procedure followed by a greedy substitution procedure then followed by

another greedy adding procedure) is called the greedy adding and substitution heuristic

(GADS-H).

2.7.3 Genetic Algorithm Based Heuristic

Genetic algorithm (GA) hased heuristics have been widely used to solve combinatorial

optimization problems. A GA imitates the mechanism of natural selection and natural

genetics. Generally, a GA starts with an initial set of solutions called a population. Each

member of the population is called a chromosome, representing an encoded version of a

solution to the optimization problem at hand. The goal of an encoding is to translate a

solution into a string of genes that make up the chromosome. There is a fitness function

that evaluates the quality of a chromosome at each iteration, called a generation. The

generation evolves through several operators: crossover, reproduction, immigration, and

mubstion. The crossover operator is a process to produce one or more offspring from the

current generation. Reproduction is simply a process that copies the best solutions from

the previous generation to the next. The immigration operator is to randomly create

certain new chromosome in each generation. The mutation operator is to randomly change

the genes in a chromosome to introduce randomness in each generation. The population

size through each generation is kept constant. After several generations, the best solutions

converge to an optimal or sub-optimal solution to the problem. Several comprehensive

treatments of GAs are given in [11, 14, 39].

In principle, a GA can he applied to any optimization problem. But there is

no generic GA since it requires ]rn lw: design decisions, such as the encoding of the

chromosome, the selection of parents, the method of the crossover operator. In this

section, we describe a GA that is suitable for URFLP.










The chromosome or the solution of the URFLP model is represented as a bit stream

with one position for every candidate location. We will use chromosome and solution

interchangeably. A "1" in position k is interpreted as that candidate site k is located

to open, while a "O" indicates that it is not. Since it is optimal to assign demand

nodes to facilities based on the distance between the demand node and the facility as

indicated in Proposition 2.1, we do not need an explicit encoding of the demand-to-facility

assignments. For model URFLP-MFP, an additional element is encoded to represent the

level at which we invest in an open facility. Table 2-2 shows the encoding for a system

with 10 candidate sites for model URFLP-MFP, for example, with open facilities at nodes

3, 4 and 8 at investment levels 2, 1 and 3 respectively.

Table 2-2. Sample chromosome for model URFLP-MFP
Candidate site 1 2 3 4 5 6i 7 8 9 10
Open? 0 0 1 1 0 0 0 1 0 0
Investment level 0 0 2 1 0 0 0 3 0 0


ClsInin..~ -nin 4~!~ are evaluated based on the value of objective function. A chromosome

with a smaller objective value is fitter than one with a larger objective value. The

followingf parameters are emploi-- II in our description of heuristic GA-H.

Table 2-3. GA-H parameters
Parameter Notation
Population size NVp
Maximum number of generations Noe
Maximum number of generations without improvement 1NMr
Number of reproduction NiR
Number of immigration NI
Mutation probability P


The initial population is randomly generated. For each solution of the population,

sites are selected randomly and the solution is checked against all other solutions in the

emerging population to ensure that the solution is unique. If it is, the solution is added to

the emerging population, if it is not, the solution is rejected and a new random solution is

generated. This process stops until NVp distinct solutions are populated.










Once we have a generation of solutions or chromosomes, we employ several operators

to create the next generation whose initial population size is zero. Reproduction carries

forward the best NVR solutions from the current generation to the next one. Naturally,

NVR < 1VP. Immigration (Teates NI solutions randomly, such that each new solution is

different from any of the solutions already in the emerging population. This immigration

process is identical to the process used for generating the initial solutions.

The main operator is crossover. Two solutions are selected at random from the

population at the current generation, with a hias toward the better solutions. The

probability of selecting the jt^ hest solution is given by z" /2.Ji~ Note rNp is the

number of solutions in the population. The denominator, NVp(N~p + 1)/2, is the sum from

1 to NVp. The numerator, NVp + 1 .), is the reverse order of j's fitness among all NVp

solutions. These values are listed in Table 2-4 as the weight in probability evaluation. As

we can see that the better a solution is, the higher weight it is assigned.

Table 2-4. Selection probabilities for a population with NVp solutions
Solution Rank Weight in Probability Evaluation Probability
1 NV
P ~Np(Np+1)/2
2 N ~ 1 >,-l
P ~Np(Np+1)/2


.1 P Np(Np+1)/2


Np 1 Np(Np+1)/2
Total N~p(N~p + 1)/2 1


After two non-identical parents are selected, a one-point crossover position in the

list of genes (candidate sites) is randomly selected. A child solution is constructed using

the genes (candidate sites) to the left of the crossover position from parent 1 and to the

right of the crossover position from parent 2. This process is depicted in Figure 2-1 for two

solutions with the one-point crossover position at 3. Encoding of the child solution in this

example is the following: values of positions 1 to 3 are from those in the same positions










as in parent 1; and value of positions 4 to 10 are from those in the same positions as in

parent 2.

Candidate site 1 2 3 4 5 6 7 8 9 10
Locate? 0 0O 1 1 0 0 0 O 1 0 0
Parent 1
Investment level 0 0 2 1 0 0 0 3 0 0


Candidate site 1 2 3 4 5 6 7 8 9 10
Crossover 4 Locate? 0 0 1 0 1 1 0 1 1 0 Child
Investment level 0 0 2 0 1 1 0 2 1 0


Candidate site 1 2 3 4 5 6 7 8 9 10
Locate? 1 0 0 0 1 1 0 1 1 0 Parent 2
Investment level 3 0 0 0 1 1 0 2 1 0



Figure 2-1. Example of crossover operation at position 3


After a child solution has been constructed in the manner outlined above, with

probability Pn/r, the solution is mutated. 1\utation is accomplished by randomly selecting

two candidate sites: one at which a facility opens and one at which a facility closes; then

swapping their states: from open ("1") to close ("O"), and from close ("O") to open ("1").

In the case of model URFLP-SFP, a randomly selected investment level is associated with

the newly open facility site.

If the child solution generated in this manner differs from all other solutions in the

emerging population, it is added to the population; if it does not, the entire process (of

parent selection, crossover, and mutation) is repeated. We continue adding solutions to the

population until the population contains NVp total solutions. In other words, the size of

each generation is maintained to be the same.

The whole process is repeated until one of the following termination criteria is met:

(1) the algorithm reaches NoG generations, or (2) it fails to improve the best-known

solution in 1Nz; generations.









2.8 Computational Results

In this section, we compare the computational performance of the four heuristics: the

sample average approximation heuristic (SAA-H), the greedy adding heuristic (GAD-H),

the greedy adding and substitution heuristic (GADS-H), and the genetic algorithm based

heuristic (GA-H). In order to evaluate the performance of these four heuristics, we first

apply them to solve URFLP-IP, where each facility has the identical failure probability.

The reason is that URFLP-IP admits a linear integer programming formulation that

can be solved to optimality by using commercial solvers such as CPLEX, so that we can

compare the heuristic results with the exact solutions, which helps to better evaluate the

performance of each heuristic.

The test dataset is generated as follows. Coordinates of the sites were drawn from

U[0, 1] x U[0, 1], demand of each site was drawn from U[0, 1000] and rounded to the

nearest integer, fixed facility costs were drawn from U[500, 1500] and rounded to the

nearest integer, and penalty costs were drawn from U[0, 15]. Further, the transportation

cost cay is set to be the Euclidean distance between points i and j. The number of sites

varies from 10 to 100. The dataset is available in Appendix B.

All the algorithms were coded in C++ and tested on a Dell Optiplex GX620

computer running the Windows XP operating system with a Pentium IV 3.6 GHz

processor and 1.0 GB RAM.

2.8.1 Sample Average Approximation Heuristic

We first test how the sample size (NV) affects (1) the quality of the solution, and (2)

the efficiency of the program, for a 50-node dataset with the uniform failure probabilities

varying from 0 to 1. Table 2-5 lists the objective values obtained from SAA-H when

M~ = 1 and the sample size varies from 10 to 200.

It is clear from Table 2-5 that the solution quality can be improved by increasing the

sample size. The ratios of the objective value obtained from SAA-H with sample sizes 10,

50, 100, 150, 200, and 250 to the optimal value are plotted in Figure 2-2.










Table 2-5. Objective values front SAA-H for the 50-node dataset
Failure Probability N10 N20 N:30 N40 N50
0 7197.27 7197.27 7197.27 7197.27 7197.27
0.1 7956.03 7956.03 776:3.80 776:3.80 776:3.80
0.2 9429.49 8770.31 8425.99 8425.99 8425.99
0.3 9908.52 10059.90 9669.49 10201.50 10095.20
0.4 11546.90 1175:3.50 11984.00 1:3887.60 119:37.60
0.5 18727.50 15128.1 1:3120.60 15052.80 1:3546.40
0.6; 27429.00 18161.70 17946.40 17946.40 17946.40
0.7 :32965.30 :331:39.20 27:374.80 2:3659.30 2:3690.40
0.8 62876.90 46:387.50 :3574:3.60 :3574:3.60 :3574:3.60
0.9 84406.10 69255.40 54722.30 54728.10 55647.40
1.0 128009.00 128009.00 128009.00 128009.00 128009.00
Failure Probability N100 N150 N200 N250 Exact
0 7197.27 7197.27 7197.27 7197.27 7197.27
0.1 776:3.80 776:3.80 776:3.80 776:3.80 776:3.80
0.2 8425.99 8425.99 8425.99 8425.99 8425.99
0.3 9:387.85 9:387.85 9275.99 9275.99 9275.99
0.4 10848.20 10599.90 10566.30 10762.00 1025:3.90
0.5 1:3157. 20 1:3041.50 1:3099.70 121:34.20 1160:3.00
0.6; 17762.80 15609.50 14559.10 14559.10 1:3416.80
0.7 20740.30 20671.60 19070.90 1'71;' .00 16157.20
0.8 29218.10 267:31.10 24066.80 2:3252.20 21500.70
0.9 46546.80 45061.00 42555.50 :38:316.90 :35987.70
1.0 128009.00 128009.00 128009.00 128009.00 128009.00


Figure 2-2 shows that SAA-H obtains fairly good solutions with a small sample size

when the failure probability is low (p < 0.4). In contrast, a large sample size is needed

to achieve such quality of solutions for the cases with a higher failure probability. The

following could be a possible explanation. Note that a sample is a collection of different

scenarios. When the failure probability is low, the 1 in U. G~y of the facilities are candidate

sites for opening in each scenario, so the sets of candidate facilities are similar in different

scenarios. Thus, any individual sample can capture the characteristics of the system pretty

well, and the corresponding solution obtained front SAA-H is close to optimal. For the

extreme case where p = 0, all facilities are available to open in each scenario, so the sets

of available facilities are the same in each scenario and the SAA-H can produce the exact

solution in this case. In the case of the higher failure probability, the candidate sites for














2.8 1
-e- N50
2.6
-a- N100
o2.4 +N150
S2.2 a- N200
S _-e- N250
.2


1.6

1.4

1.2-


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1





Figure 2-2. Objective ratio across different sample sizes for the 50-node dataset

300
-* N10

250 -- 2
-a N30
N40
op 200
cg -c-Exact




O 100


50-



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1





Figure 2-:3. CPU time across different sample sizes for the 50-node dataset



opening in each scenario is relatively small and a scenario can he quite different from

another one. Thus an increased size of sample can help to capture the characteristics of

the system uncertainty.


Figure 2-3 depicts the computation time from the run with sample sizes 10, 20,


:30, and 40. It shows that the case with NV = 10 is the only one that requires slightly

less time than the exact algorithm, whereas others require more time as the sample size










increases. Another interesting pattern in Figure 2-3 is the tail effect of the CPU time

in terms of the failure probability. SAA-H spends more time to obtain a solution when

the failure probability is around 0.5. One possible way to explain this phenomena is the

following: when the failure probability is around 0.5, the constraints .r; I ylusi among

different samples are quite different. As a result, the problem size increases, so does the

computational time.

Next, we examine the effect of the replication number (Af) on the solution quality by

fixing NV = 30. Table 2-6 provides the objective values obtained when At = 5, 10, 15, 20.

From the objective values obtained in different replication numbers, we can see that the

increase of the replication number has not affected the solution quality too much. The gap

in this table is defined as "'" x 10aI' The negative numbers in the y, Ip column

reveal that v" is not ak- -l-s a lower bound for ~i,2i. However, it is a good indication of

the quality of the solution from SAA-H. In this particular case, if the gap is within +1(1' .

the obtained objective value is close to the optimal value.

In general, SAA-H is capable to produce a fairly good solution with a large sample

size for the uniform case. But it also requires a tremendous amount of time to do so and

may run out of memory due to the increase of problem size. We defer presenting the

computational results of ITRFLP-SFP to Section 2.8.4.

2.8.2 Greedy Methods: GAD-H and GADS-H

In this section, we report the computational results of GAD-H and GADS-H on the

ITRFLP-IP model.

Table 2-7 lists the computational results of a 50-node dataset when the failure

probability varies from 0 to 1. The first column, P, is the failure probability at each

facility. The y, .p" column is defined as the percentage difference between the cost of the

solution obtained by GAD-H and the optimal cost.

As we can see from Table 2-7, GAD-H finds optimal or near-optimal solutions in most

cases in less than 0.05 seconds. Compared to the exact method using CPLEX, it takes










Table 2-6. Runs from SAA-H for the 50-node dataset
M = 5 M = 10
P ijmzn a gap min"" to gap Exact
0.0 7197.27 7197.27 0.00 7197.27 7197.27 0.00 7197.27
0.1 7763.80 7687.03 1.00 7763.80 7760.14 0.05 7763.80
0.2 8425.99 8315.65 1.33 8425.99 8436.50 -0.12 8425.99
0.3 9414.40 9054.28 6.79 9378.06 9112.31 3.32 9275.99
0.4 10872.60 9740.45 23.03 10479.80 9814.09 10.79 10253.90
0.5 11932.00 10457.00 25.47 11932 10497.60 13.66 11603.00
0.6; 17825.90 11377.60 57.73 17335.50 11475.90 55.33 13416.80
0.7 27157.40 12758.80 114.563 23227.40 12816.20 111.90 16157. 20
0.8 34912.50 14758.30 142.19 31284.20 14761.90 136.50 21500.70
0.9 54722.30 19703.50 177.73 5 !<.li 40 19428.60 181.66 35987. 70
1.0 128009.00 128009.00 0.00 128009.00 128009.00 0.00 128009.00
M~ = 15 M~ = 20
P Omin"" is gap Omin v20 gap Exact
0.0 7197.27 7197.27 0.00 7197.27 7197.27 0.00 7197.27
0.1 7763.80 7784.17 -0.26; 7763.80 7768.5 -0.06; 7763.80
0.2 8425.99 8484.24 -0.6;9 8425.99 8453.66 -0.33 8425.99
0.3 9378.06 9138.80 2.6;2 9275.99 9152.36 2.47 9275.99
0.4 10479.80 9826.75 6.65 10259.90 9842.41 6.48 10253.90
0.5 11932.00 10507.40 13.56 11932.00 10530.80 13.31 11603.00
0.6 17335.50 11523.50 50.44 17291.50 11563.00 49.92 13416.80
0.7 22894.20 12766.10 81.95 22894.20 12747. 70 79.59 16157. 20
0.8 31284.20 14736.80 112.29 31284.20 14746.00 112.15 21500.70
0.9 5 !<.2 40 19752.00 176.57 53343.90 19811.80 175.74 35987. 70
1.0 128009.00 128009.00 0.00 128009.00 128009.00 0.00 128009.00


much less time. The greedy adding algorithm seems to perform better when the facility

failure probability is high. It actually finds optimal solutions when the failure probability

exceeds 0.5. This is in contrast to the performance of the SAA-H, which works better

when the failure probability is low.

As we pointed out in Section 2.7.2, the solution quality of GAD-HI can be further

improved by GADS-HI. This is clearly demonstrated in the following computational results.

GADS-HI actually finds the optimal solutions for all instances in Table 2-7 and the CPU

times are comparable with those reported by GAD-HI. The results are summarized in

Table 2-8.











Greedy Adding Heuristic Exact Algorithm
P Objective Time (s) Objective Time (s) gap(~ .)
0.0 7551.02 0.00 7197.27 6.94 4.92
0.1 8053.11 0.00 7763.80 7.61 3.73
0.2 8637.46 0.00 8425.99 8.94 2.51
0.3 9309.50 0.00 9275.99 10.62 0.36;
0.4 10253.90 0.02 10253.90 10.38 0.00
0.5 11622.80 0.02 11603.00 10.86 0.17
0.6; 13416.80 0.02 13416.80 12.84 0.00
0.7 16157. 20 0.02 16157.20 13.47 0.00
0.8 21500.70 0.02 21500.70 14.08 0.00
0.9 35987. 70 0.05 35987.70 14.27 0.00
1.0 128009.00 0.00 128009.00 9.27 0.00


GADS-H Exact Algorithm Gap
P Objective Time (s) Objective Open Facilities Time (s) (.
0.0 7197.27 0.02 7197.27 15 31 40 41 48 6.94 0.00
0.1 7763.80 0.02 7763.80 15 22 31 40 41 48 7.61 0.00
0.2 8425.99 0.03 8425.99 15 22 31 40 41 48 8.94 0.00
0.3 9275.99 0.02 9275.99 15 22 31 40 41 48 10.62 0.00
0.4 10253.90 0.02 10253.90 15 22 31 40 41 42 48 10.38 0.00
0.5 11603.00 0.03 11603.00 15 22 31 35 40 41 42 48 10.86 0.00
0.6; 13416.80 0.06; 13416.80 2 12 15 22 31 35 40 41 43 48 12.84 0.00
0.7 16157. 20 0.09 16157. 20 2 12 14 15 19 22 31 35 40 41 43 13.47 0.00
45 48
0.8 21500.70 0.20 21500.70 2 12 14 15 19 20 21 22 26 31 35 14.08 0.00
36 40 41 43 45 48
0.9 35987. 70 0.48 35987. 70 1 2 10 11 12 14 15 17 19 20 21 22 14.27 0.00
23 24 2627 31 353640 4143 45
48 49
1.0 128009.00 0.00 128009.00 No open facility 9.27 0.00


It is interesting to compare the sets of open facilities in Table 2-8. One might

conclude that more facilities should be open as the facilities get more vulnerable, that is,

when the failure probability increases. Although this claim is usually valid, it is not ahr-7- 0

true. An extreme case is when the failure probability is 1 so that no facility should open.

One can also consider the following counter example where there is only one single facility

to open. If fl + dilrivi < divil, then this facility should be open. If pl increases to p'I such


Table 2-7. Fifty-node uniform case: greedy adding and


exact solution


Table 2-8. Fifty-node uniform


case: GADS-H and exact solution










that fl + dllrlp', > dulrl, then this facility should not be open. In this example, when the

failure probability increases, fewer facilities should be open.

2.8.3 Genetic Algorithm Based Heuristic

In all the GA-H tests, the values in Table 2-9 were used for the parameters in GA-H

described in Section 2.7.3.

Table 2-9. Values of the GA-H parameters
Parameter Value
Population size NVp 100
Maximum number of generations Noe 200
Maximum number of generations without improvement 1NM 100
Number of reproduction NsR 10
Number of immigration NI 10
Mutation probability PM 0.1


Table 2-10 lists the computational results of the 50-node dataset when the failure

probability varies from 0 to 1. The first column, P, is the failure probability at each

facility. Because the GA heuristic is a probabilistic method, two trials are performed. We

report the minimum objective it obtained and the average CPU time (in seconds) in the

second and third column respectively. The ,Ip column is defined as the percentage

difference between the cost of the solution obtained by GA and the optimal cost.

Table 2-10. Fifty-node uniform case: GA and exact solution
GA Heuristic (2 Trials) Exact Algorithm
P Min Objective Time (s) Objective Time (s) gap( ~)
0.0 7197.27 5.41 7197.27 6.94 0.00
0.1 7763.80 5.23 7763.80 7.61 110.00
0.2 8425.99 5.20 8425.99 8.94 0.00
0.3 9275.99 5.11 9275.99 10.62 0.00
0.4 10253.90 5.34 10253.90 10.38 0.00
0.5 11603.00 5.66 11603.00 10.86 0.00
0.6; 13416.80 5.52 13416.80 12.84 0.00
0.7 16157.20 5.75 16157. 20 13.47 0.00
0.8 21500.70 7.95 21500.70 14.08 0.00
0.9 35987.70 9.83 35987. 70 14.27 0.00
1.0 128009.00 3.25 128009.00 9.27 0.00










As we can see, the GA heuristic performs quite well. It can produce the exact

solution in 2 trials. While there is no guarantee that the GA solutions are optimal, it

can easily find an optimal or near optimal one by running multiple times then taking the

best solution it finds. Considering the fast speed for each run, a nmulti-trial GA is quite

attractive.

Figure 2-4 depicts the evolution of the nxininiun objective value, and the average

objective value in each generation when p = 0.5. The algorithm terminates at generation

176 after it finds the optimal solution at generation 76. In fact, in this example the GA

quickly converges to a close optimal solution after just 20 generations as shown in Figure

2-4.


50000

45000

40000

35000

30000

25000

20000

15000

10000


1 21 41 61 81 101 121 141 161
Generation


Figure 2-4. Evolution


of the solutions from GA


2.8.4 Applying Heuristics to Solve URFLP-SFP

Dropping the uniformity assumption of facility failure probabilities introduces more

challenges to solve ITRFLP exactly, due to the nonlinearity in formulation (ITRFLP-SFP).










The coninercial solvers, such as CPLEX, lack such ability to solve nonlinear integer

progranining problems. In addition, other generic global optimization solvers seems

not able to efficiently handle model ITRFLP-SFP. For example, BARON has difficulty

in solving ITRFLP-SFP with 10 demand nodes and 10 candidate facilities. The 1! in Un

difficulties come front the binary constraints on the decision variables and the large

number of nonlinear terms in model ITRFLP-SFP. In this section, we apply heuristics:

SAA-H, GAD-H, GADS-H, and GA to solve such model and compare their performance.

For some small size problems, we also provide globally optimal solutions front a simple

enumeration method.

Several datasets are derived front the 100-node dataset in Appendix B: for example,

dataset #10 is the first 10 lines from Table B-1 of Appendix B; it has 10 demand nodes

and facility sites. The other datasets are derived in the same way. Table 2-11 lists the

objective values obtained front SAA-H with different sample sizes. The colunin "Best

Obj .I :-lists the nxininiun objective value among the different sample sizes. Time is

measured in seconds with results front the sample size of 100. "-" in Table 2-11 means

that the program was out of nienory due to the surge in the problem size. "-" in Table

2-12 means that the results of the enumeration method were not obtained due to the

exponentially increased computational time. Table 2-12 suninarizes the objective values of

the solutions obtained by GAD-H, GADS-H, GA and the enumeration method, and their

corresponding computational time. The results of GA are obtained through a single run.

Comparing the results front heuristics with the globally optimal solutions in small

data sets (front #10, to #30), we can see that GADS-H and GA find optimal solutions,

whereas SAA-H and GAD-H find optimal or near-optinmal solutions. To evaluate the

quality of the solutions found by these heuristics in all datasets, we plot the objective

values in Figure 2-5. The values front SAA-H are the best ones available in Table 2-11

for each instance. Figure 2-5 shows that GADS-H and GA can find the best solutions

in all datasets, whereas SAA-H and GAD-H can find either the best known solutions or










Table 2-11. Objective values obtained front SAA-H on ITRFLP-SFP with different sample
s1Zes
Dateset I I Best Time (s)
# N50 N100 N150 N200 Objective (N100)
10 5850.47 5576.00 5128.24 5128.24 5128.24 0.4:3
15 6074.06 5:337.18 5:337.18 5:337. 18 5:337.18 2.11
20 8071.98 5761.79 5761.79 5761.79 5761.79 :3.09
25 6749.39 658:3.06 658:3.06 658:3.06 658:3.06 12.14
:30 7897.19 7622.22 7847.37 7847.37 7622.22 50.67
40 7474.92 7474.92 7474.92 7474.92 7474.92 54.20
50 8719.32 86341.28 8781.18 86341.28 86341.28 185.563
60 9:357.37 9:394.87 9:357.37 9:394.87 9:357.37 159.48
70 10:337.60 10:391.80 10:38:3.00 10:38:3.00 10:337.60 250.85
80 11054.30 11054.30 11054.30 11054.30 11054.30 :320.82
90 12405.50 12405.50 12448.70 -1 12405.50 659.57
100 1:3977.50 14028.10 -I -1 1:3977.50 2164.03


Dateset GAD-H GADS-H GA Enumeration
# Objective T (s) Objective T (s) Objective T (s) Objective T (s)
10 5128.04 0.00 5128.04 0.00 5128.04 1.62 5128.04 0.02
15 5:305.04 0.00 5:305.04 0.00 5:305.04 1.81 5:305.04 14.75
20 5761.79 0.00 5761.79 0.00 5761.79 1.81 5761.79 206.51
25 64:39.87 0.00 64:39.87 0.00 64:39.87 1.82 64:39.87 1ml.1
:30 7420.86 0.00 7:382.04 0.00 7:382.04 2.22 7:382.04 188276.35
40 7474.92 0.00 7474.92 0.02 7474.92 4.09
50 876:3.75 0.00 86341.28 0.00 86341.28 4.634
60 9:357.37 0.00 9:357.37 0.02 9:357.37 5.14
70 10:337.60 0.00 10:337.60 0.03 10:337.60 5.84
80 11054.30 0.00 11054.29 0.05 11054.29 5.563
90 1:30:30.90 0.00 12405.50 0.06 12405.50 10.75
100 1446:3.40 0.02 1:3820.87 0.09 1:3820.87 11.2:3


close to the best known ones. In terms of computational efficiency, SAA-H takes much

more time to achieve its solutions than all other three heuristics. Between GADS-H and

GA, GADS-H spends considerably less CPIT time than GA: GADS-H takes less than

0.1 seconds to get the best result in each instance. Overall, these results -II---- -r that

GADS-H is the best one among all four heuristics for model ITRFLP-SFP in terms of both

solution quality and computational time.


Table 2-12. Computational performance on
enumeration method


ITRFLP-SFP: GAD-H, GADS-H, GA and the












14000


12500 -H _-SAA-H

ia11000

9500
o
8000


6500

5000
10 15 20 25 30 40 50 60 70 80 90 100
Dataset #


Figure 2-5. Comparison of objective values from GAD-H, GADS-H, GA, and SAA-H


2.8.5 URFLP-MFP: GADS-H vs. GA

In this part, we apply GADS-H and GA to solve ITRFLP-1\FP, which is considerably

more difficult than ITRFLP-SFP. The dataset is similar to the one used in the previous

sections but with each facility having :3 levels of failure probability. The full dataset is

presented in Table B-2 of Appendix B. The first half of Table 2-1:3 reports the results

from GA, which has been run for 5 times with different random seeds. Eight datasets

have been generated for testing. The best open sites and their optimal levels are listed

in the second column. The best and worst results obtained in the 5 trials are listed in

the third and fourth columns respectively. The average time in seconds are reported in

the last column. The results of GADS-H are shown in the bottom half of Table 2-1:3. In

only one case (the 80-node problem) does the GA find a better solution than GADS-H. In

that case, the objective function value is 11782.85 compared to 11859.40, which represents

only a 0.10' improvement. But GA takes more time than GADS-H does to get this small

improvement. In all other cases, GADS-H finds the same solution as GA but with much

less time. Overall, GADS-H is more favorable than GA in solving model ITRFLP-1\FP.









Table 2-1:3. Computational results for ITRFLP-MFP using GADS-H and GA
GA, 5 Trials
Dataset Best Worst Average
# Best Sites (Levels) Result Result Time (s)
20 2(2) 7(2) 5214.55 5214.55 :3.1:3
:30 2(2) 20(1) 21(2) 6484.74 6484.74 :3.7:3
40 2(2) 20(1) :35(2) 7194.88 7194.88 4.52
50 2(2) 5(1) 20(1) :35(1) 8827.95 8827.95 5.71
6;0 2(2) 20(1) :35(2) 59(1) 9964.84 9964.84 7.20
70 2(2) 1:3(2) 20(1) :35(2) 108:37.65 108:37.65 9.06;
80 2(2) 20(1) :35(2) 59(1) 76(1) 79(1) 11782.85 11867.01 10.81
90 2(2) 20(1) :35(2) 76(1) 79(1) 88(1) 126321.95 12717.84 11.1:3
100 2(2) 1:3(2) 20(1) :35(2) 67(2) 76(1) 88(1) 1:371:3.42 1:3749.58 11.3:3
GADS-H
Sites (Levels) Result Time (s)
20 2(2) 7(2) 5214.55 0.09
:30 2(2) 20(1) 21(2) 6484.74 0.14
40 2(2) 20(1) :35(2) 7194.88 0.18
50 2(2) 5(1) 20(1) :35(1) 8827.95 0.27
6;0 2(2) 20(1) :35(2) 59(1) 9964.84 0.3:3
70 2(2) 1:3(2) 20(1) :35(2) 108:37.65 0.39
80 2(2) 1:3(2) 20(1) :35(2) 76(1) 11859.40 0.57
90 2(2) 20(1) :35(2) 76(1) 79(1) 88(1) 12621.95 1.19
100 2(2) 1:3(2) 20(1) :35(2) 67(2) 76(1) 88(1) 1:371:3.42 1.1:3

2.9 Conclusions

In this chapter, we have proposed several novel facility location models to deal with

the uncertainty in facilities. The issue arises when a facility fails is that the customers

originally assigned to it have to be reassigned to other facilities that are operational. The

impact of such uncertainty is explicitly modeled in all of our models in order to build a
reliable facility-custonler network. In particular, we use a popular scenario-based technique

to capture the uncertainty when the number of scenarios is relatively small. If the failure

probability at each facility is independent, we propose several nonlinear integer models,
ITRFLP-SFP and ITRFLP-MFP. These models greatly enrich the literature of facility

reliability.

Four heuristics, SAA-H, GAD-H, GADS-H and GA, have been proposed to solve

these problems. SAA-H is a specialized heuristic for the scenario-based model, whereas










GAD-H, GADS-H and GA can he used to solve ITRFLP-SFP and ITRFLP-1\FP. Our

computational studies show that (1) GADS-H is the best heuristics among all four in

terms of the solution quality and the efficiency, and (2) GA is also able to find the best

solution with a little more time than GADS-H.

There are several interesting research directions. We note that the 1! in r~ limitation of

the current models is the assumption that the facilities are uncapacitated. Although the

assumption itself is very coninon in the facility location models, it may be unrealistic in

practice. In the capacitated case, customer of failed facilities can he assigned to the next

level backup facilities only if they have sufficient capacity to satisfy the additional demand.

This restriction may make the capacitated model very complex. It becomes a valuable

topic of future investigation. In addition, some new measurements of the reliability

concept in the facility location problem setting are worth pursuing.










CHAPTER :3
ITNIFORM INCAPACITATED RELIABLE FACILITY LOCATION PROBLEM: A
2.674-APPROXIMATION ALGORITHM

3.1 Introduction

In this chapter, we consider a special case of the uncapacitated reliable facility

location problem in which failure probabilities are not facili'-- p;lecific. We call it the

uniform uncapacitated reliable facility location problem (ITIRFLP). Because the failure

probabilities are the same across all facilities, ITRFLP is reduced to a linear integer

progranining problem. ITIRFLP is related to the model considered in [48], where the

authors assume that some facilities are perfectly reliable while others are subject to failure

with the same probability. They formulate their nmulti-objective problem as a linear

integer program and propose a Lagrangian relaxation solution method. However, we

consider penalty cost, a factor that is missed in [48].

ITIRFLP is clearly NP-hard as it generalizes ITFLP. The focus of this chapter is

to propose and analyze an approximation algorithm with a constant worst-case bound

guarantee.

Designing approximation algorithms for the facility location problem and its

variations has recently received considerable attentions front the research coninunity.

However, to the best of our knowledge, this chapter presents the first approximation

algorithm for stochastic facility location problems with provider-side uncertainty.

The vast ill I iG~~y of approximation algorithms for the facility location problem

mainly deal with deterministic problems, e.g. [17, 21, :32, 46]. Until very recently,

approximation algorithms for ITFLP with stochastic demand have been proposed; see the

survey by Shnioys and Swanly [52]. Another related paper [6] proposes an approximation

algorithm for a facility location problem with stochastic demand and inventory. Our

approximation algorithm makes use of the ideas from several papers [17, 21, :32, 46]. In

particular, this chapter is closely related to [17], which presents a 2.41-approxiniation

algorithm for the so-called fault-tolerant facility location problem (FTFLP): every demand










point must be served by several facilities where the number is specified, and a weighted

linear combination is used to compute the connection costs. FTFLP has been motivated

by the reliability issue considered in this chapter, but the failure probabilities of facilities

are not explicitly modeled and penalty cost is not considered.

The remainder of this chapter is organized as follows. Several equivalent formulations

for UURFLP are proposed in Section 3.2, which lead us to develop a 2.674-approximation

algorithm in Section 3.3. The chapter is concluded in Section 3.4.

3.2 Formulations

The notation of this chapter follows that in C'!s Ilter 2. Recall the formulation of

UJRFLP-SFP in C'!s Ilter 2:

( URFLP-SFP)


minimize fSI/ y ~diricly (1-p)I pl
i6EF j6D k 1 iEF 16F

+ jeD +~1 nF16F; 1 dr~J

subjectto z Cz = 1, Vje D,k= 1...,|F|+1 (3-2)
i6EF t=1
x < i E F, jE D, k~ = 1, ..., | F| (3-3)

xi5 i E F,j jED (3-4)




Consider a special case of URFLP-SFP where all facilities have the same failure

probabilities, i.e., pi = p, Vi E F. This assumption simplifies formulation (URFLP-SFP)

considerably based on the following observation. Because pi = p, Vi E F, it is
stilfrai ht o w ardi th atl I p~l= P-1 Wh ich isind ep end ent of th e val ues, of xti
16F
This property is implicitly used in a multi-objective formulation proposed in [48].

Based on the above observation, we are able to reduce formulation URFLP-SFP to a

linear integer program as follows.










( URFLP-IP)


|F| |F|+1
minimize.2~ f y yey 1 p)pk-1 k-1d
i6EF j6D k=1 iEF j6D k=1

sub~jectto xz z )~=1, VjeDI, k =1,...,|F|+1
i6EF t=1


_36)


In the next section, we shall present an approximation algorithm for URFLP-IP.

We find that it is more convenient to deal with a slightly different formulation. In the

necw form~ula~tion, we introduce a. new set of decision? va~riables Of to replace zf. Define

Of = -, Note that Of are n~ot decision? va~riables when? k > |F|, Of = 1 for a~ll

k > |F| + 1. We prove that the following integer program is equivalent to formulation

(URFLP-IP), as stated in Theorem 3.1. We refer the reader to Section 3.5 for the proof.


i6EF j6D k=1 iEF j6D k=1

subject to x ~ + Of = 1, jeF 1,..., |F|
i6EF


P> (37)


xfy < yi., i; E Fj jED




Theorem 3.1. Formulation (8-6) and formulation (8-7) are equivalent.

3.3 Approximation Algorithms

In this section, we aim to propose a 2.674-approximation algforithm for the special

case where the failure probabilities are uniform. We call it a (Rf, R,, R,)-approximation










algforithm, if the algforithm produces a solution with cost no more than


RfF* + RcC* + R,P*,

where F*, C*, and P* are the optimal facility, transportation, and penalty cost, respectively.

We take advantage of several results for the fault-tolerant version of UFLP, where

every demand point j must be served by kg distinct facilities, a concept close to our level

assignment. In [17], Guha et al. propose a couple of approximation algorithms for the

fault-tolerant facility location problem using various rounding and greedy local-search

techniques.

The fault-tolerant facility location problem can be formulated as the following integer

program.

(FTFLP)


miniize eys dycyw x(3 8)
i6EF j6D k=1 iEF

subject to xfy~ > 1, Vj' e: D, < kg
i6EF




xfy, yi E {0, 1}.

Th~e notation? here follows that of model U!R~FLP-IP with som~e subtle differece~cs. wf~ is the

weighted factor at level k for demand node j. For demand j, the corresponding weights

are assumed to be w] > 2>..>w" f denotes that demand j is assigned to facility i

and facility i is the kth closest open facility to j.

One of the key results on FTFLP from Guha et al. [17] is summarized below.

Lemma 3.1. For r,:; vector (x, y) //rlifying the following inequalities (the dimension of

(x, y) should be clear from the inequalities)


xE>1 VjD-s
i6EF













x. > 0,
O< -<1,


and r,:;, as (0,1i], one can find an integer solution (x, y) ;aH itr;,i, the above inequalities

so that



i6EF j6D k=1 iEF j6D k=1 iEF

In Lemma 3.1, a~ is a parameter that we can choose to control the quality of

approximation. Indeed, the approximation ratios of the algorithms of Guha et al. are

functions of a~. One can then choose the best a~ to minimize the approximation ratio.

We are now ready to present our algorithm for UURFLP. We first solve a linear

programming relaxation of formulation (3-7).

|F| 00
minimize f eyi + dyey pk-rlZZI11 _p 39
i6EF j6D k=1 iEF j6D k=1

sub~jct to xz + f= 1, Vj D, k = 1..., |F| (310)
i6EF




Zk > 0, Ok" > 0. (3-12)


Assume that (x, y, 8) is an optimal solution to this linear program. Our algorithm rounds

the fractional solution (x, y, 8) to an integer solution (x, y, 8) that is feasible to formulation

(3-7).

The algorithm is based on a property of the optimal fractional solution (x, y, 8), which

is formalized in the following lemma. This lemma enables us to utilize known algorithms

and analysis for the fault-tolerant facility location problem.

Lemma 3.2. For each j e D, the following two statements are true.












(i) f therep exi;sts k such thatS 0 < Of < 1, then Of' = 0 anrd Of = 1 for k' < k and

k" > k.

Proof. The proof is intuitive and similar to the proof of theorem 3.1, which is thus omitted

here. O

We present our rounding procedure next. For each j E D, assume kj is the smallest

inlteger suchl that H," > 0.

The rounding procedure is carried out in two phases. The first phase rounds the

optimal fractional solution (x, y, 8) to another fractional solution (, y, 8), which is feasible

to a linear programming relaxation of an appropriately defined fault-tolerant facility

location problem. In the second phase, we use an algorithm for the fault-tolerant facility

location problem to round the fractional solution (i, y, 8) to an integer solution (x, y, 8),

which is feasible to formulation (3-7).

Phase I: Decomposition

For every j eD and i e F, let Of = Of and ifykj = xfy for all k > 1 except for k = ky.

C!. .. --- a parameter 6 E (0, 1] whose value will be fixed later.

Randomly generate a variable P that is uniformly distributed in [0, 6]. For each

.j EZanld i EF, if B*'> 8,Ithensetl

0" = my = k- 1, & f= 0


otherwise set





yi = max ~x, .
26D









Phase II: Solving Fault-Tolerant Problem

For ea~ch j eD and i e F, let = 1 and 2$ = 0 for all k: > mi, anld let = 0 for

all k < mj, where mj is defined in Phase I of the algorithm.

Use the algorithm(s) in [17] with a parameter as (0, 1] to round the solution

(x, y) to a feasible solution of a fault-tolerant facility location problem, where a set

of facilities is open such that each client j is served by at least mj distinct open

facilities.

For each i E F, set yi = 1 if facility i is open, and set yi = 0 otherwise.

For ea~ch je D,~ if i: is the k"" closest open facility to client j, then let ify = 1, where



Output the solution (x, y, 8).

This two-phase algorithm shall be referred to as Algorithm TP. It is obvious

that the solution (x, y, 8) is feasible to formulation (3-7). We now establish a worst

case approximation bound of our (randomized) algorithm, i.e., we shall show that the

(expected) total cost is no more than a constant factor times the optimal cost. We bound

the total penalty cost in Lemma 3.3, and bound the total facility and transportation costs

in Lemma 3.4.

Lemma 3.3. In Algorithm TP, the expected total I.. ..rl;ll cost is bounded from above by





Lemma 3.4. For it.;;, as (0,1i), the expected f~r ..1.:1/ cost plus the expected transportation

cost is no more than
In T ~In ~ L 3


where FLP and CLP are the total f 7.:.7//// cost and the total transportation cost, ,* i'' 1.: I;,

corresponding to the solution (x, y, 0) in the linear; r,.-ye~l,,I,,:nt:1 relaxation of formulation

(8-7).










We refer the reader to Section 3.5 for proofs of both lemmas. An immediate

consequence of these two lemmas is the following corollary.

Corollary 3.1. For Irl:;r nE (0, 1) and 8 t (0, 1), there is a ,n -~I ni

ap~prox~imation rll' .-:thm for the problem with ;,,..T. -rm probabilities.

By chloosinl au = 0.0497870517. = 0.2714781971 we derive thatl mraX( 1- 1-6 3
6 1-a' 6 1-a 6
;: I.; :1., which leads to the main result of this section.

Theorem 3.2. The uniform case of URFLP admits a 3 1I :I -ap~prox~imation rll'y..athm.

We use a technique called greedy improvement procedure (see [32] for details) to

further improve the approximation factor.

Phase III: Greedy Improvement

Apply Phase I and Phase II to an instance of the original problem where the facility

cost is scaled up by a given factor a > 1, and output a feasible solution.

Assume the costs of the current solution are (F, C, P). Pick a facility i with cost fi
such that the ratio

(C + P C Pi fi)/fi

is maximized, where Ci and Pi are the corresponding transportation cost and penalty

cost if facility i was added to the current solution. If the ratio is positive, open

facility i and repeat this step, otherwise, stop.

The greedy improvement procedure can improve the worst case bound of Algorithm

TP, as shown next. We omit the proof here as the analysis is very similar to those in [17]

and [32].

Lemma 3.5. For r: ; given (Rf, Rc,R,)-ap~prox~imation rll' y..thm for (8-7), there is a

(R, + In(A), 1 + 3II, 1 + ~i-)-approximn atio rly,. :thm,.

By choosing a~ = 0.42539606, 6 = 0.17430753, and a = 2.82899675, we obtain the
followingf theorem.

Theorem 3.3. URFLP with ,,;..[. -< I, failure probabilities admits a 2.674-ap~proximation










3.4 Conclusions

In this chapter, we employ various rounding and decomposition techniques to develop

a ;: I; :Il-approxiniation algorithm, then improve this to a 2.674-approxiniation algorithm

using greedy intprovenient procedures for the uniform uncapacitated reliable facility

location problem. To the best of our knowledge, these are the first type of approximation

algorithm for the facility location problem with uncertainties in the facility side. Whether

these bounds can he improved is an open problem for future research. Another interesting

topic is to develop an approximation algorithm for the uniform capacitated reliable facility

location problem.

3.5 Proofs

Proof. Theorem :3.1.

(1). Assume we have an optimal solution (.r, y, x) for problem (:36), then we can

construct a feasible solution (.r, y, 8) for problem (:37) with the same objective value.

L~et .r = .r anld y = y. For ealCh j D an ld k~ > 1, define~ Of = Ek=1 Z3.z Because for

ea~ch .), there is exactly on~e value of -t woullld br equlln to 1, w W~Can conclude Of t {0, 1}.

And it is straightforward that (.r, y, 8) is a feasible solution to problem (:37). Now we

check the objective value corresponding to (.r, y, 8). In fact, we only need to consider

oox no in~k,|F|+1|}

k= 1 k= 1 t= 1
|F|+1 xo

t=1 k=t
|F|+1

t=1

The first equality is due to the fact that Vt > |F| + 1,.) E D, a = 0.

(2). Now we prove the other direction; i.e., assume we have an optimal solution

(.r, y, 8) for problem (:37), then we can construct a feasible solution (.r, y, x) for problem

(:36) with the same objective value.









To that end, we first show it is without losing of generality to assume that for each

j E D, if Of = 1, then O~f' = 1. Otherwise, assume there exist do andc ko such that 0 a = 1
but Oko0+1 0. Then from the constraints, we know that Zk"o O for all i and there exists

an io such that x 0031 = .

Now we define a new solution (x, y, 8") for problem (3-7) as follows: y = y; for j / jo,

ify = xfy, Of = 8j for all i / io and k; for k / ko, ko +, 1~, i = Xfo aj = 8jo for all i / io;

x 0=1, 80 a 0, x 03 = 0 and ~foo+ = 1. It is easy to verify that (2:, y, ) isi a feasible
solution for problem (3-7). The difference between the objective value corresponding to

(x, y, 8) and the objective value corresponding to (x, y, 8) is


(clociodopko( ) jo crjopko-1~ _) jo(1,Ciolopko-~l- 1) 9 jo jop~ko P

which should be less than or equal to zero given that (x, y, 8) is an optimal solution to

problem (3-7). It follows that



If we define yet another solution (i, y, 8) such that it is the same as (x, y, 8) with the

following exceptions:

#oo,, = 0, 0 an = 1, & 003 1 = 0, 0i 00+ = 1.

Again, this solution it feasible. Also, the difference between the objective value corresponding

to (x, y, 8) and the objective value corresponding to (i, y, 8) is


djo ciojopko( _) -jo jopko (1 p) > 0,

which implies that (i, y, 8) is an optimal solution for problem (3-7) and it satisfies the

condition that if BkoU -1 then ~oU+1 i

Therefore, weit can assumet that for each je D if Of = 1, then Ofl = 1. Nowv wei ar~e

ready to find a feasible solution for problem (3-6). Let x = x, y = y. And for each j, if
Ok 1 and Ok-1 = 0, then let zk 1 and let zk ork ; f0 1, then let z 1

and zf' = for k' '> 2. Then it is clear from the definition that Of" = Ek-1z) Then we









know that (.r, y, x) is feasible for problem (:36). And front the first part of the proof, we

know the objective value corresponding to (.r, y, x) is the same as the one corresponding to

(.r, y, 8). This completes the proof. O

Proof. Lenina :3.3.
We bound the expected penalty cost for each client j e D, which depends on the

value of /9 that is randomly generated. We consider two cases.

Case 1. H0 > 79. Thl~en client .) is assigned to exactly ni;j = kj 1 facilities.

Therefore, a penalty cost will incur if all of these (ku 1) facilities failed, which happens

with probability pkj-1. Therefore, the expected total penalty cost for client .) is djrjpk,-1

Notice that Ok 1 for a~ll '> kJ + 1. Thenl



k=kj k=|F|+1




Case 2. H < 79. In this case, client .) isi assigned to exactly Tnj = kg facilities. A

penalty cost will incur if all of these kgi facilities failed, which happens with probability pk,

Therefore, the expected total penalty cost for client .) is djrjpkj which is equal to


dyyk-1 _k-1 _


If H0 > 6, then C~ase 1 h happens with probability 1. Thus the expected penalty cost of

client .) is





On the other hand, if H < 6i, then Case 1 happens with probability 4E and thuls

Case 2 happens with probability 1 4. Therefore, the expected penalty cost of client .) is










bounded by


:1 k-1
~=~j_

k=ky+1


-1
k=ky+1


1


kg
-' dyrypky-1 (

dyyky1 _


p/


This completes the proof.

Proof. Lemma 3.4.

The costs of interest depend on the value of the randomly generated P. Recall the


definition? of (i, y). For ecnch j D an id i e F, if


xfy, if k / mj; If mj k 1 then


.The latter is true only if 6 < ij. Therefore, in


23 =3 0 x if my


k
ky, then z2


both cases,


x < xx .


(3-13)


By the constraint of the linear programming relaxation of (3-7), we know that, for


each i E F and je D,


|F |


Therefore ,


-


j~~~o ~ "" ~


Z -
k=1".


(314)


To show that


i6EF
we consider the following two cases.


1 j Dk m


xty. From the constraint


*mj = k, 1:i In this case, Vj E D, k < mj, Of" = 0 and ify


3-1 &


xf = 1.i
i6EF


) :










*mj kj: when k < kj, from the proof of the first bullet, we know that Ci ify = 1.

when my, & f = 1. The last equality holds because of the
iEF 3
constraint 3-10, i.e. xr ~ 0i = 1.
i6EF
Thus & and y satisfy the following constraints:



iLEF



k=1> o


0<^ <1


Therefore, if we apply the algorithm in [17] with a parameter a~ to round the solution

(i, y), then by Lemma 3.1, we can construct a solution (x, y) such that the expected total

facility cost and total transportation cost is bounded above by

In I 0 m

iLEF j6D k=1 iEF
1 In 3
1-P 1-a~ 1-a
i6EF j6D k=1 iLEF



i6EF j6D k=1 iEF

The first inequality holds because of inequalities 3-13 and 3-14, the second one holds

because of mj < kyj.

Finally notice that was uniformly distributed in (0, 5), thus


1 11 1 1
E [1 ] =1 1p dp = In -


This completes the proof of the Lemma.









CHAPTER 4
SYSTEM RELIABILITY OPTIMIZATION AND MONOTONIC OPTIMIZATION

4.1 Introduction

The performance reliability of a system is of utmost importance in many industrial

and military systems. System reliability is a measure of how well a system meets its design

objective, and is usually expressed in terms of the reliability (a probability of successful

operations) of the subsystems or components. For example, a series system works if and

only if every component works. Such a system fails whenever any component fails. The

reliability of the series system in Figure 4-1 is



i= 1

where R, is the system reliability and ri is the reliability of component i, which is the

probability that component i successfully operates during the intended period of time. In

this chapter, we assume all components operate independently.


~- 12-- i ---- n-


Figure 4-1. Series system


System reliability can be improved in various owsi~, such as physical enhancement of

component reliability, provision of redundant components in parallel, and allocation of

interchangeable components.

Unlike in a series system, in a parallel system, not all components are necessary for

the system to work successfully. Actually, only one component in such system needs to

work properly in order for the whole system to work properly. Including a components

when only one is essential is called redundancy. The other n 1 components are included

to increase the probability that there is at least one working component. Redundancy is a

widely used technique in engineering to enhance system reliability.










For instance, in the above series system, each component i can be enhanced to a

-Ish-i--rh II. i with xi identical components in parallel. In this chapter, we use subsystem

and stage interchangeably. The reliability of such subsystem i is denoted as Ri. And

because only one component is required to make the system perform properly and all

components are independent, Ri = 1 (1 ri)"i. Therefore the improved system reliability

R, is the following.





which is increasing in each xi. The improved system is a parallel-series one as shown in

Figure 4-2.


Stage 1












Figure 4-2. Parallel-series system


Stage i


The system reliability can be maximized by choosing the right combination of xi

under certain resource constraints, denoted by gj(-), j = 1,..., m in this chapter. This

leads to a redundancy allocation optimization problem (RAOP) or a nonlinear integer

programming in general.


(RA OP): max R, = f(xl,..., x,)

subject to gyxx..,,
0 < le < Xi
xi: integfer, Vi=1..., ,


(4-1)

(4-2)

(4-3

(4-4)


Stage n










where f(-) is a general expression of the system reliability.

Another way to increase the system reliability is to simply use more reliable

components, which certainly costs more in terms of various resources. This problem

is called a reliability allocation optimization problem ([28]). Suppose there are ui

discrete choices for component reliability at stage i for i = 1,...,k (< n) and the

choice for component reliability at stage k + 1,..., n is on a continuous scale. Let

re (1), re (2),. ., re (ui) denote the component reliability choices at stage i for i = 1,. ., k

(< n). Then the continuous/discrete reliability allocation optimization problem can be

formulated as follows:


max R=f(r(x)... kk k1-** E)(4-5)

subject to gryx).. kk k1-**E)
x4 e{1,,...4} i= 1... k,(4-7)

0

where ri and u" are the lower bound and upper bound respectively for the component

reliability at stage i. If k = 0, the above mixed integer nonlinear programming formulation

reduces to a pure nonlinear programming problem.

The systems we are interested in are not limited to series and parallel systems. They

can be complex (general) systems that are non-series and non-parallel, such as the bridge

network in Figure 4-3. The system reliability of such system can be computed by the

conditional probability theory. For example, the system reliability of the five-component

bridge network in Figure 4-3 can be computed based on whether component 5 is functional

or not. We refer readers to Appendix A for the details of the following expression.


R, = (rl + T3 r.1r.3 72 + 4 r.2r4 7.5 + 71r.2 +73r4 T1r2r3r4 75 r)










If each component i is enhanced to a subsystem i with xi identical components in

parallel, the objective function of the redundancy allocation optimization problem is


R, (xl ,..., x,) = (R + R3 R1 3) 2 4 R2 4 5 (1 2 3 4


R1 2 3 4)1 5 -


Note that Ri = 1 (1 ri)"i


Figure 4-3. Five-component bridge network


Constraints of a system can be the total weight, the total cost, the total volume,

and so on. In general, such constraints are in nonlinear forms [28]. As the number of

components in each -Isth-i. b l!!/stagfe increases, more connecting equipment is required and

thus, the cost and weight may increase exponentially [53].

Next, we present a more general formulation for the system reliability optimization

problem (SROP), which includes the redundancy allocation optimization and the

reliability allocation optimization problems as special cases. Formally, SROP can be

formulated as a nonlinear mixed integer programming problem.


(SR OP) : max R =f( ,...,x;r,...,r)

subject to gyxx..,x;r,..,y < cy Vj 1 ,.m,

0 < le < xi < ui, Vi=1... q,

Ir~l rk Tu Vk=1 .p

xi: integfer, Vi=1... q,


(4-9)

(4-10)

(411)

(4-12)

(4-13)










where xi is the total number of parallel components at stage i, i = 1, .. ,q; rk is the

reliability level at stage k, k = 1,. ., p; q is not necessarily equal to p, because in some

stages, one may just have only one option to choose: allocating either redundancy or

reliability but not both. If only redundancy allocation is allowed at stage i, then xi is the

only decision variable at stage i; If only reliability allocation is allowed at stage i, then ri

is the only decision variable at stage i; If both redundancy and reliability allocations are

allowed at stage i, then both xi and ri are the decision variables at stage i. Without loss

of the generality, we assume q + p = n. f (-) is the function measuring the system reliability

that is nondecreasing in each of its variables; gj(x, r) is the consumption of resource j,

j = 1,. ., m. Naturally, gy (x, r) is assumed to be non decreasing in x and r.

The SROP model covers a vast 1 in 4 Gry of reliability optimization models discussed

in the literature. For example, model SROP reduces to model RAOP when p = 0, and

a continuous version of the reliability allocation optimization problem when q = 0. It

certainly can also model the general case of the reliability allocation optimization problem

by reinterpreting the definition of the variables, since objective functions 4-5 and 4-9 are

mathematically equivalent, so are constraints 4-6 and 4-10. In addition, model SROP

is obviously a reliability-redundancy allocation model if p = q, where at each stage the

decisions are which component reliability to choose and how much redundancy as well.

The SROP model has received tremendous research attentions over decades and has

been extensively studied and solved using many different mathematical programming

techniques and heuristic approaches. K~uo et al. [28], along with [27], provide a detailed

introduction to the models and algorithms in the reliability optimization. SROP is often

characterized by a nonlinear objective function that is neither convex nor concave over

a nonconvex feasible region. Due to the extreme difficulty of such type of problem,

the solution methods in the literature are mainly heuristics, meta-heuristics and

approximation algorithms. A comprehensive review on these methods can be found in

[27].










In this chapter, we are interested in the exact methods for the reliability optimization

problems, in particular the branch-and-hound schemes ([37], [28], [51], [29]). The

branch-and-hound algorithm is a popular approach for finding global optimal solutions

for the nonconvex problems. They differ in (i) methods of selecting a branching variable,

(ii) methods of selecting a branching node, (iii) methods of calculating upper and lower

bounds. For example, [28] provides a way to transform the integer variables into binary

variables that are branched on afterwards. [37] branches on the variables in increasing

order of the range, the difference between upper bound and lower bound, then fixes them

in that order. [51] and [29] adopt a convexification method to transform the problem

into a convex maximization problem during the process of the branch-and-hound. The

branching is done on a fractional variable. [50] follows a similar theme to [51] and [29]

except that the branch is done on the selected boundary points and it only solves the

continuous version of the monotone optimization problem.

The 1\onotonic Branch-Reduce-Bound (briefly, mBRB) algorithm presented in this

chapter follows the development of a specialized algorithm for monotonic optimization

proposed by Tuy and his collaborators in a series of papers, [40], [56], [57]. They use a

union of hyperrectangles to cover the boundary of the feasible region. The hyperrectangle

is called a polyhlock in their language so that the algorithm is called a polyhlock

algorithm. A branch-and-hound implementation of the polyhlock algorithm was mentioned

in [57] for the continuous version of the monotone optimization problem. The efficiency

of this approach is reported on various classes of global optimization problems, such as

polynomial fractional programming [59], and discrete nonlinear programming [58].

Compared to the previous branch-and-hound algorithms, the proposed method

exploits the monotonicity properties inherent in SR OP without requiring any convexity,

concavity, differentiability, and separability. Unlike [28], it does not require conversion of

the original decision variables into binary ones. The method is efficient and flexible enough

to solve pure, mixed-integer, and integer nonlinear programming problems arising from









reliability optimization. It appears to be the first of its kind in the literature on reliability

optimization.

The remainder of this chapter is organized as follows. The monotonic branch-reduce-bound

algorithm is presented in Section 4.2 with its convergence analysis. Several convergence

acceleration techniques are also discussed. The algorithm is applied to solve both the

reliability allocation and the reliability-redundancy allocation optimization problems in

Section 4.3 with a demonstration of its efficiency. The chapter is concluded in Section 4.4.

4.2 A Monotonic Branch-Reduce-Bound Algorithm

To facilitate the presentation of the monotonic branch and bound algorithm, we

recast the original SROP, i.e. (4-9), to a more general vector format by denoting x=

(x1,... Xy; r,... y), in other words, x,+i = ri, Vi = 1,..., p.


(SROP'): max R, = f (x) (4-14)

subject to x E 0 = {x| gj(x) < cy j 1.m, (4-15)

x E[x xU] (4-16)

x E 3z = {x | x E 7?", xi: integer, Vi=1 (4-17)


where [X x"] denotes a hyperrectangle with lowest boundary xL and greatest boundary
x"; fu~nctinon f andl gs are nondec~reasing for j =, 1. .,m. To ensure 0 is closed, we

assume gj are semi-continuous for j = 1, .. ., m.

Since no concavity assumption has been made on the objective function, multiple

locally optimal solutions may exist. However, from the monotonicity of the functions f

and gi, the following proposition can be easily derived.

Proposition 4.1. [56] The 11g l..lrl maximum of f(x) over 0 n 31 n [xL ,x"], if it exists, is

attained on its bette,~lr,/;,

The property of Proposition 4.1 is used by Tuy to develop the original Polyblock

Algorithm for monotonic optimization in [56]. The branch-reduce-bound implementation

of the polyblock algorithm was mentioned in [57] for the continuous version of the










monotone optimization problem. The algorithm recursively partitions the hyperrectangle

[X x"] into a smaller region that still contains the global maximizer. A variant of the

algorithm is able to handle the mixed integer version of SROP'.

In the description of Algorithm 1, S denotes a hyperrectangle partition; C is a list

of unfathomed hyperrectangles; e is a pre-defined optimality tolerance parameter; xbest

and fbest denote the current best solution and objective value respectively; UB(S) is the

upper bound of the objective function over S. Besides initialization, the 1!! r ~ steps are

described in detail as follows.


Initialization


2: if xU E g then
3: x" is the optimal solution, terminate;
4: else if xL Sf then
5: the problem is infeasible, terminate;
6: else
7: set S = [X x"], e > 0, C = {S}, Zbest = XL, best =(L)
8: end if
Select and Branch
9: if C = 0 then
10: output the current Zbest aS the solution, terminate;
11: else
12: select S = [s SU] eE such that UB(S) = maxsez{UB(S)} and <-- C\{S}
13: if (UB(S) fbest) < e then
14: output the current Zbest aS the solution, terminate;
15: else
16: select i such thaRt i = argm~axi,~ {sU a, bisect S into SI anld S2 along th~e edge i.
(case 1: i < q; case 2: i > q.)
17: end if
18: end if
Reduce and Bound
19: for k = 1, 2 do
20: reduce Sk, to S, [L" U x"] according to reduction rules










21: compute a suitable upper bound UB(Sk)
22: if UB(Sk) Ibest Or ZLr Sf 0 then
23: continue,
24: else if xU" E then
25: update Zbest and fbest, if improved.
26: else
27: update Zbest and fbest, if improved.
28: 2 <- 2U {Sk
29: end if
30: end for

31: goto Select and Branch.

Algorithm 1 Monotonic Branch-Reduce-Bound Algorithm


4.2.1 Select and Branch

The hyperrectangle with the greatest upper bound is selected for branching, as shown

in line 12 in Algorithm 1. Then the subsequent branching is carried out along the longest

side i of S, with? i = argmaxj {s~ 8 If i < ql thenl S = [s s"] is pa~rtitionecd to SI anld

S2 on the discrete variable:


is partitioned to S1 and S2


St = s L, s" af2s e



22 SL 2 s

where ei is the ith unit vector in R". It is obvious that after partition the least and

greatest elements in S1 and S2 belongs to X ,. A branchingf process is said to be exhaustive


S = s C,S s" 2 afe


S2~~S SL 2 s+ e", s,


where ei is the ith unit vector in R". If i > q, then S = [s s"]

on the continuous variable:
































S1


S2


if there exists a sequence of partitions that shrink to a singleton [19]. The longest-edge

bisection rule is exhaustive, as indicated in the proof of Theorem 4.1.

Although the longest-edge bisection rule is a popular choice for' 1.1 1,, ,11. one

can use other rules such as the longest-edge trisection, and the largest increment edge

bisection. In the largest incremental edge bisection, the edge is selected on which with the

most increment in the objective function. That is i = argmaxj { fls \8, U .sU)} where^

s\ \s, U s," denotes the value of a' except that its f"" index is replaced by s:"
4.2.2 Reduce and Bound

The next us! lb ~r step is to reduce and bound the generated partitions, S1 and S2-

The reduction process is to purge the inferior area to reduce the size of the search region

so that it helps to obtain a better feasible solution and a tighter upper bound. The bold

rectangles on the right hand side of Figure 4-4 shows S1 and S2 after reduction.


S1Reduce

S, S2



Figure 4-4. Reduce process


In the case, i > q, with




St = s ,


a reduction can be done as follows. Let


as~ ~ ~~ = ma s[,1]:8+ 2 s


s" af2si ei]






e' + a((s" s) (s"-s s)e 0n
(4-18)










Geometrically, as is thle maximum distance from point sL + U e" to the boundary

of set 0 n 31. To achieve the value of asj is a one dimensional problem can can be easily

solved by bisection search when j > q or by a sort algorithm when j < q. It helps to locate

a purge point P ='i s + ,"e" s(s (s" -s )* e") e' on thle bounldary of
set 0 n :1, the bottom-left area to P can be purged because it is inferior to the point P

and the upper-right area to P can be purged too because it is infeasible. More precisely,

S1 and S2 can be reduced to


S =s" s"- s af 'i fe e,3 s"-a 2s
j= 1



j= 1
After the reduction, the upper bound of a partition S = [s s"] can be calculated

in various v-wsi~, such as using the original polyblock algorithm in [56]. In this paper, we

simply use f(s"). A partition is discarded if its upper bound is less than the current best

solution, or if it does not cover any feasible solution, as shown in line 22 in Algorithm 1.

4.2.3 Convergence Analysis

The selection of greatest upper bound hyperrectangle and the longest-edge bisection

branch guarantee the convergence of Algorithm 1. The result is summarized in Theorem

4.1.

Theorem 4.1. Algorithm 1 either terminates after finite iterations, producing an e-

optimal solution or detecting its inf..re.T.9.9w, or it is infinite with a sequence of i,;y~ ,-

r, La..gll. < S,31 = [L l, U l] such that lim L"" lim UP = x*, where x* is an optimal
k->oo k->oo
solution.

Proof. It is obvious that if Algorithm 1 terminates within finite iterations, it produces an

e-optimal solution or detects its infeasibility. If Algorithm 1 does not terminate within

finite iterations, all the discrete variables must have been fixed. In addition, among the

infinitely generated partitions, there exists a sequence such that [L "+1, U +1] C [L" U "].










Due to the longfest-edgfe bisection branch rule,


lim? (Up~ L ) = Vj =1, 1. ..
k->oo ,

It implies that

lim L"" lim U"" = x*.
k->oo k->oo

We must also show that x* is an optimal solution. From the selection of greatest upper

bound hyperrectangle, we know that the upper bound of the partition in this sequence is

no less than any feasible solution. That is, f (U ") > f (x), Vx E g nx 3. Therefore,


lim f (L" ) = lim f (U" ) = f (x*) > f (x), Vx E g n 31z.
k->oo k->oo

It means that x* is an optimal solution. O

From Theorem 4.1, we can add the following condition after line 16 in Algorithm 1 to

terminate the algorithm in finite iterations.

if maxj {sy -c a ,f}< then

output sjL as the solution, terminate,

end if

4.2.4 Acceleration Techniques

In Algorithm 1, besides the reduction process, there are several other acceleration

techniques worth mentioning.

Preprocess. The upper bound of xi can be tightened by a number that is derived

from its lower bound. Let


As = max{A : gj(Aei) + gj(x) < cj}, (4-19)


we have all feasible xi < xC + Agei. In other words, xi < min(x xC + Xiei), which provides

a tighter upper bound for xi.









Selection and partition strategy. Algorithm 1 chooses the partition with the

greatest upper bound. Computationally, a delicate rule to periodically select the partition

between the greatest upper bound and the least upper bound helps the convergence of the

algorithm.

Algorithm 1 employs the longest-edge bisection rule. One can use other rules such

as the longest-edge trisection, and the largest incremental edge bisection to speed up

the convergence in some cases. In the largest incremental edge bisection, the edge is

selected on which with the most increment on the objective function. That is, i=

argmax,, {ir\r fs \sT` U I)} where s ~\s, UJ .s, denotes th~e va~lue of a' except that its f^" in~dex

is replaced by s, .

Improved upper bound. An improved upper bound on the hyperrectangle

accelerates convergence if it can be calculated efficiently.

4.3 Using Monotonic Branch-Reduce-Bound Algorithm to Solve System
Reliability Optimization Problems

In this section, the monotonic branch-reduce-bound algorithm (i.e. Algorithm 1) is

applied to solve various problems arising from system reliability optimization. Most of

these problems are thoroughly studied in the literature with various solution techniques,

mainly heuristics. The limitation of a heuristic is that it may find good solutions, but

without guaranteed optimality. In contrast to the heuristic techniques, the presented

mBRB Algorithm, as an exact method, can find an e-optimal solution in a finite number

of steps. The algorithm that solves all the following instances is implemented in C++ and

tested on a Dell Optiplex GX620 computer with a Pentium IV 3.6 GHz processor and 1.0

GB RAM, running under the Windows XP operating system.

4.3.1 Redundancy Allocation Optimization

The following two examples belong to the category of redundancy allocation

optimization problems.










Example 1: Five-stage series system. The following problem is to maximize the

system reliability of a five-stage series system which are subject to three nonlinear resource

constraints. This problem is widely used to demonstrate a number of optimization

techniques [28]. The problem was originally presented in [53].


max Rs = R (x4) (4 20)
i= 1

subject to gl = pg? < P. (4 21)
i= 1

g2=Ci Zi +exp( i= 1

g3 =it I e(xp < W, (423)
i= 1
xi: integer, Vi=1 ,(4-24)


where Ri = 1 (1 ri)"i is the reliability of stage i. Constraint 4-21, gl, is imposed on

the combination of weight and volume: pi is the product of weight per unit and volume

per unit. Component reliability does not usually affect the weight nor the volume, hence

91 is not a function of ri ([54]). Constraint 4-22, g2, iS the cost constraint where cixi is the
cost of all components at stage i and ce exp (\ ) i thep adirt~ional cosnt for ;,tci~n~rtelrconctn

parallel components. Constraint 4-23, g3, iS the weight constraint where I, is the weight

of all componentst at stage i. The additional factor, exp (y), is added due to the hardware

required for interconnecting components ([54]). The weight constraint is not a function of

component reliability. The coefficients are given in Table 4-1.

Table 4-1. Coefficients in Example 1
i ri pi ci I, P C W
1 0.80 1 7 7
2 0.85 2 7 8
3 0.90 3 5 8 110 175 200
4 0.65 4 9 6
5 0.75 2 4 9









It only takes the mBRB algorithm 15 milliseconds to obtain the proved optimal

solution (3, 2, 2, 3, 3) with the corresponding system reliability at 0.9044673. It is superior

to many solution techniques collected in [28].

Example 2: Four-stage series system with 2-out-of-n: G configuration.

In this example, stage 1 does not allow for component redundancy, but its reliability is

determined by choosing a component from a pool of six different components at that

stage. In other words, the reliability levels at stage 1 are discrete. The reliabilities of

stages 2 and 4 can only be enhanced by providing redundancy. However, stage 3 has a

special configuration called 2-out-of n: G configuration, which works (or is t;ood") if and

only if at least 2 of the a components work (or are good). Based on this definition, the

system reliability of stage 3 is shown in the expression of R3 X3


max Rs e4 (4-25)
i= 1

subject to gl =~ 10 exp +lx)~ 10Z2i + 3+154< 10. (4-26)

g2 xl = 10ex 4 exp(x2) + 2 3+ exp +6x< 0, 4-7

g3 =40x 6 exp~(Z2n 31. ex + x 70 (4-28)

xi: integer, Vi = 1, 2, 3, 4, (4-29)


where stage reliabilities are


Rl(xl) = 0.94, 0.95, 0.96, 0.965, 0.97, 0.975, for xi = 1, 2, .. ,6, respectively,

R2 2a) = 1 (1 0.75)""


R3 3) =(0.90)k (1 0.90)"3-k"


R4 4q) = 1 (1 0.95)24

It takes the mBRB algorithm 27 iterations to produce and verify the optimal solution

(3, 3, 5, 3) with its objective value of 0.9444472 at no CPU time, or more precisely, less










than 1 millisecond. The problem was solved by various specialized heuristics, such as

[36], [23], and [1], all of which spent more time to solve than the mBRB algorithm and

could not verify the optimality of their solutions, although they were able to produce the

same solution. As a correction to the literature, it is noted that [1] contains the wrong

expression of this example.

4.3.2 Reliability-Redundancy Allocation Optimization

Reliability-redundancy allocation optimization is a mixed integer nonlinear programming

problem that is the general form of SROP. In this setting, a design engineer can

improve the reliability of a system by increasing the component reliabilities or providing

redundancy at various stages. The following two examples are widely used in the

reliability literature.

Example 3: Five-stage series system with component reliability choice. This

example is a variant of Example 1.


max Rs e,4 (4-30)
i= 1

subject to gi = ) p,.x < P, (4-31)
i= 1


g2 ~i= In i x4 ex < C, (4-32)


g3= t'I x < W, (4-33)
i= 1
0
xi: integer, Vi=1 ,(4-35)


where the objective function and constraints are the same as those of Example 1 except

a more explicitly expression for the unit cost of component i, which is a function of the

component reliability rs. To drivew the expression of cs(ri) =i as g we follow- the
method in [54] and assume that the unit cost of component i is a decreasing function of









the component failure rate As expressed by


where asi and pi are the inherent characteristics of component i. If component i follows

the negative exponential failure law, that is, ri = exp(-Ast), then the component cost is

c4(ri) = as g where t is the duration for which component i is: requnired to operate.
The coefficients of Example 3 are given in Table 4-2.

Table 4-2. Coefficients in Example 3
i asc x 10s pi I, Pi P C W t
1 2.330 1 7 1.5
2 1.450 2 8 1.5
3 0.541 3 8 1.5 110 175 200 1000
4 8.050 4 6 1.5
5 1.950 2 9 1.5


This problem is considered difficult to solve in the literature. To the best of our

knowledge, there is no exact method being applied to solve this problem. We compare

the performance of mBRB with the THK( heuristic in [55], the GAG heuristic in [15], the

K(LXZ method in [26], the surrogate-constraints algorithm HNNN in [18], and the genetic

algorithm GA in [20]. The comparison results are summarized in Table 4-3, where the
CPU time listed in the last column is measured in seconds. The first column lists the

names of the methods. The solutions and the obtained system reliability are listed in the

second and the third column respectively. The fourth column, R,(UB), lists the upper

bound of the system reliability.

The numbers in brackets after "mBRB" in the first column is the value of e, the

pre-defined optimality tolerance. As one can see, the mBRB algorithm produces higher

quality solutions with known upper bound comparing to other algorithms, which are able

to output some feasible solutions but without upper bound guarantee. With additional









Table 4-3. Performance comparison of Example 3
Method (x, r) R, R,(UB) CPU (s)
mBRB (0.01) (3, 2, 2, 3, 3, 0.77500, 0.87500, 0.930947 0.940913 1.38
I ;' 12 ".11, 0.71500, 0.79000)
mBRB (0.001) (3, 2, 2, 3, 3, 0.78250, 0.87500, 0.931541" 0.939004 4.76
0.89938, 0.70750, 0.79000)
THK( (3, 3, 2, 2, 3, 0.78438, 0.82500, 0.915363
0.90000, 0.77500, 0.77813)
GAG (3, 2, 2, 3, 3, 0.80000, 0.86250, 0.930289
0.90156, 0.70000, 0.80000)
K(LXZ (3, 3, 2, 3, 2, 0.77960, 0.80065, 0.929750
0.90227, 0.71044, O ;;' s 7)
HNNN (3, 2, 2, 3, 3, 0.77489, 0.87007, 0.931451
0.89855, 0.71652, 0.79137)
GA (3, 2, 2, 3, 3, 0.77943, 0.51.i 18, 0.931578b
0.90267, 0.71404, 0.78689)

Terminated at iteration 2000.

b Best solution obtained in multiple trials.


CPU time, the mBRB algorithm can produce a solution of


(3, 2, 2, 3, 3, 0.77781, 0.87187, 0.90281, 0.71313, 0.751.2',)

with the system reliability at 0.931669, and its upper bound of 0.933111. This result can

serve as a benchmark for this example.

Example 4: Seven-link ARPA network. This problem is to maximize the

reliability of the following seven-link ARPA network (see [35], [51]). The derivation of the

reliability of this network, i.e. the objective function, is provided in Appendix A.


Figure 4-5. Seven-link ARPA network









max R, = R6 7 + 1 2 3 6 + 6Q 7 R1 4 7 6 2 + 2 3) (4-36)

+R3R5R67 1 1Q2 R12R5R734Q6

+R23R4R6Q157 1R34R5Q2Q67

ctf to gi = "2+ 0.5x I(1+i) x3 4 2xs+ll 03 exp (4-37)



g2 1 x + 2x2 + 1.2x3) 1n~ 2 l+ + 2x3) + 0.4x4 (38)
(0.02 0.01


0.5 < Ri < 0.99, Vi = 6, 7, (439)


subje


(4-40)


xi: integfer, Vi=1... 5,


where Ri = 1 (1 ri)"i,Vi = 1,...,5, Qi = 1 Ri,Vi = 1,...,7, and rl = 0.70, T2 =

0.90, T3 = 0.80, T4 = 0.65, rs = 0.70.

This problem was originally solved using convexification method coded in FORTRAN

in [51]. The comparison between the mBRB method and the convexification one is done

in Table 4-4, where both algorithms obtain the optimal objective value. Admittedly,

comparing CPU seconds directly does not reflect the absolute efficiency, since the

convexification method was implemented in an older computer system. However, the

table clearly shows relative efficiency of the mBRB algorithm: It takes only negligible

CPU seconds to get the optimal solution.

Table 4-4. Performance comparison of Example 4
Method (X, r) R, R,(UB) CPU (s)
mBRB (4, 1, 3, 4, 3, 0.9845, 0.9900) 0.99974 0.99974 0.01
Convexification (4, 1, 3, 4, 3, 0.9845, 0.9899) 0.99974 -38.63"


" It is measured on a SUN SPARCstation 5.


4.4 Conclusions

A monotonic branch-reduce-bound algforithm for mixed integfer nonlinear progframmingf

is presented in this chapter. Its convergence and acceleration techniques are also discussed.










The algorithm is successfully applied to solve a wide range of the system reliability

optimization problems. Computational results have been reported to show the superior

efficiency of the algorithm over existing ones.

We expect that the nionotonic branch-reduce-bound algorithm can he applied to

other classes of the problem, such as nonlinear nmultidintensional knapsack problems, and

generalized niultiplicative progranining problems. In the future research, we also would

like to compare the performance of the acceleration techniques mentioned in this chapter,

and analyze its worst-case performance theoretically and computationally.










CHAPTER 5,
FORTIFYING THE RELIABILITY OF EXISTING FACILITIES AND MONOTONIC
OPTIMIZATION

5.1 Introduction

In C'!s Ilter 2, we consider the impact of unreliability front the facilities when the

system is initially designed. In that case, we are given the option to build facilities from

scratch. However, redesigning an entire system is not ahr-l- .- an available option given

the potentially large expenses involved in closing existing facilities or opening new ones.

In many situations, methods for protecting existing infrastructure may be preferable

given limited resources available. In this chapter, we address the issues on fortifying the

reliability of existing facilities.

Am il I iG~~y of research on reliable supply chain design has been focused on the initial

system design, but not on how to improve the existing system. These works have been

well documented in [47] and surveyed in C'!s Ilter 2 as well. However, reinforcing the

components of an existing system may become more valuable and realistic considering

the increased potential disruptions and uncertainties. These disruptions and uncertainties

may evolve front the new challenges that were never faced when the initial systems were

designed. They can he nian-nmade disruptive events or natural disasters, for example, the

September 11, 2001 terrorist attack and Hurricane K~atrina in 2005.

Only a small strand of literature has been devoted to addressing the fortification of

existing facilities, which includes [4:3], [44], and [49]. These works typically focus on the

interdiction-fortification framework hased upon the p-niedian facility location problem.

The problems are generally formulated in the form of bilevel progranining. These models

can help to identify the critical facilities to protect under the events of disruption.

Another related strand of literature is on the network interdiction problems that are

mainly developed for military applications, e.g. [12], [:33], [62] and [:30]. These models

study the impact of losing one or more transportation links or network arcs based on the










network flow problem. These models can help to identify the critical arcs to protect under

the events of disruption.

In this chapter, we follow the theme of C'!s Ilter 2, but assume that the facilities have

been built and can he reinforced to be more reliable. We also assume that the facilities

are uncapacitated. We present two novel models whose objectives are to minimize the

expected connection (service) and penalty cost by allocating the limited fortification

resources to the open unreliable facilities. In the first model, the fortification efforts

are continuous, that is, the failure probability at each facility varies from 0 to 1 at the

fortification stage. We call it the Continuous Facility Fortification Model (CFFM). On

the contrary, in the second model the fortification efforts are subject to different level of

resources. The failure probability at each facility can only be chosen from a set of discrete

levels at the fortification stage. Accordingly we call it the Discrete Facility Fortification

Model (DFFM). Both models can help to identify the critical facilities to protect and

optimally determine how much resources should be allocated to achieve the objective. To

some extent, the models mathematically resemble the reliability allocation problem we

discussed in OsI I pter 4.

The remainder of this chapter is organized as follows. In Section 5.2, we present the

continuous facility fortification model and reveal its connection to the generalized linear

multiplicative programming and the inherent monotonicity. An example is presented in

Section 5.2.2 to illustrate the solution structure and properties. In Section 5.3, we present

the discrete facility fortification model and apply the monotonic branch-reduce-algorithm

to exploit the monotonicity properties inherent in the problem. The efficiency of the

algorithm is demonstrated in Section 5.3.2. Section 5.3.2 also contains an analysis of

the solution structure and tradeoff between cost deduction and fortification effort. We

conclude this chapter in Section 5.4.









5.2 Continuous Facility Fortification Model

We first introduce some common notations that will be used throughout the chapter.

Let D denote the set of clients or demand points and F denote the set of facilities. Let fi

be the facility cost to open facility i, dj be the demand of client j, and csj be the service

cost if j is serviced by facility i. For each client j e D, if it is not served by any open and

operational facility, then a penalty cost rj will be incurred.

In the reliable facility location model setting, each client is assigned to a set of backup

facilities, which is differentiated by the levels: in case of a lower level facility fails, the next

level facility, if functional, will ba~ck it up?. x@ = 1 if facility i is the k~-th? level ba~ckup-

facility of demand node j and afy = otherwise, z" 1 if j has (k 1)-th backup facility,

but has no k-th backup facility so that j incurs a penalty cost at level k. Contrary to the

models in C'!s Ilter 2, xfy and (j are not decision variables anymore. They are used to

specify the existing network.

To compute the expected failure cost, we follow the logic in C'!s Ilter 2. First of

all, we need to compute the expected failure cost at level k served by facility i. Each

demand node j is served by its level-k facility if all the lower level facilities become

non-operational. For any facility 1, if it is assigned to a lower level (i.e, less than k) for

dem~anld node j, thecn CE:r = 1, otherwise it, is zero. So th~e probabhility th~at a~ll lower

levels facilities fail is niev a And j is served at level-k by i, which has to be

operational. The probability is (1 p ). Therefore, the expected failure cost at level k

served. by facility i is dcqiif(1 -i p lF~ ) zr '. Similarly, we can calculate the penalty

cost at level k, which is nev pt ir

We can now formulate the continuous facility fortification model as follows:










(CFFM~)


|F| |F|+1
minimize~ dyey (1 ps) plP 1 51
j6D k=1 iEF 16F j6D k=1 16F
subject to gyp,., )<0 =1..m(5-2)

0 < pi < 1, Vi E F. (5-3


The objective function (5-1) is the sum of the expected failure cost and the expected

penalty cost. Constraints (5-2) denote various resource restrictions on the fortification

levels, pi,Vi E F. We assume gj (j = 1,..., m) are convex functions so that the solution

domain is convex. Constraints (5-3) are natural constraints on the failure probability.

5.2.1 Properties of the Continuous Facility Fortification Model

In this section, we show that CFFM is a special case of the following generalized

linear multiplicative programming (GLMP) problem (see [41] for details).

(GLM~P)


m~inimizej~ (:-IcX x dz3) (5-4)
j= 1 i= 1
subject to x E X (5-5)


where cij E R"n, dij E R"n, j = 1, ..,t, i = 1, ..,pj, and x E R"n, X is a nonempty

convex set. Note that without the summation sign in the objective function 5-4, GLMP is

reduced to linear multiplicative programming, another active topic in global optimization.

We first show that the objective function (5-1) is the sum of linear multiplicative

terms with positive coefficients. In other words, the objective function (5-1) can be

reduced to the form of


j i= 1
where asj are nonnegative and pij are binary.










Proposition 5.1. The objective function, 5-1, of the continuous f 7. .:1.:1; forti~p,-,,7:.w
|F|
model ~ ~ ~ ~ ~ ~ IP ca erdue oth omof a here as ar nonnegatzive and ,is are
j i= 1



Proof. To prove this proposition, it is sufficient to show that any cost term related to
|F|
a2n arbitraryi demand. nodle j has the form of ai p, Considering demand node j, it
i= 1
contributes


k=1 iEF 16F k=1 16F
to the total cost. If no facility is assigned to it, then only the penalty term left, that is
|F|
dg~ri which app~arently takes the form of n, p~ with ai = 0 and as = dgry. Otherwise,
i= 1
without loss of the generality, we assume facility i' is assigned to it at level k. So the

service cost at level k is

dycesi(1 per) p (5-7)
16F
The only negative term is


dyc r .pp .: (5 8)
16F

At level k + 1, two cases can happen:

1. No facility is assigned to demand j at level k 1 It is subject to penalty cost



16F

Because rj > ci/j in this case, adding terms 5-8 and 5-9 leads to


dj F ityp, p,
16F

|F|
which is in the form of al Ip~ Note that= x% is a binary value.
i= 1










2. Facility i" is assigned to it. And the corresponding service cost is


car,(1 p) pl (5-10)
16F

Adding terms 5-8 and 5-10, we have



16F 16F

From Proposition 2.1, we know that ci/ c ,j > 0, so the only negative term left is


yer.pl (5-12)
16F

This term can be absorbed in the next level assignment following a similar analysis at level

k. By doing this process recursively, demand j will be eventually subject to penalty cost

at certain level higher than k + 1 and only case 1 can happen. Therefore, all cost terms

related to demand node j have the form of n, np, This completes the proof. O
i= 1
As a direct result from Proposition 5.1, the following Corollary holds.

Corollary 5.1. The objective function 5-1 of the continuous f 7. .:1.:1; fori;:p,-,,7;:.w model is

monotor...: a ll;i nondecreasing.

It is obvious that Function 5-6 is a special case of Function 5-4: If pij = 0 in

function 5-6, then set cij = 0 and dij = c4, If Pij = 1 then set cij = agje", e" being

the ith unit vector in RW" and dij = 0. Therefore, the continuous facility fortification

problem is a special form of the generalized linear multiplicative programming. The

latter is multiextremal and possesses several local minima [41]. The existing algorithms

for GLMP include outer-approximation methods [25], vertex enumeration methods [19],

heuristics methods [31], among others. Corollary 5.1 also allows us to apply the monotonic

branch-reduce-bound algorithm presented in (I Ilpter 4 when the resource constraints

possess monotonicity as well.










5.2.2 An Example of the Continuous Facility Fortification Model

In this section, an example is used to illustrate the solution structure of the

continuous facility fortification model and how it helps to identify the critical facilities

to protect in the system. In this example, the following constraint is posed:


pi >~ L (5-13)
i6EF

where L is nonnegative and can be interpreted as the indication of the (un)reliability of

the whole system. In other words, with the available fortification resources the system

can be maintained at the (un)reliability level of L. If L is positive, not every facility can

operate at pi = 0. On the other hand, L = 0 means that the system is totally functional:

No facility has a possibility to fail. The CFFM model is then formulated as follows,

|F| |F|+1
minimrize Cdlce x^ (1 -~, ps p P nz, (5-14)
j6D k=1 iEF 16F j6D k=1 16F

subject to p, > L (5-15)
i6EF
0 < pi < 1, Vi E F. (516)


A network with 20 demand nodes and 5 open facilities is considered in this example.

Demands dj were drawn from U[0, 1000] and rounded to the nearest integer, and x and

y coordinates were drawn from U[0, 1]. Transportation costs cs, are set to be equal to

the Euclidean distance between i and j. The penalty cost rj were drawn from U[0, 15].

The dataset is available in Appendix C. Facilities 2, 5, 15, 18, and 20 are open in this

example. We use a vertex enumeration method to solve this problem and obtain the

results at various level of L which are shown in Table 5-1 and plotted in Figure 5-1. The

first column of Table 5-1 is the (un)reliability of the system. The second to sixth columns

report the solution of this example: the profile of the failure probability at each open

facility. It is followed by a column showing the total cost. The last column calculates











the difference of the total cost between current row and previous row. It indicates the

sensitivity of L on the objective value.

Table 5-1 and Figure 5-1 show how the system deteriorates (total cost increases) as

the resource to maintain the system reliability decreases. In addition, the last column

of Table 5-1 indicates that the curve in Figure 5-1 is piecewise linear. For example, the

difference, similarly the slope, in total cost is a constant when L changes from 1.0 to 2.0

in a step of 0.1. This is because of (1) the failure probabilities of all but one facility (20,

in this case) remain the same, the whole cost structure does not change, (2) CI lngs;th

failure probability in one facility will only affect all the service cost that related to this

particular facility.

4500

4000 -

3500-

a 3000-

o 2500-

2000-

1500-

1000
0 0.5 1 1.5 2 2.5 3 3.5 4
System (Un)reliability Level, L


Figure 5-1. Total cost at different system reliability level


There is a common pattern in Table 5-1: when L increases by 0.1, only one of the

facilities changes the failure probability accordingly. For example, when L increases

from 1.7 to 1.8, the failure probability of facility 20 jumps from 0.7 to 0.8, all the others

remaining the same. The failure probability at an individual facility usually changes by

either 0.1 or 0.0 as shown in most cases in Figure 5-2. However, there is one notable










Table 5-1. Solution of a CFFM model
Failure Probability at
L 2 5 15 18 20 Total Cost Difference
0 0 0 0 0 0 1582.93 n/a
0.1 0 0 0.1 0 0 1590.37 7.44
0.2 0 0 0.2 0 0 1597.80 7.43
0.3 0 0 0.3 0 0 1605.24 7.44
0.4 0 0 0.4 0 0 1612.67 7.43
0.5 0 0 0.5 0 0 1620.11 7.44
0.6 0 0 0.6 0 0 1627.55 7.44
0.7 0 0 0.7 0 0 16334.98 7.43
0.8 0 0 0.8 0 0 1642.42 7.44
0.9 0 0 0.9 0 0 1649.86 7.44
1.0 0 0 1 0 0 1657. 29 7.43
1.1 0 0 1 0 0.1 1689.40 32.11
1.2 0 0 1 0 0.2 1721.51 32.11
1.3 0 0 1 0 0.3 1753.62 32.11
1.4 0 0 1 0 0.4 1785.73 32.11
1.5 0 0 1I 0 0.5 1817.84 32.11
1.6 0 0 1I 0 0.6 1849.95 32.11
1.7 0 0 1I 0 0.7 1882.06 32.11
1.8 0 0 1I 0 0.8 1914. 17 32.11
1.9 0 0 1I 0 0.9 19463.28 32.11
2.0 0 0 1I 0 1 111978.39 32.11
2.1 0.1 0 1I 0 1 112050.81 72.42
2.2 0.2 0 1 0 1 2123.23 72.42
2.3 0.3 0 1 0 1 2195.65 72.42
2.4 0.4 0 1 0 1 226;8.06; 72.41
2.5 0.5 0 1 0 1 2340.48 72.42
2.6; 0.6; 0 1 0 1 2412.90 72.42
2.7 0.7 0 1 0 1 2485.31 72.41
2.8 0.8 0 1I 0 1 112557. 73 72.42
2.9 0.9 0 1I 0 1 112630.15 72.42
3.0 1 0 1I 0 1 112702.56 72.41
3.1 1 0 1 0.11 1 112863.49 160.93
3.2 1 0 1 0.2 1 113024.41 160.92
3.3 1 0 1 0.3 1 3185.34 160.93
3.4 1 0 1 0.4 1 33463.263 160.92
3.5 1 0 1 0.5 1 3507. 19 160.93
3.6; 1 0 1 0.6 1 3668.11 160.92
3.7 1 0 1 0.7 1 3829.04 160.93
3.8 0.8 1 0 1 1 3972.66 143.62
3.9 0.9 1 0 1 1 4099.09 126.43
4.0 1 1I 0 1 1 4225.52 1263.43










exception in this pattern. That is when L increases fron1:3.7 to :3.8, both facility 2 and

facility 18 change their failure probability in the optimal solution. It means that the

optimal vertex in L = :3.8 is dramatically changed from L = :3.7. This exception is

clearly signaled in Figure 5-2 if we look at the changes from L = :3.7 to L = :3.8 in

sub-figures A to D: a 0.2 dip in A, a 1.0 surge in B, a 1.0 sink in C, and a 0.3 jump in D.

This phenomenon is front the niultiextrentalness of the CFFM problem that we discuss

in Section 5.2.1. Using a bisection search, we find out that at L = 0.749842, both vertex

(1, 0, 1, 0.749842, 1) and vertex (0.749842, 1, 0, 1, 1) produce the same objective value

of :3909.2476. It means that at L = 0.749842, the optimal vertex transient front (1, 0,

1, 0.749842, 1) to (0.749842, 1, 0, 1, 1). Then from L = 0.749842 to L = 0.8, the same

pattern still holds, facility 2 is the only one that changes its failure probability.

As we point out in the introduction that CFFM model can also help to find the

key facility to fortify. In this example, the frequency that a facility is completely open is

depicted in Figure 5-:3. It shows that facility 5 is very critical in this system, because it is

chosen to be completely secured :38 times out of the total 41 cases in this example.





















09

~08
~07

05

u, 04
03
~02
01


0 05 1 15 2 25 3 35 4

System (Un)reliability Level, L


B) Facility 5




0 9 -- -
S08
'07
06


o 04 -- -
S03C ~ L~
0 2C IC(
01


---- ---



---- ---


U "" "
0 05 1 15 2 25 3 35

System (Un)reliability Level, L


0 05 1 15 2 25 3 35 4

System (Un)reliability Level, L


A) Facility 2


0 05 1 15 2 25 3

System (Un)reliability Level, L


D) Facility 18


35 4


0 05 1 15 2 25 3 35 4

System (Un)reliability Level, L


C) Facility 15


09

5 08
S07
06
-
~05
~04


,

01


E) Facility 20



Figure 5-2. Failure probability at individual facility vs system reliability Level













S30-




S15-

S10-



2 5 15 18 20
Facility


Figure 5-3. Frequency of completely open facility in Table 5-1


5.3 Discrete Facility Fortification Model

The underlying assumptions of the continuous facility fortification model in Section

5.2 are that (1) the limited fortification resource is uniformly distributed across all the

facilities, and (2) the distributed resource is dividable. However, in many cases, the

fortification resource can only be discretely distributed. That is, the fortification resource

at a facility is categorized into different levels, which are functions of available resources.

For example, 7000 units of resource may improve the failure probability of a facility to

p = 0.4, while 8000 units can improve it to p = 0.3. But there is no amount of resource

that can improve the failure probability to a number between p = 0.4 and p = 0.3. This

inspires us to consider a discrete facility fortification model (DFFM).

In DFFM, let yi be the fortification level at facility i. Naturally, we assume yi

is a positive integer and only one fortification level is allowed at each facility. The

corresponding failure probability and the amount of resource are denoted by functions

Fi~~~~~~vi)~\Yl an V4s repciey Py),i F, are nonincreasing functions of yi, whereas

IM(yi), i E F, are nondecreasing functions of yi. This is because more fortification resource

would make a facility more reliable, and consequently a smaller failure probability. Let the

upper bound of yi be Ui and the total resource be R.









As in CFFM, the objective of DFFM is to minimize the total expected transport and

fa~ilure-to-serve cost. Nota~tion xi~ anld zf = 1 keep th~e samec meaning a~s in? CFFM': x@ = 1

if facility i is the k-th level backup facility of demand node j and afy 0 otherwise,

zf = 1 if j h~as (k )-th backup facility, but h~as n~o k-th backup facility so that j in~curs a.

penalty cost at level k. The cost terms are calculated in parallel to CFFM as well, which

are shown in the following formulation.

(DFFM~)


minmie dces ( -Pi(y ))n Pl(,,)C"
j6D k=1 iEF 16F


j6D k=1 16F

subject to y) i6EF
1 < Yi < Ui, Vi E F (519)

yi: integer, Vi E F. (5-20)


The objective function (5-17) is the sum of the expected failure cost and the expected

penalty cost. Constraint (5-18) denotes the resource restrictions on the fortification.

Constraints (5-19) and (5-20) are integral constraints on the fortification levels.

5.3.1 Properties and Algorithms

In this section, we show that DFFM can be solved via the monotonic branch-reduce-bound

algorithm. It is obvious that constraint (5-18) possesses the monotonicity, since I(yi),

i E F, are nondecreasing functions of yi. The objective function of the continuous facility

fortification model is shown to be monotonically nondecreasing in Corollary 5.1. A similar
result holds in the discrete case, becauseir Ps/ys), ie~ Faennncesn uctoso

From the fact that the composite of a nondecreasing function and a nonincreasing function

is noninl i. I-; to we have the following corollary, which is parallel to Corollary 5.1.









Corollary 5.2. The objective function 5-17 of the discrete f 7. .:1.:1; fort;:p,-,,7;:.w model is

monotor...: a ll;i nonincreasing.

Therefore, DFFM is a monotonic optimization problem which can be solved by the

monotonic branch-reduce-bound algorithm presented in ('! .pter 4. To be consistent with

the general format of the monotonic optimization in this thesis, we transform DFFM

model to a maximization problem.

(DFFM~-max;)


|F|+1


j6D k=1 16F

subject to CL(s)) < Rl,
i6EF
1
yi: integer, Vi E F.


(521)


A more general format of DFFM-max can be written as the following by replacing the

decision variables yi with xi.


(MO0): max R, = f(x)

subject to x E 0 = {x| gi(x) < ci, V=1..m,

x E[x x"],


(5-25)

(5-26)

(5-27)

(5-28)


Xj: integer, V ,.. ,


where [X x"] denotes a hyperrectangle with lowest boundary xL and greatest boundary

x", functions f and gi are nondecreasing for i = 1,..., m. To ensure 0 is closed, we

assume gi are semi-continuous for i = 1, .. ., m.

For the completeness of exposition, we also include the description of the monotonic

branch-reduce-bound algorithm with slight differences than that in C'! Ilpter 4 to










accommodate this pure nonlinear integer programming. In the description of Algorithm

2, S denotes a hyperrectangle partition; C is a list of unfathomed hyperrectangles; e is a

pre-defined optimality tolerance parameter; xbest and fbest denote the current best solution

and objective value respectively; UB(S) is the upper bound of the objective function over

S. Besides initialization, the 1!! ri ~ steps are described as follows.


Initialization


2: if xU E g then
3: x" is the optimal solution, terminate;
4: else if xL Sf then
5: the problem is infeasible, terminate;
6: else
7: set S = [X x"], e > 0, C = {S}, Zbest = XL, best =(L)
8: end if
Select and Branch
9: if C = 0 then
10: output the current Zbest aS the solution, terminate;
11: else
12: select S = [s SU] eE such that UB(S) = maxse~e{UB(S)} and <-- C\{S}
13: if (UB(S) fbest) < e then
14: output the current Zbest aS the solution, terminate;
15: else
16: select, i sulch that, i argmax,(~ {sU- a bisect, S into SI and S2 atlongf the edgfe i.
17: end if
18: end if
Reduce and Bound
19: for k = 1, 2 do
20: reduce Sk, to S, [L" U x"] according to reduction rules
21: compute a suitable upper bound UB(Sk)
22: if UB(Sk) Ibest Or ZLr Sf 0 then
23: continue;
24: else if xU" E then










25: update Zbest and fbest, if improved.
26: else
27: update Zbest and fbest, if improved.
28: 2 <- 2U {Sk
29: end if
30: end for

31: goto Select and Branch.

Algorithm 2 Monotonic Branch-Reduce-Bound Algorithm


5.3.2 Computational Experiments

One advantage of using the monotonic branch-reduce-bound algorithm to solve the

discrete facility fortification problem is that no closed form of Pi(yi), i E F, and IM(yi),

i E F are required as long as they are monotonic. In this section, we use the same dataset

as the one in section 5.2.2, which is listed in Appendix C. The dataset contains 20 demand

nodes and 5 open facilities with the following specification.

Table 5-2. Input data (IM(yi) and Pi(yi)) for the 3-level model
Level 1 Level 2 Level 3
Open Facility i V P V P V P
2 0 0.55 79 0.39 6388 0.02
5 0 0.85 614 0.45 728 0.28
15 0 0.75 303 0.63 855 0.48
18 0 0.52 135 0.48 409 0.20
20 0 0.30 178 0.24 273 0.22


The monotonic branch-reduce-bound algorithm is implemented in C++. The CPU

seconds are reported from a Dell Optiplex GX620 computer with a Pentium IV 3.6 GHz

processor and 1.0 GB RAM, running under the Windows XP operating system.

The computational results are reported in Table 5-3 that shows that the fortification

level at each open facility given the resource constraints. Table 5-4 shows the results that

the fortification level is limited to 2. That is, yi < 2, Vi E F. The CPU time reported in

both tables clearly show the efficiency of the monotonic branch-reduce-bound algorithm

for this type of problem. Next, we start to analyze the computational results.










Table 5-3. Solutions of the 3-level model
Constraint Resource Fortification Level at CPUJ
R Used Objective 2 5 15 18 20 second


25000
2952
2857
2400
2305
2097
2002
1824
1710
16388
1593
1550
1415
1369
1274
1096
1095
1000
960
8635
822
687
665
487
408
391
351
256
213
78


2953
2858
2401
2306
2098
2003
1825
1711
1689
1594
1551
1416
1370
1275
1097
1096
1001
961
866
823
6388
666
488
409
392
352
257
214
79
0


2020.69
2030.01
2057.66
206;8.99
2087. 25
2100.18
2138.98
2276.06
2392.85
2408.41
2415.59
2455.11
2489.01
2516.42
2598.63
2797.01
2830.34
2841.01
2875.19
2930.35
2977.74
3308.47
3503.25
3894.44
4318.51
4331.16
4462.80
110; 43
4857. 74
5670.71


0.11
0.89
1.08
1.19
1.48
1.03
1.80
1.94
2.72
2.58
2.72
3.33
3.05
2.88
3.70
3.34
3.64
3.95
3.17
3.48
3.48
2.27
2.56
2.56
2.58
2.86;
2.09
2.09
1.97
1.64


In Figure 5-4, we plot the objective values in these two different settings (3-level

constraint and 2-level constraint) across the resource used. Unlike the result in the

continuous facility fortification model, as shown in Figure 5-1, there is no piecewise

linear property exhibited in this discrete version. Instead the curves are shown steeper

in the earlier stage. That is, the fortification efforts help to reduce a lot of total cost at

earlier stage. This indicates that reliability can be drastically improved without large










Table 5-4. Solutions of the 2-level model
Constraint Resource Fortification Level at CPUJ
R Used Objective 2 5 15 18 20 second
25000 1309 3315.22 2 2 2 2 2 0.11
1308 1174 3402.08 2 2 2 1 2 0.88
1173 1006 3461.27 2 2 1 2 2 1.03
1005 871 3557. 78 2 2 1 1 2 1.19
870 828 3666.45 2 2 1 2 1 1.02
827 693 3776.24 2 2 1 1 1 1.19
6;92 560 4193.94 2 1 2 1 2 1.50
559 392 4318.51 2 1 1 2 2 1.19
391 257 4462.80 2 1 1 1 2 1.34
2563 214 110# 43 2 1 1 2 1 1.33
213 79 4857. 74 2 1 1 1 1 1.19
78 0 5670.71 111 1 1 1 1 1.06


increases in fortification resource. After that, the whole system seems lacking the room for

improvement. In fact, when R is greater than 1500, the decrease in the total cost can not

justify the fortification resource. For example, when R increases from 1551 to 2003, the

total cost reduced from 2415.59 to 2100.18, a net loss of 136.59.


6000
5500
5000
4500
4000
3500
3000
2500
2000
1500
1000


0 500 1000 1500 2000 2500
Resource Used


3000


Figure 5-4. Tradeoff between objective and resource used










Comparing these two "obj. I I i.; lines in Figure 5-4, we also see the benefit to have a

:$-level option over a 2-level one. When the available fortification resource is very limited,

both share the same objective values as there is no enough resource to fortify the facility

to level :3. But as the available fortification resource increases, the :$-level model has more

flexibility so that it incurs much lower total cost.

Another phenomenon different from the continuous version is the fortification level

at individual facility. In the continuous version, the fortification level is more or less

monotonic as shown in Figure 5-2. However, this does not exist in the discrete version as

shown in Figure 5-5. As the fortification resource increases, the fortification level at an

individual facility does not necessarily increase as a consequence from this combinatorial

optimization. Figures D and E in Figure 5-5 clearly show this phenomenon, because

facility 18 and 20 do not admit any monotonicity pattern.

5.4 Conclusions

In this chapter, we present two optimization models (a continuous version and a

discrete one) on how to choose facilities to fortify and to what extent they should be

fortified given limited fortification resources. The objective of both models is to minimize

the sum of the expected service cost and fail-to-serve penalty cost.

In the continuous version of the model, the fortification effort is dividable. We show

that the model is a special case of the generalized linear multiplicative programming

problem. We solve an illustrative example by the vertex numeration method, which

is very effective in solving this type of problems. This example also demonstrates the

multi-extremeness of the problem: several vertices achieve the minimum. 1\anagerially, the

example shows how to identify the key facilities to fortify in a system.

The discrete facility fortification model focuses on choosing the suitable fortification

level at each facility when the fortification effort is divided into different levels. The model

is shown to be a monotonicity optimization problem since the monotonicity property

is inherent in both its objective and constraints. This model is therefore solved by the


















-* Facility 2


0 500 1000 1500 2000 2500
Resource Used


A) Facility 2


3000


0 500 1000 1500 2000 2500 3000
Resource Used



B) Facility 5


0 500 1000 1500 2000 2500 3000
Resource Used


C) Facility 15


0 500 1000 1500 2000 2500 3000
Resource Used


D) Facility 18


0 500 1000 1500 2000 2500 3000
Resource Used



E) Facility 20


Figure 5-5. Fortification level at individual facility by resource used










monotonic branch-reduce-bound algorithm. The computational results of the illustrative

example show the efficiency of the algorithm. We also analyze the tradeoff between cost

deduction and fortification effort and empirically demonstrated that the tradeoff curve is

steeper in the earlier stage, indicating major cost deduction can be achieved without large

increases in fortification resource.

The main limitation of the current models is the assumption that the facilities are

uncapacitated. Although the assumption itself is very common in the facility location

models, it may be unrealistic in practice. In the capacitated case, a 'customer' of the failed

facilities can be assigned to the next level backup facilities only if they have sufficient

capacity to satisfy the additional demands. This may make the capacitated model very

complex. We expect that the monotonic branch-reduce-bound algorithm will still be

applicable. We believe that this is a valuable topic worthy of future investigation.









CHAPTER 6
CONCLUDING RE1\ARK(S

In this chapter, we summarize the various models and algorithms discussed

throughout the dissertation, and point out directions for future research.

We study the impact of uncertainty on the decisions of facility location and demand

assignment. The uncertainty is represented by the failure probability in each facility.

Several novel models have been presented to offer solutions for both the design of initial

supply chain systems and the improvement of the existing systems. We first investigate

the uncapacitated reliable facility location model, whose objective is to minimize the total

of opening cost, expected service cost, and expected fail-to-serve penalty cost when each

facility has a site-specific failure probability. We also study a more general case that each

facility has multiple levels of failure probabilities that can he chosen. If the supply chain

system already exists, we propose two models for optimally allocating the fortification

resource to reduce the expected service and fail-to-serve penalty cost.

The algorithms presented in this dissertation include (1) four heuristics, the sample

average approximation heuristic, the greedy adding heuristic, the greedy adding and

substitution heuristic, and the genetic algorithm hased heuristic; (2) the approximation

algorithm with a worst-case bound of 2.674; (3) the monotonic branch-reduce-bound

algorithm. An in-depth theoretic treatment is provided for the approximation algorithm.

All other algorithms are thoroughly tested in the computational studies. The four

heuristics are used to solve the uncapacitated reliable facility location problem. The

monotonic branch-reduce-bound algorithm is applied to solve the facility fortification

problem as well as the system reliability problem arising from industrial or military

appli cations.

One immediate extension is to study the capacitated version of the current reliable

facility location models. Although the capacitated constraints generally pose more

challenges on findings the efficient algorithms, we expect that some of the heuristics are still










very efficient in finding the optimal or near optimal solutions. For the capacitated facility

fortification models, we believe that the nionotonic branch-reduce-bound algorithm can

still be used to find the global solution.

It would be interesting to see if there exists any approximation algorithm with a

constant worst-case bound for the capacitated uniform reliable facility location problem.

Based on the research on approximation theory of facility location problem without

considering the reliability issue, we expect that some more delicate techniques are required

to develop such approximation algorithm.

Another interesting direction is to introduce different risk/reliablity measurements

into the current models depending on the needs in reality. By doing so, the objective

functions of current models will change accordingly. For example, one may have more

interests in the cost of the worst-case scenario instead of the expected cost in the current

models.

In suninary, the current work we present serves as a useful foundation for further

research of more complicated models and delicate algorithms.





APPENDIX A
SYSTEM RELIABILITY COMPUTATION IN CHAPTER 4

We use conditional probability to derive the expressions of reliabilities of the networks

in Figure 4-3 and Figulre 4-5.

Reliability of the five-component bridge network. Reliability of the network in

Figure 4-3 can be written based on whether component 5 is functional or not.

R, = Pr(system works |component 5 works)rs+Pr(system works |component 5 fails) (1- rs).

(A-1)
When component 5 works, the original network in Figure 4-3 is reduced to Figure A-1(A),

which is a parallel-series system with a reliability of


Pr (system works |component 5 works)


(rl + T.3 T1r3 7r2 + 74 r.2r4).


(A-2)


B

Figure A-1. Configurations based on state of component 5 in Figure 4-3: A) Component 5
works, B) Component 5 fails


S2h


-
-1


3


4 "









Similarly, when component 5 fails, the original network in Figure 4-3 is reduced to

Figure A-1(B), which is a series-parallel system with a reliability of


Pr(system works | component 5 fails) = (TIT2 r3r4 T1r2r3r4). (A-3)


Substitution of equations A-2 and A-3 into equation A-1 yields the reliability of the

five-component bridge network depicted in Figure 4-3:


R, = (rl + T3 r1T3 7r 2 + 4 r2r4 75 +71r2 + 3r4 T1r2r3r4)( 5) (A-4)


Reliability of the seven-link ARPA network. Following the notation in Example

4, we assume that each block from blocks 1 to 5 in Figure 4-5 represents a subsystem.

Blocks 6 and 7 are individual components. Recall that Ri = 1 (1 riefi,Vi = 1,. .,5,

Qi = 1 Ri, Vi = 1, .., 7. Reliability of the network in Figure 4-5 can be written based on

whether -I th- i--r. 11. 4 is functional or not.


R, = Pr(system works | I I- ti--1 r ii 4 works)R4 FT SyStem works |subsystem 4 fails) (1- R4)

(A-5)

When subsystem 4 works, the original network in Figure 4-5 is reduced to Figure A-2(A),

whose reliability can be obtained by applying parallel and series reductions:


Pr(system works | subsystem 4 works) = (1 Qq6)1 &71 R31 &2 5)]} (A-6)


When subsystem 4 fails, the original network in Figure 4-5 is reduced to Figure A-2(B). A

series reduction on subsystems 1 and 2 produces a super-component, which helps to map

the topology in Figure A-2(B) to the five-component bridge network in Figure 4-3. After

the nor lpphlr we can directly use the result of equation A-4 by replacing rl with R1R2, r2











1 5 3

















Figure A-2. Configurations based on state of subsystem 4 in Figure 4-5: A) Subsystem 4
works, B) Subsystem 4 fails


with R3, 73 With R6, 74 With R7, rs with Rs:


Pr(system works |-t b..4fis

(R1R2 6g- R12R6 (3 7 R3R7 5 (1R2R3 6R7 1R2R3R67)( -5

(A-7)


Substitution of equations A-6 and A-7 into equation A-5 yields the reliability of the

five-component bridge network depicted in Figure 4-5:


R, = (1 QQ6) 7 ,[ R3( 2a5 4 (1R2 + 6 1R2R6 3 7 3R7 5

+(R1R2 3 6 7 1 2 3 6 7) R5)( -4). (A-8)


Equation A-8 can be reformulated as the following one:


R, =R6R7 1R2R3 6 6Q7) R14R7Q 62 2Q3)

+R3R5R67 1 1Q2 R12R5R734Q 6

+R23R4R6Q15 7 1R3R4R5Q2Q67 (A-9)










APPENDIX B
DATASET USED IN CHAPTER 2

The meaning of each column in Table B-1 is provided as follows: #i denotes the

facility name; (x, y) is the coordinates, di is the demand; fi is the fixed cost; ri is the

penalty cost; and pi is the failure probability.



Table B-1. Dataset of URFLP-SFP


#i x y di fi ri pi II#i x y di fi ri pi
1 10.82 0.18 957 938 5.32 0.81 1151 0.63 0.04 4863 971 13.14 0.63
2 0.54 0.7 202 6342 1.9 0.39 52 0.53 0.32 548 1023 0.31 0.94
3 0.911 0.72 186 1230 3.11 0.42 53 0.89 0.99 870 754 0.66 0.76
4 0.15 0.31 635 1008 1.83 0.36 54 0.02 0.19 335 734 0.22 0.97
5 0.74 0.16 737 1279 1.34 0.28 55 0.51 0.32 446 1249 3.93 0.91
6; 0.58 0.92 953 1431 2.3 0.83 56 0.53 0.06; 198 1371 3.77 0.31
7 0.6; 0.09 450 1187 7.96 0.98 57 0.81 0.86; 212 1489 9.92 0.58
8 0.37 0.19 188 1044 3.42 1 58 0.53 0.36 903 8633 4.06 0.29
9 0.7 0.52 206 1466 9.05 0.86 59 0.89 0.58 594 521 1.64 0.15
10 0.22 0.4 995 989 4.563 0.55 60 0.87 0.56 250 8635 5.11 0.21
11 0.5 0.45 429 948 9.87 0.58 61 0.91 0.16 472 1464 2.64 0.43
12 0.3 0.52 528 585 0.53 0.79 6;2 0.32 0.15 244 730 6.05 0.09
13 0.95 0.2 570 923 3.411 0.46 63 0.37 0.37 353 1034 4.27 0.78
14 0.65 0.07 938 758 8.98 0.15 64 0.38 0.73 183 585 6.18 0.23
15 0.53 0.11 726 552 3.53 0.48 65 0.96 0.34 749 782 9.13 0.6
16 0.95 0.95 533 1471 1.64 0.7 6;6 0.15 0.76 200 985 6;.6;3 0.42
17 0.15 0.13 565 930 1.36 0.98 67 0.15 0.48 321 6;62 7.02 0.1
18 0.31 0.4 322 1103 5.1 0.2 6;8 0.99 0 650 904 8.73 0.16
19 0.98 0.73 326; 586 1.22 0.49 6;9 0.47 0.28 946 1242 3.92 0.52
20 0.59 0.04 663 812 6.95 0.3 70 0.84 0.16 143 513 8.16 0.79
21 0.46 0.21 952 850 9.57 0.37 71 0.71 0.9 5635 1117 1.25 0.84










continued from previous page

#i :r y di fi ri pi II#i :r y di fi ri pi
22 0.77 0.44 919 561 1.49 0.38 72 0.46 0.86 11 928 4.44 0.26
23 0.87 0.79 292 750 6.95 0.6 73 0.09 0.74 374 1028 2.635 0.33
24 0.69 0.15 48 956 2.29 0.71 1174 0.71 0.78 284 522 1.45 0.35
25 0.24 0.28 581 1456 2.9 0.3 75 0.27 0.04 598 6;09 9.6;3 0.22
26; 0.84 0.73 659 764 5.05 0.99 76 0.25 0.07 720 541 3.72 0.25
27 0.49 0.24 986; 1005 8.81 0.23 77 0.57 0.18 457 1493 2.38 0.43
28 0.56 0.77 4863 1121 3 0.81 1178 0.96 0.49 213 736 7.09 0.49
29 0.38 0.05 915 1065 6.94 0.4 79 0.83 0.21 550 1452 8.3 0.43
30 0.43 0.22 282 1494 0.88 0.68 80 0.72 0.49 418 758 2.55 0.57
31 0.61 0.73 310 605 2.6;6 0.39 81 0.6;9 0.5 86;3 1147 1.53 0.93
32 0.48 0.88 980 1304 3.32 0.86 82 0.22 0.89 36;8 976 3.5 0.52
33 0.811 0.75 134 1073 0.66 0.28 83 0.37 0.88 282 1180 7.15 0.95
34 0.13 0.71 20 1107 3.08 0.62 84 0.36 0.82 8111 1045 9.07 0.3
35 0.411 0.34 151 651 2.634 0.01 1185 0.11 0.1 1866 1021 2.463 0.37
36; 0.72 0.14 615 800 3.84 0.35 86; 0.77 0.6;9 895 1165 3.19 0.44
37 0.3 0.28 36;9 1479 1.74 0.22 87 0.16 0.09 959 1250 5.52 0.6;6
38 0.02 1 875 1278 1.14 0 88 0.75 0.6;3 375 1406 5.28 0.06;
39 0.62 0.9 73 1289 2.76 0.42 89 0.16 0.41 711 515 7.34 0.46
40 0.8 0.06 776 510 3.363 0.69 90 0.01 0.21 208 923 3.99 0.87
41 0.1 0.98 342 5363 5.611 0 91 10.51 0.76 954 1378 4.06 0.31
42 0.15 0.13 929 1022 9.77 0.73 92 0.98 0.32 843 733 7.77 0.33
43 0.48 0.44 445 594 4.32 0.83 93 0.55 0.39 905 545 0.08 0.74
44 0.83 0.22 684 1466 9.61 0.28 94 0.36; 0.6;3 729 1047 8.47 0.84
45 0.82 0.39 6343 677 1.45 0.71 1195 0.18 0.75 382 13635 63.23 0.38
46 0.67 0.53 771 1334 7.46 0.74 96 0.09 0.46 91 513 0.53 0.16
47 0.03 0.73 181 1445 1.03 0.35 97 0.18 0.67 991 1338 6.62 0.92
48 0.26 0.39 926; 691 7.6 0.26 98 0.1 0.38 644 1341 3.13 0.11
49 0.59 0.56 733 1001 0.15 0.08 99 0.25 0.6;6 539 1256 0.5 0.41
50 0.22 0.6;6 326; 1244 5.93 0.98 100 0.6;8 0.49 294 1168 7.27 0.6;6



















#i x ydi ri fl p1 f2 92 p 3 p3


21 0.46 0.21 952 9.57 850 0.37 932 0.18 980 0.14
22 0.77 0.44 919 1.49 561 0.92 742 0.76 1290 0.38
23 0.87 0.79 292 6.95 750 0.6 1213 0.59 1311 0.24
24 0.69 0.15 48 2.29 6384 0.911 772 0.79 956 0.71
25 0.24 0.28 581 2.9 1040 0.62 1153 0.3 1456 0.23


The meaning of each column in Table B-2 is provided as follows: #i denotes the

facility name; (x, y) is the coordinates, di is the demand; ri is the penalty cost; and fi, p

(i = 1, 2, 3) are the investment level and its corresponding failure probability.

Table B-2. Dataset of URFLP-MFP: 3-level


0.82
0.54
0.91
0.15
0.74
0.58
0.6
0.37
0.7
0.22
0.5
0.3
0.95
0.65
0.53
0.95
0.15
0.31
0.98
0.59


0.18
0.7
0.72
0.31
0.16
0.92
0.09
0.19
0.52
0.4
0.45
0.52
0.2
0.07
0.11
0.95
0.13
0.4
0.73
0.04


957
202
186
6335
737
953
450
188
206
995
429
528
570
938
726
533
5635
322
326
6;6;3


5.32
1.9
3.11
1.83
1.34
2.3
7.96
3.42
9.05
4.56
9.87
0.53
3.41
8.98
3.53
1.64
1.36
5.1
1.22
6.95


938
642
1125
772
665
890
6;20
503
1231
989
948
551
6;82
758
552
791
930
694
586
634


0.81
0.55
0.63
1
0.85
0.9
0.98
1
0.86
1
0.87
0.84
0.46
0.78
0.75
0.7
0.98
0.52
0.95
0.3


954
721
1230
1008
1279
1034
1187
703
1278
1037
1082
585
923
987
855
1471
958
829
975
812


0.44
0.39
0.42
0.9
0.45
0.83
0.48
0.75
0.65
0.55
0.6
0.79
0.32
0.5
0.6;3
0.66
0.8
0.48
0.49
0.24


1260
1330
1355
1440
1393
1431
1394
1044
1466
1455
1422
1303
999
1497
1407
1488
1227
1103
1127
907


0.15
0.02
0.19
0.36
0.28
0.61
0.06
0.49
0.24
0.54
0.58
0.39
0.04
0.15
0.48
0.47
0.35
0.2
0.15
0.22










continued from previous page

#i x y di ri fl p1 f2 p2 f3 p3


0.84
0.49
0.56
0.38
0.43
0.61
0.48
0.81
0.13
0.41
0.72
0.3
0.02
0.62
0.8
0.1
0.15
0.48
0.83
0.82
0.67
0.03
0.26
0.59
0.22
0.63
0.53
0.89
0.02
0.51
0.53


0.73
0.24
0.77
0.05
0.22
0.73
0.88
0.75
0.71
0.34
0.14
0.28
1
0.9
0.06
0.98
0.13
0.44
0.22
0.39
0.53
0.73
0.39
0.56
0.66
0.04
0.32
0.99
0.19
0.32
0.06


659
986
486
915
282
310
980
134
20
151
615
36;9
875
73
776
342
929
445
6384
6343
771
181
926
733
326
486
548
870
335
446
198


5.05
8.81
3
6.94
0.88
2.6;6
3.32
0.66
3.08
2.64
3.84
1.74
1.14
2.76
3.36
5.61
9.77
4.32
9.61
1.45
7.46
1.03
7.6
0.15
5.93
3.14
0.31
0.66
0.22
3.93
3.77


764
910
928
1040
771
605
1106
1073
905
651
800
723
890
1104
510
536
1022
594
1130
518
892
1055
691
1001
699
971
853
754
611
1217
742


0.99
0.94
0.81
0.81
0.76
0.81
0.86
0.88
0.79
0.87
0.98
0.7
0.81
0.88
0.91
0.63
0.73
0.83
0.68
0.71
0.74
0.97
0.56
0.72
0.98
0.68
0.99
0.84
0.97
0.91
0.69


941
1005
1079
1065
1028
748
1304
1153
1107
652
1298
1351
1029
1289
981
1157
1064
1289
1192
677
1334
1156
865
1058
1244
1154
1023
787
734
1249
1025


0.2
0.25
0.72
0.4
0.6;8
0.41
0.21
0.34
0.62
0.25
0.57
0.51
0.04
0.42
0.69
0.06
0.67
0.76
0.28
0.38
0.72
0.6;3
0.26
0.61
0.42
0.63
0.94
0.76
0.83
0.88
0.49


1035
1058
1121
1108
1494
1381
1400
1315
1260
1066
1423
1479
1278
1485
1341
1351
1157
1386
1466
1072
1483
1445
1312
1450
1472
1493
1207
1271
1214
1367
1371


0.04
0.23
0.35
0.16
0.04
0.39
0.06
0.28
0.62
0.01
0.35
0.22
0
0.37
0.59
0
0.33
0.43
0.09
0.28
0.45
0.35
0.23
0.08
0.04
0.49
0.28
0.61
0.51
0.05
0.31










continued from previous page

#i x y di ri fl p1 f2 p2 f3 p3
57 0.811 0.86 212 9.92 12711 0.58 14463 0.4 1489 0.38
58 0.53 0.36 903 4.06 8633 0.75 1078 0.29 1354 0.05
59 0.89 0.58 594 1.64 521 0.64 1071 0.44 1100 0.15
6;0 0.87 0.56 250 5.11 865 0.95 1273 0.6 1348 0.21


472
244
353
183
749
200
321
650
946
143
565
11
374
284
598
720
457
213
550
418
863
3638
282
811
86;6
895
959


2.64
6.05
4.27
6.18
9.13
6;.6;3
7.02
8.73
3.92
8.16
1.25
4.44
2.65
1.45
9.63
3.72
2.38
7.09
8.3
2.55
1.53
3.5
7.15
9.07
2.46
3.19
5.52


974
689
745
585
782
561
569
880
1190
513
601
904
691
522
609
541
986
6;6;6
564
588
1147
510
598
674
568
665
980


0.43
0.58
0.78
0.78
0.6
0.42
0.5
0.26
0.58
0.48
0.84
0.32
0.33
0.6;8
0.22
0.14
0.43
0.6;6
0.43
0.57
0.77
0.52
0.73
0.4
0.37
0.44
0.68


1464
810
1034
1259
1392
1068
713
1424
1332
1355
1117
982
1291
1246
1026
1012
1493
861
1452
758
1310
1070
1180
1166
1192
1165
1412


0.91
0.32
0.37
0.38
0.96
0.15
0.15
0.99
0.47
0.84
0.71
0.46
0.09
0.71
0.27
0.25
0.57
0.96
0.83
0.72
0.69
0.22
0.37
0.36
0.11
0.77
0.16


0.16
0.15
0.37
0.73
0.34
0.76
0.48
0
0.28
0.16
0.9
0.86
0.74
0.78
0.04
0.07
0.18
0.49
0.21
0.49
0.5
0.89
0.88
0.82
0.1
0.6;9
0.09


0.92
0.88
0.89
0.79
0.84
0.93
0.68
0.79
0.64
0.79
1
0.53
0.48
0.94
0.85
0.25
0.75
0.77
0.74
0.78
0.93
0.65
0.95
0.58
0.63
0.93
0.92


1220
730
821
1026
1219
985
6;6;2
904
1242
993
945
928
1028
818
771
704
1307
736
796
619
1299
976
871
1045
1021
1015
1250


0.22
0.09
0.46
0.23
0.13
0.22
0.1
0.16
0.52
0.14
0.13
0.26
0.32
0.35
0.11
0.09
0.19
0.49
0.15
0.39
0.71
0.39
0.07
0.3
0.24
0.25
0.66










continued from previous page

#i x y di ri fl p1 f2 p2 f3 p3
88 0.75 0.63 375 5.28 522 0.59 1206 0.06 1406 0.04
89 0.16 0.41 711 7. 34 515 0.63 707 0.48 1415 0.46
90 0.011 0.21 208 3.99 923 0.87 1049 0.84 12811 0.32
91 0.51 0.76 954 4.06 739 0.79 1186 0.31 1378 0.28
92 0.98 0.32 843 7. 77 733 0.52 834 0.33 1335 0.18
93 0.55 0.39 905 0.08 545 0.74 86;9 0.61 1178 0.61
94 0.36 0.63 729 8.47 1047 0.84 1098 0.66 1460 0.6
95 0.18 0.75 382 63.23 538 0.62 613 0.59 13635 0.38
96 0.09 0.46 91 10.53 513 0.67 521 10.26 1448 0.16
97 0.18 0.67 991 6;.6;2 723 0.92 1119 0.91 1338 0.41
98 0.1 0.38 644 3.13 6;28 0.54 822 0.46 1341 0.11
99 0.25 0.66 539 0.5 848 0.49 942 0.48 1256 0.41
100 0.68 0.49 294 7.27 1069 0.99 1108 0.88 1168 0.66























#i x y di ri #i x y di ri
Open Facilities: 2, 5, 15, 18, 20


APPENDIX C
DATASET USED IN CHAPTER 5

The meaning of each column in Table C-1 is provided as follows: #i denotes the

facility name; (x, y) is the coordinates, di is the demand; fi is the fixed cost; ri is the

penalty cost.

Table C-1. Dataset of DFFM


0.63
0.53
0.89
0.02
0.51
0.53
0.81
0.53
0.89
0.87
0.91
0.32
0.37
0.38
0.96
0.15
0.15
0.99
0.47
0.84
0.71
0.46


0.04
0.32
0.99
0.19
0.32
0.06
0.86
0.36
0.58
0.56
0.16
0.15
0.37
0.73
0.34
0.76
0.48
0
0.28
0.16
0.9
0.86


0.82
0.54
0.91
0.15
0.74
0.58
0.6
0.37
0.7
0.22
0.5
0.3
0.95
0.65
0.53
0.95
0.15
0.31
0.98
0.59
0.46
0.77


0.18
0.7
0.72
0.31
0.16
0.92
0.09
0.19
0.52
0.4
0.45
0.52
0.2
0.07
0.11
0.95
0.13
0.4
0.73
0.04
0.21
0.44


957
202
186
635
737
953
450
188
206
995
429
528
570
938
726
533
565
322
326
663
952
919


5.32
1.9
3.11
1.83
1.34
2.3
7.96
3.42
9.05
4.56
9.87
0.53
3.41
8.98
3.53
1.64
1.36
5.1
1.22
6.95
9.57
1.49


486
548
870
335
446
198
212
903
594
250
472
244
353
183
749
200
321
650
946
143
5635
11


3.14
0.31
0.66
0.22
3.93
3.77
9.92
4.06
1.64
5.11
2.64
6.05
4.27
6.18
9.13
6.63
7.02
8.73
3.92
8.16
1.25
4.44










continued from previous page

#i x y di ri #i x y di ri


0.87
0.69
0.24
0.84
0.49
0.56
0.38
0.43
0.61
0.48
0.81
0.13
0.41
0.72
0.3
0.02
0.62
0.8
0.1
0.15
0.48
0.83
0.82
0.67
0.03
0.26
0.59


0.79
0.15
0.28
0.73
0.24
0.77
0.05
0.22
0.73
0.88
0.75
0.71
0.34
0.14
0.28
1
0.9
0.06
0.98
0.13
0.44
0.22
0.39
0.53
0.73
0.39
0.56


292
48
581
659
986
486
915
282
310
980
134
20
151
615
369
875
73
776
342
929
445
6384
643
771
181
926
733


6.95
2.29
2.9
5.05
8.81
3
6.94
0.88
2.66
3.32
0.6;6
3.08
2.634
3.84
1.74
1.14
2.76
3.36
5.61
9.77
4.32
9.61
1.45
7.46
1.03
7.6
0.15
5.93


0.09
0.71
0.27
0.25
0.57
0.96
0.83
0.72
0.69
0.22
0.37
0.36
0.11
0.77
0.16
0.75
0.16
0.01
0.51
0.98
0.55
0.36
0.18
0.09
0.18
0.1
0.25


0.74
0.78
0.04
0.07
0.18
0.49
0.21
0.49
0.5
0.89
0.88
0.82
0.1
0.69
0.09
0.63
0.41
0.21
0.76
0.32
0.39
0.63
0.75
0.46
0.67
0.38
0.66


374
284
598
720
457
213
550
418
863
36;8
282
811
866 i
895
959
375
711
208
954
843
905
729
382
91
991
6344
539


2.635
1.45
9.63
3.72
2.38
7.09
8.3
2.55
1.53
3.5
7.15
9.07
2.463
3.19
5.52
5.28
7.34
3.99
4.06
7.77
0.08
8.47
6;.23
0.53
6;.6;2
3.13
0.5


0.22 | 0.66 | 326


0.68 | 0.49 | 294 | 7.27









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BIOGRAPHICAL SKETCH

Lezhou Zhan was born in Zh.~ i; I>.- China, in the year of the Horse. He is also

known as Roger whose pronunciation is similar to Lezhou in his hometown dialect.

Prior to college, he graduated from Yueqing Middle School in 1997. He received his

Bachelor of Science in Applied Mathematics and his Bachelor of Science in Business

Administration and Engineering Management from Clue~~~, 11png University in 2001.

Before he transferred to the University of Florida in the fall of 2002, he studied scientific

computation at the Hong K~ong University of Science and Technology in a Master of

Philosophy program. He served as Secretary-General of FACSS, a Chinese student

association at ITF, from 2003 to 2004. He earned his Master of Science and Doctor of

Philosophy in Industrial and Systems Engineering from the University of Florida in

May, 2004 and August, 2007 respectively. His current research interests include reliable

supply chain design, auction mechanism design, operations research models in airline

applications, and system reliability optimization. His work has been presented in various

conferences, book chapters, and journals, including Proceedings of the 2005 Winter

Simulation C'onference, Proceedings of the 2005 IIE Research C'onference and Production

and Op~erations _Ifor..~ry.; n.;.1 He is a member of INFORMS, SIAM, and IIE.





PAGE 1

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IwouldneverbeabletoadequatelythankDr.Zuo-JunMaxShen,mysupervisorandmentor,forhelpingmeindevelopingmyresearchtopics.Heallowedmetoworkfreely,buthewasalwaystherewhenIneededadviceorguidance.Iwanttothankhimnotonlyforhistremendousguidanceandencouragementthroughoutmystudy,butalsohisendlesstrustandunderstanding.Dr.PanosPardalosdeservesmanysincerethanks.HerecruitedmetotheUniversityofFloridaandwasalwaysverysupportivethroughmyyearsthere.IthankDr.JosephP.GeunesandDr.EdwinRomeijnfortheirmanygoodsuggestionsinthisresearchaswellastheirwonderfulcoursesIhavetakenwiththem.Ialsothankanothermemberofmycommittee,Dr.JuanFeng,forhertimeandguidanceinthelastthreeyears.Iamgratefultobothofmyco-authorsDr.JiaweiZhang,andDr.MarkDaskinfortheirincisiveinsights.IalsothankProf.H.TuyforintroducinghismonotonicoptimizationmethodologytomewhilehevisitedGainesvilleinAugust,2003.IacknowledgeallthehelpandnancialsupportfromtheDepartmentofIndustrialandSystemsEngineering.Inparticular,IthankDr.HearnforassigningmetoteachvariouscoursesattheUniversityofFlorida.Mydeepestappreciationgoestomyentirefamily,includingmywife,Ngana,mybrother,Leping,myparentsandin-laws.Icouldnothavenishedthisresearchwithouttheirsupport,love,andencouragement.Mostspecially,IwanttothankmywifeforbeingwithmeinnumerouslatenightsinGainesville,Miami,andGreenwich,andhelpingtoproofreadmymanuscript.Lastbutnottheleast,Ithankmyfriends,Leon,Lian,Shu,Gang,Bin,Jean,Chunhua,Jun,Altannar,Jie,Jonathon,Cindy,Silu,Junmin,andmanyotherswhomademyexperienceattheUniversityofFloridamemorable. 4

PAGE 5

page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 9 CHAPTER 1INTRODUCTION .................................. 11 2RELIABLEFACILITYLOCATIONPROBLEM:MODELSANDHEURISTICS 15 2.1Introduction ................................... 15 2.2LiteratureReview ................................ 17 2.3NotationsandAcronyms ............................ 19 2.4UncapacitatedReliableFacilityLocationProblem:aScenario-BasedModel 20 2.5UncapacitatedReliableFacilityLocationProblemwithaSingle-levelFailureProbability ................................... 22 2.5.1NonlinearIntegerProgrammingModel ................ 22 2.5.2ModelProperties ............................ 23 2.5.3ASpecialCase:UniformFailureProbabilities ............ 25 2.6UncapacitatedReliableFacilityLocationProblemwithMulti-levelFailureProbabilities ................................... 26 2.7SolutionMethodologies ............................. 28 2.7.1SampleAverageApproximationHeuristic ............... 28 2.7.2GreedyMethods ............................. 29 2.7.3GeneticAlgorithmBasedHeuristic .................. 31 2.8ComputationalResults ............................. 35 2.8.1SampleAverageApproximationHeuristic ............... 35 2.8.2GreedyMethods:GAD-HandGADS-H ................ 38 2.8.3GeneticAlgorithmBasedHeuristic .................. 41 2.8.4ApplyingHeuristicstoSolveURFLP-SFP .............. 42 2.8.5URFLP-MFP:GADS-Hvs.GA .................... 45 2.9Conclusions ................................... 46 3UNIFORMUNCAPACITATEDRELIABLEFACILITYLOCATIONPROBLEM:A2.674-APPROXIMATIONALGORITHM .................... 48 3.1Introduction ................................... 48 3.2Formulations .................................. 49 3.3ApproximationAlgorithms ........................... 50 3.4Conclusions ................................... 56 5

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...................................... 56 4SYSTEMRELIABILITYOPTIMIZATIONANDMONOTONICOPTIMIZATION 61 4.1Introduction ................................... 61 4.2AMonotonicBranch-Reduce-BoundAlgorithm ............... 67 4.2.1SelectandBranch ............................ 69 4.2.2ReduceandBound ........................... 70 4.2.3ConvergenceAnalysis .......................... 71 4.2.4AccelerationTechniques ........................ 72 4.3UsingMonotonicBranch-Reduce-BoundAlgorithmtoSolveSystemReliabilityOptimizationProblems ............................. 73 4.3.1RedundancyAllocationOptimization ................. 73 4.3.2Reliability-RedundancyAllocationOptimization ........... 76 4.4Conclusions ................................... 79 5FORTIFYINGTHERELIABILITYOFEXISTINGFACILITIESANDMONOTONICOPTIMIZATION ................................... 81 5.1Introduction ................................... 81 5.2ContinuousFacilityForticationModel .................... 83 5.2.1PropertiesoftheContinuousFacilityForticationModel ...... 84 5.2.2AnExampleoftheContinuousFacilityForticationModel ..... 87 5.3DiscreteFacilityForticationModel ...................... 92 5.3.1PropertiesandAlgorithms ....................... 93 5.3.2ComputationalExperiments ...................... 96 5.4Conclusions ................................... 99 6CONCLUDINGREMARKS ............................. 102 APPENDIX ASYSTEMRELIABILITYCOMPUTATIONINCHAPTER4 .......... 104 BDATASETUSEDINCHAPTER2 ......................... 107 CDATASETUSEDINCHAPTER5 ......................... 113 REFERENCES ....................................... 115 BIOGRAPHICALSKETCH ................................ 120 6

PAGE 7

Table page 2-1Acronyms ....................................... 19 2-2SamplechromosomeformodelURFLP-MFP .................... 32 2-3GA-Hparameters ................................... 32 2-4SelectionprobabilitiesforapopulationwithNPsolutions ............ 33 2-5ObjectivevaluesfromSAA-Hforthe50-nodedataset ............... 36 2-6RunsfromSAA-Hforthe50-nodedataset ..................... 39 2-7Fifty-nodeuniformcase:greedyaddingandexactsolution ............ 40 2-8Fifty-nodeuniformcase:GADS-Handexactsolution ............... 40 2-9ValuesoftheGA-Hparameters ........................... 41 2-10Fifty-nodeuniformcase:GAandexactsolution .................. 41 2-11ObjectivevaluesobtainedfromSAA-HonURFLP-SFPwithdierentsamplesizes .......................................... 44 2-12ComputationalperformanceonURFLP-SFP:GAD-H,GADS-H,GAandtheenumerationmethod ................................. 44 2-13ComputationalresultsforURFLP-MFPusingGADS-HandGA ......... 46 4-1CoecientsinExample1 .............................. 74 4-2CoecientsinExample3 .............................. 77 4-3PerformancecomparisonofExample3 ....................... 78 4-4PerformancecomparisonofExample4 ....................... 79 5-1SolutionofaCFFMmodel .............................. 89 5-2Inputdata(Vi(yi)andPi(yi))forthe3-levelmodel ................ 96 5-3Solutionsofthe3-levelmodel ............................ 97 5-4Solutionsofthe2-levelmodel ............................ 98 B-1DatasetofURFLP-SFP ............................... 107 B-2DatasetofURFLP-MFP:3-level .......................... 109 C-1DatasetofDFFM ................................... 113 7

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Figure page 1-1Researchstructure .................................. 13 2-1Exampleofcrossoveroperationatposition3 .................... 34 2-2Objectiveratioacrossdierentsamplesizesforthe50-nodedataset ....... 37 2-3CPUtimeacrossdierentsamplesizesforthe50-nodedataset .......... 37 2-4EvolutionofthesolutionsfromGA ......................... 42 2-5ComparisonofobjectivevaluesfromGAD-H,GADS-H,GA,andSAA-H .... 45 4-1Seriessystem ..................................... 61 4-2Parallel-seriessystem ................................. 62 4-3Five-componentbridgenetwork ........................... 64 4-4Reduceprocess .................................... 70 4-5Seven-linkARPAnetwork .............................. 78 5-1Totalcostatdierentsystemreliabilitylevel .................... 88 5-2FailureprobabilityatindividualfacilityvssystemreliabilityLevel ........ 91 5-3FrequencyofcompletelyopenfacilityinTable 5-1 ................. 92 5-4Tradeobetweenobjectiveandresourceused ................... 98 5-5Forticationlevelatindividualfacilitybyresourceused .............. 100 A-1Congurationsbasedonstateofcomponent5inFigure 4-3 :A)Component5works;B)Component5fails ............................. 104 A-2Congurationsbasedonstateofsubsystem4inFigure 4-5 :A)Subsystem4works;B)Subsystem4fails ............................. 106 8

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Uncertaintyisoneoftheelementsthatmakethisworldsofascinatinganddynamic.However,theexistenceofuncertaintyalsoposesagreatchallengetoreliablesystemdesign.Ourstudyusesvariousmodelsandalgorithmstoaddressreliabilityissuesinthecontextof(1)theuncapacitatedfacilitylocationproblemwherefacilitiesarevulnerable,and(2)thesystemreliabilityproblemwherecomponentsaresubjecttofail. Werststudytheuncapacitatedreliablefacilitylocationprobleminwhichthefailureprobabilitiesaresite-specic.Theproblemisformulatedasatwo-stagestochasticprogramandthenanonlinearintegerprogram.Severalheuristicsthatcanproducenear-optimalsolutionsareproposedforthiscomputationallydicultproblem.Theeectivenessoftheheuristicsistestedthroughextensivecomputationalstudies.Thecomputationalresultsalsoleadtosomemanagerialinsights.Forthespecialcasewherethefailureprobabilityateachfacilityisaconstant(independentofthefacility),weprovideanapproximationalgorithmwithaworst-caseboundof2.674. Anotherpartofourresearchisrelatedtothedevelopmentandapplicationofamonotonicbranch-reduce-boundalgorithm,apowerfultooltoobtaingloballyoptimalsolutiontoproblemsinwhichboththeobjectivefunctionandconstraintspossessmonotonicity.Wetailorthealgorithmtosolveamixedintegernonlinearprogrammingproblem.Itsconvergenceanalysisandaccelerationtechniquesarealsodiscussed.Thealgorithmisthensuccessfullyappliedtosolvesystemreliabilityoptimizationproblems 9

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Wealsodevelopseveralmodelsthatcanbeusedtofortifythereliabilityoftheexistingfacilities.Theyaretheextensionstothemodelsintherstpartofthedissertationandoerinsightsonwhichfacilitytochooseandtowhatextentitshouldbefortied.Thepropertiesandsolutionmethodologiesofthemodelsarediscussed.Inparticular,amonotonicbranch-reduce-boundalgorithmisusedtosolveoneofthesemodels.Theeciencyofthealgorithmisdemonstratedinthecomputationalresults. 10

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Ourstudyfocusesonreliabilityissuesarisinginfacilitylocationdesignproblemsandcomplexsystems.Intheformercase,weconsidermathematicalmodelsthatminimizethesumoffacilityopeningcostsandexpectedserviceandpenaltycostswhenfacilitiesaresubjecttofailfromtimetotime.Thesefailuresmaycomefromdisruptiveevents(e.g.laborstrikes,supplierbusinessfailures,terroristattacks),ornaturaldisasters(e.g.hurricanes,earthquake).Facility-specicfailureprobabilitiesareexplicitlyconsideredinourmodels.Tothebestofourknowledge,theseappeartobetherstsuchmodelsintheliterature.Thesemodelshelptomakedecisionsinthesystemdesignphase.Severalheuristicsandanapproximationalgorithmareproposedforsolvingthesemodels. Iffacilitieshavebeenbuiltbutarestillsubjecttofail,weconsidermodelstofortifythereliabilityoftheexistingsystemgivenlimitedforticationresources.Thesemodelscanbereducedtoaspecialclassofglobaloptimizationproblems,calledmonotonicoptimization,inwhichboththeobjectivefunctionandconstraintspossessmonotonicity.Aspecializedmonotonicbranch-reduce-boundalgorithmisdevelopedtoecientlysolvetheseproblems. Wealsoexaminereliabilityissuesincomplexindustrialandmilitarysystems.Thereliabilityofsuchasystemismeasuredbytheprobabilityofsuccessfuloperation.Weaddresstheissueofallocatingunreliablecomponentsinthesystemtoachievethemaximumprobabilityofsuccessfuloperation,adierentobjectivefromthatusedinthefacilitylocationmodel.Theproblemisgenerallycategorizedasasystemreliabilityoptimizationproblem,includingtheclassesofredundancyallocationandreliability-redundancyallocationoptimizationproblems.Inredundancyallocationoptimization,oneisgiventheoptiontoallocatetheappropriatelevelsofredundancytomaximizereliabilityorminimizethecostofasystemgiventhedesignconstraints.Forexample,ifacomponentofreliabilitylevelat0.9isassignedinparalleltobackup 11

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Thecontributionsofourstudyencompasstheoreticaldevelopments,computationalalgorithmsandpracticalapplications.Inparticular,wemakethefollowingcontributions: 12

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ThestructureofourresearchisdepictedinFigure 1-1 .InChapter 2 ,wepresentseveralmodelsfortheuncapacitatedreliablefacilitylocationprobleminwhichsomefacilitiesaresubjecttofailurefromtimetotime.Thesemodelsarethefoundationofourresearch.Besidesthegeneralscenario-basedmodel,theyincludethecaseinwhicheachfacilityhasasite-specicfailureprobability,andthecaseinwhicheachfacilityhasmulti-levelfailureprobabilities.Thepropertiesanddierentformulationsofthemodelsarethoroughlydiscussed.Severalheuristicsarepresentedalongwiththecomputationalresults. Figure1-1. Researchstructure InChapter 3 ,wepresentatighterapproximationalgorithmwithaworst-caseboundof2.674foraspecialcaseoftheuncapacitatedreliablefacilitylocationproblem,whereallfailureprobabilitiesareidentical. InChapter 4 ,wepresentamonotonicbranch-reduce-boundalgorithmforaspecialcaseofthenonlinearmixed/pureintegerprogrammingproblemwhereboth 13

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BasedonthemodelsinChapter 2 ,wedeveloptwomodelsthatareusedtofortifythereliabilityoftheexistingfacilities.Thepropertiesandsolutionmethodologiesofthemodelsarediscussed.Inparticular,themonotonicbranch-reduce-boundalgorithmpresentedinChapter 4 isusedtosolveoneofthesemodels.Theeciencyofthealgorithmisdemonstratedthroughthecomputationalresults. ThisdissertationisconcludedinChapter 6 withadiscussiononfutureresearchdirections. 14

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34 ],[ 8 ],and[ 45 ]).Mostmodelsintheliteraturehavetreatedfacilitiesasiftheywouldneverfail;inotherwords,theywerecompletelyreliable.Inthischapter,werelaxthisassumptiontomodelamorerealisticcase. Thereliabilityissueweconsiderisundertheframeworkoftheso-calleduncapacitatedfacilitylocationproblem(UFLP).InUFLP,wearegivenasetofdemandpoints,asetofcandidatesites,thecostofopeningafacilityateachlocation,andthecostofconnectingeachdemandpointtoanyfacility.Theobjectiveistoopenasetoffacilitiesfromthecandidatesitesandassigneachdemandpointtoanopenfacilitysoastominimizethetotalfacilityopeningandconnectioncosts. UFLPanditsgeneralizationsareNP-hard,i.e.,unlessP=NPtheydonotadmitpolynomial-timealgorithmstondanoptimalsolution.ThereisavastliteratureontheseNP-hardfacilitylocationproblemsandmanysolutionapproacheshavebeendevelopedinthelastfourdecades,includingintegerprogramming,meta-heuristics,andapproximationalgorithms.Onecommonassumptioninthisliteratureisthattheinputparametersoftheproblems(costs,demands,facilitycapacities,etc.)aredeterministic.However,suchassumptionsmaynotbevalidinmanyrealisticsituationssincemanyinputparametersinthemodelareuncertainduringthedecision-makingprocess. 15

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47 ]).Thecommonfeatureofthereceiver-sideandin-betweenuncertaintiesisthattheuncertaintydoesnotchangethetopologyoftheprovider-receivernetworkoncethefacilitieshavebeenbuilt.However,thisisnotthecaseifthebuiltfacilitiesaresubjecttofail(provider-sideuncertainty).Ifafacilityfails,customersoriginallyassignedtoithavetobereassignedtoother(operational)facilities,andthustheconnectioncostchanges(usuallyincreases). Wefocusonthereliabilityissueofprovider-sideuncertaintyinthischapter.Theuncertaintyismodeledusingtwodierentapproaches:1)byasetofscenariosthatspecifywhichsubsetofthefacilitieswillbecomenon-operational;or2)byanindividualandindependentfailureprobabilityinherentineachfacility.Althougheachdemandpointneedstobeservedbyoneoperationalfacilityonly,itshouldbeassignedtoagroupoffacilitiesthatareorderedbylevels:intheeventofthelowestlevelfacilitybecomingnon-operational,thedemandcanthenbeservedbythenextlevelfacilitythatisoperational;andsoon.Ifalloperationalfacilitiesaretoofarawayfromademandpoint,onemaychoosenottoservethisdemandpointbypayingapenaltycost.Theobjectiveisthustominimizethefacilityopeningcostplustheexpectedconnectionandpenaltycosts.Thisproblemwillbereferredtoastheuncapacitatedreliablefacilitylocationproblem(URFLP). Inparticular,twovariantsofURFLPareconsideredinthischapterintermsofthecharacteristicsofthefailureprobabilityateachfacility.Intherstone,weassumethatthereisonlyonesite-specicfailureprobabilityateachfacility.We 16

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URFLPisclearlyNP-hardasitgeneralizesUFLP.WeproposeseveralheuristicstosolveURFLP.Theyincludethesampleaverageapproximationheuristicforthescenario-basedmodel,thegreedyaddingheuristics,thegreedyaddingandsubstitutionheuristics,andthegeneticalgorithmforthenonlinearintegerprogrammingmodel. Therestofthischapterisorganizedasfollows.InSection 2.2 ,wereviewtherelatedliteratureandprovidesomebasicbackgroundforourmodels.ThenotationandacronymsareintroducedinSection 2.3 .InSection 2.4 ,ascenario-basedmodelisproposed,whichisfollowedbythenonlinearintegermodelforURFLP-SPFinSection 2.5 .Section 2.6 containsthenonlinearintegermodelforURFLP-SPF.ThethreeheuristicsarepresentedinSection 2.7 .InSection 2.8 ,weconductcomputationalstudiesontheperformanceoftheheuristics.InSection 2.9 ,weconcludethechapterbysuggestingseveralfutureresearchdirections. 38 47 ]).However,aswepointedoutintheIntroduction,amajorityofthecurrentliteraturemainlydealswiththereceiver-sideand/orin-betweenuncertainties.Thisincludes[ 63 ],[ 10 ],[ 9 ],[ 7 ]and[ 42 ]amongothers. Thefollowingtwopapers,[ 48 ]and[ 5 ],arecloselyrelatedtothischapter.In[ 48 ],theauthorsassumethatsomefacilitiesareperfectlyreliablewhileothersaresubjecttofailurewiththesameprobability.Onthecontrary,weassumethatthefailureprobability 17

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5 ],whichisbasedonthep-medianproblemratherthantheframeworkofUFLP. Thereisalsoasmallstrandofliteraturedevotedtoaddressingtheforticationofreliabilityforexistingfacilities,whichincludes[ 43 ],[ 44 ],and[ 49 ].Thesemodelstypicallyfocusontheinterdiction-forticationframeworkbaseduponthep-medianfacilitylocationproblem.Theyaregenerallyformulatedasbilevelprogrammingmodels.Theirmainfocusistoidentifytheexistingcriticalfacilitiestoprotectundertheeventsofdisruption. Inourmodel,thefailureprobabilitiesaresite-specic,whichsignicantlycomplicatestheproblemwhenformulatingitasamathematicalprogram.ThemodelisfurtherextendedtoURFLP-MFP,whereeachfacilityisallowedtohavemultiplelevelsofthefailureprobabilities.Weproposetwodierentmodelingapproaches:ascenario-basedstochasticprogrammingapproachandanonlinearintegerprogrammingapproach.Thescenario-basedmodelisattractiveduetoitsstructuralsimplicityanditsabilitytomodeldependenceamongrandomparameters.Butthemodelbecomescomputationallyexpensiveasthenumberofscenariosincreases.Ifthenumberofscenariosistoolarge,thenonlinearintegerprogrammingapproachprovidesanalternativewaytotackletheproblem. Thesampleaverageapproximationmethodiswidelyusedforsolvingcomplicatedstochasticdiscreteoptimizationproblems,e.g.,[ 24 ],[ 42 ],and[ 61 ].Thebasicideaofthismethodistouseasampleaveragefunctiontoestimatetheexpectedvaluefunction.Thustheoriginalproblemistransformedtotheonethatcanbeecientlysolved. Othertypesofthecommonlyusedheuristicsinoptimizationarethelocalsearchanditerativeimprovementalgorithms([ 13 ],[ 16 ],[ 4 ]and[ 3 ]).Theseheuristicsstartwithanemptysetandrepeatedlyconsideraddingapotentialfacilityintothesolution.Ortheystartwithannon-emptysetandrepeatedlyconsiderdeletingorsubstitutingafacilityin 18

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60 ]. Wealsoapplyageneticalgorithm(GA)basedheuristictosolvethemodelswedevelop.GAsimitatethenaturalselectioninbiologicalevolution.Assolutiontechniques,theymaintainalargenumberofsolutions,calledthepopulation,andalloweachmemberofthepopulation(calledachromosome)toevolveiterativelyintogoodones.SomegooddescriptionsofGAsareprovidedin[ 11 14 39 ]. GAshavebeenusedtosolvemanycombinatorialoptimizationproblemswithsuccess,includingvariousfacilitylocationproblems,e.g.[ 22 ]and[ 2 ].Inthischapter,wedesignaspecializedGAforthemodelsinwhichweareinterested.Acomputationalstudyonallthreeheuristicsisalsoprovided. TheacronymslistedinTable 2-1 arefrequentlyusedinthischapter. Table2-1. Acronyms Acronym Meaning UFLP Uncapacitatedfacilitylocationproblem URFLP-SFP Uncapacitatedreliablefacilitylocationproblemwithasingle-levelfailureprobability URFLP-MFP Uncapacitatedreliablefacilitylocationproblemwithmulti-levelfailureprobabilities GA Geneticalgorithm SAA-H Sampleaverageapproximationheuristic GAD-H Greedyaddingheuristic GADS-H Greedyaddingandsubstitutionheuristic 19

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LetSbethesetofscenarios.ForanyA2S,letpAbetheprobabilitythatscenarioAhappens.ThenURFLPcanbeformulatedasthefollowingtwo-stagestochasticprogram. minimizeXi2Ffiyi+XA2SpAgA(y)subjecttoyi2f0;1g; wheregA(y)=minXj2DXi2FdjcijxAij+Xj2DdjrjzAj s.t.Xi2FxAij+zAj=1;8j2D Intheaboveformulation,thebinaryvariableyiindicatesiffacilityiisopenedintherststage.ParameterIA;iindicatesiffacilityiisoperationalunderscenarioA,whichisaninputregardlessofthevalueofyi.VariablexAijistheassignmentvariablewhichindicates 20

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2{1 ,istominimizethesumofthexedcostandtheexpectedsecondstagecost.Theobjectiveofthesecondstage,i.e. 2{2 ,istominimizetheserviceandpenaltycost.Constraints( 2{3 )ensurethatclientjiseitherassignedtoafacilityorsubjecttoapenaltyateachlevelkinScenarioA.Constraints( 2{4 )and( 2{5 )makesurethatnoclientisassignedtoanunopenfacilityoranonfunctionalfacilityrespectively. Itisstraightforwardtoshowthattheformulation( 2{1 )isequivalenttothefollowingmathematicalprogram. (URFLP-SP)minimizeXi2Ffiyi+XASpAXj2DXi2FdjcijxAij+Xj2DdjrjzAj!s.t.Xi2FxAij+zAj=1;8j2D;ASxAijyiIA;i;8i2F;j2D;ASyi;xAij;zAj2f0;1g: However,whenthefailureprobabilitiesareindependent,thepossiblenumberofscenarioscanbeextremelylarge.Therefore,thenumberofvariablesandconstraintsinURFLP-SPisexponentiallylargeaccordingly,whichmakesitextremelydiculttosolve.Underthissituation,weproposeseveralalternativenonlinearintegerprogrammingformulationsandecientsolutionalgorithms.Wediscussthesealternativeformulationsinthenexttwosections. 21

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2.5.1NonlinearIntegerProgrammingModel 48 ]toamoregeneralsetting.Comparingto[ 48 ],URFLP-SFPcanbeinterpretedslightlydierentlyasfollows.Eachclientshouldbeassignedtoasetoffacilitiesinitially.Thefacilitiesassignedtoanyclientcanbedierentiatedbythelevels:incasealowerlevelfacilityfails,thenextlevelfacility,ifoperational,willprovideserviceinstead. Mathematically,wedenetwotypesofnewbinaryvariablesxkij;zkjtocapturedierentleveloffacilitiesforaclientj.Inparticular,xkij=1iffacilityiisthek-thlevelbackupfacilityofclientjandotherwise,xkij=0.zkj=1ifjhas(k1)-thbackupfacility,buthasnok-thbackupfacilitysothatjincursapenaltycostatlevelk. Giventhevariablesxkij;zkj,onecancomputetheexpectedtotalservicecostasfollows.Consideraclientjanditsexpectedservicecostatitslevel-kfacility.Clientjisservedbyitslevel-kfacilityonlyifallitsassignedfacilitiesatlowerlevelsbecomenon-operational.Ontheotherhand,foranyfacilityl,ifitisonthelowerlevel(i.e,lessthank)fordemandnodej,thenPk1s=1xslj=1,otherwisePk1s=1xslj=0.Itfollowsthatforclientj,theprobabilitythatallitslowerlevelfacilitiesfailisQl2FpPk1s=1xsljl.Ifjisseveredbyfacilityi,asj'slevel-kbackupfacility,thenfacilityihastobeoperationalwhichoccurswithprobability(1pi).Therefore,theexpectedservicecostofclientjatlevelkisPi2Fdjcijxkij(1pi)Ql2FpPk1s=1xsljl.Similarly,wecancalculatetheexpectedpenaltycostofclientjatlevelk,whichisQl2FpPk1s=1xsljldjrjzkj. 22

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minimizeXi2Ffiyi+Xj2DjFjXk=1Xi2Fdjcijxkij(1pi)Yl2FpPk1s=1xsljl+Xj2DjFj+1Xk=1Yl2FpPk1s=1xsljldjrjzkj subjecttoXi2Fxkij+kXt=1ztj=1;8j2D;k=1;:::;jFj+1 (2{8) Thedecisionvariablesxkij;zkjaredenedearlier.Theindicatorvariableyi=1iffacilityiisopenintherststage;otherwiseyi=0.Theobjectivefunction( 2{7 )isthesummationofthefacilitycost,theexpectedservicecost,andtheexpectedpenaltycost.Constraints( 2{8 )ensurethatclientjiseitherassignedtoafacilityorsubjecttoapenaltyateachlevelk.Constraints( 2{9 )makesurethatnoclientisassignedtoanunopenfacility.Constraints( 2{10 )prohibitaclientfrombeingassignedtoaspecicfacilityatmorethanonelevel.Notethatconstraints( 2{9 )and( 2{10 )canbetightenedas 23

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Proof. Inparticular,ifwesetxk+1uj=1andxkvj=1withthevaluesofothervariablesunchanged,wecancomputethenewobjectivevalue.Thedierencebetweenthenewobjectivevalueandtheoriginaloneis AnimplicationofProposition 2.1 isthatifthesetofopenfacilitiesisdetermined,thenitistrivialtosolvethelevelassignmentproblemforeachclient:assigninglevelsaccordingtotherelativedistancesofdierentfacilitiestotheclient.Ifatsomelevelthedistanceisbeyondthepenaltycost,thennofacilitywillbeassignedatthislevel(andhigherones)andthedemandnodesimplytakesthe(cheaper)penalty. Wewouldliketopointouttherelationshipbetweenformulation(URFLP-SP)andformulation(URFLP-SFP).Sincethesetwoformulationsarejusttwowaysofmodelingthesameproblem,theyshouldhavethesameminimumcostaslongastheinputstothetwomodelsareconsistent.Informulation(URFLP-SFP),eachfacilityihasindependentfailureprobabilitypi.Thisimpliesthatthereare2jFjscenariosandtheprobabilitythat 24

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48 ]. Basedontheaboveobservation,weareabletoreduceformulation(URFLP-SFP)toalinearintegerprogramasfollows. (URFLP-IP) minimizeXi2Ffiyi+Xj2DjFjXk=1Xi2Fdjcijxkij(1p)pk1+Xj2DjFj+1Xk=1pk1djrjzkj subjecttoXi2Fxkij+kXt=1ztj=1;8j2D;k=1;:::;jFj+1jFjXk=1xkijyi;8i2F;j2Dxkij;zkj;yi2f0;1g: 2.8 25

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Givendierentinvestmentlevelsatfacilityi,theoutputofthefailureprobabilityatfacilityi,P0i(yti),isdeterminedbyyti: (2{14) withf0i=0,P(0)=1,andP()isadecreasingfunction.f0i=0andP(0)=1implythatifthereisnoinvestmentatfacilityi,thenitiscompletelynonfunctional. Afterthefailureprobabilityatfacilityiisdetermined,URFLP-MFPisessentiallynodierentfromURFLP-SFP.Inthefollowingformulation,piintheobjectivefunction 2{7 ofURFLP-SFPisreplacedbyP0i(yti). 26

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minimizeXi2FXt(ftiyti)+Xj2DjFjXk=1Xi2Fdjcijxkij(1P0i(yti))Yl2F(P0l(ytl))Pk1s=1xslj+Xj2DjFj+1Xk=1Yl2F(P0l(ytl))Pk1s=1xsljdjrjzkj subjecttoXi2Fxkij+kXt=1ztj=18j;k=1;:::;jFj+1 (2{16) whereconstraints( 2{18 )ensureatmostoneinvestmentlevelisallowedateachfacilityandalltheotherconstraintsaresimilartotheonesinURFLP-SFP.NotethatProposition 2.1 stillholdsinthismodel. Theaboveformulation,URFLP-MFP-1,isabinarymodel.URFLP-MFPcanalsobemodeledasaregularintegermodelbyreinterpretingthedenitionofyiastheinvestmentlevelatfacilityi.Thus,yiisnotbinaryanymore;0yiUi,whereUiisthehighestlevelatwhichfacilityicanbepossiblybuilt.ThecorrespondingfailureprobabilityandthexedcostatfacilityiaredenotedbyfunctionsPi(yi)andFi(yi)respectively.Pi(yi);i2F,arenonincreasingfunctionsofyiandPi(0)=1,whereasFi(yi);i2F,arenondecreasingfunctionsofyiandFi(0)=0.ThenURFLP-MFPcanbemodeledasfollows. 27

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minimizeXi2FFi(yi)+Xj2DjFjXk=1Xi2Fdjcijxkij(1Pi(yi))Yl2F(Pl(yl))Pk1s=1xslj+Xj2DjFj+1Xk=1Yl2F(Pl(yl))Pk1s=1xsljdjrjzkj subjecttoXi2Fxkij+kXt=1ztj=18j;k=1;:::;jFj+1 (2{22) 0yiUi;8i2F; whereconstraints( 2{23 )ensurenoclientisassignedtoanunopenfacility.Thatis,whenyi=0,xkij=0,8i;j;k=1;:::;jFj.Becauseofthebinaryconstraints( 2{26 )onxkij,therighthandsideofconstraints( 2{23 )canbereplacedbyyiwithoutaectingthefeasibledomain. 2.7.1SampleAverageApproximationHeuristic 24 ],[ 42 ],and[ 61 ]).Thebasicideaofthismethodistorandomlygeneratesamples,thenuseasampleaveragefunctiontoestimatethetrueexpectedvaluefunction.Bydoingso,theoriginalproblemisreducedtoarelativelysmallproblemthatcanberepeatedlysolved.Suchanapproachhasbeenusedbyvariousauthorsovertheyears.WeapplythefollowingprocedurestosolvemodelURFLP-SP. 28

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minimizeXi2Ffiyi+NXs=11 RepeatthisstepMtimes.Foreachm=1;2;;M,letymandvmbethecorrespondingoptimalsolutionanditsoptimalobjectivevaluerespectively.InviewofProposition 2.1 ,thelevelassignmentdecision(thesecondstagedecision)canbesolelydeterminedbyym.Computethetrueobjectivevalue^vmusingtheformulationofURFLP-SFPforeachym,m=1;2;;M. Tworemarksareinorder.First,inastandardSAAapproach([ 42 ]and[ 61 ]),oneadditionalindependentsampleisneededtoestimatethetrueexpectedvalue^vm.Butinourcase,ananalyticalformulaisreadyforestimatingthetrueexpectedvalue.Second,theaverageofthevmvalues,i.e.vM=PMm=1vm,doesnotprovideastatisticallowerboundfortheoptimalvalue.Thisisdierentfromtheresultin[ 42 ]wheretheuncertaintyonlycomesfromdemand-side.Inthecurrentmodel,dierentsamplesmayleadtodierentsolutionspacesoftheproblem,duetotheprovider-sideuncertainty.ThereforetheaveragevaluevMisnolongerunbiasedtothetrueexpectedvalue.Nonetheless,vMmaystillserveasagoodindicationofthequalityofthesolutionfromtheSAAapproach,aswewillillustrateinthecomputationaltests. 29

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2.1 ensuresthatthelevelassignmentscanbeeasilyderivedforagivensetofopenfacilities.Therefore,onecanconcentrateonselectingasetofopenfacilitieswithoutworryingtoomuchonthedecisionsoflevelassignment.Letv(T)denotetheobjectivefunctionvaluegivenbythesetofopenfacilities,T.LetTtbethesetofopenfacilitiesatstept,andbetheemptyset. Ingeneral,aswecanseefromthecomputationaltestslater,thegreedyaddingheuristicisabletondahighqualitysolutionveryeciently.ThecomplexityofthisheuristicisO(n4logn),wheren=jFj.GivenTt1,ittakesO(nlogn)todothelevelassignmentsforeachnode,mainlybecauseitinvolvesasortprocessthatisincomplexityofO(nlogn).Thereisnsuchnodes,soittakesO(n2logn)toevaluatethevalueofv(Tt1).Intheworstcase,ittakesnsuchevaluationstogetthemostcosteectivefacility,jtatstept.Thegreedyaddingprocessiteratesatmostntimes,whichleadstothecomplexityofO(n4logn)forGAD-H. Afterthegreedyaddingheuristic,weperformthefollowinggreedysubstitutionheuristictofurtherimprovethesolution:ateachiteration,asubstitutefacilityischosentoreplacetheexistingopenfacilityifdoingsoreducesthetotalcostthemost.Thisprocedureisrepeateduntilnosubstitutefacilitycanbefoundtofurtherreducethetotalcost.Thesubstitutioncanbeanullfacility.Replacinganopenfacilitywithanull 30

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11 14 39 ]. Inprinciple,aGAcanbeappliedtoanyoptimizationproblem.ButthereisnogenericGAsinceitrequiresmanydesigndecisions,suchastheencodingofthechromosome,theselectionofparents,themethodofthecrossoveroperator.Inthissection,wedescribeaGAthatissuitableforURFLP. 31

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2.1 ,wedonotneedanexplicitencodingofthedemand-to-facilityassignments.FormodelURFLP-MFP,anadditionalelementisencodedtorepresentthelevelatwhichweinvestinanopenfacility.Table 2-2 showstheencodingforasystemwith10candidatesitesformodelURFLP-MFP,forexample,withopenfacilitiesatnodes3,4and8atinvestmentlevels2,1and3respectively. Table2-2. SamplechromosomeformodelURFLP-MFP Candidatesite 1 2 3 4 5 6 7 8 9 10 Open? 0 0 1 1 0 0 0 1 0 0 Investmentlevel 0 0 2 1 0 0 0 3 0 0 Chromosomesareevaluatedbasedonthevalueofobjectivefunction.Achromosomewithasmallerobjectivevalueistterthanonewithalargerobjectivevalue.ThefollowingparametersareemployedinourdescriptionofheuristicGA-H. Table2-3. GA-Hparameters Parameter Notation Populationsize 32

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Themainoperatoriscrossover.Twosolutionsareselectedatrandomfromthepopulationatthecurrentgeneration,withabiastowardthebettersolutions.TheprobabilityofselectingthejthbestsolutionisgivenbyNP+1j NP(NP+1)=2.NoteNPisthenumberofsolutionsinthepopulation.Thedenominator,NP(NP+1)=2,isthesumfrom1toNP.Thenumerator,NP+1j,isthereverseorderofj'stnessamongallNPsolutions.ThesevaluesarelistedinTable 2-4 astheweightinprobabilityevaluation.Aswecanseethatthebetterasolutionis,thehigherweightitisassigned. Table2-4. SelectionprobabilitiesforapopulationwithNPsolutions SolutionRank WeightinProbabilityEvaluation Probability 1 NP ... ... NP+1j NP(NP+1)=2 ... ... 2-1 fortwosolutionswiththeone-pointcrossoverpositionat3.Encodingofthechildsolutioninthisexampleisthefollowing:valuesofpositions1to3arefromthoseinthesamepositions 33

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Figure2-1. Exampleofcrossoveroperationatposition3 Afterachildsolutionhasbeenconstructedinthemanneroutlinedabove,withprobabilityPM,thesolutionismutated.Mutationisaccomplishedbyrandomlyselectingtwocandidatesites:oneatwhichafacilityopensandoneatwhichafacilitycloses;thenswappingtheirstates:fromopen(\1")toclose(\0"),andfromclose(\0")toopen(\1").InthecaseofmodelURFLP-SFP,arandomlyselectedinvestmentlevelisassociatedwiththenewlyopenfacilitysite. Ifthechildsolutiongeneratedinthismannerdiersfromallothersolutionsintheemergingpopulation,itisaddedtothepopulation;ifitdoesnot,theentireprocess(ofparentselection,crossover,andmutation)isrepeated.WecontinueaddingsolutionstothepopulationuntilthepopulationcontainsNPtotalsolutions.Inotherwords,thesizeofeachgenerationismaintainedtobethesame. Thewholeprocessisrepeateduntiloneofthefollowingterminationcriteriaismet:(1)thealgorithmreachesNGgenerations,or(2)itfailstoimprovethebest-knownsolutioninNMgenerations. 34

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Thetestdatasetisgeneratedasfollows.CoordinatesofthesitesweredrawnfromU[0;1]U[0;1],demandofeachsitewasdrawnfromU[0;1000]androundedtothenearestinteger,xedfacilitycostsweredrawnfromU[500;1500]androundedtothenearestinteger,andpenaltycostsweredrawnfromU[0;15].Further,thetransportationcostcijissettobetheEuclideandistancebetweenpointsiandj.Thenumberofsitesvariesfrom10to100.ThedatasetisavailableinAppendix B AllthealgorithmswerecodedinC++andtestedonaDellOptiplexGX620computerrunningtheWindowsXPoperatingsystemwithaPentiumIV3.6GHzprocessorand1.0GBRAM. 2-5 liststheobjectivevaluesobtainedfromSAA-HwhenM=1andthesamplesizevariesfrom10to200. ItisclearfromTable 2-5 thatthesolutionqualitycanbeimprovedbyincreasingthesamplesize.TheratiosoftheobjectivevalueobtainedfromSAA-Hwithsamplesizes10,50,100,150,200,and250totheoptimalvalueareplottedinFigure 2-2 35

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ObjectivevaluesfromSAA-Hforthe50-nodedataset FailureProbability N10 N20 N30 N40 N50 0 7197.27 7197.27 7197.27 7197.27 7197.27 0.1 7956.03 7956.03 7763.80 7763.80 7763.80 0.2 9429.49 8770.31 8425.99 8425.99 8425.99 0.3 9908.52 10059.90 9669.49 10201.50 10095.20 0.4 11546.90 11753.50 11984.00 13887.60 11937.60 0.5 18727.50 15128.1 13120.60 15052.80 13546.40 0.6 27429.00 18161.70 17946.40 17946.40 17946.40 0.7 32965.30 33139.20 27374.80 23659.30 23690.40 0.8 62876.90 46387.50 35743.60 35743.60 35743.60 0.9 84406.10 69255.40 54722.30 54728.10 55647.40 1.0 128009.00 128009.00 128009.00 128009.00 128009.00 FailureProbability N100 N150 N200 N250 7197.27 7197.27 7197.27 7197.27 7197.27 0.1 7763.80 7763.80 7763.80 7763.80 7763.80 0.2 8425.99 8425.99 8425.99 8425.99 8425.99 0.3 9387.85 9387.85 9275.99 9275.99 9275.99 0.4 10848.20 10599.90 10566.30 10762.00 10253.90 0.5 13157.20 13041.50 13099.70 12134.20 11603.00 0.6 17762.80 15609.50 14559.10 14559.10 13416.80 0.7 20740.30 20671.60 19070.90 17289.00 16157.20 0.8 29218.10 26731.10 24066.80 23252.20 21500.70 0.9 46546.80 45061.00 42555.50 38316.90 35987.70 1.0 128009.00 128009.00 128009.00 128009.00 128009.00 Figure 2-2 showsthatSAA-Hobtainsfairlygoodsolutionswithasmallsamplesizewhenthefailureprobabilityislow(p0:4).Incontrast,alargesamplesizeisneededtoachievesuchqualityofsolutionsforthecaseswithahigherfailureprobability.Thefollowingcouldbeapossibleexplanation.Notethatasampleisacollectionofdierentscenarios.Whenthefailureprobabilityislow,themajorityofthefacilitiesarecandidatesitesforopeningineachscenario,sothesetsofcandidatefacilitiesaresimilarindierentscenarios.Thus,anyindividualsamplecancapturethecharacteristicsofthesystemprettywell,andthecorrespondingsolutionobtainedfromSAA-Hisclosetooptimal.Fortheextremecasewherep=0,allfacilitiesareavailabletoopenineachscenario,sothesetsofavailablefacilitiesarethesameineachscenarioandtheSAA-Hcanproducetheexactsolutioninthiscase.Inthecaseofthehigherfailureprobability,thecandidatesitesfor 36

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Objectiveratioacrossdierentsamplesizesforthe50-nodedataset Figure2-3. CPUtimeacrossdierentsamplesizesforthe50-nodedataset openingineachscenarioisrelativelysmallandascenariocanbequitedierentfromanotherone.Thusanincreasedsizeofsamplecanhelptocapturethecharacteristicsofthesystemuncertainty. Figure 2-3 depictsthecomputationtimefromtherunwithsamplesizes10,20,30,and40.ItshowsthatthecasewithN=10istheonlyonethatrequiresslightlylesstimethantheexactalgorithm,whereasothersrequiremoretimeasthesamplesize 37

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2-3 isthetaileectoftheCPUtimeintermsofthefailureprobability.SAA-Hspendsmoretimetoobtainasolutionwhenthefailureprobabilityisaround0.5.Onepossiblewaytoexplainthisphenomenaisthefollowing:whenthefailureprobabilityisaround0.5,theconstraintsxsijyiIAs;iamongdierentsamplesarequitedierent.Asaresult,theproblemsizeincreases,sodoesthecomputationaltime. Next,weexaminetheeectofthereplicationnumber(M)onthesolutionqualitybyxingN=30.Table 2-6 providestheobjectivevaluesobtainedwhenM=5;10;15;20.Fromtheobjectivevaluesobtainedindierentreplicationnumbers,wecanseethattheincreaseofthereplicationnumberhasnotaectedthesolutionqualitytoomuch.Thegapinthistableisdenedas^vminvM Ingeneral,SAA-Hiscapabletoproduceafairlygoodsolutionwithalargesamplesizefortheuniformcase.Butitalsorequiresatremendousamountoftimetodosoandmayrunoutofmemoryduetotheincreaseofproblemsize.WedeferpresentingthecomputationalresultsofURFLP-SFPtoSection 2.8.4 Table 2-7 liststhecomputationalresultsofa50-nodedatasetwhenthefailureprobabilityvariesfrom0to1.Therstcolumn,P,isthefailureprobabilityateachfacility.The\gap"columnisdenedasthepercentagedierencebetweenthecostofthesolutionobtainedbyGAD-Handtheoptimalcost. AswecanseefromTable 2-7 ,GAD-Hndsoptimalornear-optimalsolutionsinmostcasesinlessthan0.05seconds.ComparedtotheexactmethodusingCPLEX,ittakes 38

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RunsfromSAA-Hforthe50-nodedataset P ^vmin ^vmin Exact 0.0 7197.27 7197.27 0.00 7197.27 7197.27 0.00 7197.27 0.1 7763.80 7687.03 1.00 7763.80 7760.14 0.05 7763.80 0.2 8425.99 8315.65 1.33 8425.99 8436.50 -0.12 8425.99 0.3 9414.40 9054.28 6.79 9378.06 9112.31 3.32 9275.99 0.4 10872.60 9740.45 23.03 10479.80 9814.09 10.79 10253.90 0.5 11932.00 10457.00 25.47 11932 10497.60 13.66 11603.00 0.6 17825.90 11377.60 57.73 17335.50 11475.90 55.33 13416.80 0.7 27157.40 12758.80 114.56 23227.40 12816.20 111.90 16157.20 0.8 34912.50 14758.30 142.19 31284.20 14761.90 136.50 21500.70 0.9 54722.30 19703.50 177.73 54628.80 19428.60 181.66 35987.70 1.0 128009.00 128009.00 0.00 128009.00 128009.00 0.00 128009.00 P ^vmin ^vmin Exact 0.0 7197.27 7197.27 0.00 7197.27 7197.27 0.00 7197.27 0.1 7763.80 7784.17 -0.26 7763.80 7768.5 -0.06 7763.80 0.2 8425.99 8484.24 -0.69 8425.99 8453.66 -0.33 8425.99 0.3 9378.06 9138.80 2.62 9275.99 9152.36 2.47 9275.99 0.4 10479.80 9826.75 6.65 10259.90 9842.41 6.48 10253.90 0.5 11932.00 10507.40 13.56 11932.00 10530.80 13.31 11603.00 0.6 17335.50 11523.50 50.44 17291.50 11563.00 49.92 13416.80 0.7 22894.20 12766.10 81.95 22894.20 12747.70 79.59 16157.20 0.8 31284.20 14736.80 112.29 31284.20 14746.00 112.15 21500.70 0.9 54628.80 19752.00 176.57 53343.90 19811.80 175.74 35987.70 1.0 128009.00 128009.00 0.00 128009.00 128009.00 0.00 128009.00 muchlesstime.Thegreedyaddingalgorithmseemstoperformbetterwhenthefacilityfailureprobabilityishigh.Itactuallyndsoptimalsolutionswhenthefailureprobabilityexceeds0.5.ThisisincontrasttotheperformanceoftheSAA-H,whichworksbetterwhenthefailureprobabilityislow. AswepointedoutinSection 2.7.2 ,thesolutionqualityofGAD-HcanbefurtherimprovedbyGADS-H.Thisisclearlydemonstratedinthefollowingcomputationalresults.GADS-HactuallyndstheoptimalsolutionsforallinstancesinTable 2-7 andtheCPUtimesarecomparablewiththosereportedbyGAD-H.TheresultsaresummarizedinTable 2-8 39

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Fifty-nodeuniformcase:greedyaddingandexactsolution GreedyAddingHeuristic ExactAlgorithm P Objective Time(s) Objective Time(s) gap(%) 0.0 7551.02 0.00 7197.27 6.94 4.92 0.1 8053.11 0.00 7763.80 7.61 3.73 0.2 8637.46 0.00 8425.99 8.94 2.51 0.3 9309.50 0.00 9275.99 10.62 0.36 0.4 10253.90 0.02 10253.90 10.38 0.00 0.5 11622.80 0.02 11603.00 10.86 0.17 0.6 13416.80 0.02 13416.80 12.84 0.00 0.7 16157.20 0.02 16157.20 13.47 0.00 0.8 21500.70 0.02 21500.70 14.08 0.00 0.9 35987.70 0.05 35987.70 14.27 0.00 1.0 128009.00 0.00 128009.00 9.27 0.00 Table2-8. Fifty-nodeuniformcase:GADS-Handexactsolution GADS-H ExactAlgorithm Gap P Objective Time(s) Objective OpenFacilities Time(s) (%) 0.0 7197.27 0.02 7197.27 0.00 0.1 7763.80 0.02 7763.80 0.00 0.2 8425.99 0.03 8425.99 0.00 0.3 9275.99 0.02 9275.99 0.00 0.4 10253.90 0.02 10253.90 0.00 0.5 11603.00 0.03 11603.00 0.00 0.6 13416.80 0.06 13416.80 0.00 0.7 16157.20 0.09 16157.20 0.00 0.8 21500.70 0.20 21500.70 0.00 0.9 35987.70 0.48 35987.70 0.00 1.0 128009.00 0.00 128009.00 0.00 ItisinterestingtocomparethesetsofopenfacilitiesinTable 2-8 .Onemightconcludethatmorefacilitiesshouldbeopenasthefacilitiesgetmorevulnerable,thatis,whenthefailureprobabilityincreases.Althoughthisclaimisusuallyvalid,itisnotalwaystrue.Anextremecaseiswhenthefailureprobabilityis1sothatnofacilityshouldopen.Onecanalsoconsiderthefollowingcounterexamplewherethereisonlyonesinglefacilitytoopen.Iff1+d11r1p1
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2-9 wereusedfortheparametersinGA-HdescribedinSection 2.7.3 Table2-9. ValuesoftheGA-Hparameters Parameter Value PopulationsizeNP MaximumnumberofgenerationsNG MaximumnumberofgenerationswithoutimprovementNM NumberofreproductionNR NumberofimmigrationNI MutationprobabilityPM Table 2-10 liststhecomputationalresultsofthe50-nodedatasetwhenthefailureprobabilityvariesfrom0to1.Therstcolumn,P,isthefailureprobabilityateachfacility.BecausetheGAheuristicisaprobabilisticmethod,twotrialsareperformed.WereporttheminimumobjectiveitobtainedandtheaverageCPUtime(inseconds)inthesecondandthirdcolumnrespectively.The\gap"columnisdenedasthepercentagedierencebetweenthecostofthesolutionobtainedbyGAandtheoptimalcost. Table2-10. Fifty-nodeuniformcase:GAandexactsolution GAHeuristic(2Trials) ExactAlgorithm P MinObjective Time(s) Objective Time(s) gap(%) 0.0 7197.27 5.41 7197.27 6.94 0.00 0.1 7763.80 5.23 7763.80 7.61 0.00 0.2 8425.99 5.20 8425.99 8.94 0.00 0.3 9275.99 5.11 9275.99 10.62 0.00 0.4 10253.90 5.34 10253.90 10.38 0.00 0.5 11603.00 5.66 11603.00 10.86 0.00 0.6 13416.80 5.52 13416.80 12.84 0.00 0.7 16157.20 5.75 16157.20 13.47 0.00 0.8 21500.70 7.95 21500.70 14.08 0.00 0.9 35987.70 9.83 35987.70 14.27 0.00 1.0 128009.00 3.25 128009.00 9.27 0.00 41

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Figure 2-4 depictstheevolutionoftheminimumobjectivevalue,andtheaverageobjectivevalueineachgenerationwhenp=0:5.Thealgorithmterminatesatgeneration176afteritndstheoptimalsolutionatgeneration76.Infact,inthisexampletheGAquicklyconvergestoacloseoptimalsolutionafterjust20generationsasshowninFigure 2-4 Figure2-4. EvolutionofthesolutionsfromGA 42

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Severaldatasetsarederivedfromthe100-nodedatasetinAppendix B :forexample,dataset#10istherst10linesfromTable B-1 ofAppendix B ;ithas10demandnodesandfacilitysites.Theotherdatasetsarederivedinthesameway.Table 2-11 liststheobjectivevaluesobtainedfromSAA-Hwithdierentsamplesizes.Thecolumn\BestObjective"liststheminimumobjectivevalueamongthedierentsamplesizes.Timeismeasuredinsecondswithresultsfromthesamplesizeof100.\-"inTable 2-11 meansthattheprogramwasoutofmemoryduetothesurgeintheproblemsize.\-"inTable 2-12 meansthattheresultsoftheenumerationmethodwerenotobtainedduetotheexponentiallyincreasedcomputationaltime.Table 2-12 summarizestheobjectivevaluesofthesolutionsobtainedbyGAD-H,GADS-H,GAandtheenumerationmethod,andtheircorrespondingcomputationaltime.TheresultsofGAareobtainedthroughasinglerun. Comparingtheresultsfromheuristicswiththegloballyoptimalsolutionsinsmalldatasets(from#10,to#30),wecanseethatGADS-HandGAndoptimalsolutions,whereasSAA-HandGAD-Hndoptimalornear-optimalsolutions.Toevaluatethequalityofthesolutionsfoundbytheseheuristicsinalldatasets,weplottheobjectivevaluesinFigure 2-5 .ThevaluesfromSAA-HarethebestonesavailableinTable 2-11 foreachinstance.Figure 2-5 showsthatGADS-HandGAcanndthebestsolutionsinalldatasets,whereasSAA-HandGAD-Hcanndeitherthebestknownsolutionsor 43

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ObjectivevaluesobtainedfromSAA-HonURFLP-SFPwithdierentsamplesizes Dateset Best Time(s) # N50 N100 N150 N200 Objective (N100) 10 5850.47 5576.00 5128.24 5128.24 5128.24 0.43 15 6074.06 5337.18 5337.18 5337.18 5337.18 2.11 20 8071.98 5761.79 5761.79 5761.79 5761.79 3.09 25 6749.39 6583.06 6583.06 6583.06 6583.06 12.14 30 7897.19 7622.22 7847.37 7847.37 7622.22 50.67 40 7474.92 7474.92 7474.92 7474.92 7474.92 54.20 50 8719.32 8641.28 8781.18 8641.28 8641.28 185.56 60 9357.37 9394.87 9357.37 9394.87 9357.37 159.48 70 10337.60 10391.80 10383.00 10383.00 10337.60 250.85 80 11054.30 11054.30 11054.30 11054.30 11054.30 320.82 90 12405.50 12405.50 12448.70 12405.50 659.57 100 13977.50 14028.10 13977.50 2164.03 Table2-12. ComputationalperformanceonURFLP-SFP:GAD-H,GADS-H,GAandtheenumerationmethod Dateset GAD-H GADS-H GA Enumeration # Objective T(s) Objective T(s) Objective T(s) Objective T(s) 10 5128.04 0.00 5128.04 0.00 5128.04 1.62 5128.04 0.02 15 5305.04 0.00 5305.04 0.00 5305.04 1.81 5305.04 14.75 20 5761.79 0.00 5761.79 0.00 5761.79 1.81 5761.79 206.51 25 6439.87 0.00 6439.87 0.00 6439.87 1.82 6439.87 6806.57 30 7420.86 0.00 7382.04 0.00 7382.04 2.22 7382.04 188276.35 40 7474.92 0.00 7474.92 0.02 7474.92 4.09 50 8763.75 0.00 8641.28 0.00 8641.28 4.64 60 9357.37 0.00 9357.37 0.02 9357.37 5.14 70 10337.60 0.00 10337.60 0.03 10337.60 5.84 80 11054.30 0.00 11054.29 0.05 11054.29 5.56 90 13030.90 0.00 12405.50 0.06 12405.50 10.75 100 14463.40 0.02 13820.87 0.09 13820.87 11.23 closetothebestknownones.Intermsofcomputationaleciency,SAA-Htakesmuchmoretimetoachieveitssolutionsthanallotherthreeheuristics.BetweenGADS-HandGA,GADS-HspendsconsiderablylessCPUtimethanGA:GADS-Htakeslessthan0.1secondstogetthebestresultineachinstance.Overall,theseresultssuggestthatGADS-HisthebestoneamongallfourheuristicsformodelURFLP-SFPintermsofbothsolutionqualityandcomputationaltime. 44

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ComparisonofobjectivevaluesfromGAD-H,GADS-H,GA,andSAA-H B-2 ofAppendix B .ThersthalfofTable 2-13 reportstheresultsfromGA,whichhasbeenrunfor5timeswithdierentrandomseeds.Eightdatasetshavebeengeneratedfortesting.Thebestopensitesandtheiroptimallevelsarelistedinthesecondcolumn.Thebestandworstresultsobtainedinthe5trialsarelistedinthethirdandfourthcolumnsrespectively.Theaveragetimeinsecondsarereportedinthelastcolumn.TheresultsofGADS-HareshowninthebottomhalfofTable 2-13 .Inonlyonecase(the80-nodeproblem)doestheGAndabettersolutionthanGADS-H.Inthatcase,theobjectivefunctionvalueis11782.85comparedto11859.40,whichrepresentsonlya0.6%improvement.ButGAtakesmoretimethanGADS-Hdoestogetthissmallimprovement.Inallothercases,GADS-HndsthesamesolutionasGAbutwithmuchlesstime.Overall,GADS-HismorefavorablethanGAinsolvingmodelURFLP-MFP. 45

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ComputationalresultsforURFLP-MFPusingGADS-HandGA GA,5Trials Dataset Best Worst Average # BestSites(Levels) Result Result Time(s) 20 2(2)7(2) 5214.55 5214.55 3.13 30 2(2)20(1)21(2) 6484.74 6484.74 3.73 40 2(2)20(1)35(2) 7194.88 7194.88 4.52 50 2(2)5(1)20(1)35(1) 8827.95 8827.95 5.71 60 2(2)20(1)35(2)59(1) 9964.84 9964.84 7.20 70 2(2)13(2)20(1)35(2) 10837.65 10837.65 9.06 80 2(2)20(1)35(2)59(1)76(1)79(1) 11782.85 11867.01 10.81 90 2(2)20(1)35(2)76(1)79(1)88(1) 12621.95 12717.84 11.13 100 2(2)13(2)20(1)35(2)67(2)76(1)88(1) 13713.42 13749.58 11.33 GADS-H Sites(Levels) Result Time(s) 20 2(2)7(2) 5214.55 0.09 30 2(2)20(1)21(2) 6484.74 0.14 40 2(2)20(1)35(2) 7194.88 0.18 50 2(2)5(1)20(1)35(1) 8827.95 0.27 60 2(2)20(1)35(2)59(1) 9964.84 0.33 70 2(2)13(2)20(1)35(2) 10837.65 0.39 80 2(2)13(2)20(1)35(2)76(1) 11859.40 0.57 90 2(2)20(1)35(2)76(1)79(1)88(1) 12621.95 1.19 100 2(2)13(2)20(1)35(2)67(2)76(1)88(1) 13713.42 1.13 Fourheuristics,SAA-H,GAD-H,GADS-HandGA,havebeenproposedtosolvetheseproblems.SAA-Hisaspecializedheuristicforthescenario-basedmodel,whereas 46

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Thereareseveralinterestingresearchdirections.Wenotethatthemajorlimitationofthecurrentmodelsistheassumptionthatthefacilitiesareuncapacitated.Althoughtheassumptionitselfisverycommoninthefacilitylocationmodels,itmaybeunrealisticinpractice.Inthecapacitatedcase,customeroffailedfacilitiescanbeassignedtothenextlevelbackupfacilitiesonlyiftheyhavesucientcapacitytosatisfytheadditionaldemand.Thisrestrictionmaymakethecapacitatedmodelverycomplex.Itbecomesavaluabletopicoffutureinvestigation.Inaddition,somenewmeasurementsofthereliabilityconceptinthefacilitylocationproblemsettingareworthpursuing. 47

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48 ],wheretheauthorsassumethatsomefacilitiesareperfectlyreliablewhileothersaresubjecttofailurewiththesameprobability.Theyformulatetheirmulti-objectiveproblemasalinearintegerprogramandproposeaLagrangianrelaxationsolutionmethod.However,weconsiderpenaltycost,afactorthatismissedin[ 48 ]. UURFLPisclearlyNP-hardasitgeneralizesUFLP.Thefocusofthischapteristoproposeandanalyzeanapproximationalgorithmwithaconstantworst-caseboundguarantee. Designingapproximationalgorithmsforthefacilitylocationproblemanditsvariationshasrecentlyreceivedconsiderableattentionsfromtheresearchcommunity.However,tothebestofourknowledge,thischapterpresentstherstapproximationalgorithmforstochasticfacilitylocationproblemswithprovider-sideuncertainty. Thevastmajorityofapproximationalgorithmsforthefacilitylocationproblemmainlydealwithdeterministicproblems,e.g.[ 17 21 32 46 ].Untilveryrecently,approximationalgorithmsforUFLPwithstochasticdemandhavebeenproposed;seethesurveybyShmoysandSwamy[ 52 ].Anotherrelatedpaper[ 6 ]proposesanapproximationalgorithmforafacilitylocationproblemwithstochasticdemandandinventory.Ourapproximationalgorithmmakesuseoftheideasfromseveralpapers[ 17 21 32 46 ].Inparticular,thischapteriscloselyrelatedto[ 17 ],whichpresentsa2.41-approximationalgorithmfortheso-calledfault-tolerantfacilitylocationproblem(FTFLP):everydemand 48

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Theremainderofthischapterisorganizedasfollows.SeveralequivalentformulationsforUURFLPareproposedinSection 3.2 ,whichleadustodevelopa2.674-approximationalgorithminSection 3.3 .ThechapterisconcludedinSection 3.4 2 .RecalltheformulationofURFLP-SFPinChapter 2 : (URFLP-SFP) minimizeXi2Ffiyi+Xj2DjFjXk=1Xi2Fdjcijxkij(1pi)Yl2FpPk1s=1xsljl+Pj2DPjFj+1k=1Ql2FpPk1s=1xsljldjrjzkj subjecttoXi2Fxkij+kXt=1ztj=1;8j2D;k=1;:::;jFj+1 (3{2) ConsideraspecialcaseofURFLP-SFPwhereallfacilitieshavethesamefailureprobabilities,i.e.,pi=p;8i2F.Thisassumptionsimpliesformulation(URFLP-SFP)considerablybasedonthefollowingobservation.Becausepi=p;8i2F,itisstraightforwardthatYl2FpPk1s=1xsljl=pk1,whichisindependentofthevaluesofxslj.Thispropertyisimplicitlyusedinamulti-objectiveformulationproposedin[ 48 ]. Basedontheaboveobservation,weareabletoreduceformulationURFLP-SFPtoalinearintegerprogramasfollows. 49

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minimizeXi2Ffiyi+Xj2DjFjXk=1Xi2Fdjcijxkij(1p)pk1+Xj2DjFj+1Xk=1pk1djrjzkj subjecttoXi2Fxkij+kXt=1ztj=1;8j2D;k=1;:::;jFj+1jFjXk=1xkijyi;8i2F;j2Dxkij;zkj;yi2f0;1g: 3.1 .WereferthereadertoSection 3.5 fortheproof. minimizeXi2Ffiyi+Xj2DjFjXk=1Xi2Fdjcijxkijpk1(1p)+Xj2D1Xk=1djrjkjpk1(1p) (3{7) subjecttoXi2Fxkij+kj=1;8j2F;k=1;:::;jFjjFjXk=1xkijyi;8i2F;j2Dxkij;kj;yi2f0;1g: 3{6 )andformulation( 3{7 )areequivalent. 50

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Wetakeadvantageofseveralresultsforthefault-tolerantversionofUFLP,whereeverydemandpointjmustbeservedbykjdistinctfacilities,aconceptclosetoourlevelassignment.In[ 17 ],Guhaetal.proposeacoupleofapproximationalgorithmsforthefault-tolerantfacilitylocationproblemusingvariousroundingandgreedylocal-searchtechniques. Thefault-tolerantfacilitylocationproblemcanbeformulatedasthefollowingintegerprogram. (FTFLP) minimizeXi2Ffiyi+Xj2DkjXk=1Xi2Fdjcijwkjxkij subjecttoXi2Fxkij1;8j2D;kkjkjXk=1xkijyi;8i2F;j2Dxkij;yi2f0;1g: OneofthekeyresultsonFTFLPfromGuhaetal.[ 17 ]issummarizedbelow.

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1Xj2DkjXk=1Xi2Fwkjdjcijxkij: 3.1 ,isaparameterthatwecanchoosetocontrolthequalityofapproximation.Indeed,theapproximationratiosofthealgorithmsofGuhaetal.arefunctionsof.Onecanthenchoosethebesttominimizetheapproximationratio. WearenowreadytopresentouralgorithmforUURFLP.Werstsolvealinearprogrammingrelaxationofformulation( 3{7 ). minimizeXi2Ffiyi+Xj2DjFjXk=1Xi2Fdjcijxkijpk1(1p)+Xj2D1Xk=1djrjkjpk1(1p) (3{9) subjecttoXi2Fxkij+kj=1;8j2D;k=1;:::;jFj Assumethat(x;y;)isanoptimalsolutiontothislinearprogram.Ouralgorithmroundsthefractionalsolution(x;y;)toanintegersolution(x;y;)thatisfeasibletoformulation( 3{7 ). Thealgorithmisbasedonapropertyoftheoptimalfractionalsolution(x;y;),whichisformalizedinthefollowinglemma.Thislemmaenablesustoutilizeknownalgorithmsandanalysisforthefault-tolerantfacilitylocationproblem.

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(ii).Ifthereexistsksuchthat0k. Proof. 3.1 ,whichisthusomittedhere. Wepresentourroundingprocedurenext.Foreachj2D,assumekjisthesmallestintegersuchthatkjj>0. Theroundingprocedureiscarriedoutintwophases.Therstphaseroundstheoptimalfractionalsolution(x;y;)toanotherfractionalsolution(^x;^y;^),whichisfeasibletoalinearprogrammingrelaxationofanappropriatelydenedfault-tolerantfacilitylocationproblem.Inthesecondphase,weuseanalgorithmforthefault-tolerantfacilitylocationproblemtoroundthefractionalsolution(^x;^y;^)toanintegersolution(x;y;),whichisfeasibletoformulation( 3{7 ).

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17 ]withaparameter2(0;1]toroundthesolution(^x;^y)toafeasiblesolutionofafault-tolerantfacilitylocationproblem,whereasetoffacilitiesisopensuchthateachclientjisservedbyatleastmjdistinctopenfacilities. Thistwo-phasealgorithmshallbereferredtoasAlgorithmTP.Itisobviousthatthesolution(x;y;)isfeasibletoformulation( 3{7 ).Wenowestablishaworstcaseapproximationboundofour(randomized)algorithm,i.e.,weshallshowthatthe(expected)totalcostisnomorethanaconstantfactortimestheoptimalcost.WeboundthetotalpenaltycostinLemma 3.3 ,andboundthetotalfacilityandtransportationcostsinLemma 3.4 1 1CLP; 3{7 ).

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3.5 forproofsofbothlemmas.Animmediateconsequenceofthesetwolemmasisthefollowingcorollary. 1 1 1;1 1 1 1;1 32 ]fordetails)tofurtherimprovetheapproximationfactor. ThegreedyimprovementprocedurecanimprovetheworstcaseboundofAlgorithmTP,asshownnext.Weomittheproofhereastheanalysisisverysimilartothosein[ 17 ]and[ 32 ]. 3{7 ),thereisa(Rf+ln();1+Rc1 ;1+Rp1 )-approximationalgorithm.

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3.1 (1).Assumewehaveanoptimalsolution(x;y;z)forproblem( 3{6 ),thenwecanconstructafeasiblesolution(x;y;)forproblem( 3{7 )withthesameobjectivevalue. Letx=xandy=y.Foreachj2Dandk1,denekj=Pkt=1ztj.Becauseforeachj,thereisexactlyonevalueofztjwouldbeequalto1,wecanconcludekj2f0;1g.Anditisstraightforwardthat(x;y;)isafeasiblesolutiontoproblem( 3{7 ).Nowwechecktheobjectivevaluecorrespondingto(x;y;).Infact,weonlyneedtoconsider1Xk=1pk1(1p)kj=1Xk=1pk1(1p)minfk;jFj+1jgXt=1ztj=jFj+1Xt=11Xk=tpk1(1p)ztj=jFj+1Xt=1pt1ztj 3{7 ),thenwecanconstructafeasiblesolution(x;y;z)forproblem( 3{6 )withthesameobjectivevalue. 56

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Nowwedeneanewsolution(~x;~y;~)forproblem( 3{7 )asfollows:~y=y;forj6=j0,~xkij=xkij,~kj=jforalli6=i0andk;fork6=k0;k0+1,~xkij0=xkij0,~kj0=j0foralli6=i0;xk0i0j0=1,k0j0=0,xk0+1i0j0=0andk0+1j0=1.Itiseasytoverifythat(~x;~y;~)isafeasiblesolutionforproblem( 3{7 ).Thedierencebetweentheobjectivevaluecorrespondingto(x;y;)andtheobjectivevaluecorrespondingto(~x;~y;~)isdj0ci0j0pk0(1p)+dj0rj0pk01(1p)dj0ci0j0pk01(1p)+dj0rj0pk0(1p) 3{7 ).Itfollowsthatrj0ci0j0: 3{7 )anditsatisestheconditionthatif^k0j0=1then^k0+1j0=1. Therefore,wecanassumethatforeachj2D,ifkj=1,thenk+1j=1.Nowwearereadytondafeasiblesolutionforproblem( 3{6 ).Letx=x;y=y.Andforeachj,ifkj=1andk1j=0,thenletzkj=1andletzk0j=0fork06=k;if1j=1,thenletz1j=1andzk0j=0fork02.Thenitisclearfromthedenitionthatkj=Pkt=1ztj.Thenwe 57

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3{6 ).Andfromtherstpartoftheproof,weknowtheobjectivevaluecorrespondingto(x;y;z)isthesameastheonecorrespondingto(x;y;).Thiscompletestheproof. 3.3 Weboundtheexpectedpenaltycostforeachclientj2D,whichdependsonthevalueofthatisrandomlygenerated.Weconsidertwocases. 58

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3.4 Thecostsofinterestdependonthevalueoftherandomlygenerated.Recallthedenitionof(^x;^y).Foreachj2Dandi2F,^xkij=xkij,ifk6=mj;Ifmj=kj1,then^xkjij=0xkjij;ifmj=kj,then^xkjij=xkjij ^xkij1 1xkij:(3{13) Bytheconstraintofthelinearprogrammingrelaxationof( 3{7 ),weknowthat,foreachi2Fandj2D,jFjXk=1xkijyi: ^yi=maxj2DkjXk=1^xkijmaxj2D1 1kjXk=1xkij1 1yi:(3{14) ToshowthatXi2F^xkij=18j2D;kmj; 3{10 ,Xi2F^xkij=Xi2Fxkij=1. 59

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3{10 ,i.e.Xi2Fxkij+kjj=1. Thus^xand^ysatisfythefollowingconstraints:Xi2F^xkij=18j2D;kmjmjXk=1^xkij^yi8i2F;j2D^xkij0;0^yi1 Therefore,ifweapplythealgorithmin[ 17 ]withaparametertoroundthesolution(^x;^y),thenbyLemma 3.1 ,wecanconstructasolution(x;y)suchthattheexpectedtotalfacilitycostandtotaltransportationcostisboundedabovebyln1 1Xj2DmjXk=1Xi2Fdjcij^xkijpk1(1p)1 1ln1 1Xj2DmjXk=1Xi2Fdjcijxkijpk1(1p)!1 10@ln1 1Xj2DkjXk=1Xi2Fdjcijxkijpk1(1p)1A: 3{13 and 3{14 ;thesecondoneholdsbecauseofmjkj. Finallynoticethatwasuniformlydistributedin(0;),thusE[1 1]=Z01 11 1: 60

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4-1 isRs=nYi=1ri; Figure4-1. Seriessystem Systemreliabilitycanbeimprovedinvariousways,suchasphysicalenhancementofcomponentreliability,provisionofredundantcomponentsinparallel,andallocationofinterchangeablecomponents. Unlikeinaseriessystem,inaparallelsystem,notallcomponentsarenecessaryforthesystemtoworksuccessfully.Actually,onlyonecomponentinsuchsystemneedstoworkproperlyinorderforthewholesystemtoworkproperly.Includingncomponentswhenonlyoneisessentialiscalledredundancy.Theothern1componentsareincludedtoincreasetheprobabilitythatthereisatleastoneworkingcomponent.Redundancyisawidelyusedtechniqueinengineeringtoenhancesystemreliability. 61

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4-2 Figure4-2. Parallel-seriessystem Thesystemreliabilitycanbemaximizedbychoosingtherightcombinationofxiundercertainresourceconstraints,denotedbygj(),j=1;:::;minthischapter.Thisleadstoaredundancyallocationoptimizationproblem(RAOP)oranonlinearintegerprogrammingingeneral. (RAOP):maxRs=f(x1;:::;xn) (4{1) subjecttogj(x1;:::;xn)cj;8j=1;:::;m; 0lixiui;8i=1;:::;n; 62

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Anotherwaytoincreasethesystemreliabilityistosimplyusemorereliablecomponents,whichcertainlycostsmoreintermsofvariousresources.Thisproblemiscalledareliabilityallocationoptimizationproblem([ 28 ]).Supposethereareuidiscretechoicesforcomponentreliabilityatstageifori=1;:::;k(n)andthechoiceforcomponentreliabilityatstagek+1;:::;nisonacontinuousscale.Letri(1);ri(2);:::;ri(ui)denotethecomponentreliabilitychoicesatstageifori=1;:::;k(n).Thenthecontinuous/discretereliabilityallocationoptimizationproblemcanbeformulatedasfollows: maxRs=f(r1(x1);:::;rk(xk);rk+1;:::;rn) (4{5) subjecttogj(r1(x1);:::;rk(xk);rk+1;:::rn)cj;8j=1;:::;m; 0rliriuui;8i=k+1;:::;n; whererlianduuiarethelowerboundandupperboundrespectivelyforthecomponentreliabilityatstagei.Ifk=0,theabovemixedintegernonlinearprogrammingformulationreducestoapurenonlinearprogrammingproblem. Thesystemsweareinterestedinarenotlimitedtoseriesandparallelsystems.Theycanbecomplex(general)systemsthatarenon-seriesandnon-parallel,suchasthebridgenetworkinFigure 4-3 .Thesystemreliabilityofsuchsystemcanbecomputedbytheconditionalprobabilitytheory.Forexample,thesystemreliabilityoftheve-componentbridgenetworkinFigure 4-3 canbecomputedbasedonwhethercomponent5isfunctionalornot.WereferreaderstoAppendix A forthedetailsofthefollowingexpression.Rs=(r1+r3r1r3)(r2+r4r2r4)r5+(r1r2+r3r4r1r2r3r4)(1r5):

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Figure4-3. Five-componentbridgenetwork Constraintsofasystemcanbethetotalweight,thetotalcost,thetotalvolume,andsoon.Ingeneral,suchconstraintsareinnonlinearforms[ 28 ].Asthenumberofcomponentsineachsubsystem/stageincreases,moreconnectingequipmentisrequiredandthus,thecostandweightmayincreaseexponentially[ 53 ]. Next,wepresentamoregeneralformulationforthesystemreliabilityoptimizationproblem(SROP),whichincludestheredundancyallocationoptimizationandthereliabilityallocationoptimizationproblemsasspecialcases.Formally,SROPcanbeformulatedasanonlinearmixedintegerprogrammingproblem. (SROP):maxRs=f(x1;:::;xq;r1;:::;rp) (4{9) subjecttogj(x1;:::;xq;r1;:::;rp)cj;8j=1;:::;m; 0lixiui;8i=1;:::;q; 0rlkrkruk;8k=1;:::;p; 64

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TheSROPmodelcoversavastmajorityofreliabilityoptimizationmodelsdiscussedintheliterature.Forexample,modelSROPreducestomodelRAOPwhenp=0,andacontinuousversionofthereliabilityallocationoptimizationproblemwhenq=0.Itcertainlycanalsomodelthegeneralcaseofthereliabilityallocationoptimizationproblembyreinterpretingthedenitionofthevariables,sinceobjectivefunctions 4{5 and 4{9 aremathematicallyequivalent,soareconstraints 4{6 and 4{10 .Inaddition,modelSROPisobviouslyareliability-redundancyallocationmodelifp=q,whereateachstagethedecisionsarewhichcomponentreliabilitytochooseandhowmuchredundancyaswell. TheSROPmodelhasreceivedtremendousresearchattentionsoverdecadesandhasbeenextensivelystudiedandsolvedusingmanydierentmathematicalprogrammingtechniquesandheuristicapproaches.Kuoetal.[ 28 ],alongwith[ 27 ],provideadetailedintroductiontothemodelsandalgorithmsinthereliabilityoptimization.SROPisoftencharacterizedbyanonlinearobjectivefunctionthatisneitherconvexnorconcaveoveranonconvexfeasibleregion.Duetotheextremedicultyofsuchtypeofproblem,thesolutionmethodsintheliteraturearemainlyheuristics,meta-heuristicsandapproximationalgorithms.Acomprehensivereviewonthesemethodscanbefoundin[ 27 ]. 65

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37 ],[ 28 ],[ 51 ],[ 29 ]).Thebranch-and-boundalgorithmisapopularapproachforndingglobaloptimalsolutionsforthenonconvexproblems.Theydierin(i)methodsofselectingabranchingvariable,(ii)methodsofselectingabranchingnode,(iii)methodsofcalculatingupperandlowerbounds.Forexample,[ 28 ]providesawaytotransformtheintegervariablesintobinaryvariablesthatarebranchedonafterwards.[ 37 ]branchesonthevariablesinincreasingorderoftherange,thedierencebetweenupperboundandlowerbound,thenxestheminthatorder.[ 51 ]and[ 29 ]adoptaconvexicationmethodtotransformtheproblemintoaconvexmaximizationproblemduringtheprocessofthebranch-and-bound.Thebranchingisdoneonafractionalvariable.[ 50 ]followsasimilarthemeto[ 51 ]and[ 29 ]exceptthatthebranchisdoneontheselectedboundarypointsanditonlysolvesthecontinuousversionofthemonotoneoptimizationproblem. TheMonotonicBranch-Reduce-Bound(briey,mBRB)algorithmpresentedinthischapterfollowsthedevelopmentofaspecializedalgorithmformonotonicoptimizationproposedbyTuyandhiscollaboratorsinaseriesofpapers,[ 40 ],[ 56 ],[ 57 ].Theyuseaunionofhyperrectanglestocovertheboundaryofthefeasibleregion.Thehyperrectangleiscalledapolyblockintheirlanguagesothatthealgorithmiscalledapolyblockalgorithm.Abranch-and-boundimplementationofthepolyblockalgorithmwasmentionedin[ 57 ]forthecontinuousversionofthemonotoneoptimizationproblem.Theeciencyofthisapproachisreportedonvariousclassesofglobaloptimizationproblems,suchaspolynomialfractionalprogramming[ 59 ],anddiscretenonlinearprogramming[ 58 ]. Comparedtothepreviousbranch-and-boundalgorithms,theproposedmethodexploitsthemonotonicitypropertiesinherentinSROPwithoutrequiringanyconvexity,concavity,dierentiability,andseparability.Unlike[ 28 ],itdoesnotrequireconversionoftheoriginaldecisionvariablesintobinaryones.Themethodisecientandexibleenoughtosolvepure,mixed-integer,andintegernonlinearprogrammingproblemsarisingfrom 66

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Theremainderofthischapterisorganizedasfollows.Themonotonicbranch-reduce-boundalgorithmispresentedinSection 4.2 withitsconvergenceanalysis.Severalconvergenceaccelerationtechniquesarealsodiscussed.Thealgorithmisappliedtosolveboththereliabilityallocationandthereliability-redundancyallocationoptimizationproblemsinSection 4.3 withademonstrationofitseciency.ThechapterisconcludedinSection 4.4 4{9 ),toamoregeneralvectorformatbydenotingx=(x1;:::;xq;r1;:::;rp),inotherwords,xq+i=ri,8i=1;:::;p. (SROP'):maxRs=f(x) (4{14) subjecttox2G=fxjgj(x)cj;8j=1;:::;mg; where[xL;xU]denotesahyperrectanglewithlowestboundaryxLandgreatestboundaryxU;functionsfandgiarenondecreasingforj=1;:::;m.ToensureGisclosed,weassumegjaresemi-continuousforj=1;:::;m. Sincenoconcavityassumptionhasbeenmadeontheobjectivefunction,multiplelocallyoptimalsolutionsmayexist.However,fromthemonotonicityofthefunctionsfandgi,thefollowingpropositioncanbeeasilyderived. 56 ]Theglobalmaximumoff(x)overG\Xz\[xL;xU],ifitexists,isattainedonitsboundary. 4.1 isusedbyTuytodeveloptheoriginalPolyblockAlgorithmformonotonicoptimizationin[ 56 ].Thebranch-reduce-boundimplementationofthepolyblockalgorithmwasmentionedin[ 57 ]forthecontinuousversionofthe 67

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InthedescriptionofAlgorithm 1 ,Sdenotesahyperrectanglepartition;Lisalistofunfathomedhyperrectangles;isapre-denedoptimalitytoleranceparameter;xbestandfbestdenotethecurrentbestsolutionandobjectivevaluerespectively;UB(S)istheupperboundoftheobjectivefunctionoverS.Besidesinitialization,themajorstepsaredescribedindetailasfollows. SelectandBranch ReduceandBound 68

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12 inAlgorithm 1 .ThenthesubsequentbranchingiscarriedoutalongthelongestsideiofS,withi=argmaxjfsUjsLjg.Ifiq,thenS=[sL;sU]ispartitionedtoS1andS2onthediscretevariable:S1=sL;sUsUisLi 69

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19 ].Thelongest-edgebisectionruleisexhaustive,asindicatedintheproofofTheorem 4.1 Althoughthelongest-edgebisectionruleisapopularchoiceforbranching,onecanuseotherrulessuchasthelongest-edgetrisection,andthelargestincrementedgebisection.Inthelargestincrementaledgebisection,theedgeisselectedonwhichwiththemostincrementintheobjectivefunction.Thatisi=argmaxjff(sLnsLj[sUj)g,wheresLnsLj[sUjdenotesthevalueofsLexceptthatitsjthindexisreplacedbysUj. 4-4 showsS1andS2afterreduction. Figure4-4. Reduceprocess Inthecase,i>q,withS1=sL;sUsUisLi 70

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56 ].Inthispaper,wesimplyusef(sU).Apartitionisdiscardedifitsupperboundislessthanthecurrentbestsolution,orifitdoesnotcoveranyfeasiblesolution,asshowninline 22 inAlgorithm 1 1 .TheresultissummarizedinTheorem 4.1 1 eitherterminatesafterniteiterations,producingan-optimalsolutionordetectingitsinfeasibility;oritisinnitewithasequenceofhyper-rectanglesSpk=[Lpk;Upk]suchthatlimk!1Lpk=limk!1Upk=x,wherexisanoptimalsolution. Proof. 1 terminateswithinniteiterations,itproducesan-optimalsolutionordetectsitsinfeasibility.IfAlgorithm 1 doesnotterminatewithinniteiterations,allthediscretevariablesmusthavebeenxed.Inaddition,amongtheinnitelygeneratedpartitions,thereexistsasequencesuchthat[Lpk+1;Upk+1][Lpk;Upk]. 71

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FromTheorem 4.1 ,wecanaddthefollowingconditionafterline 16 inAlgorithm 1 toterminatethealgorithminniteiterations. 4.2.4AccelerationTechniques 1 ,besidesthereductionprocess,thereareseveralotheraccelerationtechniquesworthmentioning. wehaveallfeasiblexixLi+iei.Inotherwords,ximin(xUi;xLi+iei),whichprovidesatighterupperboundforxi. 72

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1 choosesthepartitionwiththegreatestupperbound.Computationally,adelicateruletoperiodicallyselectthepartitionbetweenthegreatestupperboundandtheleastupperboundhelpstheconvergenceofthealgorithm. Algorithm 1 employsthelongest-edgebisectionrule.Onecanuseotherrulessuchasthelongest-edgetrisection,andthelargestincrementaledgebisectiontospeeduptheconvergenceinsomecases.Inthelargestincrementaledgebisection,theedgeisselectedonwhichwiththemostincrementontheobjectivefunction.Thatis,i=argmaxjff(sLnsLj[sUj)g,wheresLnsLj[sUjdenotesthevalueofsLexceptthatitsjthindexisreplacedbysUj. 1 )isappliedtosolvevariousproblemsarisingfromsystemreliabilityoptimization.Mostoftheseproblemsarethoroughlystudiedintheliteraturewithvarioussolutiontechniques,mainlyheuristics.Thelimitationofaheuristicisthatitmayndgoodsolutions,butwithoutguaranteedoptimality.Incontrasttotheheuristictechniques,thepresentedmBRBAlgorithm,asanexactmethod,canndan-optimalsolutioninanitenumberofsteps.ThealgorithmthatsolvesallthefollowinginstancesisimplementedinC++andtestedonaDellOptiplexGX620computerwithaPentiumIV3.6GHzprocessorand1.0GBRAM,runningundertheWindowsXPoperatingsystem. 73

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28 ].Theproblemwasoriginallypresentedin[ 53 ]. maxRs=5Yi=1Ri(xi) (4{20) subjecttog1=5Xi=1pix2iP; whereRi=1(1ri)xiisthereliabilityofstagei.Constraint 4{21 ,g1,isimposedonthecombinationofweightandvolume:piistheproductofweightperunitandvolumeperunit.Componentreliabilitydoesnotusuallyaecttheweightnorthevolume,henceg1isnotafunctionofri([ 54 ]).Constraint 4{22 ,g2,isthecostconstraintwherecixiisthecostofallcomponentsatstageiandciexpxi 4{23 ,g3,istheweightconstraintwherewixiistheweightofallcomponentsatstagei.Theadditionalfactor,expxi 54 ]).Theweightconstraintisnotafunctionofcomponentreliability.ThecoecientsaregiveninTable 4-1 Table4-1. CoecientsinExample1 74

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28 ]. maxRs=4Yi=1Ri(xi) (4{25) subjecttog1=10exp0:02 1R1(x1)+10x2+6x3+15x4150; wherestagereliabilitiesare 75

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36 ],[ 23 ],and[ 1 ],allofwhichspentmoretimetosolvethanthemBRBalgorithmandcouldnotverifytheoptimalityoftheirsolutions,althoughtheywereabletoproducethesamesolution.Asacorrectiontotheliterature,itisnotedthat[ 1 ]containsthewrongexpressionofthisexample. maxRs=5Yi=1Ri(xi) (4{30) subjecttog1=5Xi=1pix2iP; 0ri1;8i=1;:::;5; wheretheobjectivefunctionandconstraintsarethesameasthoseofExample1exceptamoreexplicitlyexpressionfortheunitcostofcomponenti,whichisafunctionofthecomponentreliabilityri.Toderivetheexpressionofci(ri)=it 54 ]andassumethattheunitcostofcomponentiisadecreasingfunctionof 76

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4-2 Table4-2. CoecientsinExample3 Thisproblemisconsidereddiculttosolveintheliterature.Tothebestofourknowledge,thereisnoexactmethodbeingappliedtosolvethisproblem.WecomparetheperformanceofmBRBwiththeTHKheuristicin[ 55 ],theGAGheuristicin[ 15 ],theKLXZmethodin[ 26 ],thesurrogate-constraintsalgorithmHNNNin[ 18 ],andthegeneticalgorithmGAin[ 20 ].ThecomparisonresultsaresummarizedinTable 4-3 ,wheretheCPUtimelistedinthelastcolumnismeasuredinseconds.Therstcolumnliststhenamesofthemethods.Thesolutionsandtheobtainedsystemreliabilityarelistedinthesecondandthethirdcolumnrespectively.Thefourthcolumn,Rs(UB),liststheupperboundofthesystemreliability. Thenumbersinbracketsafter\mBRB"intherstcolumnisthevalueof,thepre-denedoptimalitytolerance.Asonecansee,themBRBalgorithmproduceshigherqualitysolutionswithknownupperboundcomparingtootheralgorithms,whichareabletooutputsomefeasiblesolutionsbutwithoutupperboundguarantee.Withadditional 77

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PerformancecomparisonofExample3 Method (x;r) CPU(s) mBRB(0.01) (3,2,2,3,3,0.77500,0.87500,0.89250,0.71500,0.79000) 0.930947 0.940913 1.38 mBRB(0.001) (3,2,2,3,3,0.78250,0.87500,0.89938,0.70750,0.79000) 0.931541 4.76 THK (3,3,2,2,3,0.78438,0.82500,0.90000,0.77500,0.77813) 0.915363 GAG (3,2,2,3,3,0.80000,0.86250,0.90156,0.70000,0.80000) 0.930289 KLXZ (3,3,2,3,2,0.77960,0.80065,0.90227,0.71044,0.85947) 0.929750 HNNN (3,2,2,3,3,0.77489,0.87007,0.89855,0.71652,0.79137) 0.931451 GA (3,2,2,3,3,0.77943,0.86848,0.90267,0.71404,0.78689) 0.931578 CPUtime,themBRBalgorithmcanproduceasolutionof(3;2;2;3;3;0:77781;0:87187;0:90281;0:71313;0:78625) withthesystemreliabilityat0.931669,anditsupperboundof0.933111.Thisresultcanserveasabenchmarkforthisexample. 35 ],[ 51 ]).Thederivationofthereliabilityofthisnetwork,i.e.theobjectivefunction,isprovidedinAppendix A Figure4-5. Seven-linkARPAnetwork 78

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(4{36) +R3R5R6Q7(Q1+R1Q2)+R1R2R5R7Q3Q4Q6+R2R3R4R6Q1Q5Q7+R1R3R4R5Q2Q6Q7subjecttog1=x1x2+0:5x1ln(1+x3)+x4+2x5+0:3exp0:02 1R6 +0:3exp0:01 1R727;g2=(x1+2x2+1:2x3)ln(1+x1+x2+2x3)+0:4x4 +0:2x5exp0:02 1R6+0:5exp0:01 1R729;0:5Ri0:99;8i=6;7; whereRi=1(1ri)xi;8i=1;:::;5;Qi=1Ri;8i=1;:::;7;andr1=0:70;r2=0:90;r3=0:80;r4=0:65;r5=0:70: 51 ].ThecomparisonbetweenthemBRBmethodandtheconvexicationoneisdoneinTable 4-4 ,wherebothalgorithmsobtaintheoptimalobjectivevalue.Admittedly,comparingCPUsecondsdirectlydoesnotreecttheabsoluteeciency,sincetheconvexicationmethodwasimplementedinanoldercomputersystem.However,thetableclearlyshowsrelativeeciencyofthemBRBalgorithm:IttakesonlynegligibleCPUsecondstogettheoptimalsolution. Table4-4. PerformancecomparisonofExample4 Method(x;r)RsRs(UB)CPU(s) mBRB(4,1,3,4,3,0.9845,0.9900)0.999740.999740.01Convexication(4,1,3,4,3,0.9845,0.9899)0.99974-38.63 a 79

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Weexpectthatthemonotonicbranch-reduce-boundalgorithmcanbeappliedtootherclassesoftheproblem,suchasnonlinearmultidimensionalknapsackproblems,andgeneralizedmultiplicativeprogrammingproblems.Inthefutureresearch,wealsowouldliketocomparetheperformanceoftheaccelerationtechniquesmentionedinthischapter,andanalyzeitsworst-caseperformancetheoreticallyandcomputationally. 80

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2 ,weconsidertheimpactofunreliabilityfromthefacilitieswhenthesystemisinitiallydesigned.Inthatcase,wearegiventheoptiontobuildfacilitiesfromscratch.However,redesigninganentiresystemisnotalwaysanavailableoptiongiventhepotentiallylargeexpensesinvolvedinclosingexistingfacilitiesoropeningnewones.Inmanysituations,methodsforprotectingexistinginfrastructuremaybepreferablegivenlimitedresourcesavailable.Inthischapter,weaddresstheissuesonfortifyingthereliabilityofexistingfacilities. Amajorityofresearchonreliablesupplychaindesignhasbeenfocusedontheinitialsystemdesign,butnotonhowtoimprovetheexistingsystem.Theseworkshavebeenwelldocumentedin[ 47 ]andsurveyedinChapter 2 aswell.However,reinforcingthecomponentsofanexistingsystemmaybecomemorevaluableandrealisticconsideringtheincreasedpotentialdisruptionsanduncertainties.Thesedisruptionsanduncertaintiesmayevolvefromthenewchallengesthatwereneverfacedwhentheinitialsystemsweredesigned.Theycanbeman-madedisruptiveeventsornaturaldisasters,forexample,theSeptember11,2001terroristattackandHurricaneKatrinain2005. Onlyasmallstrandofliteraturehasbeendevotedtoaddressingtheforticationofexistingfacilities,whichincludes[ 43 ],[ 44 ],and[ 49 ].Theseworkstypicallyfocusontheinterdiction-forticationframeworkbaseduponthep-medianfacilitylocationproblem.Theproblemsaregenerallyformulatedintheformofbilevelprogramming.Thesemodelscanhelptoidentifythecriticalfacilitiestoprotectundertheeventsofdisruption. Anotherrelatedstrandofliteratureisonthenetworkinterdictionproblemsthataremainlydevelopedformilitaryapplications,e.g.[ 12 ],[ 33 ],[ 62 ]and[ 30 ].Thesemodelsstudytheimpactoflosingoneormoretransportationlinksornetworkarcsbasedonthe 81

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Inthischapter,wefollowthethemeofChapter 2 ,butassumethatthefacilitieshavebeenbuiltandcanbereinforcedtobemorereliable.Wealsoassumethatthefacilitiesareuncapacitated.Wepresenttwonovelmodelswhoseobjectivesaretominimizetheexpectedconnection(service)andpenaltycostbyallocatingthelimitedforticationresourcestotheopenunreliablefacilities.Intherstmodel,theforticationeortsarecontinuous,thatis,thefailureprobabilityateachfacilityvariesfrom0to1attheforticationstage.WecallittheContinuousFacilityForticationModel(CFFM).Onthecontrary,inthesecondmodeltheforticationeortsaresubjecttodierentlevelofresources.Thefailureprobabilityateachfacilitycanonlybechosenfromasetofdiscretelevelsattheforticationstage.AccordinglywecallittheDiscreteFacilityForticationModel(DFFM).Bothmodelscanhelptoidentifythecriticalfacilitiestoprotectandoptimallydeterminehowmuchresourcesshouldbeallocatedtoachievetheobjective.Tosomeextent,themodelsmathematicallyresemblethereliabilityallocationproblemwediscussedinChapter 4 Theremainderofthischapterisorganizedasfollows.InSection 5.2 ,wepresentthecontinuousfacilityforticationmodelandrevealitsconnectiontothegeneralizedlinearmultiplicativeprogrammingandtheinherentmonotonicity.AnexampleispresentedinSection 5.2.2 toillustratethesolutionstructureandproperties.InSection 5.3 ,wepresentthediscretefacilityforticationmodelandapplythemonotonicbranch-reduce-algorithmtoexploitthemonotonicitypropertiesinherentintheproblem.TheeciencyofthealgorithmisdemonstratedinSection 5.3.2 .Section 5.3.2 alsocontainsananalysisofthesolutionstructureandtradeobetweencostdeductionandforticationeort.WeconcludethischapterinSection 5.4 82

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Inthereliablefacilitylocationmodelsetting,eachclientisassignedtoasetofbackupfacilities,whichisdierentiatedbythelevels:incaseofalowerlevelfacilityfails,thenextlevelfacility,iffunctional,willbackitup.xkij=1iffacilityiisthek-thlevelbackupfacilityofdemandnodejandxkij=0otherwise;zkj=1ifjhas(k1)-thbackupfacility,buthasnok-thbackupfacilitysothatjincursapenaltycostatlevelk.ContrarytothemodelsinChapter 2 ,xkijandzkjarenotdecisionvariablesanymore.Theyareusedtospecifytheexistingnetwork. Tocomputetheexpectedfailurecost,wefollowthelogicinChapter 2 .Firstofall,weneedtocomputetheexpectedfailurecostatlevelkservedbyfacilityi.Eachdemandnodejisservedbyitslevel-kfacilityifallthelowerlevelfacilitiesbecomenon-operational.Foranyfacilityl,ifitisassignedtoalowerlevel(i.e,lessthank)fordemandnodej,thenPk1s=1xslj=1,otherwiseitiszero.SotheprobabilitythatalllowerlevelsfacilitiesfailisQl2FpPk1s=1xsljl.Andjisservedatlevel-kbyi,whichhastobeoperational.Theprobabilityis(1pi).Therefore,theexpectedfailurecostatlevelkservedbyfacilityiisdjcijxkij(1pi)Ql2FpPk1s=1xsljl.Similarly,wecancalculatethepenaltycostatlevelk,whichisQl2FpPk1s=1xsljldjrjzkj. Wecannowformulatethecontinuousfacilityforticationmodelasfollows: 83

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minimizeXj2DjFjXk=1Xi2Fdjcijxkij(1pi)Yl2FpPk1s=1xsljl+Xj2DjFj+1Xk=1Yl2FpPk1s=1xsljldjrjzkj subjecttogj(p1;:::;pjFj)0;j=1;:::;m 0pi1;8i2F: Theobjectivefunction( 5{1 )isthesumoftheexpectedfailurecostandtheexpectedpenaltycost.Constraints( 5{2 )denotevariousresourcerestrictionsontheforticationlevels,pi;8i2F.Weassumegj(j=1;:::;m)areconvexfunctionssothatthesolutiondomainisconvex.Constraints( 5{3 )arenaturalconstraintsonthefailureprobability. 41 ]fordetails). (GLMP) minimizetXj=1kjYi=1(cTijx+dij) (5{4) subjecttox2X wherecij2Rn,dij2Rn,j=1;:::;t;i=1;:::;pj,andx2Rn,Xisanonemptyconvexset.Notethatwithoutthesummationsignintheobjectivefunction 5{4 ,GLMPisreducedtolinearmultiplicativeprogramming,anotheractivetopicinglobaloptimization. Werstshowthattheobjectivefunction( 5{1 )isthesumoflinearmultiplicativetermswithpositivecoecients.Inotherwords,theobjectivefunction( 5{1 )canbereducedtotheformof wherejarenonnegativeandijarebinary. 84

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5{1 ,ofthecontinuousfacilityforticationmodelcanbereducedtotheformofXjjjFjYi=1pijiwherejarenonnegativeandijarebinary. Proof. Theonlynegativetermis Atlevelk+1,twocasescanhappen: 1. Nofacilityisassignedtodemandjatlevelk+1.Itissubjecttopenaltycost Becauserjci0jinthiscase,addingterms 5{8 and 5{9 leadstodj(rjci0j)pi0Yl2FpPk1s=1xsljl; 85

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Facilityi00isassignedtoit.Andthecorrespondingservicecostis Addingterms 5{8 and 5{10 ,wehave FromProposition 2.1 ,weknowthatci00jci0j0,sotheonlynegativetermleftis Thistermcanbeabsorbedinthenextlevelassignmentfollowingasimilaranalysisatlevelk.Bydoingthisprocessrecursively,demandjwillbeeventuallysubjecttopenaltycostatcertainlevelhigherthank+1andonlycase1canhappen.Therefore,allcosttermsrelatedtodemandnodejhavetheformofjjFjYi=1piji.Thiscompletestheproof. AsadirectresultfromProposition 5.1 ,thefollowingCorollaryholds. 5{1 ofthecontinuousfacilityforticationmodelismonotonicallynondecreasing. 5{6 isaspecialcaseofFunction 5{4 :Ifij=0infunction 5{6 ,thensetcij=0anddij=j;Ifij=1,thensetcij=jei,eibeingtheithunitvectorinRnanddij=0.Therefore,thecontinuousfacilityforticationproblemisaspecialformofthegeneralizedlinearmultiplicativeprogramming.Thelatterismultiextremalandpossessesseverallocalminima[ 41 ].TheexistingalgorithmsforGLMPincludeouter-approximationmethods[ 25 ],vertexenumerationmethods[ 19 ],heuristicsmethods[ 31 ],amongothers.Corollary 5.1 alsoallowsustoapplythemonotonicbranch-reduce-boundalgorithmpresentedinChapter 4 whentheresourceconstraintspossessmonotonicityaswell. 86

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whereLisnonnegativeandcanbeinterpretedastheindicationofthe(un)reliabilityofthewholesystem.Inotherwords,withtheavailableforticationresourcesthesystemcanbemaintainedatthe(un)reliabilitylevelofL.IfLispositive,noteveryfacilitycanoperateatpi=0.Ontheotherhand,L=0meansthatthesystemistotallyfunctional:Nofacilityhasapossibilitytofail.TheCFFMmodelisthenformulatedasfollows, minimizeXj2DjFjXk=1Xi2Fdjcijxkij(1pi)Yl2FpPk1s=1xsljl+Xj2DjFj+1Xk=1Yl2FpPk1s=1xsljldjrjzkj subjecttoXi2FpiL 0pi1;8i2F: Anetworkwith20demandnodesand5openfacilitiesisconsideredinthisexample.DemandsdjweredrawnfromU[0;1000]androundedtothenearestinteger,andxandycoordinatesweredrawnfromU[0;1].TransportationcostscijaresettobeequaltotheEuclideandistancebetweeniandj.ThepenaltycostrjweredrawnfromU[0;15].ThedatasetisavailableinAppendix C .Facilities2,5,15,18,and20areopeninthisexample.WeuseavertexenumerationmethodtosolvethisproblemandobtaintheresultsatvariouslevelofLwhichareshowninTable 5-1 andplottedinFigure 5-1 .TherstcolumnofTable 5-1 isthe(un)reliabilityofthesystem.Thesecondtosixthcolumnsreportthesolutionofthisexample:theproleofthefailureprobabilityateachopenfacility.Itisfollowedbyacolumnshowingthetotalcost.Thelastcolumncalculates 87

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Table 5-1 andFigure 5-1 showhowthesystemdeteriorates(totalcostincreases)astheresourcetomaintainthesystemreliabilitydecreases.Inaddition,thelastcolumnofTable 5-1 indicatesthatthecurveinFigure 5-1 ispiecewiselinear.Forexample,thedierence,similarlytheslope,intotalcostisaconstantwhenLchangesfrom1.0to2.0inastepof0.1.Thisisbecauseof(1)thefailureprobabilitiesofallbutonefacility(20,inthiscase)remainthesame,thewholecoststructuredoesnotchange;(2)Changingthefailureprobabilityinonefacilitywillonlyaectalltheservicecostthatrelatedtothisparticularfacility. Figure5-1. Totalcostatdierentsystemreliabilitylevel ThereisacommonpatterninTable 5-1 :whenLincreasesby0.1,onlyoneofthefacilitieschangesthefailureprobabilityaccordingly.Forexample,whenLincreasesfrom1.7to1.8,thefailureprobabilityoffacility20jumpsfrom0.7to0.8,alltheothersremainingthesame.Thefailureprobabilityatanindividualfacilityusuallychangesbyeither0.1or0.0asshowninmostcasesinFigure 5-2 .However,thereisonenotable 88

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SolutionofaCFFMmodel FailureProbabilityat 5 15 18 20 TotalCost Dierence 0 0 0 0 0 0 1582.93 n/a 0.1 0 0 0.1 0 0 1590.37 7.44 0.2 0 0 0.2 0 0 1597.80 7.43 0.3 0 0 0.3 0 0 1605.24 7.44 0.4 0 0 0.4 0 0 1612.67 7.43 0.5 0 0 0.5 0 0 1620.11 7.44 0.6 0 0 0.6 0 0 1627.55 7.44 0.7 0 0 0.7 0 0 1634.98 7.43 0.8 0 0 0.8 0 0 1642.42 7.44 0.9 0 0 0.9 0 0 1649.86 7.44 1.0 0 0 1 0 0 1657.29 7.43 1.1 0 0 1 0 0.1 1689.40 32.11 1.2 0 0 1 0 0.2 1721.51 32.11 1.3 0 0 1 0 0.3 1753.62 32.11 1.4 0 0 1 0 0.4 1785.73 32.11 1.5 0 0 1 0 0.5 1817.84 32.11 1.6 0 0 1 0 0.6 1849.95 32.11 1.7 0 0 1 0 0.7 1882.06 32.11 1.8 0 0 1 0 0.8 1914.17 32.11 1.9 0 0 1 0 0.9 1946.28 32.11 2.0 0 0 1 0 1 1978.39 32.11 2.1 0.1 0 1 0 1 2050.81 72.42 2.2 0.2 0 1 0 1 2123.23 72.42 2.3 0.3 0 1 0 1 2195.65 72.42 2.4 0.4 0 1 0 1 2268.06 72.41 2.5 0.5 0 1 0 1 2340.48 72.42 2.6 0.6 0 1 0 1 2412.90 72.42 2.7 0.7 0 1 0 1 2485.31 72.41 2.8 0.8 0 1 0 1 2557.73 72.42 2.9 0.9 0 1 0 1 2630.15 72.42 3.0 1 0 1 0 1 2702.56 72.41 3.1 1 0 1 0.1 1 2863.49 160.93 3.2 1 0 1 0.2 1 3024.41 160.92 3.3 1 0 1 0.3 1 3185.34 160.93 3.4 1 0 1 0.4 1 3346.26 160.92 3.5 1 0 1 0.5 1 3507.19 160.93 3.6 1 0 1 0.6 1 3668.11 160.92 3.7 1 0 1 0.7 1 3829.04 160.93 3.8 0.8 1 0 1 1 3972.66 143.62 3.9 0.9 1 0 1 1 4099.09 126.43 4.0 1 1 0 1 1 4225.52 126.43 89

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5-2 ifwelookatthechangesfromL=3:7toL=3:8insub-guresAtoD:a0.2dipinA,a1.0surgeinB,a1.0sinkinC,anda0.3jumpinD.ThisphenomenonisfromthemultiextremalnessoftheCFFMproblemthatwediscussinSection 5.2.1 .Usingabisectionsearch,wendoutthatatL=0:749842,bothvertex(1,0,1,0.749842,1)andvertex(0.749842,1,0,1,1)producethesameobjectivevalueof3909.2476.ItmeansthatatL=0:749842,theoptimalvertextransientfrom(1,0,1,0.749842,1)to(0.749842,1,0,1,1).ThenfromL=0:749842toL=0:8,thesamepatternstillholds,facility2istheonlyonethatchangesitsfailureprobability. AswepointoutintheintroductionthatCFFMmodelcanalsohelptondthekeyfacilitytofortify.Inthisexample,thefrequencythatafacilityiscompletelyopenisdepictedinFigure 5-3 .Itshowsthatfacility5isverycriticalinthissystem,becauseitischosentobecompletelysecured38timesoutofthetotal41casesinthisexample. 90

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FailureprobabilityatindividualfacilityvssystemreliabilityLevel 91

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FrequencyofcompletelyopenfacilityinTable 5-1 5.2 arethat(1)thelimitedforticationresourceisuniformlydistributedacrossallthefacilities;and(2)thedistributedresourceisdividable.However,inmanycases,theforticationresourcecanonlybediscretelydistributed.Thatis,theforticationresourceatafacilityiscategorizedintodierentlevels,whicharefunctionsofavailableresources.Forexample,7000unitsofresourcemayimprovethefailureprobabilityofafacilitytop=0:4,while8000unitscanimproveittop=0:3.Butthereisnoamountofresourcethatcanimprovethefailureprobabilitytoanumberbetweenp=0:4andp=0:3.Thisinspiresustoconsideradiscretefacilityforticationmodel(DFFM). InDFFM,letyibetheforticationlevelatfacilityi.Naturally,weassumeyiisapositiveintegerandonlyoneforticationlevelisallowedateachfacility.ThecorrespondingfailureprobabilityandtheamountofresourcearedenotedbyfunctionsPi(yi)andVi(yi)respectively.Pi(yi),i2F,arenonincreasingfunctionsofyi,whereasVi(yi),i2F,arenondecreasingfunctionsofyi.Thisisbecausemoreforticationresourcewouldmakeafacilitymorereliable,andconsequentlyasmallerfailureprobability.LettheupperboundofyibeUiandthetotalresourcebeR. 92

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(DFFM) minimizeXj2DjFjXk=1Xi2Fdjcijxkij(1Pi(yi))Yl2FPl(yl)Pk1s=1xslj+Xj2DjFj+1Xk=1Yl2FPl(yl)Pk1s=1xsljdjrjzkj subjecttoXi2FVi(yi)R; 1yiUi;8i2F Theobjectivefunction( 5{17 )isthesumoftheexpectedfailurecostandtheexpectedpenaltycost.Constraint( 5{18 )denotestheresourcerestrictionsonthefortication.Constraints( 5{19 )and( 5{20 )areintegralconstraintsontheforticationlevels. 5{18 )possessesthemonotonicity,sinceVi(yi),i2F,arenondecreasingfunctionsofyi.TheobjectivefunctionofthecontinuousfacilityforticationmodelisshowntobemonotonicallynondecreasinginCorollary 5.1 .Asimilarresultholdsinthediscretecase,becausePi(yi),i2F,arenonincreasingfunctionsofyi.Fromthefactthatthecompositeofanondecreasingfunctionandanonincreasingfunctionisnonincreasing,wehavethefollowingcorollary,whichisparalleltoCorollary 5.1 93

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5{17 ofthediscretefacilityforticationmodelismonotonicallynonincreasing. 4 .Tobeconsistentwiththegeneralformatofthemonotonicoptimizationinthisthesis,wetransformDFFMmodeltoamaximizationproblem. (DFFM-max) maxmizeXj2DjFjXk=1Xi2Fdjcijxkij(1Pi(yi))Yl2FPl(yl)Pk1s=1xsljXj2DjFj+1Xk=1Yl2FPl(yl)Pk1s=1xsljdjrjzkj subjecttoXi2FVi(yi)R; 1yiUi;8i2F AmoregeneralformatofDFFM-maxcanbewrittenasthefollowingbyreplacingthedecisionvariablesyiwithxi. (MO):maxRs=f(x) (5{25) subjecttox2G=fxjgi(x)ci;8i=1;:::;mg; where[xL;xU]denotesahyperrectanglewithlowestboundaryxLandgreatestboundaryxU,functionsfandgiarenondecreasingfori=1;:::;m.ToensureGisclosed,weassumegiaresemi-continuousfori=1;:::;m. Forthecompletenessofexposition,wealsoincludethedescriptionofthemonotonicbranch-reduce-boundalgorithmwithslightdierencesthanthatinChapter 4 to 94

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2 ,Sdenotesahyperrectanglepartition;Lisalistofunfathomedhyperrectangles;isapre-denedoptimalitytoleranceparameter;xbestandfbestdenotethecurrentbestsolutionandobjectivevaluerespectively;UB(S)istheupperboundoftheobjectivefunctionoverS.Besidesinitialization,themajorstepsaredescribedasfollows. SelectandBranch ReduceandBound

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5.2.2 ,whichislistedinAppendix C .Thedatasetcontains20demandnodesand5openfacilitieswiththefollowingspecication. Table5-2. Inputdata(Vi(yi)andPi(yi))forthe3-levelmodel Level1 Level2 Level3 OpenFacilityi V P V P V P 0 0.55 79 0.39 688 0.02 5 0 0.85 614 0.45 728 0.28 15 0 0.75 303 0.63 855 0.48 18 0 0.52 135 0.48 409 0.20 20 0 0.30 178 0.24 273 0.22 Themonotonicbranch-reduce-boundalgorithmisimplementedinC++.TheCPUsecondsarereportedfromaDellOptiplexGX620computerwithaPentiumIV3.6GHzprocessorand1.0GBRAM,runningundertheWindowsXPoperatingsystem. ThecomputationalresultsarereportedinTable 5-3 thatshowsthattheforticationlevelateachopenfacilitygiventheresourceconstraints.Table 5-4 showstheresultsthattheforticationlevelislimitedto2.Thatis,yi2;8i2F.TheCPUtimereportedinbothtablesclearlyshowtheeciencyofthemonotonicbranch-reduce-boundalgorithmforthistypeofproblem.Next,westarttoanalyzethecomputationalresults. 96

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Solutionsofthe3-levelmodel Constraint Resource ForticationLevelat CPU Objective 5 15 18 20 25000 2953 2020.69 3 3 3 3 3 0.11 2952 2858 2030.01 3 3 3 3 2 0.89 2857 2401 2057.66 3 3 2 3 3 1.08 2400 2306 2068.99 3 3 2 3 2 1.19 2305 2098 2087.25 3 3 1 3 3 1.48 2097 2003 2100.18 3 3 1 3 2 1.03 2002 1825 2138.98 3 3 1 3 1 1.80 1824 1711 2276.06 3 2 1 3 1 1.94 1710 1689 2392.85 3 3 1 1 3 2.72 1688 1594 2408.41 3 3 1 1 2 2.58 1593 1551 2415.59 3 3 1 2 1 2.72 1550 1416 2455.11 3 3 1 1 1 3.33 1415 1370 2489.01 3 1 1 3 3 3.05 1369 1275 2516.42 3 1 1 3 2 2.88 1274 1097 2598.63 3 1 1 3 1 3.70 1096 1096 2797.01 3 1 1 2 3 3.34 1095 1001 2830.34 3 1 1 2 2 3.64 1000 961 2841.01 3 1 1 1 3 3.95 960 866 2875.19 3 1 1 1 2 3.17 865 823 2930.35 3 1 1 2 1 3.48 822 688 2977.74 3 1 1 1 1 3.48 687 666 3308.47 2 1 1 3 2 2.27 665 488 3503.25 2 1 1 3 1 2.56 487 409 3894.44 1 1 1 3 1 2.56 408 392 4318.51 2 1 1 2 2 2.58 391 352 4331.16 2 1 1 1 3 2.86 351 257 4462.80 2 1 1 1 2 2.09 256 214 4688.43 2 1 1 2 1 2.09 213 79 4857.74 2 1 1 1 1 1.97 78 0 5670.71 1 1 1 1 1 1.64 InFigure 5-4 ,weplottheobjectivevaluesinthesetwodierentsettings(3-levelconstraintand2-levelconstraint)acrosstheresourceused.Unliketheresultinthecontinuousfacilityforticationmodel,asshowninFigure 5-1 ,thereisnopiecewiselinearpropertyexhibitedinthisdiscreteversion.Insteadthecurvesareshownsteeperintheearlierstage.Thatis,theforticationeortshelptoreducealotoftotalcostatearlierstage.Thisindicatesthatrealiabilitycanbedrasticallyimprovedwithoutlarge 97

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Solutionsofthe2-levelmodel Constraint Resource ForticationLevelat CPU Objective 5 15 18 20 25000 1309 3315.22 2 2 2 2 2 0.11 1308 1174 3402.08 2 2 2 1 2 0.88 1173 1006 3461.27 2 2 1 2 2 1.03 1005 871 3557.78 2 2 1 1 2 1.19 870 828 3666.45 2 2 1 2 1 1.02 827 693 3776.24 2 2 1 1 1 1.19 692 560 4193.94 2 1 2 1 2 1.50 559 392 4318.51 2 1 1 2 2 1.19 391 257 4462.80 2 1 1 1 2 1.34 256 214 4688.43 2 1 1 2 1 1.33 213 79 4857.74 2 1 1 1 1 1.19 78 0 5670.71 1 1 1 1 1 1.06 increasesinforticationresource.Afterthat,thewholesystemseemslackingtheroomforimprovement.Infact,whenRisgreaterthan1500,thedecreaseinthetotalcostcannotjustifytheforticationresource.Forexample,whenRincreasesfrom1551to2003,thetotalcostreducedfrom2415.59to2100.18,anetlossof136.59. Figure5-4. Tradeobetweenobjectiveandresourceused 98

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5-4 ,wealsoseethebenettohavea3-leveloptionovera2-levelone.Whentheavailableforticationresourceisverylimited,bothsharethesameobjectivevaluesasthereisnoenoughresourcetofortifythefacilitytolevel3.Butastheavailableforticationresourceincreases,the3-levelmodelhasmoreexibilitysothatitincursmuchlowertotalcost. Anotherphenomenondierentfromthecontinuousversionistheforticationlevelatindividualfacility.Inthecontinuousversion,theforticationlevelismoreorlessmonotonicasshowninFigure 5-2 .However,thisdoesnotexistinthediscreteversionasshowninFigure 5-5 .Astheforticationresourceincreases,theforticationlevelatanindividualfacilitydoesnotnecessarilyincreaseasaconsequencefromthiscombinatorialoptimization.FiguresDandEinFigure 5-5 clearlyshowthisphenomenon,becausefacility18and20donotadmitanymonotonicitypattern. Inthecontinuousversionofthemodel,theforticationeortisdividable.Weshowthatthemodelisaspecialcaseofthegeneralizedlinearmultiplicativeprogrammingproblem.Wesolveanillustrativeexamplebythevertexnumerationmethod,whichisveryeectiveinsolvingthistypeofproblems.Thisexamplealsodemonstratesthemulti-extremenessoftheproblem:severalverticesachievetheminimum.Managerially,theexampleshowshowtoidentifythekeyfacilitiestofortifyinasystem. Thediscretefacilityforticationmodelfocusesonchoosingthesuitableforticationlevelateachfacilitywhentheforticationeortisdividedintodierentlevels.Themodelisshowntobeamonotonicityoptimizationproblemsincethemonotonicitypropertyisinherentinbothitsobjectiveandconstraints.Thismodelisthereforesolvedbythe 99

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Forticationlevelatindividualfacilitybyresourceused 100

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Themainlimitationofthecurrentmodelsistheassumptionthatthefacilitiesareuncapacitated.Althoughtheassumptionitselfisverycommoninthefacilitylocationmodels,itmaybeunrealisticinpractice.Inthecapacitatedcase,a`customer'ofthefailedfacilitiescanbeassignedtothenextlevelbackupfacilitiesonlyiftheyhavesucientcapacitytosatisfytheadditionaldemands.Thismaymakethecapacitatedmodelverycomplex.Weexpectthatthemonotonicbranch-reduce-boundalgorithmwillstillbeapplicable.Webelievethatthisisavaluabletopicworthyoffutureinvestigation. 101

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Inthischapter,wesummarizethevariousmodelsandalgorithmsdiscussedthroughoutthedissertation,andpointoutdirectionsforfutureresearch. Westudytheimpactofuncertaintyonthedecisionsoffacilitylocationanddemandassignment.Theuncertaintyisrepresentedbythefailureprobabilityineachfacility.Severalnovelmodelshavebeenpresentedtooersolutionsforboththedesignofinitialsupplychainsystemsandtheimprovementoftheexistingsystems.Werstinvestigatetheuncapacitatedreliablefacilitylocationmodel,whoseobjectiveistominimizethetotalofopeningcost,expectedservicecost,andexpectedfail-to-servepenaltycostwheneachfacilityhasasite-specicfailureprobability.Wealsostudyamoregeneralcasethateachfacilityhasmultiplelevelsoffailureprobabilitiesthatcanbechosen.Ifthesupplychainsystemalreadyexists,weproposetwomodelsforoptimallyallocatingtheforticationresourcetoreducetheexpectedserviceandfail-to-servepenaltycost. Thealgorithmspresentedinthisdissertationinclude(1)fourheuristics,thesampleaverageapproximationheuristic,thegreedyaddingheuristic,thegreedyaddingandsubstitutionheuristic,andthegeneticalgorithmbasedheuristic;(2)theapproximationalgorithmwithaworst-caseboundof2.674;(3)themonotonicbranch-reduce-boundalgorithm.Anin-depththeoretictreatmentisprovidedfortheapproximationalgorithm.Allotheralgorithmsarethoroughlytestedinthecomputationalstudies.Thefourheuristicsareusedtosolvetheuncapacitatedreliablefacilitylocationproblem.Themonotonicbranch-reduce-boundalgorithmisappliedtosolvethefacilityforticationproblemaswellasthesystemreliabilityproblemarisingfromindustrialormilitaryapplications. Oneimmediateextensionistostudythecapacitatedversionofthecurrentreliablefacilitylocationmodels.Althoughthecapacitatedconstraintsgenerallyposemorechallengesonndingtheecientalgorithms,weexpectthatsomeoftheheuristicsarestill 102

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Itwouldbeinterestingtoseeifthereexistsanyapproximationalgorithmwithaconstantworst-caseboundforthecapacitateduniformreliablefacilitylocationproblem.Basedontheresearchonapproximationtheoryoffacilitylocationproblemwithoutconsideringthereliabilityissue,weexpectthatsomemoredelicatetechniquesarerequiredtodevelopsuchapproximationalgorithm. Anotherinterestingdirectionistointroducedierentrisk/reliablitymeasurementsintothecurrentmodelsdependingontheneedsinreality.Bydoingso,theobjectivefunctionsofcurrentmodelswillchangeaccordingly.Forexample,onemayhavemoreinterestsinthecostoftheworst-casescenarioinsteadoftheexpectedcostinthecurrentmodels. Insummary,thecurrentworkwepresentservesasausefulfoundationforfurtherresearchofmorecomplicatedmodelsanddelicatealgorithms. 103

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WeuseconditionalprobabilitytoderivetheexpressionsofreliabilitiesofthenetworksinFigure 4-3 andFigure 4-5 4-3 canbewrittenbasedonwhethercomponent5isfunctionalornot. Whencomponent5works,theoriginalnetworkinFigure 4-3 isreducedtoFigure A-1 (A),whichisaparallel-seriessystemwithareliabilityof FigureA-1. Congurationsbasedonstateofcomponent5inFigure 4-3 :A)Component5works;B)Component5fails 104

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4-3 isreducedtoFigure A-1 (B),whichisaseries-parallelsystemwithareliabilityof Substitutionofequations A{2 and A{3 intoequation A{1 yieldsthereliabilityoftheve-componentbridgenetworkdepictedinFigure 4-3 : (A{4) 4-5 representsasubsystem.Blocks6and7areindividualcomponents.RecallthatRi=1(1ri)xi;8i=1;:::;5;Qi=1Ri;8i=1;:::;7:ReliabilityofthenetworkinFigure 4-5 canbewrittenbasedonwhethersubsystem4isfunctionalornot. Whensubsystem4works,theoriginalnetworkinFigure 4-5 isreducedtoFigure A-2 (A),whosereliabilitycanbeobtainedbyapplyingparallelandseriesreductions: 4-5 isreducedtoFigure A-2 (B).Aseriesreductiononsubsystems1and2producesasuper-component,whichhelpstomapthetopologyinFigure A-2 (B)totheve-componentbridgenetworkinFigure 4-3 .Afterthemapping,wecandirectlyusetheresultofequation A{4 byreplacingr1withR1R2,r2

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Congurationsbasedonstateofsubsystem4inFigure 4-5 :A)Subsystem4works;B)Subsystem4fails withR3,r3withR6,r4withR7,r5withR5: (A{7) Substitutionofequations A{6 and A{7 intoequation A{5 yieldsthereliabilityoftheve-componentbridgenetworkdepictedinFigure 4-5 : Equation A{8 canbereformulatedasthefollowingone: 106

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ThemeaningofeachcolumninTable B-1 isprovidedasfollows:#idenotesthefacilityname;(x;y)isthecoordinates,diisthedemand;fiisthexedcost;riisthepenaltycost;andpiisthefailureprobability. TableB-1.DatasetofURFLP-SFP #i x y di x y di 0.82 0.18 957 938 5.32 0.81 51 0.63 0.04 486 971 3.14 0.63 2 0.54 0.7 202 642 1.9 0.39 52 0.53 0.32 548 1023 0.31 0.94 3 0.91 0.72 186 1230 3.11 0.42 53 0.89 0.99 870 754 0.66 0.76 4 0.15 0.31 635 1008 1.83 0.36 54 0.02 0.19 335 734 0.22 0.97 5 0.74 0.16 737 1279 1.34 0.28 55 0.51 0.32 446 1249 3.93 0.91 6 0.58 0.92 953 1431 2.3 0.83 56 0.53 0.06 198 1371 3.77 0.31 7 0.6 0.09 450 1187 7.96 0.98 57 0.81 0.86 212 1489 9.92 0.58 8 0.37 0.19 188 1044 3.42 1 58 0.53 0.36 903 863 4.06 0.29 9 0.7 0.52 206 1466 9.05 0.86 59 0.89 0.58 594 521 1.64 0.15 10 0.22 0.4 995 989 4.56 0.55 60 0.87 0.56 250 865 5.11 0.21 11 0.5 0.45 429 948 9.87 0.58 61 0.91 0.16 472 1464 2.64 0.43 12 0.3 0.52 528 585 0.53 0.79 62 0.32 0.15 244 730 6.05 0.09 13 0.95 0.2 570 923 3.41 0.46 63 0.37 0.37 353 1034 4.27 0.78 14 0.65 0.07 938 758 8.98 0.15 64 0.38 0.73 183 585 6.18 0.23 15 0.53 0.11 726 552 3.53 0.48 65 0.96 0.34 749 782 9.13 0.6 16 0.95 0.95 533 1471 1.64 0.7 66 0.15 0.76 200 985 6.63 0.42 17 0.15 0.13 565 930 1.36 0.98 67 0.15 0.48 321 662 7.02 0.1 18 0.31 0.4 322 1103 5.1 0.2 68 0.99 0 650 904 8.73 0.16 19 0.98 0.73 326 586 1.22 0.49 69 0.47 0.28 946 1242 3.92 0.52 20 0.59 0.04 663 812 6.95 0.3 70 0.84 0.16 143 513 8.16 0.79 21 0.46 0.21 952 850 9.57 0.37 71 0.71 0.9 565 1117 1.25 0.84 107

PAGE 108

x y di x y di 0.77 0.44 919 561 1.49 0.38 72 0.46 0.86 11 928 4.44 0.26 23 0.87 0.79 292 750 6.95 0.6 73 0.09 0.74 374 1028 2.65 0.33 24 0.69 0.15 48 956 2.29 0.71 74 0.71 0.78 284 522 1.45 0.35 25 0.24 0.28 581 1456 2.9 0.3 75 0.27 0.04 598 609 9.63 0.22 26 0.84 0.73 659 764 5.05 0.99 76 0.25 0.07 720 541 3.72 0.25 27 0.49 0.24 986 1005 8.81 0.23 77 0.57 0.18 457 1493 2.38 0.43 28 0.56 0.77 486 1121 3 0.81 78 0.96 0.49 213 736 7.09 0.49 29 0.38 0.05 915 1065 6.94 0.4 79 0.83 0.21 550 1452 8.3 0.43 30 0.43 0.22 282 1494 0.88 0.68 80 0.72 0.49 418 758 2.55 0.57 31 0.61 0.73 310 605 2.66 0.39 81 0.69 0.5 863 1147 1.53 0.93 32 0.48 0.88 980 1304 3.32 0.86 82 0.22 0.89 368 976 3.5 0.52 33 0.81 0.75 134 1073 0.66 0.28 83 0.37 0.88 282 1180 7.15 0.95 34 0.13 0.71 20 1107 3.08 0.62 84 0.36 0.82 811 1045 9.07 0.3 35 0.41 0.34 151 651 2.64 0.01 85 0.11 0.1 866 1021 2.46 0.37 36 0.72 0.14 615 800 3.84 0.35 86 0.77 0.69 895 1165 3.19 0.44 37 0.3 0.28 369 1479 1.74 0.22 87 0.16 0.09 959 1250 5.52 0.66 38 0.02 1 875 1278 1.14 0 88 0.75 0.63 375 1406 5.28 0.06 39 0.62 0.9 73 1289 2.76 0.42 89 0.16 0.41 711 515 7.34 0.46 40 0.8 0.06 776 510 3.36 0.69 90 0.01 0.21 208 923 3.99 0.87 41 0.1 0.98 342 536 5.61 0 91 0.51 0.76 954 1378 4.06 0.31 42 0.15 0.13 929 1022 9.77 0.73 92 0.98 0.32 843 733 7.77 0.33 43 0.48 0.44 445 594 4.32 0.83 93 0.55 0.39 905 545 0.08 0.74 44 0.83 0.22 684 1466 9.61 0.28 94 0.36 0.63 729 1047 8.47 0.84 45 0.82 0.39 643 677 1.45 0.71 95 0.18 0.75 382 1365 6.23 0.38 46 0.67 0.53 771 1334 7.46 0.74 96 0.09 0.46 91 513 0.53 0.16 47 0.03 0.73 181 1445 1.03 0.35 97 0.18 0.67 991 1338 6.62 0.92 48 0.26 0.39 926 691 7.6 0.26 98 0.1 0.38 644 1341 3.13 0.11 49 0.59 0.56 733 1001 0.15 0.08 99 0.25 0.66 539 1256 0.5 0.41 50 0.22 0.66 326 1244 5.93 0.98 100 0.68 0.49 294 1168 7.27 0.66 108

PAGE 109

B-2 isprovidedasfollows:#idenotesthefacilityname;(x;y)isthecoordinates,diisthedemand;riisthepenaltycost;andfi;pi(i=1;2;3)aretheinvestmentlevelanditscorrespondingfailureprobability. TableB-2.DatasetofURFLP-MFP:3-level #i x y di 0.82 0.18 957 5.32 938 0.81 954 0.44 1260 0.15 2 0.54 0.7 202 1.9 642 0.55 721 0.39 1330 0.02 3 0.91 0.72 186 3.11 1125 0.63 1230 0.42 1355 0.19 4 0.15 0.31 635 1.83 772 1 1008 0.9 1440 0.36 5 0.74 0.16 737 1.34 665 0.85 1279 0.45 1393 0.28 6 0.58 0.92 953 2.3 890 0.9 1034 0.83 1431 0.61 7 0.6 0.09 450 7.96 620 0.98 1187 0.48 1394 0.06 8 0.37 0.19 188 3.42 503 1 703 0.75 1044 0.49 9 0.7 0.52 206 9.05 1231 0.86 1278 0.65 1466 0.24 10 0.22 0.4 995 4.56 989 1 1037 0.55 1455 0.54 11 0.5 0.45 429 9.87 948 0.87 1082 0.6 1422 0.58 12 0.3 0.52 528 0.53 551 0.84 585 0.79 1303 0.39 13 0.95 0.2 570 3.41 682 0.46 923 0.32 999 0.04 14 0.65 0.07 938 8.98 758 0.78 987 0.5 1497 0.15 15 0.53 0.11 726 3.53 552 0.75 855 0.63 1407 0.48 16 0.95 0.95 533 1.64 791 0.7 1471 0.66 1488 0.47 17 0.15 0.13 565 1.36 930 0.98 958 0.8 1227 0.35 18 0.31 0.4 322 5.1 694 0.52 829 0.48 1103 0.2 19 0.98 0.73 326 1.22 586 0.95 975 0.49 1127 0.15 20 0.59 0.04 663 6.95 634 0.3 812 0.24 907 0.22 21 0.46 0.21 952 9.57 850 0.37 932 0.18 980 0.14 22 0.77 0.44 919 1.49 561 0.92 742 0.76 1290 0.38 23 0.87 0.79 292 6.95 750 0.6 1213 0.59 1311 0.24 24 0.69 0.15 48 2.29 684 0.91 772 0.79 956 0.71 25 0.24 0.28 581 2.9 1040 0.62 1153 0.3 1456 0.23 109

PAGE 110

x y di 0.84 0.73 659 5.05 764 0.99 941 0.2 1035 0.04 27 0.49 0.24 986 8.81 910 0.94 1005 0.25 1058 0.23 28 0.56 0.77 486 3 928 0.81 1079 0.72 1121 0.35 29 0.38 0.05 915 6.94 1040 0.81 1065 0.4 1108 0.16 30 0.43 0.22 282 0.88 771 0.76 1028 0.68 1494 0.04 31 0.61 0.73 310 2.66 605 0.81 748 0.41 1381 0.39 32 0.48 0.88 980 3.32 1106 0.86 1304 0.21 1400 0.06 33 0.81 0.75 134 0.66 1073 0.88 1153 0.34 1315 0.28 34 0.13 0.71 20 3.08 905 0.79 1107 0.62 1260 0.62 35 0.41 0.34 151 2.64 651 0.87 652 0.25 1066 0.01 36 0.72 0.14 615 3.84 800 0.98 1298 0.57 1423 0.35 37 0.3 0.28 369 1.74 723 0.7 1351 0.51 1479 0.22 38 0.02 1 875 1.14 890 0.81 1029 0.04 1278 0 39 0.62 0.9 73 2.76 1104 0.88 1289 0.42 1485 0.37 40 0.8 0.06 776 3.36 510 0.91 981 0.69 1341 0.59 41 0.1 0.98 342 5.61 536 0.63 1157 0.06 1351 0 42 0.15 0.13 929 9.77 1022 0.73 1064 0.67 1157 0.33 43 0.48 0.44 445 4.32 594 0.83 1289 0.76 1386 0.43 44 0.83 0.22 684 9.61 1130 0.68 1192 0.28 1466 0.09 45 0.82 0.39 643 1.45 518 0.71 677 0.38 1072 0.28 46 0.67 0.53 771 7.46 892 0.74 1334 0.72 1483 0.45 47 0.03 0.73 181 1.03 1055 0.97 1156 0.63 1445 0.35 48 0.26 0.39 926 7.6 691 0.56 865 0.26 1312 0.23 49 0.59 0.56 733 0.15 1001 0.72 1058 0.61 1450 0.08 50 0.22 0.66 326 5.93 699 0.98 1244 0.42 1472 0.04 51 0.63 0.04 486 3.14 971 0.68 1154 0.63 1493 0.49 52 0.53 0.32 548 0.31 853 0.99 1023 0.94 1207 0.28 53 0.89 0.99 870 0.66 754 0.84 787 0.76 1271 0.61 54 0.02 0.19 335 0.22 611 0.97 734 0.83 1214 0.51 55 0.51 0.32 446 3.93 1217 0.91 1249 0.88 1367 0.05 56 0.53 0.06 198 3.77 742 0.69 1025 0.49 1371 0.31 110

PAGE 111

x y di 0.81 0.86 212 9.92 1271 0.58 1446 0.4 1489 0.38 58 0.53 0.36 903 4.06 863 0.75 1078 0.29 1354 0.05 59 0.89 0.58 594 1.64 521 0.64 1071 0.44 1100 0.15 60 0.87 0.56 250 5.11 865 0.95 1273 0.6 1348 0.21 61 0.91 0.16 472 2.64 974 0.92 1220 0.43 1464 0.22 62 0.32 0.15 244 6.05 689 0.88 730 0.58 810 0.09 63 0.37 0.37 353 4.27 745 0.89 821 0.78 1034 0.46 64 0.38 0.73 183 6.18 585 0.79 1026 0.78 1259 0.23 65 0.96 0.34 749 9.13 782 0.84 1219 0.6 1392 0.13 66 0.15 0.76 200 6.63 561 0.93 985 0.42 1068 0.22 67 0.15 0.48 321 7.02 569 0.68 662 0.5 713 0.1 68 0.99 0 650 8.73 880 0.79 904 0.26 1424 0.16 69 0.47 0.28 946 3.92 1190 0.64 1242 0.58 1332 0.52 70 0.84 0.16 143 8.16 513 0.79 993 0.48 1355 0.14 71 0.71 0.9 565 1.25 601 1 945 0.84 1117 0.13 72 0.46 0.86 11 4.44 904 0.53 928 0.32 982 0.26 73 0.09 0.74 374 2.65 691 0.48 1028 0.33 1291 0.32 74 0.71 0.78 284 1.45 522 0.94 818 0.68 1246 0.35 75 0.27 0.04 598 9.63 609 0.85 771 0.22 1026 0.11 76 0.25 0.07 720 3.72 541 0.25 704 0.14 1012 0.09 77 0.57 0.18 457 2.38 986 0.75 1307 0.43 1493 0.19 78 0.96 0.49 213 7.09 666 0.77 736 0.66 861 0.49 79 0.83 0.21 550 8.3 564 0.74 796 0.43 1452 0.15 80 0.72 0.49 418 2.55 588 0.78 619 0.57 758 0.39 81 0.69 0.5 863 1.53 1147 0.93 1299 0.77 1310 0.71 82 0.22 0.89 368 3.5 510 0.65 976 0.52 1070 0.39 83 0.37 0.88 282 7.15 598 0.95 871 0.73 1180 0.07 84 0.36 0.82 811 9.07 674 0.58 1045 0.4 1166 0.3 85 0.11 0.1 866 2.46 568 0.63 1021 0.37 1192 0.24 86 0.77 0.69 895 3.19 665 0.93 1015 0.44 1165 0.25 87 0.16 0.09 959 5.52 980 0.92 1250 0.68 1412 0.66 111

PAGE 112

x y di 0.75 0.63 375 5.28 522 0.59 1206 0.06 1406 0.04 89 0.16 0.41 711 7.34 515 0.63 707 0.48 1415 0.46 90 0.01 0.21 208 3.99 923 0.87 1049 0.84 1281 0.32 91 0.51 0.76 954 4.06 739 0.79 1186 0.31 1378 0.28 92 0.98 0.32 843 7.77 733 0.52 834 0.33 1335 0.18 93 0.55 0.39 905 0.08 545 0.74 869 0.61 1178 0.61 94 0.36 0.63 729 8.47 1047 0.84 1098 0.66 1460 0.6 95 0.18 0.75 382 6.23 538 0.62 613 0.59 1365 0.38 96 0.09 0.46 91 0.53 513 0.67 521 0.26 1448 0.16 97 0.18 0.67 991 6.62 723 0.92 1119 0.91 1338 0.41 98 0.1 0.38 644 3.13 628 0.54 822 0.46 1341 0.11 99 0.25 0.66 539 0.5 848 0.49 942 0.48 1256 0.41 100 0.68 0.49 294 7.27 1069 0.99 1108 0.88 1168 0.66 112

PAGE 113

ThemeaningofeachcolumninTable C-1 isprovidedasfollows:#idenotesthefacilityname;(x;y)isthecoordinates,diisthedemand;fiisthexedcost;riisthepenaltycost. TableC-1.DatasetofDFFM #i x y di x y di 2,5,15,18,20 1 0.82 0.18 957 5.32 51 0.63 0.04 486 3.14 2 0.54 0.7 202 1.9 52 0.53 0.32 548 0.31 3 0.91 0.72 186 3.11 53 0.89 0.99 870 0.66 4 0.15 0.31 635 1.83 54 0.02 0.19 335 0.22 5 0.74 0.16 737 1.34 55 0.51 0.32 446 3.93 6 0.58 0.92 953 2.3 56 0.53 0.06 198 3.77 7 0.6 0.09 450 7.96 57 0.81 0.86 212 9.92 8 0.37 0.19 188 3.42 58 0.53 0.36 903 4.06 9 0.7 0.52 206 9.05 59 0.89 0.58 594 1.64 10 0.22 0.4 995 4.56 60 0.87 0.56 250 5.11 11 0.5 0.45 429 9.87 61 0.91 0.16 472 2.64 12 0.3 0.52 528 0.53 62 0.32 0.15 244 6.05 13 0.95 0.2 570 3.41 63 0.37 0.37 353 4.27 14 0.65 0.07 938 8.98 64 0.38 0.73 183 6.18 15 0.53 0.11 726 3.53 65 0.96 0.34 749 9.13 16 0.95 0.95 533 1.64 66 0.15 0.76 200 6.63 17 0.15 0.13 565 1.36 67 0.15 0.48 321 7.02 18 0.31 0.4 322 5.1 68 0.99 0 650 8.73 19 0.98 0.73 326 1.22 69 0.47 0.28 946 3.92 20 0.59 0.04 663 6.95 70 0.84 0.16 143 8.16 21 0.46 0.21 952 9.57 71 0.71 0.9 565 1.25 22 0.77 0.44 919 1.49 72 0.46 0.86 11 4.44 113

PAGE 114

x y di x y di 0.87 0.79 292 6.95 73 0.09 0.74 374 2.65 24 0.69 0.15 48 2.29 74 0.71 0.78 284 1.45 25 0.24 0.28 581 2.9 75 0.27 0.04 598 9.63 26 0.84 0.73 659 5.05 76 0.25 0.07 720 3.72 27 0.49 0.24 986 8.81 77 0.57 0.18 457 2.38 28 0.56 0.77 486 3 78 0.96 0.49 213 7.09 29 0.38 0.05 915 6.94 79 0.83 0.21 550 8.3 30 0.43 0.22 282 0.88 80 0.72 0.49 418 2.55 31 0.61 0.73 310 2.66 81 0.69 0.5 863 1.53 32 0.48 0.88 980 3.32 82 0.22 0.89 368 3.5 33 0.81 0.75 134 0.66 83 0.37 0.88 282 7.15 34 0.13 0.71 20 3.08 84 0.36 0.82 811 9.07 35 0.41 0.34 151 2.64 85 0.11 0.1 866 2.46 36 0.72 0.14 615 3.84 86 0.77 0.69 895 3.19 37 0.3 0.28 369 1.74 87 0.16 0.09 959 5.52 38 0.02 1 875 1.14 88 0.75 0.63 375 5.28 39 0.62 0.9 73 2.76 89 0.16 0.41 711 7.34 40 0.8 0.06 776 3.36 90 0.01 0.21 208 3.99 41 0.1 0.98 342 5.61 91 0.51 0.76 954 4.06 42 0.15 0.13 929 9.77 92 0.98 0.32 843 7.77 43 0.48 0.44 445 4.32 93 0.55 0.39 905 0.08 44 0.83 0.22 684 9.61 94 0.36 0.63 729 8.47 45 0.82 0.39 643 1.45 95 0.18 0.75 382 6.23 46 0.67 0.53 771 7.46 96 0.09 0.46 91 0.53 47 0.03 0.73 181 1.03 97 0.18 0.67 991 6.62 48 0.26 0.39 926 7.6 98 0.1 0.38 644 3.13 49 0.59 0.56 733 0.15 99 0.25 0.66 539 0.5 50 0.22 0.66 326 5.93 100 0.68 0.49 294 7.27 114

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LezhouZhanwasborninZhejiang,China,intheyearoftheHorse.HeisalsoknownasRogerwhosepronunciationissimilartoLezhouinhishometowndialect.Priortocollege,hegraduatedfromYueqingMiddleSchoolin1997.HereceivedhisBachelorofScienceinAppliedMathematicsandhisBachelorofScienceinBusinessAdministrationandEngineeringManagementfromChongqingUniversityin2001.BeforehetransferredtotheUniversityofFloridainthefallof2002,hestudiedscienticcomputationattheHongKongUniversityofScienceandTechnologyinaMasterofPhilosophyprogram.HeservedasSecretary-GeneralofFACSS,aChinesestudentassociationatUF,from2003to2004.HeearnedhisMasterofScienceandDoctorofPhilosophyinIndustrialandSystemsEngineeringfromtheUniversityofFloridainMay,2004andAugust,2007respectively.Hiscurrentresearchinterestsincludereliablesupplychaindesign,auctionmechanismdesign,operationsresearchmodelsinairlineapplications,andsystemreliabilityoptimization.Hisworkhasbeenpresentedinvariousconferences,bookchapters,andjournals,includingProceedingsofthe2005WinterSimulationConference,Proceedingsofthe2005IIEResearchConferenceandProductionandOperationsManagement.HeisamemberofINFORMS,SIAM,andIIE. 120