Nonlinear Approximation Techniques to Solve Network Flow Problems with Nonlinear Arc Cost Functions

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Nonlinear Approximation Techniques to Solve Network Flow Problems with Nonlinear Arc Cost Functions
NAHAPETYAN, ARTYOM ( Author, Primary )
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Algorithms ( jstor )
Heuristics ( jstor )
Integers ( jstor )
Linear programming ( jstor )
Mathematical vectors ( jstor )
Objective functions ( jstor )
Pricing ( jstor )
Tolls ( jstor )
Transportation ( jstor )
Travel time ( jstor )

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University of Florida
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Copyright 2006


Artyom Nahapetyan

I dedicate this work to my parents, Nina Hovhanni- i, and Garush Nahapetyan,

who alv--x supported me in my studies.


I would like to thank my chair and cochair, Prof. Siriphong L ,i.--1 i,.! i ich

and Prof. Donald W. Hearn, for their valuable advice, support and guidance during

my studies. Our meetings and discussions were ahv--l- very helpful.

Also I would like to express my sincere gratitude to the committee members

Prof. Panos Pardalos, Prof. William Hager, and Prof. Ravindra Al!n I, for their

encouragement. Especially, I am grateful to Prof. Panos Pardalos for his valuable

si.-.- -I ir 1 and advice on the supply chain problems I have worked on.

The tremendous support from my parents is invaluable, and there are no words

to express my appreciation for that.

Finally, I would like to thank all my friends and collaborators who made my

studies enjov- 1'l1' and productive.


ACKNOWLEDGMENTS ................... ...... iv

LIST OF TABLES ................... .......... viii

LIST OF FIGURES ................................ x

ABSTRACT ....................... ........... xi


1 INTRODUCTION .................... ....... 1


2.1 Introduction to the C!i ipter ................... ... 5
2.2 A Bilinear Reduction Technique for the Concave Piecewise Linear
Network Flow Problem ........... ..... ....... .. 7
2.3 Concave Piecewise Linear Problems with Separable Objective
Functions .............................. 11
2.4 Dynamic Cost Updating Procedure ................ 13
2.5 On the Dynamic Slope Scaling Procedure ....... ....... 16
2.6 Numerical Experiments ......... ................ 19
2.7 Concluding Remarks ............... ....... .. 21


3.1 Introduction to the C!i lpter ................... ... 22
3.2 Approximation of the Fixed CI irge Network Flow Problem by a
Two-Piece Linear Concave Network Flow Problem . ... 24
3.3 Adaptive Dynamic Cost Updating Procedure . . 27
3.4 On the Dynamic Slope Scaling Procedure . . ..... 30
3.5 Numerical Experiments .................. ..... .. 31
3.6 Concluding Remarks .................. ....... .. 34


4.1 Introduction to the C!i lpter .................. .... 35
4.2 Problem Description .................. ....... .. 37

4.3 A Bilinear Reduction Based Algorithm for Solving AC'\ Il)P Problem 43
4.4 Numerical Experiments .................. ... .. 46
4.5 Concluding Remarks .................. ....... .. 49


5.1 Introduction to the C'!i lter .................. .... 50
5.2 Periodic Planning Horizon ......... .... .... ....... 53
5.3 Discrete-Time Dynamic Traffic Assignment Problem with Periodic
Time Horizon ................... ....... 54
5.4 Bounds for the DTDTA Problem .................. .. 67
5.5 Numerical Experiments .................. ...... .. 71
5.6 Concluding Remarks .................. ....... .. 76

FOR THE UPPER-BOUND PROBLEM ....... ......... 78

6.1 Introduction to the C!i lpter ................... ... 78
6.2 Nonlinear Relaxation of DTDTA-U Problem ............. 83
6.3 Nonlinear Relaxation Based Heuristic Algorithm . .... 86
6.4 Numerical Experiments .................. ...... .. 89
6.5 Concluding Remarks .................. ....... .. 91


7.1 Introduction to the C!i lpter ................... ... 92
7.2 The Reduced Time-Expanded Network and UE Solution . 97
7.3 The Dynamic Toll Set .................. ..... .. 103
7.4 Dynamic Toll Pricing Problems .............. .. .. 108
7.5 Illustrative Examples .................. .. .... .. .. 110
7.6 Concluding Remarks .................. ..... .. 113








REFERENCES .. .... ............................ 131

BIOGRAPHICAL SKETCH ........ ........ ............ 140

Table page

5-1 Demand patterns .................. ............ .. 72

5-2 Optimal solutions to the two-arc problem. ............... 73

5-3 Solutions from the lower and upper-bound problems: linear travel cost
function .................. ................. .. 75

5-4 Solutions from the lower and upper-bound problems: quadratic travel
cost function. .................. .............. .. 75

5-5 Quality of refined upper and lower-bound solutions: linear travel cost
function .................. ................. .. 76

5-6 Quality of refined upper and lower-bound solutions: quadratic travel cost
function. .................. ................. .. 76

6-1 Equivalent objective functions .................. .... .. 88

6-2 Distributions of parameters of randomly generated travel time functions 89

7-1 Additional constraints .................. ....... .. .. 110

7-2 Distributions of parameters of randomly generated travel time functions 111

A-1 Set of problems. .................. ............. .. 117

A-2 Computational results of sets 1-18: quality of the solution and the CPU
times .................. .................. .. 118

A-3 Computational results of sets 1-18: DSSP vs. DCUP. . .... 119

A-4 Computational results for sets 19-30. .............. .. 120

A-5 Computational results for the combined mode. . . ...... 120

B-1 Set of problems. .................. ............. .. 121

B-2 Computational results of groups G1 and G2: quality of the solutions and
the CPU times. .................. ............ 122

B-3 Computational results of groups G1 and G2: the percentage of problem
where one of the algorithms finds a better solution than another one. 123

B-4 Computational results of groups G3 and G4 ............... ..124

C-1 The quality of the solution: Procedure 6. ................ 125

C-2 The quality of the solution: Procedure 5. ................ 126

C-3 The CPU time of the procedures. ................ ..... 126

D-1 Computational results of the experiments. ............... 127

D-2 Computational results of the combined mode. ............. ..128

E-1 The total collected toll and the total cost for each problem and parameter
S.. ........................................ .. 129

E-2 The number of toll collecting centers for each problem and parameter e.. 129

Figure page

3-1 Approximation of function fa(xa). ............ . .. 25

3-2 ,a (xa) and 0a (X,) functions. .................. ..... 28

4-1 The price and the revenue functions. . ......... 38

5-1 Linear versus circular intervals. .................. .... 53

5-2 Events occurring in two consecutive planning horizons. . .... 54

5-3 Three-node network. ............... ..... .... 55

5-4 Time expansion of arc (1, 2) at t = ................ .. 56

5-5 Time-expansion of the three-node network. ................ 58

5-6 Oa(Xa(t)) C ( 6, s] versus Xa(t) E (1( 6), 1(6)] . . 68

5-7 Two-arc network ................ ........... .. 72

5-8 Four-node network. ............... .......... 74

6-1 Two feasible solutions. ............... ...... 80

7-1 4-Node network and traffic demand. ............. .. 92

7-2 User equilibrium flows and travel times. ................ 93

7-3 System optimum flows and travel times. ................ 93

7-4 Tolled user equilibrium flows and travel times. .. . ..... 94

7-5 The value of at. ............... ............ .. 111

D-1 Two Networks. ............... ............ 127

E-1 9-node network. .................. .. ...... ...... 129

E-2 The toll vector for different values of E in the MinRev(E) problem. 130

E-3 The toll vector for different values of E in the MinCost(E) problem. 130

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



Artyom Nahapetyan

August 2006

C('! ,': Siriphong L i.v !. wi! )panich
Cochair: Donald W. Hearn
Major Department: Industrial and Systems Engineering

In this dissertation we investigate network flow problems with nonlinear arc

cost functions. The first group of problems consists of concave piecewise linear

network flow, fixed charge network flow, and dynamic pricing problems that

arise in the areas of supply chain management and logistics. Based on the MIP

formulation, we construct bilinear reduction problems, in which the global solution

of the latter is a solution of the initial formulation. To solve the reduction problem,

we propose some heuristic algorithms. In the experiments, we compare the solution

provided by our algorithm with an exact solution as well as a solution provided by

other heuristic algorithms in the literature. Numerical experiments on randomly

generated instances confirm the quality of the algorithms.

The second group of problems is related to the dynamic traffic assignment

problem. In particular, we consider a periodic discrete time dynamic traffic

assignment problem (DTDTA), in which the travel time is a function of the number

of cars on the road, and the planning horizon is circular. The mathematical

formulation belongs to the class of nonlinear mixed integer problems. To obtain an

appropriate solution to the problem, we construct a linear mixed integer problem

for providing an upper bound and discuss an approximation scheme based on the

bounding problem. However, the bounding problem involves binary variables, and

when the problem is large, it is hard to solve. To overcome these difficulties, we

propose a heuristic algorithm based on a bilinear relaxation of the problem. Using

an approximate solution, in the dissertation we develop a toll pricing framework for

the dynamic case. In particular, based on a feasible vector of DTDTA we describe

a set of valid tolls and discuss several toll pricing problems. By constructing

an appropriate time-expanded network, one may consider a similar toll pricing

framework for other solutions obtained, for example, from a simulation.


Network flow problems are minimization/maximization problems with

underlying network structure. Although there are different representations of the

network, perhaps the most popular one is based on flow conservation constraints

via a node-arc incidence matrix. Apart from the flow conservation constraints,

many problems have additional restrictions on the variables, e.g., non-negativity

and lower/upper boundary constraints. Based on the objective function and other

additional constraints the problems can be classified as linear or nonlinear, where

the latter can be further decomposed into convex, concave, or other problems.

The linear problems assume that the constraints as well as the objective

function are linear. Polynomial time algorithms for solving the problems are

well known. Some classical examples of the network flow problems with linear

constraints include shortest path, minimum spanning tree, minimum cut, maximum

flow, minimum cost network flow, and other problems. Based on the optimality

conditions and other properties of the problem, several algorithms have been

proposed to solve the problems. For details on the linear network flow problems,

see Al!,i et al. [2].

Despite the nice theoretical results developed for the linear problems, most of

the practical problems are not linear, i.e.; the objective function and/or some of the

constraints are nonlinear. If it is a convex minimization (concave maximization)

problem with a convex feasible region then any local minimum (maximum)

is a global solution of the problem, and appropriate algorithms such that the

Frank-Wolf algorithm, gradient based and direction finding methods, can be used

to solve the problem. A large variety of algorithms for solving convex minimization

(concave maximization) problems can be found in Bazaraa et al. [5]. When the
objective function is not convex (concave) and/or the feasible region is not convex,

these algorithms do not necessarily converge to a global optimal solution. Finding

a global solution is a hard task, and global optimization techniques are required to

solve the problem (see, e.g., Horst et al. [54] and Horst and Tuy [55]).

In this dissertation, we consider two groups of non-convex network flow

problems. The first group includes piecewise linear network flow, fixed charge

network flow, and dynamic pricing problems (see C'! lpters 2, 3, and 4) that have

a large variety of applications in the production 1p1 ,liiiir.- scheduling, investment

decision, network design, location of plants and distribution centers, pricing

policy, and many other practical problems that arise in supply chains, logistics,

transportation science, and computer networks. It is well known that the problems

in their general form are NP-Hard; therefore, there are no polynomial time

algorithms to solve the problems unless P = NP. Although the mathematical

formulation belongs to the class of linear mixed integer problems, solving large

problems requires a large amount of CPU time and memory. On the other hand,

one can consider approximation techniques that are able to provide a good-quality

solution using less computer resources. Many of these techniques employ a linear

relaxation of the problem. Unlike those in the literature, we propose nonlinear

reduction techniques to solve the problems. In particular, in all three problems

we develop a method to reduce the problem to a bilinear one and propose a

heuristic algorithm to solve the resulting problem. In the numerical experiments we

compare the results with an exact solution (or the best feasible solution) provided

by MIP solvers, as well as with Dynamic Slope Scaling Procedure (DSSP), since

it is known to be one of the best heuristic algorithms to solve such problems.

Numerical experiments on randomly generated problems confirm the quality of the

solutions provided by our algorithms. In particular, it outperforms the DSSP in

the quality of the solution as well as in the computational time. In addition, we

transform the problems into alternative continuous network flow problems with flow

dependent cost function and prove that a global solution of the resulting problem is

a solution of the initial MIP formulation. Despite an unusual structure of the cost

function, the mathematical formulation of the problems is similar to the system

optimum problems arising in the traffic assignment modelling. Using the same cost

function, we also construct a variational inequality problem similar to those in the

transportation literature and prove that the DSSP converges to a solution of the

resulting problem; i.e., it provides an equilibrium solution. However, the problem

requires finding a system optimum solution, and the algorithms we propose finds an

approximate solution to the problem.

The second group of problems is related to the dynamic traffic assignment

problem. Unlike the static case, where the travel time is a function of the arc flow,

the dynamic models involve three variables: inflow rate, outflow rate, and density,

and the travel time can be a function of all three variables. In the literature several

continuous and discrete time models have been proposed for different travel time

functions. The model in this dissertation assumes that the travel time is a function

of the density, and all cars that enter an arc at the same point of time experience

the same traffic conditions; therefore, they leave the arc at the same time. In

addition, the models in the literature assume that the network is empty at the

beginning and the end of a planning horizon. In the case when some cars are

present in the network, the time to enter the network for those cars is unknown,

and it is hard to model the propagation of the cars in the network. Unlike other

models in the literature, we consider a periodic planning horizon and assume

that the processes repeat themselves from one period to another (see C'! plter

5). The mathematical formulation of the problem minimizes the total delay and

belongs to the class of nonlinear mixed integer problems, a hard problem to solve.

By linearizing the objective function and the constraints, we construct linear

mixed integer problems that provide upper and lower bounds. The solution of

the bounding problems can be made arbitrarily close to a solution of the initial

formulation by decreasing the discretization parameter. However, the bounding

problems involve binary variables, and it is hard to solve large problems using

MIP solvers. In ('!i lpter 6 we discuss a heuristic algorithm based on a nonlinear

relaxation of the problem. In particular, we construct a continuous bilinear

problem, which provides a tighter lower bound than the LP relaxation. Using the

bilinear relaxation, the heuristic algorithm aims to find an integer solution, which

has an objective function value close to the one provided by the relaxation problem.

Another problem of interest is the toll pricing framework for the dynamic

traffic assignment problem (see ('!i lpter 7). Similar to the static case, we construct

a set of valid toll vectors such that a system optimum solution is a solution of

the tolled user equilibrium problem. The latter is a user equilibrium problem

where the arc cost functions include tolls in addition to the travel times. A key

component in the development of such technique is the reduced time-expanded

(RTE) network constructed based on a feasible vector. Using the network, we show

that a feasible vector is a user equilibrium solution if and only if it is a solution of

a linear problem with an underlying RTE network structure. The latter allows the

construction of a set of valid tolls and formulation of a toll pricing problem with a

secondary objective, and we provide several examples of such problems.


2.1 Introduction to the Chapter

The cost functions in most of the network flow problems considered in the

literature are assumed to be linear or convex. However, this assumption might not

hold in practical real-world problems. In fact, the costs often have the structure of

a concave or piecewise linear concave function (see Guisewite and Pardalos [45] and

Geunes and Pardalos [41]). We consider the concave piecewise linear network flow

problem (CPLNF), which has diverse applications in supply chain management,

logistics, transportation science, and telecommunication networks. In addition,

the CPLNF problem can be used to find an approximate solution for network flow

problems with a continuous concave cost function. It is well known that these

problems are NP-hard (see Guisewite and Pardalos [45]).

This chapter deals with a nonlinear reduction technique for the linear mixed

integer formulation of the CPLNF problem. In particular, the problem is reduced

to a continuous one with linear constraints and a bilinear objective function. The

reduction has an economical interpretation and its solution is proven to be the

solution of the CPLNF problem. Based on the reduction, we propose an algorithm

for finding a local minimum of the problem, which we refer to as the dynamic cost

updating procedure (DCUP). In the chapter, we show that DCUP converges in a

finite number of iterations.

The theoretical results presented in this chapter can be extended to a more

general concave minimization problem with a separable piecewise linear objective

function and linear/nonlinear constraints. It should be emphasized that Horst

et al. [54] (see also Horst and Tuy [55]) discuss a bilinear program with disjoint

feasible regions and prove that the problem is equivalent to a subclass of piecewise

linear concave minimization problems. The results in this chapter show that

any concave minimization problem with a separable concave piecewise linear

objective function is equivalent to a bilinear program. It is well known that an

optimal solution of a general jointly constrained bilinear program belongs to

the boundary of the feasible region and is not necessarily a vertex (see Horst et

al. [54]). However, the reduction technique presented in this chapter has a jointly

constrained feasible region with a special structure and it is still equivalent to a

concave piecewise linear program. From the latter it follows that two parts of a

solution of the problem are vertices of two different polytopes that are "joined"

by a set of constraints. In that sense, these types of problems are n, .'/ 1.; joined

bilinear programs.

The CPLNF problem can be transformed into an equivalent network flow

problem with flow dependent costs function (NFPwFDCF). Using NFPwFDCF,

it can be shown that the dynamic slope scaling procedure (DSSP) (see Kim and

Pardalos [61] and [62]) converges to an equilibrium solution of NFPwFDCF.

Although DSSP provides a solution, which can be quite close to the system

solution, it is well known that the equilibrium and the system solutions in general

are not the same. On the other hand, DCUP converges to a local minimum of the

problem. In the numerical experiments, we solve different problems using DCUP

and DSSP and compare the quality of the solution as well as the running time.

Computational results show that DCUP often provides a better solution than

DSSP and uses fewer iterations and less CPU time. Since DCUP starts from a

feasible vector and converges to a local minimum, one considers first solving DSSP

and then improving the solution using DCUP. The numerical experiments using

this combined mode are provided as well.

For the remainder, Section 2.2 discusses the nonlinear reduction technique

for the CPLNF problem. Section 2.3 generalizes the results from Section 2.2 for a

concave piecewise linear problem with a separable objective function. Section 2.4

describes DCUP and theoretical results on the convergence and the solution of the

procedure. In Section 2.5 we prove that the solution of the DSSP is an equilibrium

solution of a network flow problem with flow dependent cost functions. The results

of numerical experiments on DCUP and DSSP are provided in Section 2.6, and

finally, Section 2.7 concludes the chapter.

2.2 A Bilinear Reduction Technique for the Concave Piecewise Linear
Network Flow Problem

Let G(N, A) represent a network where N and A are the sets of nodes and

arcs, respectively. The following is the mathematical formulation of the concave

piecewise linear network flow problem (CPLNF)

min fa(x)

s.t. Bx = b (2-1)

xa e [A, A"'] Va EA (2-2)

where B is the node-arc incident matrix of the network G, and fa(xa) are piecewise

linear concave functions, i.e.,

fa(X + xa ( f/(Xa)) Xa e [A2, A')

fa(xa) = ... ...

cjaXa + aS( ffa(xa)) xa G [A/a 1, al]

with ca > c > > c. Let Ka = {1,2,..., na}. Notice that because of the

concavity of fa(Xa), the function can be written in the following alternative form

fa(Xa) = min{f(xa)} = min{c xa + s}.
kEKa kEKa

Using binary variables, y, k e Ka, one can formulate the CPLNF problem as

the following linear mixed integer program (CPLNF-IP).

min a ac 5 E kak
aEA kEKa aEA EkEKa

s.t. Bxr b (2-3)

Sx= X Va e A (2-4)
a A <1Y Xa< < A Va c A (2-5)
kEKa kEKa

S 1 Va A (2-6)
xk < My Va E A (2-7)

S> 0, y {0,1} Va e A and k E Ka (2-8)

where M is a sufficiently large number.

In the above formulation, equality (2-6) makes sure that Va E A, there is only

one E Ka such that y = 1 and y = 0, Vk e Ko, k / The correct choice of

( depends on the value of Xa and has to satisfy constraint (2-5). In particular, if

Xa E [A-1, A4] then from constraints (2-5) and (2-6), it follows that y 1. As
for the rest of the constraints, inequality (2-7) ensures that xr = 0 if y = 0, and

qualities (2-3) and (2-4) make sure that the demand is satisfied and the sum of xk

over all indices k E Ka is equal to the flow on arc a. In addition, it is easy to show

that the objective of the problem is equivalent to the objective of CPLNF and one

concludes that the CPLNF and the CPLNF-IP problems are equivalent.

Consider a relaxation of the CPLNF-IP problem where constraint (2-7) and

the integrality of yj are replaced by

rk k= Xa
a ata


and y > 0, respectively. Observe that in the resulting problem constraint (2-4)

is redundant and follows from (2-6) and (2-9); therefore, it can be removed from

the formulation. In addition, notice that one can remove the variable xk from

the formulation as well by substituting (2-9) into the objective function. The

mathematical description of the resulting problem is provided below and we refer to

the relaxation problem as CPLNF-R.

min g(x, y) [: Zc xa > y f as
aEA LkEKa aEA kEKa aEA kEKa

s.t. Bx = b (2-10)

a y-1a < Xa< < A Va A (2-11)
kEKa kEKa

S 1 Va A (2-12)
x_ > 0, y_ >0 Va A and k Ka (2-13)

Lemma 2.2.1. Any feasible vector of the CPLNF-IP problem is feasible to the


Proof: Observe that constraints (2-10)-(2-13) are present in the CPLNF-IP

problem. Therefore, any feasible vector of the CPLNF-IP problem satisfies

constraints (2-10)-(2-13). U

Lemma 2.2.2. Any local optimum of the CPLNF-R problem is either feasible to

the CPLNF-IP or or ca be used to construct a feasible vector of CPLNF-IP with the

same objective function value.

Proof: Let (x*, y*) denote a local minimum of the CPLNF-R problem. From the

local optimality it follows that g(x*, y*) < g(x*, y) in the c-neighborhood of y*.

However, observe that by fixing the value of the vector x to x* in CPLNF-R, the

problem reduces to a linear one; therefore y* is a global minimum of the resulting

LP. In addition, notice that the problem can be decomposed into IA| problems of

the following form

min c x X + s k [c X + s]y
aya a aya a a a* a
keKa kEKa kEKa
s.t. A k-1 y < x A ky (2-14)

kEKa kEKa

y =a 1 (2-15)
y >0 Vk c K (2-16)

Let x e [A*- Ak*]. As we have mentioned before, fa(x,) = minkEKafa(xa)};


f(x*) min{fk(x*)} min {c + s}.
EKKa keKa
Observe that by assigning y = 1 and yk = 0, Vk e Ka, k / k*, the resulting

vector ya (i) satisfies constraint (2-14) because xa e [A '*-, A*] and (ii) ya

argmin{ [cKa[ + s kKk = 1,ac > 0}. Based on the above, one

concludes that ya is an optimal solution of the problem. If x* C (A -1, A *) then

y is the unique solution of the problem because cxk + s > c *x + s Vk e Ka,

k / k*; therefore, y* y. If x A 1 or x A, there are exactly two binary

solutions of the problem, and both have the same objective function value. As a

result, either one can be used to construct a binary solution y. A similar result

holds for all arcs a E A. Regarding variable xa, given (x*, y*), the only feasible one

is x a = x* and x = 0, Vkc Ka, k / k*. U

The following theorem is a direct consequence of the above two lemmas.

Theorem 2.2.1. A 1 ,l.',l optimum of the CPLNF-R problem is a solution or ca be

used to construct a solution of the CPLNF-IP .

Proof: From Lemma 2.2.2, it follows that a global optimum of CPLNF-R is either

feasible to CPLNF-IP or can be used to construct a feasible solution with the same

objective function value. Since all feasible vectors of CPLNF-IP are feasible to

CPLNF-R (see Lemma 2.2.1), one concludes that a global solution of CPLNF-R

leads to a solution of CPLNF-IP. U

From Theorem 2.2.1, it follows that solving the CPLNF-IP problem is

equivalent to finding a global optimum of the bilinear problem CPLNF-R. If

the solution of CPLNF-R is not feasible to CPLNF-IP then the proof of Lemma

2.2.2 provides an easy way to construct a feasible solution with the same objective

function value. Other properties of local minima of the CPLNF-R problem are

discussed in Section 2.4.

It is noticed that the CPLNF-R problem has the following economic

interpretation. Observe that because of equality (2-12), yJ E [0, 1], and one

can interpret the variables yS as weights. Under this assumption, one can view

the objective function as the sum of the weighted averages of the variable costs

multiplied by the flow, [LkEK, c>ya] Xa, and the fixed costs, Ek:K sak In other

words, the objective function consists of the weighted averages of functions fa(xa).

However, the weights have to satisfy constraint (2-11), where the flow, Xa, is

bounded by the weighted averages of the left and the right ends of the intervals

[Ak-1, A ], k E Ka. According to Lemma 2.2.2, a local (global) optimum leads to a

solution where the weights are either equal 0 or 1.

2.3 Concave Piecewise Linear Problems with Separable Objective

Consider the following generalization of the CPLNF problem where constraint

(2-1) is replaced by a requirement x E X C R".

min f(xi)
s.t. xEX

Xi E [A, An1

where the fi(xi) are piecewise linear concave functions, i.e.,

ci Xi s+slj=f (x)) x [A',\1)

fi(xi) =

Cii "nn i ni 1
ci xi + s( f-'(xi)) ax E [An 1 ,An']

with ci > c > > c'. Let Ki = {1, 2,..., ni}. Although the theoretical results

in the previous section are derived for the concave piecewise linear network flow

problem, one can replace constraint (2-10) by x e X, i.e.,

i=1 kEKi
s.t. xEX

E k-1 k< xi < k k
Ai Yi A
kEKi kEKi

xi > 0, yk > 0,

and the bilinear reduction technique used before, as well as Lemmas 2.2.1 and

2.2.2, and Theorem 2.2.1, are still valid. As a result, one concludes that the

CPLPwSOF and the CPLPwSOF-R problems are equivalent in the sense that

a solution of the CPLPwSOF problem can be easily constructed from a global

solution of the CPLPwSOF-R problem.

If the set X is a polytope then CPLPwSOF-R is a bilinear program with a

jointly constrained linear feasible region. Let Y = {y EkeK y 1, y? > 0}

and X+ = {xlx e X,xi e [A, Ai ]}. Denote by V(X+) and V(Y) the sets

of vertices of the polytopes X+ and Y, respectively. Notice that the sets X+

and Y are "j.- 1I 1 by the constraints >kEKi 1i Y < i _< kEK, A-" It is

well known that an optimal solution of a general bilinear program with jointly

constrained feasible region occurs at the boundary of the feasible region and is not

necessarily a vertex (see Section 3.2.2, Horst et al. [54] and the related problem

set). However, CPLPwSOF-R is equivalent to CPLPwSOF. In particular, if (x*, y*)

is a global solution of CPLPwSOF-R then from Theorem 2.2.1, it follows that x* is

a solution of CPLPwSOF. The latter is a concave minimization problem where the

feasible region is a polytope. It is well known that the solution of such a concave

minimization problem is one of the vertices of the polytope; therefore x* e V(X+).

In addition, from the theorem it follows that there exists y E V(Y) such that

(x*, ) is a global solution of CPLPwSOF-R problem. In that sense, CPLPwSOF-R
is a ;, ., l./; joined bilinear program. The above discussion is summarized in the

following two theorems.

Theorem 2.3.1. A concave minimization problem with a separable piecewise linear

objective function, CPLPwSOF, is equivalent to a bilinear 1i. '",'r CPLPwSOF-R.

Theorem 2.3.2. Let (x*, y*) be a solution of problem CPLPwSOF-R. If the

set X+ is a ..1;,/. pe, then (i) x* E V(X+), and (ii) 3 y E V(Y) such that

Zi=fl kKi J (iI'l :i"l EkeKi Jik(i' .
2.4 Dynamic Cost Updating Procedure

In this section, we discuss an algorithm for solving the CPLNF-R problem. In

general, bilinear programs similar to CPLNF-R are not separable in the sense that

the feasible sets of variables x and y are joined by common constraints. By fixing

one of the variables to a particular value, the resulting problem transforms into a

linear one.

Consider the following two linear problems which we refer to as LP(y) and

LP(x), where y (x) denote the parameter of the problem LP(y) (LP(x)), i.e., fixed

to a particular value.



s.t. Bx b

Xa C [0, \a]


min [caxa + sa
aEA kEKa

S.t. A k-1 < ky k A
kEKa kEKa

S> 0

In the LP(y) problem we assume that variables y4 are given, and the Xa are

the only decision variables. Similarly, in the LP(x) problem, the Xa are given,

and the y are the decision variables. As we have shown in the proof of Lemma

2.2.2, problem LP(x) can be decomposed into IA| problems and the solutions

of the decomposed problems are binary vectors, which satisfy constraint (2-14).

Therefore, given vector x, a solution of the LP(x) problem can be found by a

simple search technique where y = 1 if Xa E [a -1, A].

We propose a dynamic cost updating procedure (DCUP), where one considers

solving the problems LP(x) and LP(y) iteratively, using the solution of one

problem as a parameter for the other (see Procedure 1). Although in the procedure

the initial vector yo, is such that yl0 = 1 and y0 = 0, Vk E Ka, k / 1, one can

choose any other binary vector, that satisfies constraint (2-12). It is noticed that

a similar iterative procedure has been used for solving a bilinear program with

a disjoint feasible region (see, e.g., Horst et al. [54] and Horst and Tuy [55]). In

Procedure 1 : Dynamic Cost Updating Procedure
Step 1: Let yo denote the initial vector of 0o, where y0 = 1 and o = 0, Vk e
Ka, k / 1. m -- 1.
Step 2: Let xm = argmin{LP(ym-l)}, and y = argmin{LP(x')}.
Step 3: If y"' y"-1 then stop. Otherwise, m -- m + 1 and go to Step 2.

the DCUP, LP(y) does not include constraint (2-11). In other words, L(x) is the

CPLNF-R problem with fixed variable y and relaxed constraint (2-11). The latter

allows using the iterative procedure to solve bilinear programs with a ; .,../;/l; joined

feasible region. Let V represent the feasible region of CPLNF-R and (x*, y*) be the

solution of the DCUP.

Theorem 2.4.1. The solution of the DCUP -.,I.f the following .! ,il.:l;'

min g(x*,y) g(x*,y*) min g(x,y*).
(x*,y)EV (x,y*)EV

Proof: Because of the stopping criteria, x* = argmin{LP(y*)} and y*

argmin{LP(x*)}. From the latter it follows that x* = argmin(x,y*)evg(x, y*) and

y* = argmin(X*,y)Evg(x*,y).

Theorem 2.4.2. If y* is a unique solution of the LP(x*) problem then (x*,y*) is a

local minimum of CPLNF-R.

Proof: See the proof of Proposition 3.3, Horst et al. [54]. 0

In the proof of Lemma 2.2.2 we have shown that y* is not unique if and

only if one of the components of vector x*, x*, is equal to the value of one of the

breakpoints A However, observe that x* = argmin{LP(y*)}, and the feasible

region of problem LP(y*) does not involve breakpoints. As a result, in practice it is

unlikely that x* is equal to one of the breakpoints.

Theorem 2.4.3. Given ,i.; initial binary vector yO that .,/I.:.- constraint (2-12),

DCUP converges in a finite number of iterations.

Proof: Let yo denote the initial binary vector, x1 = argmin{LP(yo)} and

y = argmin{LP(x1)}. According to the assumption of the theorem, Va E A,

there exists only one ( e Ka such that yo = 1 and y0 = 0, Vk e Ka, k / $ .

If x cE [A--1, A ] then the corresponding components of the vector yo do not

change their values in the next iteration, i.e. yl = 1 and yl = 0, Vk E Ka,

k $ IHowever, if x1 e [A-1A ], then y = 1 and yl = 0,
/ 0 0. Hoevr if0'EO

Vk E Ka, k / (. In addition, notice that c4xk + s > cix + s. As a result,

g(xl, yo) > g(xl, y') > g(x2, y) and one concludes that the objective function

value of the CPLNF-R problem does not increase in the next iteration. To

prove this by induction, assume that the objective function does not increase

until iteration m. Similar to the above, one can show that g(xm+l, y') >

g(xm+l, ym+l) > g(xm+2, ym+1). The constructed non-increasing sequence, i.e.,

{g(xO yo), g(xl, y'), g (x2, y),..., g(xm+, ym) g(xm+l, m+), g(xm+2, ym+1)...},
is bounded from below by the optimal objective function value, and one concludes

that the algorithm converges.

Observe that in each iteration the procedure changes the binary vector

y. If ym = t-1 then the procedure stops and g(xm, y- 1) g(xm, y")
g(xm+l, y'). If there exist mT and m2 such that mi 1 > m2 and y"l yp2, then

xm1+1 = argminLP(yp )} = argminLP(y 2)} x= +1, i.e., g(xm2+1, m2)

g(xm2+l, yml) = g(xml+, ym). From the non-increasing property of the sequence
it follows that g(x"2+1, y-2) = g(x2+l, y-2+1); therefore, yp2 = y2+1 and the

algorithm must stop on iteration m2. From the latter it follows that all vectors y",

constructed by the procedure before it stops, are different. Since the set of binary

vectors y is finite, one concludes that the procedure converges in a finite number of

iterations. U

2.5 On the Dynamic Slope Scaling Procedure

In this section, we discuss another equivalent formulation of the CPLNF

problem with a slightly different objective function. Using the new formulation, we

prove some properties regarding the solution of Dynamic Slop Scaling Procedure

(DSSP) (see Kim and Pardalos [61] and [62]).

Although in the CPLNF problem there are no restrictions on the values of

parameters sa and Ao, by subtracting s' from function fa(Xa) and replacing the
variable Xa by Xa = x A', one can transform the problem into an equivalent one

where s, = 0 and A = 0. Therefore, without loss of generality, we assume that
s = 0 and A 0.
a a
To investigate DSSP, let

1 Xa (0, A1]

Ca X E (A,2
(^ x,) > 0
F (x,) -
Fa(Xa) X > 0 ... ...
M Xa = 0
na a (A n-1 na]

M a 0

where M is a sufficiently large number. Consider the following network flow

problem with flow dependent cost functions Fa(xa) (NFPwFDCF).

min FT(x)x

s.t. Bx = b (2-17)

Xa E [0, A] (2-18)

where F(x) is the vector of functions Fa(Xa).

Theorem 2.5.1. The NFPwFDCF problem is equivalent to the CPLNF problem.

Proof: Observe that both problems have the same feasible region. Let x be a

feasible vector. If Xa > 0 then Fa(xa)xa = f Xaz fa(za). On the other

hand, if Xa = 0 then Fa(xa)xa = 0 = fa(xa). From the above it follows that

FT(x)x = aEA fa(Xa), and one concludes that NFPwFDCF is equivalent to


Let NFPwFDCF(F) denote the NFPwFDCF problem where the cost

vector function, F(x), is fixed to the value of the vector F. Notice that the

NFPwFDCF(F) problem is a minimum network flow problem with arc costs Fa

and flow upper bounds A=. DSSP starts with initial costs F = ca + s" /Aa and

solves the NFPwFDCF(Fo) problem (see Procedure 2). Then it iteratively updates

Procedure 2 : Dynamic Slope Scaling Procedure
Step 1: Let F = c"a + s~a/A~a be the initial arc costs and
x1 = argmin{NFPwFDCF(F0)}. m <- 2
Step 2: Update the arc costs, F <-- Fa(x'1), and let
xm+l argmMinNFPwFDCF(Fm-1)}.
Step 3: If x"+l xm then stop. Otherwise, m <- m + 1 and go to Step 2.

the value of the cost vector using the function Fa(xa), F' = Fa(x''1), and solves

the resulting NFPwFDCF(F") problem. In the cost updating procedure, different

variations of the algorithm use different values for M. In particular, one may

consider replacing M by F~-1 or maxn
Theorem 2.5.2. The solution of the DSSP is a user equilibrium solution of the

network flow problem with the flow dependent cost functions Fa(xa).

Proof: Assume that DSSP stops on the m*-th iteration, and let x* denote

the solution of the procedure. From the stopping criteria it follows that x*

argmin{(Fm)rx Bx = b, x E [0, A"]}. If xr / 0 then F7* = Fa(x*). On the

other hand, if x = 0 then one can replace Fa* by a sufficiently large M without

changing the optimality of x*. As a result, x* = argmin{FT(x*)xBx = b,Xa E

[0, An] }. From the latter it follows that x* is a solution of the following variational
inequality problem

find feasible x* such that FT(x*)(x x*) > 0, VXa E [0, A-], Bx = b,

and one concludes that x* is an equilibrium solution of the network flow problem

with arc cost functions Fa(xa). U

If the arc costs are not constant, it is well known that the equilibrium and

system optimum solutions may not be the same (see e.g., Pigou [87], Dafermos

and Sparrow [29], and N 1,;, n.:y [77]). However, in the NFPwFDCF problem we

are interested in the system optimum solution, x*, where FT(x*)x* < FT(x)x,

VXa C [0, AX"], Bx b.

2.6 Numerical Experiments

In this section, we provide computational results for the dynamic cost

updating procedure and compare the solution of the DCUP with solutions provided

by DSSP and CPLEX.

The set of problems is divided into five groups that correspond to the networks

with different sizes and numbers of supply/demand nodes. For each group we

randomly generate three types of demand; U[10, 20], U[20, 30], or U[30, 40],

and consider 5 or 10 linear pieces (see Table A-1, Appendix A). In Kim and

Pardalos [62] the authors consider increasing concave piecewise linear cost functions

for experiments. Although the bilinear reduction technique as well as DCUP are

valid for any concave piecewise linear function, to remain impartial for comparison

we generate similar increasing cost functions. Doing so, first for each arc we

randomly generate a concave quadratic function of the form g(x) = -ax2 + 3x.

Notice that the maximum of the function is reached at the point x = %. Next

we divide the interval [0, 2] into na equal pieces, i.e. [A -1, A ] [(k-i) 2, ]

k E Ka {1, 2,..., n}. Finally, we construct the function fa(xa) by approximating

the function g(x) in the breakpoints A\, i.e. fa() -(A) + 3\A. There

are 30 problems generated for each choice of the group, the demand distribution

and the number of linear pieces. We use the GAMS environment to construct the

model and CPLEX 9.0 to solve the problems. Computations are made on a Unix

machine with dual Pentium4 3.2Ghz processors and 6GB of memory. All results are

tabulated in the Appendix A.

Sets 1-18 have a relatively small network size, and it is possible to solve

them exactly using CPLEX (see Table A-2). The relative errors for those sets are

computed using the following formulas

RED p fDCUP exact
REDCUP ) xact

In addition to the relative errors, we compare the results of DCUP versus DSSP.

In Table A-3, columns B, C, and D describe the percentage of problems where

DCUP is better than DSSP, DSSP is better than DCUP, and they are the same,

respectively. The numbers in column A are the averages (maximum values) of the

numbers REDSSP REDCUP, given REDSSP REDCUP > 0. According to the

numerical experiments DCUP provides a better solution than DSSP in about 41.

of the problems and the same solution in 3:'.. of problems. Also notice that DCUP

requires fewer iterations to converge and consumes less CPU time. Regarding

CPLEX, the computational time varies from several seconds in the sets 1-6 to

several thousands of seconds in the sets 13-18.

In the case of the problem sets 19-30, CPLEX is not able to find an exact

solution within 10,000 seconds of CPU time, and the best found solution is not

better than the one provided by the heuristics; therefore, we compare the results

of DCUP versus DSSP. In Table A-4, columns B and D describe the percentage

of problems where DCUP is better than DSSP, and DSSP is better than DCUP,

respectively. The numbers in columns A and C are computed based on the formula

fDSSP-DCUP given fDP fDCUP > 0, and DCU-fDSSP, given fDCUP fDss > 0,
respectively. Observe that the percentage of problems where DCUP is better

than DSSP is higher in the problems with large demands. On average, in 59' of

problems DCUP finds a better solution than DSSP using fewer iterations and less

CPU time.

In the above numerical experiments, we have used the vector yo (Va E A,

0o = 1 and y0 = 0, Vk E Ka, k / 1) as an initial binary vector. However,

DCUP can start from any other binary vector that satisfies constraint (2-12). In

particular, one can considers the solution of DSSP as an initial vector and use

DCUP to improve the solution. Table A-5 compares the results of DCUP versus

DSSP where column A is similar to the one in Table A-4 (i.e. the numbers in the

column are computed based on the formula fDSSP-fDCUP given fDSSP-fDCUP > 0),
and column B is the percentage of problems that have been improved. Observe

that the percentage of problems where DCUP improves the solution of the DSSP

increases with the size of the network and the demand. Overall, the DCUP

improves the solution of the DSSP in about i'- of the problems.

2.7 Concluding Remarks

In this chapter, we have shown that the concave piecewise linear network flow

problem is equivalent to a bilinear program. Because of the special structure of

the feasible region of the reduced problem, we are able to prove that the optimum

is attained on a vertex of the di-i- iiil parts of the feasible region. In addition, we

have shown that the results are valid for a general concave minimization problem

with a piecewise linear separable objective function.

Based on the theoretical results, we have developed a finite convergent

algorithm to find a local minimum of the bilinear relaxation. The computational

results show that the dynamic cost updating procedure is able to find a near

optimum or an exact solution of the problem using less of CPU time than CPLEX.

In addition, we compare the quality of the solution and the running time with

the dynamic slope scaling procedure. Since DCUP is fast, one can aim to find the

global minimum by randomly generating the initial binary vector and running

DCUP. In addition, DCUP can be used in cutting plane algorithms for finding an

exact solution.


3.1 Introduction to the Chapter

During the twentieth century due to a variety of applications many researchers

focused their attention on the fixed charge network flow problem (FCNF). In

particular, production p11 iii:"i' scheduling, investment decision, network design,

location of plants and distribution centers, pricing policy, and many other practical

problems that arise in the supply chain, logistics, transportation science, and

computer networks can be modeled as a FCNF problem (see, e.g., Geunes and

Pardalos [41]).

The FCNF problem is well known to be NP-Hard and belongs to the class

of concave minimization problems. The problem can be modeled as a 0-1 mixed

integer linear program (see Hirsch and Dantzig [50]) and most solution approaches

utilize branch-and-bound techniques to find an exact solution (see Barr et al. [4],

Cabot and Erenguc [12], Gray [44], Kennington and Unger [59], and Palekar et

al. [83]). Since the concave minimization problem attains a solution at one of the

vertices of the feasible region, Murty [76] proposed a vertex ranking procedure to

solve the problem. However, finding an exact solution is computationally expensive

and it is not practical for solving large problems. Some heuristic procedures

are discussed in Cooper and Drebes [27], Diaby [31], Khang and Fujiwara [60],

and Kuhn and Baumol [63]. Recently Kim and Pardalos [61] (see also Kim and

Pardalos [62]) proposed a heuristic algorithm, Dynamic Slope Scaling Procedure

(DSSP), to solve the fixed charge network flow problem. The procedure solves a

sequence of linear problems, where the slope of the cost function is updated based

on the solution of the previous iteration. The algorithm is known to be one of the

best heuristic procedures to sole FCNF problems.

Note that all approaches to solve the problem are based on linear approximation

techniques. Instead, we approximate FCNF by a concave piecewise linear network

flow problem (CPLNF), where the cost functions have two linear pieces. A proper

choice of the approximation parameter ensures the equivalence between FCNF

and the resulting CPLNF problem. However, finding the proper parameter is

computationally expensive; therefore, we propose an algorithm that solves a

sequence of CPLNF problems by gradually decreasing the parameter of the

problem. We prove that the stopping criteria of the algorithm is consistent in the

sense that a solution of the last CPLNF problem in the sequence is a solution of

the FCNF problem.

Despite the above mentioned theoretical results, the algorithm requires finding

exact solutions of the CPLNF problems, which are NP-Hard (see Guisewite and

Pardalos [45]). In C!i plter 2 (see also Nahapetyan and Pardalos [79]), we have

shown that the CPLNF problem is equivalent to a bilinear program. In addition,

we have proposed a finite convergent dynamic cost updating procedure (DCUP)

to find a local minimum of the resulting bilinear program. To solve the FCNF

problem, in the algorithm one transforms the CPLNF problems into equivalent

bilinear programs and uses the DCUP to solve the resulting problems. We refer

to the combined algorithm as the adaptive dynamic cost updating procedure


Similar to the result presented in the C(i plter 2, we prove that the solution

provided by DSSP is a solution to a variational inequality problem, which is

formulated based on the feasible region of the FCNF problem. Although in general

an equilibrium solution and a system solution are not the same, the difference

between the objective function values of the solutions can be fairly small. On

the other hand, ADCUP is a heuristic procedure for finding a system optimum

solution. To compare these two procedures, we conduct numerical experiments

on 36 problems sets for different networks and choices of cost functions. There

are 30 randomly generated problems for each problem set. In the experiments,

we compare ADCUP versus DSSP in terms of the quality of the solution as

well as CPU time. In addition, for small networks we find an exact solution of

the problems using MIP solvers of CPLEX and compute relative errors. The

computational results show that ADCUP provides a near optimum solution using

a negligible amount of CPU time. In addition, the procedure outperforms DSSP in

the quality of the solution as well as CPU time. The difference between solutions is

more noticeable in the cases of small general slopes and large fixed costs.

For the remainder, Section 3.2 discusses the approximation technique and

establishes the equivalence between the FCNF problem and a CPLNF problem

with a special structure. A solution algorithm for solving the FCNF problem is

provided in Section 3.3. Some properties of the DSSP are introduced in Section 3.4.

The results of numerical experiments on ADCUP are summarized in Section 3.5,

and finally, Section 3.6 concludes the chapter.

3.2 Approximation of the Fixed Charge Network Flow Problem by a
Two-Piece Linear Concave Network Flow Problem

This section discusses an approximation of FCNF by concave piecewise linear

network flow problems (CPLNF). In particular, by choosing a sufficiently small

approximation parameter one can guarantee the equivalence between the FCNF

and the CPLNF problems.

Consider a general fixed charge network flow problem constructed on a

network G(N, A), where N and A denote the sets of nodes and arcs, respectively.

Let f,(a() denote the cost function of arc a E A, and


Figure 3-1. Approximation of function fa,(.).

fa (Xa) CaXa + Sa Xa E (0, a,]
0 Xa= 0

where Aa is the capacity of the arc. The fixed charge network flow problem can be

stated as follows.


min f(x) fa(/a)
s.t. Bx b,

XaC [0, Aa, Va A,

where B is the node-arc incident matrix of network G, and b is a supply/demand


Observe that the cost function is discontinuous at the origin and linear on the

interval (0, Aa]. Although we assume that the flows on the arcs are bounded by Aa,

the bounds can be replaced by a sufficiently large M, and the problem transforms

into an unbounded one.

Let aE E (0, Ao], and

X) CaXa + Sa Xa C [Fa, Aal
Caa Xa Xa C [0, a)

where ca = Ca + Sa/Ea Observe that ,' (xa) = fa(Xa), Vx, E {0} U[Fa, A] and

' (Xa) < fa(xa), VXa E (0, E) (see Figure 3-1). Consider the following concave
piecewise linear network flow problem


min 0(x) = (Xa)
s.t. Bx b,

XaG [0, Aa, Va e A,

where E denotes the vector of Ea. The function 0((x) as well as problem CPLNF(E)
depend on the vector E. In the discussion below, we refer to 0"(x) and CPLNF(E)

as E-approximations of the function f(x) and the FCNF problem, respectively.

Denote by x* and x" the optimal solutions of the FCNF and the CPLNF(E)

problems, respectively. Let V represent the set of vertices of the feasible region,

and 6 minf{xal x" E V, a c A, x > 0}. That is, 6 is the minimum among all
positive components of all vectors x" e V; therefore, 6 > 0.

Theorem 3.2.1. For all E such that Ea e (0, al] for all a e A, 0~(x") < f(x*).
Proof: Notice that (x*) < fa(x), Va c A; therefore, "(x*) < f(x*). Since

x' = argmin{CPLNF(E)}, the statement of the theorem follows. U

Theorem 3.2.2. For all E such that Ea e (0,6] for all a c A, O~(x") = f(x*).

Proof: To prove the theorem by contradiction, assume that 0"(x") < f(x*).

Observe that CPLNF(E) is a concave minimization problem; therefore, solutions

of the problem attain on a vertice of the feasible region. From the latter it follows

that xa > 6 > E, or a = 0, Va c A. As a result, (xa) =fa,(x), Va E A,

Q ^(x ) f(xE) < f(x*). The latter contradicts the optimality of x*, and one
concludes that "'(x") = f(x*). U

From Theorem 3.2.1 it follows that for all E such that Ea e (0, Al] for all a c A,

CPLNF(E) provides a lower bound for the FCNF problem. In addition, Theorem

3.2.2 makes sure that by choosing a sufficiently small e > 0 (e.g., Fa = 6, Va E A),

both problems have the same solution; therefore, FCNF is equivalent to a concave

piecewise linear network flow problem.

3.3 Adaptive Dynamic Cost Updating Procedure

In this section we discuss an algorithm for finding a solution of the fixed

charge network flow problem. As we have shown in the previous section, the

problem is equivalent to CPLNF(E), where E is such that aE E (0, 6], for all a E A.

However, it is computationally expensive to find the value of 6. Instead, we propose

solving a sequence of CPLNF(E) problems by gradually decreasing the values of Ea.

Consider Procedure 3. In Step 1, the procedure assigns initial values for

E,. Step 2 solves the resulting CPLNF problem. Notice that CPLNF(e1) is

indeed a linear problem, because [ a] = {Aa}. If 3a E A such that x" e

(0, F7), we decrease the value of aE to a where a is a constant from the open
interval (0, 1). Assume that the procedure stops at iteration k and let xk


Lemma 3.3.1. For all e such that 0 < F < Va E A, problems CPLNF(E) and

CPLNF(k) have the same set of optimal solutions.

Proof: Let x" = argmin{CPLNF()e}. Consider the following sequence of


_(xk) > 0X>) > kX) > O{X (31)

The first and the third inequalities follow from the optimality of x" and xk in the

CPLNF(E) and the CPLNF(Ek) problems, respectively. Since ,a < Ek, Va e A, from

Procedure 3
Step 1: Let e <-- Aa. m -- 1.
Step 2: Solve problem CPLNF(E") and let x" = argmin{CPLNF(')}.
Step 3: If 3a e A such that x~ e (0, E-) then Fe+1 <-- a e, m m + 1, and go
to step 2. Otherwise, stop.

Figure 32. (xa) and (x) functions.

Figure 3 2. 't (xa) and aa (Xa) functions.

the definition of E-approximation it follows that (xa) > (Xa), Va E A and

Xa e [0, Aa] (see Figure 3-2), and the second inequality follows.
Observe that because of the stopping criteria, x = 0 or x E [e a].
k 'k
Since aE < Fa, (Xa) a (Xa:r), Va e A and Xa {0}U[ aAa]; therefore,

Q (xk) = O(xk). The latter together with (3-1) insures that Q(xE) = 0(k).
Since both problems, CPLNF(E) and CPLNF(ek), have the same objective function

value at xr and xk, one concludes that they are equivalent. U

Theorem 3.3.1. A solution of the CPLNF(Ek) is a solution of the FCNF problem.

Proof: From Lemma 3.3.1 it follows that VE such that 0 < aE < KE, Va E A, the

CPLNF(E) and the CPLNF(Ek) problems have the same set of solutions. On the

other hand, by choosing 0 < aE < min{E, )}, CPLNF(E) is equivalent to the FCNF

problem (see Theorem 3.2.2), and the statement of the theorem follows. U

From Theorem 3.3.1 it follows the consistency of the stopping criteria in

the sense that an optimal solution of the resulting problem, CPLNF(Ek), is an

optimal solution of the FCNF problem. Observe that Procedure 3 requires solving

a sequence of concave piecewise linear network flow problems, which are NP-Hard.

To overcome this difficulty, one considers a bilinear relaxation technique to solve

the CPLNF problems (see C'! lpter 2). In Section 2.2, we have shown that the

CPLNF problem is equivalent to a bilinear program. To solve the bilinear problem,

we propose the dynamic cost updating procedure (DCUP), which finds a local

minimum of the problem and can be used in Step 2 of Procedure 3 to find a

solution of the CPLNF(E") problem. The resulting algorithm is summarized in

Procedure 4, which we refer to as adaptive dynamic cost updating procedure

(ADCUP). Below we provide the mathematical formulation of the bilinear problem,

which is equivalent to CPLNF( '"). For details on the formulation of the problem,

finite convergence and other properties of the DCUP we refer to C'! Ilpter 2.

Problem CPLNF-R( ') is defined by:

minm [cI -y + cat.] a + s~ g

s.t. Bx b,

?Ya < Xa < ?'I a" + AaYa, Va E A,

Y a + Ya = 1, Va e A,

Xa > 0, Y > > 0, and ga > 0, Va e A.

The ADCUP is a heuristic algorithm to find a solution to the FCNF problem.

Note that the choice of a has a direct influence on the CPU time of the procedure

as well as the quality of the solution. In particular if the value of the a is close

to one, the value of the F decreases slowly and the procedure requires a large

number of iterations to stop. On the other hand, small values of the parameter can

worsen the quality of the solution. In the numerical experiments, we use a = 0.5

because in our test problems the procedure with that parameter provides a fairly

high-quality solution using 1-4 seconds of CPU time (see Section 3.5).

Procedure 4 : Adaptive Dynamic Cost Updating Procedure (ADCUP)
Step 1: Let \ <-- Aa, 0 = 0, y 0, andy 1. m -- 1.
Step 2: Run DCUP to solve the CPLNF-R("') problem with initial vector
(4r1n- 1). Let (,- ,) be the solution that is returned by DCUP.
Step 3: If 3a e A such that e (0, F-) then eF+1 -- 7a', m -- n + 1, and go
to step 2. Otherwise, stop.

3.4 On the Dynamic Slope Scaling Procedure

This section discusses some properties of the dynamic slope scaling procedure

proposed by Kim and Pardalos [61] (see also Kim and Pardalos [62]). In the paper,

the authors discuss four variations of DSSP based on the choice of the initial vector

and the slope updating scheme. However, regardless of the initial vector and the

updating scheme, DSSP provides an equilibrium type of solution. To prove the

statement, we first transform FCNF into an alternative problem then prove that

the solution provided by DSSP is a solution of a variational inequality problem

constructed based on the new formulation. The theoretical results provided below

are very similar to those in C'! lpter 2, where we have shown that the property

holds for the concave piecewise linear network flow problem.

'X)- Xa > 0aiC a +a s,, c (0, Aa,
M X"- 0 M X= 0

where M is a sufficiently large number. Consider the following network flow

problem with flow dependent cost functions F,(xa).


min FT (x)

s.t. Bx = b, (3-2)

Xa[0,X"], VacA, (3-3)

where F(x) is the vector of functions F,(Xa).

Theorem 3.4.1. The NFPwFDCF problem is equivalent to the FCNF problem.

Proof: (See proof of Theorem 3.4.1, ('! Ilpter 2) U

Let NFPwFDCF(F) denote the NFPwFDCF problem, where the vector

function F(x) is fixed to the value of the vector F. In the first step, DSSP

solves NFPwFDCF(F) problem with an initial vector F = F0. Let xm

argmin{NFPwFDCF(Fm)}. Next, DSSP iteratively updates the value of the

vector F using the solution x", i.e., F"+1 = F,(xj), and solves the resulting

NFPwFDCF(Fm+1) problem. The procedure stops if xm+l = xm. In Kim and

Pardalos [61], the authors propose different updating schemes, where they replace

M by a smaller value. However, the next theorem proves that regardless of the

initial vector F0 and the updating scheme, the final solution provided by DSSP is a

solution of a variational inequality problem.

Theorem 3.4.2. The solution of DSSP is the solution of the following variational

.,' .,. ;l,.;i problem

find x* feasible to (3 2) and (3 3) such that FT(x*)(x x*) > 0, Vx feasible to

(3-2) and (3-3)

Proof: Assume that DSSP stops on iteration k, and let x* = argmin{(Fk)TxlBx

b, xa E [0, OA ]}. From the stopping criteria it follows that x* = xk. If x* > 0

then Ff = Fa(x*). On the other hand, if x* 0 then one can replace Ff

by a sufficiently large M without changing the optimality of x*. As a result,

x* = argmin{FT(x*)x Bx = b,Xa E [0, A'-]}. From the latter it follows that

FT(x*)(x x*) > 0, for all feasible x. U

From Theorem 3.4.2 it follows that the solution of DSSP, x*, satisfies the

inequality FT(x*)x > FT(x*)x*, Vx feasible to (3-2) and (3-3), i.e., x* is an

equilibrium solution. However, since NFPwFDCF is equivalent to the FCNF

(see Theorem 3.4.1), one is interested in finding a feasible x such that FT(x)x >
FT(x)x, Vx feasible to (3-2) and (3-3), i.e., x is a system optimum solution. Notice

that the equilibrium and the system optimum solutions may not be the same,

unless Fa(xa) is constant.

3.5 Numerical Experiments

This section discusses numerical experiments on the adaptive dynamic cost

updating procedure. We solve all problem sets using ADCUP as well as DSSP (see

Kim and Pardalos [61]). To compare the results of the heuristic procedures, in the

case of small problems we find an exact solution using CPLEX MIP solver and

compute relative errors. In the case of large problems, CPLEX is not able to find

an exact solution within 5,000 seconds of CPU time; therefore, we compare the

results of DCUP versus DSSP.

In the experiments, we solve problems using all four variations of DSSP and

choose the best solution to compare with the solution provided by the ADCUP.

In addition to the final solution (the solution that the algorithm returns when it

stops), during the iterative process DSSP as well as ADCUP construct feasible

vectors that might have a better objective function value. In the procedures, we

record those vectors and choose the best one. The comparison between the best

solutions of both algorithms is also provided. With regard to the computational

time, we compare the CPU time of ADCUP versus the best CPU time among four

variations of DSSP.

There are four groups of test problems based on the size of the network and

the number of supply/demand nodes (see Table B 1, Appendix B). For each group,

we construct different types of functions fa(x,), where the slope and the fixed cost

are generated randomly according to the specified distributions. In total, there

are nine sets of problems (nine types of function f,(xa)) for each group, i.e., one

set of problems for each choice of distribution for the slope and the fixed cost.

There are 30 randomly generated problems for each problem set. The components

of the supply/demand vector are generated uniformly between 30 and 50 units.

The model is constructed using the GAMS environment and solved by CPLEX

9.0. Computations were made on a Unix machine with dual Pentium 4 3.2Ghz

processors and 6GB of memory. All results are tabulated in the Appendix B.

Tables B-2 and B-3 illustrate the computational results for groups G1 and G2.

Since the size of those problems is not big, we have solved the problems exactly

using the CPLEX MIP solver. The relative errors are computed based on the

following formulas
From the results it follows that on average ADCUP provides a better solution

than DSSP using less CPU time. Notice that ADCUP outperforms DSSP in the

comparison of the final solutions as well as the best solutions. Although there

are some problems where DSSP provides a better solution than ADCUP, the

percentage of those problems is fairly small and decreases with the increase of the

size of the network. In addition, observe that the quality of the solutions provided

by both algorithms changes with the choice of the distributions for the slopes and

the fixed costs. In particular, both algorithms provide a near optimum solution

for the problems with a larger slope and smaller fixed cost. Although the relative

error of both algorithms increases with the decrease of the slope and the increase of

the fixed cost, observe that the quality of the solutions provided by DSSP changes

more than those provided by ADCUP.

In the case of groups G3 and G4, we compare ADCUP versus DSSP using the

following formula

min{fDssp, fADCUP}

The computational results on those groups are summarized in Table B-4. Similar

to the previous two groups, one observes that on average ADCUP provides a

better solution than DSSP. Notice that DSSP consumes much more CPU time

before termination than ADCUP. In addition, the percentage of problems where

ADCUP provides a better solution than DSSP is higher than in the previous cases.

Similar to groups G1 and G2, the difference between the solutions provided by

both algorithms is small for the problem sets with a larger slope and smaller fixed

cost. When the slope decreases (or the fixed cost increases), ADCUP provides a

perceptibly better solution than DSSP.

3.6 Concluding Remarks

Unlike other models in the literature, we consider concave piecewise linear

network flow problems to solve fixed charge network flow problems. A proper

choice of parameter a guarantees the equivalence between the CPLNF(E) and

the FCNF problems. Based on the theoretical results, we propose a solution

algorithm for the FCNF problem, where it is required to solve a sequence of

CPLNF(E) problems. To find a solution of CPLNF(E), the algorithm employs the

dynamic cost updating procedure. The computational results show that ADCUP

outperforms DSSP in the quality of the solution as well as CPU time. Although

in the computations we choose a = 0.5, one can use a higher value in an attempt

to improve the quality of the solution. Note that a large value of the parameter

increases the CPU time of the procedure. Although ADCUP often provides an

exact solution, it is not guaranteed because DCUP converges to a local minimum of

the bilinear relaxation problem, CPLNF-R(F').

In the numerical experiments, we have shown that the relative error of the

solutions of both procedures increases in the cases of small slopes and large fixed

costs. To explain this phenomena, observe that by decreasing the value of the slope

the angle between function fa(xa) and the first linear piece of function ,a' (xa)

increases (see Figure 3-1). As a result, ,' (x,) does not approximate the function

f,(ax) as well as in the case of large variable costs. The same discussion applies to
the case of a large slope.


4.1 Introduction to the Chapter

Supply chain problems with fixed costs and production planning problems

involving lot-sizing have been active research topics during resent decades. Many

research papers have addressed single-item problems with additional important

features such as backlogging, constant and varying capacities, and different

cost functions (see, e.g., Gilbert [43], Florian and Klein [36], van Hoesel and

Wagelmans [53], Loparic et al. [70], and Loparic et al. [71]). It is well known that

incapacitated problems can be reduced to a shortest path problem. Florian and

Klein [36] studied capacitated single-item problems, where they characterized

the optimal solution and proposed a simple dynamic programming algorithm for

problems in which the capacities are the same in every period. The single-item

problems with varying capacities are known to be NP-hard. A classification

of different problems and a survey on existence of a polynomial algorithm for

solving problems for different classes can be found in Wolsey [100] and Pochet and

Wolsey [88]. Tight formulations for polynomially solvable problems are discussed in

Miller and Wolsey [75] and Pochet and Wolsey [88].

Almost all practical problems involve multiple items, machines and/or levels,

and polynomial results for those problems are limited. Using binary variables,

one can construct a mixed integer linear programming (\!IP) formulation

of the problem with an imbedded network structure. To solve the problem,

branch-and-bound and cutting plane algorithms have been used (see, e.g., Barr

et al. [4], Cabot and Erenguc [12], Gray [44], Kennington and Unger [59], Palekar

et al. [83], Marchand et al. [72], and Wolsey [100]). In addition, several heuristic

algorithms have been proposed (see, e.g., Cooper and Drebes [27], Diaby [31],

Khang and Fujiwara [60], Kuhn and Baumol [63], van Hoesel and Wagelmans [52],

Kim and Pardalos [61] and [62], Nahapetyan and Pardalos [79] and [80]).

In this chapter we discuss a capacitated multi-item dynamic pricing (C' \I)P)

problem where one maximizes the profit by choosing a proper production level as

well as pricing policy for each product. In the problem, the demand is a decision

variable, and in order to satisfy a higher demand one needs to reduce the price of

the product. On the other hand, reducing the price can decrease the revenue, which

is the product of the demand and the price. In addition, the problem includes

an inventory and production cost for each product, where the latter involves a

setup cost. The objective of the problem is to find an optimal production strategy,

which maximizes the profit subject to production capacities that are -!I ied"

by the products. Different variations of a single-item uncapacitated problem

with deterministic demands are discussed by Gilbert [43], Loparic et al. [71], and

Thomas [96]. A capacitated single-item problem with time invariant capacities is

discussed in Geunes et al. [42]. The polynomial algorithms proposed by the authors

are based on the corresponding results for the lot-sizing problems.

In C'!i lpters 2 and 3 (see also Nahapetyan and Pardalos [79] and [80]) we have

proposed a bilinear reduction technique, which can be used to find an approximate

solution of concave piecewise linear and fixed charge network flow problems. A

similar technique is proposed to solve the C' I )P problem. In particular, we

consider a bilinear reduction technique of the problem and prove that solving the

C'\ I)P problem is equivalent to finding a global maximum of the bilinear problem.

The latter belongs to the class of bilinear problems with disjoint feasible region,

and one considers a heuristic algorithm to find a solution of the problem. The

heuristic algorithm employs a well known iterative procedure for finding a local

maximum of the problem (see, e.g., Horst et al. [54] and Horst and Tuy [55]).
Numerical experiments on randomly generated problems confirm the efficiency of

the algorithm.

For the remainder, Section 4.2 provides a linear mixed integer formulation

of the problem and discusses a bilinear reduction of the problem. We prove that

solving the C'\ 1 )P problem is equivalent to finding a global maximum of the

bilinear reduction. In Section 4.3 we propose a heuristic algorithm for solving the

bilinear problem. Numerical experiments on the algorithm are provided in Section

4.4, and finally, Section 4.5 concludes the chapter.

4.2 Problem Description

In this section we provide a nonlinear mixed integer formulation of the

problem. Using some standard linearization techniques, the problem can be

simplified. To solve the problem, we propose a bilinear reduction technique and

prove some properties of the bilinear problem.

Let P and A represent the set of products and discrete times, respectively.

In addition, let f(p,j)(d) denote the price of product p at time j as a function

of the demand d, and g(pj)(d) = f(p,j)(d)d, i.e., g(pj)(d) represents the revenue

obtained from selling d amount of product p at time j. In the problem, we assume

that f(pj)(d) and g(p,j)(d) are nonincreasing and concave functions, respectively

(see Figures 4-1). If f(p,j)(d) is a concave function, then it is easy to show that
concavity of g(p,j)(d) follows.
Let x(p,i,j) denote an amount of product p that is produced at time i to satisfy

the demand at time j, and y(p,i) represent a binary variable, which equals one

if product p is produced at time i and zero otherwise. Assume that inventory

costs, c(, production costs, c ,, and setup costs, c as well as production

capacities, Ci, are given, where p, i, and j represent the product, producing time,

(,j)( g(pj)(d)
A quadratic function

A linear function

dmax d 7 dmax d
d(PJ) d(pj)

Figure 4-1. The price and the revenue functions.

and selling time, respectively. The following is the mathematical formulation of the

C'\ I )P problem.


max .P ,. l (p,i) Y(13i)
pEP jE'e ieAi|ji pEP idjEA|ij pep iEAz

s.t. X(p) pEP jeAzli
SX(p,i,j) < CiY(p,i), Vp C P and i c A,
X(p,i,j) > 0, Y(p,i) E {0, 1}, Vp P and i,j e A.

The objective function of the problem maximizes the profit, which is the

difference between the revenue and the costs. The latter includes the inventory

as well as the production costs. The first constraint ensures the satisfaction of

the capacity restrictions, and the second one makes sure that y(p,i) equals one if

jeA|jE' 0.

Although the above formulation belongs to the class of nonlinear mixed integer

programs, using standard techniques one can approximate the revenue function by

a piecewise linear one and linearize the objective function. Doing so, observe that

from the concavity of the revenue function it follows that there exists a point, d(pj),

where the function reaches its maximum (see Figure 4-1). As a result, producing

and selling more than d(pj) is not profitable, and at optimality xiEAi
d(p,). To linearize the revenue function, divide 0, d(pj) into N intervals of equal

length. Let 4k i) denote the end points of the intervals, i.e., 4, i = kd(pj)/N,

Vk c {1,..., N} U{0} = K U{0}, and g Pj) represents the value of the revenue

function at the point 4k ), i.e., g = gP, )) ) ,' Using those

parameters, construct the function



A (pj),

where it is required that

(P'') k k d k
k 0 k 1 (pj)'

Vp C P and j E A,

A k 1, A ) >0, Vp P and je A,
and Ak(p,) / 0 for at most two consecutive indices k. Observe that g(p,j)(d) is a

concave nondecreasing function on the interval 0, d(pj) ; therefore, its piecewise

linear approximation, i.e., g(p,j)(A(p,j)), preserves the same property. From the

latter, it follows that in the maximization problem the requirement on A) being

positive for at most two consecutive indices k can be removed from the formulation,

and it is satisfied at optimality (for details see C'! plter 11, Bazaraa et al. [5]). The

following is the mathematical formulation of the approximation problem:

max A3P) > >1 KcU + .] Cr, X(pYi ~ ji
x,y,A p ptE
pEP jEA kEK pEP i,jEAli
s.t. 3 Y X(ZPij) < C' V1 e A,
.(p,ij) < Cy(pi), Vp c P and i c A,
X (pij) dk k,) VpcPand j A,

9(pj) (A(p,j))

A(py) = 1,
x(p,i,j) > O, A j > 0, Y(p,i) e {0, 1},

Vp E P and j e A,

Vp E P, i,j E A and k e KU {0}.

Next, we simplify the formulation using nonnegative variables x kj), k c K,

instead of x(p,ij), where x (j)k represents the amount of product p that is produced

at time i and sold at time j using unit price g kj)/ i) fk,) (pj)(

Doing so, the third constraint in the above formulation can be replaced by the

following one:

k1 k X dk Ak
xP(y) d(PJ) (pj),

Vp c P,j c A and k E K.

The latter can be used to remove the variable Ak from the formulation. In

particular, the fourth constraint is replaced by inequalities

S_< 1,
kEK iEAli

Vp P and jE cA,

and in the objective

g()) A (3) ~ (3 x( ,,) Vp j c A and k K,

where fj) f1,=)k (k, i)). The following is the resulting alternative formulation of

the approximation problem, which we refer to as AC\ i )P:


max Y (P j)X(P l)
pEP iEA jeA|i<_j kEK

s.t. X < ci,
pEP jEAti<_j kEK

x kj) < Cy(p,), Vp
jEA|ije kEK

c(pi) (p,i)

Vi eA,

- P and i E A,



S< 1, Vp P andj A, (4-3)
X _k) > 0, ,i) {0,1}, Vp e P, i,j A and k e K, (4-4)

where qg, j) = () c" ,. Observe, that at optimality x(,j) 0 for

all indices such that qk < 0, and those variables can be removed from the

formulation. Therefore, without lost of generality, in the analysis below, we assume

that kq,) > 0.

Define X = {xlx > 0 and x(pi,j) are feasible to (4-1) and (4-3)}, and

Y = [0, 1]IPlHl. Consider the following bilinear program:


max i'j)X(pi) -(pi) Y(pi) (Xy)
pEP izA jEAli<_j kEK

s.t. x EX and y E Y.

Theorem 4.2.1. Any local maximum of the A('CI)P-B problem is feasible or can

provide a feasible solution of the AC 'II)P problem with the same objective function


Proof: Let (x*, y*) denote a local maximum of the AC I 1)P-B problem. Observe

that by fixing x to the value of the vector x*, the AC' I I)P-B problem reduces to

a linear one, and y* is an optimal solution of the resulting problem. Assume that

Ep E P and i E A such that y*i) E (0, 1), i.e., y*, is a fractional number. If

yEI<3 ZkEK q(p,ij) pi,) (p,) < 0 or 2Ejij< ZkEK q( ip,)Xi,) ,) (p,) > 0
then it is possible to improve the objective function value by assigning y*, = 0

or y*,i 1, respectively. The latter contradicts the optimality of (x*,y*). On the

other hand, if jElij -kEkK q pij)(P,) i ,. = 0 then by changing the value of

the variable y*,) to zero the objective function value remains the same. Construct

a vector y, where y(p,) = [YP,* Note that (x*, y) is feasible to constraints (4-1),

(4-3) and (4-4). If (x*, y) violates constraint (4-2) then 3p E P and i c A such

that E |<7 zkEKKij) > 0 and y(p,) = 0. From the local optimality of (x*, )

it follows that CeAli
produce product p at time i. Furthermore, by assigning x*,) = 0, Vj E A and

k e K, the objective function value of the AC\ I I)P-B problem remains the same.

Let x denote the resulting vector, i.e.,

k 0 if ECa|< ECkEK q(pj)xk() "- cs 0 -5
(*, i f k *k k- CSt > 0
( i(pj) .,
( Xj) if ZjAzJiZkEK q Cij) ,i,) >

The vector (x, y) is feasible to the AC'\ I)P as well as the AC' \I)P-B problem and

has the same objective function value as (x*, y*). *

Theorem 4.2.2. A 11..1l maximum of the AC('iP-B problem is a solution or can

be used to construct a solution of the AC('DP problem.

Proof: Observe that any feasible solution of the AC' I 1)P is feasible to the

AC'i\ )P-B problem. Furthermore, if (x, y) is feasible to the AC'i\I)P problem then

j) (p i' j) ~ (p, )yP(p, )= z z ^(Pij) (P ,) -- c(p,i) Y(p,i)
From the latter it follows that the AC\ lI)P-B problem can be obtained from

AC\ I)P by replacing the objective function by

a q ( ) -- Y(p,i),
xpP iEz jEAli<_j kEK K

removing constraint (4-2) and relaxing the integrality of the variable y(p,i). In

other words, the AC\ i )P-B problem is a relaxation of the AC'\ I )P problem. From

Theorem 4.2.1 it follows that a global solution of AC'\ I )P-B is a solution (or leads

to a solution) of the AC' \ 1)P problem. U

The above two theorems prove that solving the AC'i\I)P problem is equivalent

to finding a global maximum of the AC' I 1)P-B problem. In particular, one can

solve the AC'\ i I)P-B problem and if the solution is not feasible to the AC'\ iI )P

problem, then use the method described in the proof of Theorem 4.2.1 to construct

a feasible one with the same objective function value.

4.3 A Bilinear Reduction Based Algorithm for Solving ACMDP

In this section we discuss a heuristic algorithm for solving the ACi\ I)P-B

problem, which employs a well known iterative procedure for finding a local

maximum of the AC'\ i I)P-B problem.

Observe that the problem belongs to the class of bilinear programs. By fixing

vector x or y to a particular value, the problem can be reduced to a linear one. Let

LP(x) and LP(y) denote the corresponding linear programs, i.e.,
LP(x) : ma ZpeP EieA [LEjA|i-< kEK C ,i p j st ] y(p,), and

LP(y) : maX pEP iEA jEA|i KCeK ['pi'j) (ip) ,i,j)'
Notice that the solution of the LP(x) is easy to obtain. In particular,

k k -st < 0
{ O if Zeli V(Ip,i) -
1 ) if YjEAlJi 0

is an optimal solution of the problem. The Procedure 5 describes a well known

algorithm, which starts from an initial binary vector and converges to a local

maximum of the AC'\ I)P-B problem in a finite number of iterations (see Horst and

Tuy [55] or Horst et al. [54]).

However, the procedure has the following disadvantage. Let (x", y') represent

the solution obtained on iteration m, and assume that 3p E P and i E A such that

y =,) 0. As a result, in the LP(ym) problem q(P,
Procedure 5
Step 1: Let yo denote an initial binary vector of y(p,i). m -- 1.
Step 2: Let xm ar,,,i., {LP(y -1)}, and ym = ,I.nt.,, {LP(xm)}.
Step 3: If y" y"-1 then stop. Otherwise, m <-- m + 1 and go to Step 2.

k E K, and perturbations of the values of the corresponding variables xzk ) do not

change the objective function value. Furthermore, because the products -I! ire"

the capacity and other products can have a positive cost in the LP(ym) problem, it

is likely that the value of the variable ximk decreases in the next iteration. From

the latter it follows that y'()1 = 0. To summarize, if y"(,i) 0 then it is likely that

(i) y( =,) 0, Vn > m, and (ii) the final solution is far from being a global one.
To overcome those difficulties, next we propose an approximation problem, which

avoids having zero costs in the objective of the LP(y) problem.

Let )(x(pL)) = E ECK pij)pi) and

(p ,) ( ( ,j) (pij),
1! jEAjti
where (p,) > 0 and x(p,,) is the vector of xk .). It is easy to show that both

functions have the same value, (pi), on the hyperplane CEAl
E(p,t) +c ,t Furthermore, p (i((p)) > (xi) ,) if ZE A|iz kEK 1i p,) >

E(p,i) + and p(p,)(X(p,)) < (pX)(X(p,) if EAi (pi) + cst Define

S(Xy) 0 >1 > LYp,)(x(p,t))u(p,t) + (#5pt)(X(p,t))(1 1(p,i))]

where E denotes the vector of E(p,t). The function op(x, y) depends on the value

of the vector E, and "(x, y) > p(x, y), for all E > 0 and (x, y) feasible to

the AC\Ii)P-B problem. Observe that if E 0 then Y(ti)(x(p,i)) 0, and

P(x, y) p(x, y). In that sense, p"(x, y) is an E-approximation of p(x, y), and
it approximates the function from above. By replacing the objective function of

the AC\ I )P-B problem by the function (x, y), we refer to the resulting problem

as E-approximation of the AC\ I )P-B problem and denote by AC\ I )P-B(E).

Construct the corresponding LP"(x) and LP"(y) linear problems by fixing vectors

x and y in the ACi\I)P-B(E) problem to a particular value, i.e.,

Procedure 6 :
Step 1: Let (p,i) be a sufficiently large number, and yo be such that y = 1,
VpE P and i E A. m 0.
Step 2: Construct the E-approximation problem ACi\ I)P-B(E) and run
Procedure 5 to find a local maximum of the problem, where y" is an initial
binary vector. Let (x"m +l1) denote the local maximum.
Step 3: If 3p E P and i e A such that EY:Ei K p, and EjEA|i 0 then <-- aE, m <-- m + 1 and go to Step 2.
Otherwise, stop.

LP(x) : max pEp Ei (, p,)(x(p,i))y(p,i) + () (x((p,))(1 y(p,,)), and
LP"(y) : maX YpP,i, EkEK,jEAj i) (p) ,+ 1 Yp))i X))
As before, the solution of the LP"(x) problem is easy to obtain by assigning

0 jf Zke K V k k cst <
i(p,i) --
0 if jE zjiA| j Y kEfiK q(pi,j) (p,ij) > '(pl)

{ 0 if j )(X(P,)) < ) (X(,))

Sif ) X(p)) > X ))

The heuristic procedure (see Procedure 6) starts with a sufficiently large E and

finds a local maximum of the resulting E-approximation problem. If the stopping

criteria is not satisfied in Step 3 then it decreases the value of the vector E to aE,

where a is a constant from the open interval (0, 1), and the process continues using

a new E-approximation problem. Observe that Procedure 6 uses vector y' from the

previous iteration as an initial vector.

The procedure depends on two parameters: the initial vector E and the value

of a. The value of E(p,t) depends on parameters of the problem, and one can

consider them equal to the maximum profit, which can be obtained by producing

only product p at time i. Although such maximization problem is easy to solve

using standard LP solvers, for large IPI and |A| one finds computationally

expensive solving the problem for all pairs (p, i) c P x A. Instead, we propose

an algorithm for finding the values of (p,i) (see Procedure 7). Observe that

x, : j*) -< i ), and the maximum additional profit that can be obtained using the
variable x j)is q(i .k i). Using this property, for all pairs (p, i) the procedure

iteratively finds the maximum among q ,dk i), and assigns the demand (or the

remaining of the capacity) to the corresponding variable xij).k The value of (p,i)

is computed based on the formula E(p,) = jeA|z, < k eK C ,ij)Xpij) cst .. As for

the parameter a, its larger value increases the computational time of the procedure,

and it is likely to provide a better solution.

4.4 Numerical Experiments

In this section we discuss numerical experiments conducted on randomly

generated problems. The problems are solved by Procedure 5 as well as Procedure

6 using different values for the parameter a. The latter procedure employs

Procedure 7 to find an initial value for the vector E. In addition, we solve the

problems by the MIP solver of CPLEX using the ACl\ I)P formulation. In the

cases where the MIP solver is not able to solve large problems within posted CPU

and memory limitations, we compare the solutions of the procedures with the

best solutions found by CPLEX. The main purpose of the computations is the

performance of the procedures for different capacities.

Procedure 7
Assign xzi) = 0, Vp e P, i,j e A, and k e K
for all p e P and i c A do
C C ,j) q~,) max = max{q(~p, i)k c K, j A,j > i}
while C / 0 and qmax / 0 do
Let jmax and kmax are such that qmax = q.m~ma,.m
Assign xl, ,max, = min{C',., ), Ca C X jmax p,ijmax = 0, Vk
K, and qmax max{q 4k'j), i),k e K,j A,j > i}
end while
(pi) jEa|J end for

In the numerical experiments we consider problem sets with different numbers
of products, |P| = 5, 10, or 20, and time horizons, |A| = 12 or 52. For each
problem set we randomly generate capacities for all i E A using the formula C

PI U, where U is a random number uniformly generated from interval [10,100],
[50,150], [100, 200], or [150,250]. Note that all intervals allow generating capacities
that are tight at optimality with respect to the revenue function discussed below.
In addition, using term IPI one generates capacities that depend on the number
of products. The latter allows comparing of results across different numbers
of products. As for the costs, we generate the production costs cp' and the
inventory costs c" i) according to the uniform distributions U[20, 40] and U[4, 8],
respectively. Observe that on average the inventory cost is equal to 211' i of the
production cost. Finally the setup cost c, i)is generated uniformly from interval
[600, 1000].
In the experiments we restrict ourself by considering only linear price functions
of the form f(p,)(d) = fa f_ /d, I4 d. To avoid generating functions that at
optimality result in unrealistically large profits, we introduce an index /, where
Vk _^ LJ < YV (if() cPr \) X$,J) cst il* 1i)

__^ V^ fv^ in + (,Pr ,,*k + st *
EpeP Ez [ 'e j YkK C i) L Z,)(p,i,j) + .+ t (p,i)j

That is, the index measures the amount of the profit per unit of investment and
it is computed based on the optimal or the best solution provided by CPLEX.
By generating fma' and nC i according to the uniform distributions U[70, 90]
and U[500, 1000], respectively, at optimality (i) / E [0.7, 1.3], (ii) all capacities
considered above are tight, and (iii) in most of the cases the satisfied demand
is less than d(pj) (see Figure 4-1). In addition, the proposed price function and
distributions of the costs and capacities allow generating problems that have an
optimal objective function value ranging from hundreds of thousands to several

millions. Finally, in the construction of the piecewise linear approximation of the

revenue function we use N = 10.

The model is constructed using the GAMS environment and solved by CPLEX

9.0 with a CPU restriction of 2000 sec and a memory restriction of 2Gb, where the

latter is the memory that is required to store the tree in the branch-and-bound

algorithm. Computations are made on a Unix machine with dual Pentium 4 3.2Ghz

processors and 6GB of memory. The results are tabulated in the Appendix C.

In the experiments we solve 10 randomly generated problems for each problem

set and capacity. Tables C 1 and C-2 compare the results provided by CPLEX

with the solutions provided by both procedures. The relative error is computed

using the formula
ObjCPLEX ObjProc.6(5)
RE( ,)=

In the Table C 1, column A indicates the number of problems where the heuristic

procedure finds a better solution than CPLEX. Note that CPLEX is able to

provide an exact solution for all capacities from the problem set 5-12. In all other

cases, the solver stops after reaching the CPU limit or the memory limit and

returns the best found solution. Although the relative optimality gap of the final

solutions of those problem sets varies from _' to 5'. we believe that the solution

is an optimal or close to an optimal one, and the large optimality gap is due to

imperfect lower bounds. The fact that the heuristic procedures provide a slightly

better solution in the case with |A| = 52 than |A| = 12 partially confirms our


The relative errors in the Table C 1 confirms the effectiveness of the heuristic

procedure. In particular, in the 1i I ii iiy of the problems the heuristic algorithm is

able to provide a solution within 1 from the optimal one or the best one provided

by CPLEX. Observe that the larger value of a provides a better solution and the

number of problems where the heuristic procedure finds a better solution than

CPLEX is increasing with the size of the problem. By comparing with the solutions

provided by Procedure 5 (see Table C-2) one notices that Procedure 6 outperforms

the Procedure 5, and it is more stable with changes in the capacities. As for

the CPU time (see Table C-3), the heuristic procedures require fewer resources

than CPLEX. In addition, unlike CPLEX the heuristic procedures do not require

gigabytes of memory to store the tree.

4.5 Concluding Remarks

We have discussed a bilinear reduction scheme for the capacitated multi-item

dynamic pricing problem, where solving the latter is equivalent to finding a global

solution of the former. Based on theoretical results of the reduction problem, two

procedures have been proposed to find a global maximum of the problem. The

first one is a well known technique and has been intensively used to solve other

bilinear problems. Because of the reasons discussed in Section 4.3, in the very

beginning of the iterative process the procedure eliminates some products from the

further consideration. The latter worsen the quality of the solution returned by

the procedure. In the second procedure we construct approximate problems and

gradually decrease parameters of the problems. As a result, during the iterative

process the costs of the eliminated products remain positive and the procedure

considers them again if need be. Although the second procedure requires more

CPU time to stop than the first one, it provides a higher-quality solution.


5.1 Introduction to the Chapter

Since Merchant and Nemhauser (see, Merchant and Nemhauser [73] and

[74]) first proposed their model in 1978, there have been a number of papers

(see, e.g., Carey and Subrahmanian [23], Carey [15], Carey [13], Carey [14],
Friesz [37], Friesz [38], Wie et al. [98], C('. ,1 and Hsueh [25], Janson [57], Ho [51],

Ziliaskopoulos [104], Drissi-Kaitouni and Hameda-Benchekroun [32], Li et al. [66],

Kaufman et al. [58], Boyce et al. [11], Ran and Boyce [90], Ran et al. [92], and

Wie et al. [99]) discussing variational inequality or mathematical programming

formulations for the dynamic traffic assignment problem with the assumption that

the planning horizon is a set of discrete points instead of a continuous interval.

_M iw, of these papers use a dynamic or time-expanded network (see, e.g., Al!mi et

al. [2]) to simultaneously capture the topology of the transportation network and

the evolution of traffic over time. Implicitly or otherwise, these papers typically

assume that there is no traffic at the beginning of the planning horizon (or at time

zero) and that all trips must exit the network prior to the end. When there are cars

at the time zero, the times at which these cars enter the network must be known in

order to determine when they will exit the arcs on which they were travelling. In

practice, data with such details do not generally exist.

There are two main factors that distinguish the models in papers referenced

above. First, some (e.g., Merchant and Nemhauser [73], Carey and Subrahmanian

[23], Ho [51], Carey [15], Ziliaskopoulos [104], Kaufman et al. [58], Garcia et
al. [40]) seek a system optimal solution and others (e.g., Janson [57], Wie et

al. [98], C'!, i, and Hsueh [25], and Drissi-Kaitouni and Hameda-Benchekroun [32])

compute a user equilibrium instead. The other factor is the travel cost function

used by these models. Among other parameters (physical or otherwise), a travel

time or cost function may depend on the number of cars on the link and the input

and output rates. 1T ,tn (e.g., Carey and Ge [20], Carey and McCartney [18],

Carey [16], Carey [17], Lin and Lo [69], Han and Heydecker [46], Daganzo [28])

have analyzed the effects of travel cost functions on various models. Some (e.g.,

Lin and Lo [69] and Han and Heydecker [46]) have shown that some travel cost

functions are not consistent with the models that use them.

Similar to Carey and Srinivasan [21], Carey and Subrahmanian [23], Carey

[15], C('! i and Hsueh [25] and Kaufman et al. [58], the model in this chapter

is based on the time-expanded network. However, instead of assuming that the

network is empty at the beginning or at the end, we treat the planning horizon

as a circular interval instead of linear. For example, consider the interval [0, 24],

i.e. a 24-hour planning horizon. When viewed in a linear fashion, it is typically

assumed that there is no car in the network at times 0 and 24. In turn, this implies

there is no travel demand after time k < 24. Otherwise, cars that enter the

network after time k cannot reach their destinations by time 24, thereby leaving

cars in the network at the end of the horizon. On the other hand, if there is a car

entering a street at 23:55h (11:55 PM) and exiting at 24:06h (12:06 AM, the next

day) in a circular planning horizon, the exit time of this car would be treated as

00:06h instead. When accounted for in this manner, it is possible to determine the

exit time for every car that is in the network at time zero without requiring any

additional data. Additionally, models that view the planning horizon in a circular

fashion are more general in that they include those with a linear planning horizon.

By setting the travel demands and other variables during an appropriate time

interval to zero, models with a circular planning horizon effectively reduce to ones

with a linear horizon.

It is often argued that the number of cars at the beginning and the end of

the horizon are small and solutions to DTA are not drastically affected by setting

them to zero. When the paths that these cars use do not overlap, the argument

is valid. However, when these cars have to traverse the same arc in reaching their

destinations, the number of cars on the arc may be significant and ignoring it may

lead to a solution significantly different from the one that accounts for all cars.

This chapter makes two main assumptions. One requires the link travel time

at time t to be a function of only the number of cars on the link at that time.

Carey and Ge [20] show that the solutions of models using functions of this type

converge to the solution of the Lighthill-Whitham-Richards model (see Lighthill

and Whitham [68] and Richards [93]) as the discretization of links into smaller

segments is refined. Because minimizing the total travel time or delay mitigates

its occurrence, models discussed herein do not explicitly address spillback. On

the other hand, the models can be extended to handle spillback using a technique

similar to the one in Lieberman [67] or an alternative travel time function that

includes the effect of spillback (see, e.g., Perakis and Roels [86]). However, as

indicated in the reference, using such a function may not lead to a model with a


For the remainder, Section 5.2 defines the concept of periodic planning

horizon. Section 5.3 formulates the system version of the discrete-time dynamic

traffic assignment problem with periodic planning horizon or DTDTA and prove

that a feasible solution exists under a relatively mild condition. To our knowledge,

there are only four papers (Brotcorne [10], Smith [95], Wie et al. [98], and Zhu and

Marcotte [103]) that address the existence issue and some (see, e.g., Smith [95] and

Zhu and Marcotte [103]) consider this small number to be lacking. All four deal

I I vs
0 T

Figure 5-1. Linear versus circular intervals.

with user equilibrium problems instead of system optimal. Section 5.4 describes

two linear integer programs that provide bounds for DTDTA. Section 5.5 presents

numerical results for small test problems and, finally, Section 5.6 concludes the


5.2 Periodic Planning Horizon

The models in this chapter assume that the planning horizon is a half-open

interval of length T, i.e., [0, T). Instead of viewing this interval in a linear fashion,

the interval is treated in a circular manner as shown in Figure 5-1. In doing

so, time 0 and T are the same instant. For example, time 0:00h and 24:00h (or

midnight) are the same instant in a 24-hour d-iv. For this reason, T is excluded and

the planning horizon is half-open. To make the discussion herein more intuitive, we

often refer to the planning horizon as a 24-hour d-4v, i.e., T = 24. In theory, the

planning horizon can be of any length as long as events occur in a periodic fashion.

If an event (e.g., five cars enter a street) occurs at time t, then the same event also

occurs at time t + kT, for all integer k > 1.

Because the planning horizon is circular, events occurring tomorrow are

assumed to occur in the same interval that represents tod-iv. For example, consider

a car that enters a street at tl = 23:00h (or 11 PM) todci- and traverses the street

until it leaves at t2 = 01:00h (or 1 AM) tomorrow. (See Figure 5-2.) In a circular

planning horizon, these two events, a car entering and leaving a street, occur at

t O/T

0 ti T t2

Figure 5-2. Events occurring in two consecutive planning horizons.

time 23:00h and 01:00h in the same interval [0, 24). In general, if a car enters a

street at time t1 < T and takes r < T units of time to traverse, then the two events

are assumed to occur at t1 and mod{ti + 7, T} on the interval [0, T).

5.3 Discrete-Time Dynamic Traffic Assignment Problem with Periodic
Time Horizon

Although, it is possible to formulate the dynamic traffic assignment problem

with a periodic time horizon as an optimal control problem, solving it is typically

troublesome (see, e.g., Peeta and Ziliaskopoulos [85]). This section presents a

discrete-time version of the problem in which the interval [0, T) is represented as

a set of discrete points, i.e., A {0, 6, 26,. T 6}, where 6 = and N is

a positive integer. (In general, the subdivision of the planning horizon need not

be uniform. For example, the subdivision during the period between 22:00h to

06:00h may be coarser than the one for the period between 06:00h to 22:00h.) In

order to avoid using fractional numbers in the set of indices and to simplify our

presentation, we typically assume that 6 = 1.

To formulate the problem, let G(N, A) represent the underlying transportation

network where N and A denote the set of nodes and arcs in the network,

respectively. It is convenient to refer to elements of A either as a single index a

or a pair of indices (i,j). The latter is used when it is necessary to reference the

two ends of an arc explicitly. Furthermore, C is a set of origin-destination (OD)

Figure 5-3. Three-node network.

pairs and the travel demand for OD pair k during the time interval [t, t + 6], t E A,

is ht.

There is also a travel time function associated with each arc in the network. In

the literature (see, e.g., Wu et al. [101], Ran and Boyce [89] and Carey et al. [19]),

these functions can depend on a number of factors such as in-flow and out-flow

rates and traffic densities. We assume in this formulation that 0a, the travel time

associated with arc a, depends only on the number of cars on the arc. Furthermore,

Oa is continuous, non-decreasing and bounded by T, i.e., 0 < a((w) < T,

Vw E [0, .1,], where .i, is a sufficiently large upper bound for the range of Oa(w)

and there is no feasible solution whose flow on arc a can exceed .i,. In particular,

a(0) represents the free-flow travel time on arc a.
We use the dynamic or time-expanded (TE) network (see, e.g., Section 19.6

in A!,li et al. [2]) to determine the state of vehicular traffic in the system at

each time t c A. To illustrate the concept of time expansion, consider the static

network with three nodes shown in Figure 5-3 or the three-node network. In

this network, all arcs have the same upper bound value, ., = M, and there

is only one OD pair, (1,3). Let the planning horizon be the interval [0, 5) and

6 = 1. Thus, A {0, 1, 2, 3, 4}. The travel time function for every arc is Q and

1.5 < O(w) < 4,Vw c [0, M]. To construct the TE network, the travel time also

needs to be discretized. In general, the set of possible discrete travel times for arc a

1 (2

1 2

Figure 5-4. Time expansion of arc (1, 2) at t = 1.

is F, { { s : 0 = ], 0 < w < 1, }. For our example, the set of possible discrete

travel times for each arc is F, {2, 3, 4},Va.

To incorporate the time component in the TE network, every node in the

static network (or static node) is 'expanded' or replicated once for each t E A.
For the three-node network, static node 1 is transformed into five TE nodes, one

for each t E A, in the TE network. For example, node 1 is expanded into nodes

lo, 11, 12, 13, and 14 in the TE network. (See Figure 5-4.) Similarly, each arc (i,j)
in the static network (or static arc) is replicated once for each pair of (t, s), where

t E A and s E F(cj). Consider arc (1, 2) in the three-node network. Cars that
enter this arc at time 1 can take 2, 3, or 4 units of time to traverse depending

(as assumed earlier) on the number of cars on the arc at t = 1. To allow all
possibilities, arc (1,2) is expanded into three TE arcs (11, 23), (1, 24), and (11,20).
The latter represents a car that enters arc (1,2) at time 1, takes 4 units of time to

traverse, and leaves the arc at time 5 or time 0 (or mod(1 + 4, 5)) of the following

day. Similar expansion applies to each t E A. In general, each static arc (i,j)

expands into |A| x IP(yj) T TE arcs of the form (it, jmod (t+s,T)),V t E A, s cE (j).

Figure 5-5 di-pl--'i the complete time expansion of the three-node network. In

addition to the time-expanded nodes and arcs, the figure also di pl!-- the travel

demand at the origin TE nodes (i.e., node 1t, Vt c A) and decision variables g(k)t

representing number of cars arriving at the destination node d(k) of OD pair k at

time t, i.e., at node 3t,Vt E A.

To reference flows on TE arcs, let y>(t,s) denote the amount of flow for

commodity k that enters static arc a at time t E A, takes s E Fa units of time to

traverse it, and then exits the arc at time mod{t + s, T}. In particular, if a = (i,j),

then the subscript a(t, s) refers to TE arcs of the form (it,j mod (t+s,T)),V t E A, s E

F(i,). In addition, Ya(t,s) EkEC (t,s) represents the total flow on arc a(t, s).
To compute the time to traverse a static arc at time t, let

a(t,) = {(, S) : 7 = [t 1]r, [t 2]r, [t S]r, S c Fa .

[ q if q > 0

T+q ifq < 0

In words, Qa(t) contains pairs of entrance, T, and travel times, s, for static arc a

such that, if a car enters static arc a at time T and takes s time units to traverse it,

the car will still be on the arc at time t. For example, if t = ll:00h and the time to

traverse arc a is five hours for the previous five consecutive time periods, then cars

entering arc a at time T 10:00h, 9:00h, 8:00h, 7:00h, and 6:00h will be on the arc

at ll:00h. (We assume here that cars entering arc a at, e.g., 6:00h are still on the

arc at ll:00h even though it is scheduled or expected to leave at ll:00h.) When t

is relatively near the beginning of the planning horizon, the notation [-]T accounts

for cars on the arc at time t that enter it from the previous d4 i. Continuing with

the foregoing example, let t = 3:00h instead. Then, cars entering arc a at time -

Figure 5-5. Time-expansion of the three-node network.

2:00h, 1:00h, 0:00h, 23:00h, and 22:00h are still on the arc at 3:00h. Using the set

Qo(t), the total amount of flow on static arc a at time t or Xa(t) is (T,s)Ea(t) Ya(r,s).
There are two additional sets of decision variables. One set consists of za(t,s),

a binary variable that equals one if it takes between (s 6) and s units of time

to traverse arc a at time t. In the formulation below, the value of Za(t,s) depends

on Xa(t) and, for each t, Za(t,s) = 1 for only one s E Fa. The other set consists of

g9, a vector with a component for each node in the TE network. Component it

of gk is set to zero if i is not the destination node of OD pair k. Otherwise, g(k)t

where d(k) denotes the destination node of OD pair k, is a decision variable that

represents the amount of flow for commodity k that reaches its destination, d(k), at

time t.

Below is a mathematical formulation of the discrete-time dynamic traffic

assignment problem with periodic planning horizon (DTDTA).

mm in a(a(t)) Yat
(x,y,z,g) t aA I sEFa

s.t. Byk +gk bk Vk C

gd(k)t ht Vk G C

a(t,s) E (t Vt e A, a c A and se c


Xa(t) Y Ya(r,s) Vt

YE Za(t,s) 1 Vt E z

)oa(t,s) < oa(Xa(t)) < Y. SZa(t,s)

Ya(t,s) < 1 ,,.,)

E A and a c A

N and a c A

Vt c A and a E A

Vt c A,a E A and a c Fa






a(t,s), 9g(k)t X(t) > 0, za(t,s) E {0, 1}

Vt e A,a E A, s Fa and k C (5-8)

In the objective function, serP Ya(t,s) represents the number of cars that enter arc

a at time t and, based on our assumption, these cars experience the same travel

time, Qa(xa(t)). Thus, the goal of this problem is to minimize the total travel time

or delay. Using constraint (7-10), the objective function can be equivalently written


min MYY)
( tEA aEA (T,s)EQ(t) SEFa
or, more compactly, as

min () fY,

where Y and K(Y) are vectors of arc flows (Ya(t,s)) and travel times

(a((,Ts)Ean( Ya(r,s))) whose components are defined so that their inner product is
consistent with the summations.

Constraint (7-8) ensures that flows are balanced at each node in the TE

network. In this constraint, B denotes the node-arc incidence matrix of the TE

network and bk is a constant vector with a component for each TE node and

defined as follows:

k 0 if i o(k)
hk if i = o(k)

where o(k) denotes the origin node of OD pair k. Constraint (7-9) guarantees

that the number of cars arriving at the destination node d(k) equals the total

travel demand of OD pair k during the planning horizon. Then, constraint (7-10)

computes the total flow on each TE arc and (5-4) determines the number of cars

that are still on static arc a at time t.

In combination, the next three constraints, i.e., constraints (5-5) (5-7),

compute the travel time for the cars that enter arc a at time t and only allow flows

to traverse the corresponding arc in the TE network. In particular, constraint

(5-5), in conjunction with (5-6), chooses one (discretized) travel time s E Fa

that best approximates ,a(Xa(t)), i.e., a(x,(t)) E (s 6, s]. When a represents

arc (i,j), constraint (5-7) only allows arc (it, imod(t+s,T)) to have a positive flow.

Otherwise, (7) forces flows on arc (it, imod(t+7,T)), for T E Fa and T / s, to be zero.

Finally, constraint (7-11) makes sure that appropriate decision variables are either

nonnegative or binary.

As formulated above, the travel time associated with Za(t,s) in equation (5-6)

can only take on discrete values from the set Fa while the travel time in the

objective function varies continuously. Although it may be more consistent to use

discrete values of travel times in the objective function, the above model would

provide a better solution because the true travel time is used to calculate the

total delay. The model also has interesting properties discussed in Section 5.4.

In addition, the treatments of travel times in both the objective function and

constraints can be made consistent by solving the (approximation) refinement

problem also discussed in the same section.

Under a relatively mild sufficient condition, we show below that DTDTA has

a solution by constructing a feasible solution. In fact, the solution we construct

below is generally far from being optimal. However, it suffices for the purpose of

proving existence. Let Ra(t) be a set of discrete times at which a car enters arc a

and still remains on the arc at time t. Below, we refer to Ra(t) as the enter-remain

set. Given xa(t), Ra(t) C A is a union of two sets, i.e.,

Ra(t) ={w A: w < (t 1),w+ L(xi(w)) > t}U

{w : w > (t + 1),w+ ((xw) >t}.

In addition, let Ua(t) denote the total flow into arc a at time t. When Ua(t) is

given for each t E A, the lemma below shows that a set of Xa(t), Ya(t,s), and Za(t,s)

consistent with constraints (5-4)-(5-7) and relevant conditions in (7-11) exists

when if is sufficiently large.

Lemma 5.3.1. Assume that Ua(t) is known for a given a E A and all t E A. If 31,

is suff:.' i.l i o then there exists a set of Xa(t), Ya(t,s) and Za(t,s) that -.,l:fi.

constraints (5-4) to (5-7) and the relevant conditions in (7 11).

Proof: (Because a is given, we discard the subscript a in places where there

is no confusion in order to simplify our notation.) Below, we construct sequences

{sm}, {zm(,S)}, {(Y~t,)}, {x(t)} and {R~} whose limits yield the set of decision
variables feasible to constraints stated above.

For m = 1, let

s = [(a(0)/6l, i.e., s' is the discretized free flow travel time for arc a,
z(t,) 1 and (ts) 0, Vs e A, s / s1
1 andY 0t
Y1(t,s) a() at)and Y(t,) 0, Vs A, s / s}
a t )at ,
Ri e :w < (t -),+s >t}U{ A : > (t+1),w+s T T>t},

Si I
(t)- ZWERa "oiy)
As defined above, R1 is the enter-remain set based on the travel time s1, a vector of

1,Vw E A. For m > 2, let

zat,s-) tand -- a (s A, t
s( /

n ,) 1 and z ,) 0, Vs e A, s / sm

(,)) d() a (nd Y = Vs A,
Rm {w A : < (t ),w+s > t}U{ A :w > (t+1),w+s -T>t},


(t) ZwERYa a(,s)

Sequences {st}, {x t)} and {R-} constructed above are monotonically

non-decreasing. Consider the sequence {s-}. Observe that s2 > sl,V t c A because

Xi(t) > 0, V t A, and as assumed earlier O a() is non-decreasing. It follows that,

for any t A, + s2 > > t and + s- T > + s1- T>t. Thus, uw R
for~ WntA W3s ss- s _

implies that w E R2, i.e., R1 C R2 for all t E A. The latter, and the fact that Ua(t)

is nonnegative, imply that xa(t) -= Yw ERY (,) > ERt a(w,s) X t) t E A.

Assume that the claim is true up to some fixed m. For all t E A, s'l

[ra(x t))/1 > [a(X1t) )/] s, because ) > x-1 and a(.) is
non-decreasing. Using an argument similar to above, R C C R+1 and x"+1 > X "
a(t) a (t)
Thus, the three sequences are monotonically non-decreasing. In addition, all three

sequences are bounded, i.e., s' < T, R' C A, and xZ < tea U) and, therefore,

convergent. Let s', R', and xt) be their limits. Based on our construction,
s = [Oa(x(t))/6], ats) = 1 and z"t,s) 0, Vs E A, s / s'. In combination,

these ensure that constraints (5-5) and (5-6) are satisfied. Our construction also

implies that Y'sO) U a() and Yo -(t, 0, Vs e A, s / s". Because if, is

sufficiently large, Y~t,,) satisfies constraint (5-7).

In the limit, R' = {w E A : u < (t 1), u + s > t} U { EA :

w > (t + 1),w + s- T > t}. Thus, R' is consistent with s' and x't)
yoo because Y'
EER aR w,s) (T xt) satisfies (5-4). Furthermore, xz and Y ) are both nonnegative and z"t,)is
(t a(t) (t,)nonnegative a is

binary. Thus, the proof is complete. U

In the above proof, if, for some m, s' is larger than the maximum travel time

for arc a, i.e., max{s :s E Fa} (or, equivalently, xZ > .1,), then Ua(t) is infeasible

or not compatible with the upper bound i. ,.

To establish the existence of a feasible solution to DTDTA, recall that

G(N, A) denotes the (static) transportation network. For the theorem below,

assume without loss of generality that each node in N can be either an origin or

destination, but not both. If node i is both an origin and a destination, then we

create a dummy node i' and use node i as the origin node and i' as a destination.

For example, consider OD pairs (i,j) and (j, i). In this case, i and j are both

origins and destinations. When the dummy nodes are added, the two OD pairs

become (i,j') and (j, i'). Let pk denote a path in G(N, A) connecting the OD pair

k, i.e., pk E pk. The set of these paths, {pk : k E C}, induces a subgraph

G(N, A), where N C N and A C A denote the sets of nodes and arcs, respectively,

belonging to the paths in T. For each i E N, define [i+] = {(i,j) : (i,j) A} and

[i-] = {(j,i) : (j, i) c A}. In words, [i+] and [i-] are the sets of arcs in G(N,A)
that emanate from and terminate at node i, respectively. Also, let order(i) denote

a topological order of node i (see Al!ni et al. [2]). If (i,j) A and G(N,A) can be

topologically ordered, then order(i) < order(j).

Theorem 5.3.1. Assume that if, is suff. .ii:l, '1,n'/ for all a E A and a node

can be either an origin or a destination, but not both. Then, DTDTA has a feasible

solution, if there exists a path pk for each k E C such that the -.l'.q'rl, they induce
is i. ,. 1.:.

Proof: Let T be a set of paths, one per OD pair, such that the subgraph,

G(N, A), it induces has no cycle. Thus, N can be ordered topologically. (See

Al!li et al. [2].) Below, we construct a feasible solution one arc at a time and in

a topological order using Lemma 5.3.1 and only the paths in T. The latter implies
that Ya(t,s) = (t,s) =x(t) 0 for all a A.

Let node i E N be of topologicall) order 1 and, for each arc a in [i+], define

Q(a) to be the set of paths in T that contain or use arc a, i.e., Q(a) = {k :
a E pk, k E C}. (It is not necessary to index Q(a) with i because each arc a
can belong to only one [i+].) For each k E Q(a), arc a must be the first arc in

path pk because node i is of order 1. Let Ua(t) = ckQ(a) hk. Because .1, is

sufficiently large, Lemma 5.3.1 ensures that there exist Xa(t), Ya(t,s), and Za(t,s)

feasible to (5-4) (5-7) and the relevant conditions in (7 11). Let y(t, ()) = h
and /(ts) = 0,V s c A, s / s o(t). So constructed, these y(t,s)'s are consistent with

Ya(t,s) and satisfy the flow balance equation (7-8) for node i.

To construct the variables Xa(t), Ya(ts), Za(t,s) and y(ts) for arcs emanating

from nodes of higher order, assume that the decision variables for arcs emanating

from nodes with order m or less have been constructed and let node i be of order

(m + 1).

Case 1: The set [i+] is empty. Then, i must be a destination node for some

commodity k, i.e., i = d(k). For a c [i-], k c Q(a) and t c A, set

(ta )a(t)
{*: t+-", -t} {f: t+-", -T-t}

For each k E Q(a), every demand hk uses arc a. Thus, rk, as constructed must

satisfy the appropriate constraints in (7-8) and (7-9).

Case 2: The set [i+] is not empty. Let a E [i-], also a nonempty set. Assume

that a (q, i). Then, order(q) < order(i) and, by the above assumption, xa(t),

Ya(t,s), Za(t,s), and k(t,s) are available.

Consider an arc a c [i+]. For each a c [i-], define Q(a, a) = {k : a E pk a

pk, k e C} and, for each k E Q(a, a), let U(,k) denote the flow into arc a at time t
for OD pair k. Then,

> tj")= Ec)+ nE > : sc
e(t) )(t)
ti t\' ," t}{ t+ )-T=t}

and the total flow into arc a at time t is Ua(t) = keQ(,a) U(t)I Because I, is

sufficiently large, Lemma 5.3.1 ensures that Xa(t), Ya(t,s), ya(t,s) and za(t,s) feasible to

relevant constraints exist.

Thus, when carried out in the topological order for every arc in A, the above

process must produce a feasible solution to DTDTA. U

The theorem above assumes that each .[ is sufficiently large so that it is

feasible to send the entire flow for each OD pair along a single path. Although this

assumption appears to be stringent, it can be made less so by allowing the flow for

each OD pair to traverse over several paths as long as they do not induce cycles in

G(N, A). With more cumbersome notation, the above argument can be extended to

the case with multiple paths per OD pair as well.

When applied to the above example in which the OD pairs (i,j) and (j, i)

become (i, j') and (j, i'), the ... i- lic subgraph assumption implies that the paths

from i to j' and from j to i' cannot form a cycle. Intuitively, this means that

there must exist two routes between the original nodes i (e.g., home) and j (e.g.,

work) with no road in common. These routes need not be optimal and there is no

requirement in our formulation or algorithms to use them. They are used only to

established the existence in Theorem 5.3.1.

The First-In-First-Out (FIFO) condition requires that cars entering an arc at

time t must leave the arc before those entering after time t. In the literature, many

(see, e.g., Ran et al. [91], Zhu and Marcotte [103], and Parakis and Roels [86])

assume that the travel cost function satisfied certain conditions to ensure FIFO.

To avoid making additional assumptions, we ensure FIFO by adding the following

constraints to DTDTA instead. Doing so may make the problem harder to solve

because of the additional constraints.

t + E sza(t,s) < t + E sza(t,s), Va Al and t,t E A : (t + 6) < t
t+ SZa(t,s) < (t + T) + Y SZa(t,s), Va c A1 and t,t E A: (t + J) < t

When t and t represent two instances of time on the same d4i-, the first inequality

ensures that cars entering arc a at time t leave the arc before those that enter at

time t > t. On the other hand, t and t may refer to times on consecutive d-iv- e.g.,

t = 08:00h todci- and t = 09:00h yesterday. Because of our periodic assumption,

these two times are on the same interval [00:00h, 24:00h) and t (incorrectly)

appears to be an earlier time than t. To distinguish times on consecutive di-4

and preserve FIFO, the second equation represents tod i-'s time t (e.g., 08:00h of

tod(vI) as (t + T) (e.g., as 08:00h of yesterday plus T) and forces cars entering the

arc at this time to depart after those that enter at yesterday's time t (e.g., 09:00h


5.4 Bounds for the DTDTA Problem

As formulated in the previous section, DTDTA is a nonlinear optimization

problem with binary decision variables, a difficult class of problems to solve. This

section describes mixed integer programs for obtaining an approximate solution to

DTDTA as well as bounds for the optimal delay.

Except for constraint (5-6), the constraints for DTDTA are linear. To develop

a linear version of (5-6), assume that the travel time function, Q0, is invertible for

all a E A. For example, if Oa is a continuous and increasing function, then Oa1

exists on the interval [,(0), ,(a ,)]. (See Figure 5-6.) Under this assumption,

Qa(Xa(t)) C (s 6, s] if and only if Xa(t) C (a 1(s 6), oa1()]. Thus, the requirement
(s 6)Za(t,s) < Oa(Xa(t)) < sza(t,s) is equivalent to 01(s 6)Za(t,s) < Xa(t) <

al(s)z(t,,). Recall that Fa {s :s ['='], 0 < w < 11,}. Let si].
Then, (si 6) g [a (0), O,(a(,)] and al'(sl ) is not well defined. (In Figure 5-6,

a1(si 6) = Oa(1) is not well defined.) In this thesis, we set a1'(si 6) = 0.
Using this convention, constraint (5-6) can be replaced by the following linear


Sa-1s -I )Za(t,s) < Xa(t) Q-l(s)za(t,s) Vt c A and a c A (5 9)

The following lemma implies that there exist linear functions that approximate

the objective function of DTDTA.

Lemma 5.4.1. There exist vectors ql and q, such that qY < (Y)TY < qfY for

all Y feasible to DTDTA.

Oa a(Xt0
2 -----------

S1 --

1 (2) a1 (3) a( (4)


Figure 5-6. Oa(xa(t)) E (s s] versus Xa(t) E ((-1(S 6), a-1()].

Proof: As defined earlier, )(Y)TY = Y ~(Xa(t)) Y Ya(t,s) From constraint
tE aEA Lsea
(5-6), the following hold for any feasible solution to DTDTA:

y ( 6)za(t)] [ Y(ts) )j ( te aEA sEFera sera


N(Y)TY< S Sa(t) E(a(ts)
ted EA LseF, seFa

The summand of the last set of summations (i.e., K sZa(t,s) s8 Ya(t,s) ) can
LsrFa I LsEFa J
be simplified. Constraint (5-7) implies that Ya(t,s) > 0 only if Za(t,s) = 1. In

addition, constraint (5-5) ensures that, for each pair (a, t), Za(t,-) = 1 for some

s E Fa and za(t,s) = 0,V s E Fa,s / s. This implies that Ya(t,s) > 0 and

Ya(t,s) 0, Vs F,, / s and

Y SZa(ts) Ya(t, s) SYa(t,s) sYa(t,s)-
-sera sera I sera

A similar result holds for the first set of summations. Thus, the above inequalities
reduce to the following

E SE (s -J)Y(6,s)< tEz aEA sEFa tEz aEA sEFa
Let ql and q, be two constant vectors with a component for each arc in the
TE network such that [q]a(t,s) = (s 6) and [q,]a(t,s) = s, respectively, for all

a A, tE A, and sE Fa. Then,

q1T 555 (s 6)ya(ts) < qjy)Ty < 55 SYa(t,) qY. U
tEz aEA sEFa tEz aEA sEFa

Let S(6) denote the feasible region defined by linear constraints (7-8) (5-5),

(5-9), (5-7), and (7-11) and, for convenient, (Y, Z) represents an element in S(6).
In addition, let (Y', ZI), (Y*, Z*), and (Y", Z") be solutions to the lower-bound
problem (or min{qTy : (Y, Z) E S()}), the original problem (or min{j(Yy)T

(Y, Z) c S(6)}), and the upper-bound problem (or min{qY (Y, Z) e S(6)}),
respectively. Then, the following lemma holds.
Lemma 5.4.2. For i,., 6 > 0, qTY1 < (Y*)T* < qTU < qTy.

Proof: In following sequence of inequalities, the first one holds because Y* is
feasible to the lower-bound problem and the second follows from Lemma 5.4.1:

qYlV < qlY* < F(Y*)TY*.

Similarly, the following sequence also holds

q(Y*)TY* < (Yu)Ty" < qY".

Combining the above two sequences yield the first two inequalities in the lemma.

Finally, the last inequality holds because yl is not necessarily optimal to
min{qTY: (Y, Z) e S()}. U

In view of the above lemma, the solutions to the upper and lower-bound

problems are approximations of the solution to the original problem. The theorem

below states that the approximation can be made arbitrarily close to the original

problem by choosing a sufficiently small 6.

Theorem 5.4.1. Given c > 0, there exists 6 > 0 such that qTV" {Y*)Y* < e

and (Y*)Ty* (Y-
Proof: By construction, qu = qi + 6e, where e is (1, 1, 1)T. Let Hk denote the

travel demand for OD pair k during the entire planning horizon, i.e., Hk = teA ht

and set H = Ekec Hk. For each t c A, sa(t) is such that sa(t) e Fa and

z(ts ) 1. In words, Sa(t) is the approximate travel time for (static) arc a at

time t in the optimal solution (Y', ZI).

Then, the following sequence must hold:

0 < (q- qi)Tyl


E E E y y s)
aEA kEC tEA sEra

< 6 EE Hk
-6H E 1

The first inequality follows from Lemma 5.4.2. Then, the above relationship

between q1 and q, and letting EckEC Y(t,s) denote individuals components of YI

yield the first two qualities. The third equality follows from the definition of

Sa(t). Following this, the second inequality holds because the total amount of

flow on (static) arc a for OD pair k during the entire planning horizon cannot

exceed Hk. The sum of the latter is H, a constant that can be factored out of

the summation over A. This validates the penultimate equality. Finally, the last

equality follows from the fact that ZaEA 1 simply denotes the number of elements

in the set A. Choosing 6 = HA guarantees that qY qfY' < e. When combined

with the results in Lemma 2, the latter implies that qTY" N(Y*)TY* < c and

_(y*)Ty* q Ty
The approximate solution Yu can be improved by solving an additional

nonlinear program. In particular, consider the approximation refinement problem

min{t(Y)TY : (Y,Z") E S(6)}, i.e., this is the original problem with Z = Z".

Doing so makes it possible to remove TE arcs corresponding to z (t,) 0 from the

TE network and discard decision variables Xa(t) and constraints (5-5) and (5-7)

from the problem. In DTDTA, we use Xa(t), the number of cars on arc a at time t,

to compute the travel time on arc a and, subsequently, to select which TE arc to

use or which Za(t,s) to set to one. Thus, when Z is given, Xa(t) becomes unnecessary.

Additionally, let s(t) be such that z"ts()) = 1 for each t E A. Then, constraint

(5-9), originally (5-6), reduces to requiring E(rs)EQ Ya(T,s) to be in the interval

(s(t) 6, s(t)]. In other words, the original problem with Z = Zu is a nonlinear
multi-commodity flow problem with the latter as side constraints.

Let Y" be an optimal solution to min{)(Y)TY : (Y, Z") e S(6)}. Then, the

following corollary shows that Y" better approximates Y*.

Corollary 5.4.1. (Y*)TY* < +(Y")TY" < ((yu)TY" < qTY"

Proof: In the above sequence of inequalities, the first one follows because Y* is

optimal to the original problem and Yu is only feasible. The second holds because

Y" is feasible to min{)(Y)TY : (Y, Z") e S(6)}. Finally, the last is due to Lemma

5.4.1. U

5.5 Numerical Experiments

We conducted numerical experiments using small test networks to empirically

verify our understanding of DTDTA as well as to evaluate the efficiency and

effectiveness of the approximation schemes discussed in previous sections.

Table 5-1. Demand patterns

Traffic Intensity 0 1 2 3 4 5 6 7 8 9 Total
Low 20 25 30 35 40 40 35 30 25 20 300
Medium 30 35 40 45 50 50 45 40 35 30 400
High 40 45 50 55 60 60 55 50 45 40 500

In all problems, the planning horizon is [0, 10) and the travel cost functions

are either linear, i.e., O(w) = 1.5 + 2.5(1-'), or quadratic, i.e. O(w) = 1.5 + 2.5(1o )2,

where w is the number of cars on the arc. We consider the three different demand

patterns di-p'1l ,i'1 in Table 5-1. In all three patterns, travel demands at discrete

points increases gradually until time 4, levels off briefly, and then decreases

gradually after time 5. The individual demands in the three patterns are different

and represent three traffic intensities: low, medium, and high. We used GAMS [39]

to implement and solve all problems using NEOS Server of Optimization [82]. In

particular, we used SBB [94] to solve our nonlinear integer programming problem,

i.e., DTDTA, XPress-XP [102] to solve our linear integer programs, i.e., the lower

and upper-bound problems, and CONOPT [26] to solve our linearly constrained

optimization problems, i.e., the approximation refinement problems. All CPU times

reported herein are from the NEOS server.

To empirically verify that DTDTA problem is not convex, we first consider

the two-arc network in Figure 5-7 that has one OD pair. We let 6 = 1. Thus, the

discrete-time planning horizon is A = {0, 1, ,9}. We use the above quadratic

Figure 5-7. Two-arc network.

Table 5-2. Optimal solutions to the two-arc problem.

Solution 1 Solution 2
Inflow Travel time Inflow Travel time
Time al a2 al a2 al a2 al a2
0 0 20 1.600 1.500 20 0 1.500 1.600
1 25 0 1.500 1.600 0 25 1.600 1.500
2 0 30 1.656 1.500 30 0 1.500 1.656
3 35 0 1.500 1.725 0 35 1.725 1.500
4 0 40 1.806 1.500 40 0 1.500 1.806
5 40 0 1.500 1.900 0 40 1.900 1.500
6 0 35 1.900 1.500 35 0 1.500 1.900
7 30 0 1.500 1.806 0 30 1.806 1.500
8 0 25 1.725 1.500 25 0 1.500 1.725
9 20 0 1.500 1.656 0 20 1.656 1.500

travel time function for both arcs and the function yields travel times in the

interval [1.5, 4.0]. Because 6 = 1, the set of discrete travel times is F {2, 3, 4}.

Using the low traffic intensity demand pattern in Table 5-1, we solved DTDTA

using SBB and terminated it when the relative optimality gap is less than 0.005 (or

0.5'.-). There are two optimal solutions (see Table 5-2) to the two-arc problem with

an optimal total d. 1 iv of 450.

Consider the first solution, labelled 'Solution 1', in the Table 5-2. At time 0,

there are 20 cars to travel from node 1 to node 2. At this time, there are also 20

cars already on arc al. These cars enter the arc at time 9 and have not reached

their destination at time 0. Because DTDTA assumes that the time to traverse

arc al depends on the number of cars on the arc at the entrance time, the travel

time for arc al at time 0 is 1.5 + 2.5( 0)2 1.6. On the other hand, there is

no car on a2 at time 0. Cars that enter the arc at time 8 already left the arc by

time 0. Thus, the travel time for a2 at time 0 is 1.5, the free-flow travel time. To

minimize the travel time, all 20 cars entering the network at time 0 must travel on

a2. In fact, every car in Solution 1 travels at the free-flow travel time of 1.5. Thus,

there cannot be any solution with less total d, 1 iv and Solution 1 must be optimal.

Figure 5-8. Four-node network.

Because of the symmetry in the data, switching the flows between the two arcs in

the network yields Solution 2, another optimal solution. Furthermore, it is easy to

verify that every convex combination of these two solutions is feasible to DTDTA

and yields, on the other hand, a larger total delay, thereby confirming empirically

that the objective function is not convex.

Additionally, the "extreme" travel behavior di -p i 1 in Table 5-2 may not

be intuitive. This is due to the assumption that the system operator is extremely

sensitive to the difference in travel times and is willing to switch routes in order to

save a minute amount of travel time.

When the network is large, it would be too time-consuming to solve DTDTA

optimally or otherwise. In our experiments, we consider four approximate solutions

to DTDTA: (Yl, Z), (Y, ZU), (Yl, Z), and (YU, Z), where the last two are

refinements of the first two. To evaluate the quality and the computation times

of these solutions, we consider the four-node network in Figure 5-8 with two OD

pairs, (1, 4) and (2, 4). In our experiments, both OD pairs have the same demand

pattern and all arcs have the same travel cost function, linear or quadratic, as

specified above.

First, we solved the lower and upper-bound problems with using two levels of

discretization, 6 = 1 and 6 = 0.5. As before, when 6 = 1, the discrete-time planning

horizon is A = {0, 1, ,9}. On the other hand, when 6 = 0.5, A becomes

Table 5-3. Solutions from the lower and upper-bound problems: linear travel cost

Traffic = 1 6 = 0.5
Intensity ql Y q, Y" Gap ql Y q Y'" Gap
low 820.0 1580.0 760.0 1187.5 1560.0 372.5
medium 1200.0 2230.0 1030.0 1705.0 2230.0 525.0
high 1500.0 2875.0 1375.0 2187.5 2870.0 682.5

{0, 0.5, 1, 1.5, ... ,9, 9.5}. For the comparison below (see Tables 5-3 and 5-4), we

assume that, when 6 = 0.5, there is no demand at fractional times ( e.g., at 0.5,

1.5, 2.5, etc.) and the demands at integral times (i.e., 1, 2, 3, etc.) are as shown in

Table 5-1.

For both types of travel cost functions, the size of the optimality gap (i.e.,

quY" q1Yy) decreases by approximately 5('. as 6 decreases from 1 to 0.5.

However, the results in Tables 5-3 and 5-4 sl--'-, -1 that the reduction in the gap

is due mainly to the improvement in the solution, yl, of the lower-bound problem.

The approximate travel d-1i, as estimated by Yu change relatively little for the

two values of 6.

Tables 5-5 and 5-6 compare the solutions from DTDTA, (Y*, Z*), against

two approximations, (Y", Z") and (Yi, ZI). As in the two-node problem, we solve

DTDTA using SBB to obtain a (integer) solution (Y*, Z*) with less than 0.5'.

relative optimality gap. To obtain (Y", Z"), we first solve the upper-bound problem

using XPress-MP to obtain (Y", Z"), a (integer) solution with less than 0.5'.

optimality gap, and, then, solve the approximation refinement problem (with

Table 5-4. Solutions from the lower and upper-bound problems: quadratic travel
cost function.

Traffic = 1 6 = 0.5
Intensity q1 Y qu Y" Gap qf Y q Y" Gap
Low 600.0 1200.0 600.0 900.0 1200.0 300.0
Medium 822.2 1644.5 822.2 1233.3 1644.5 411.1
High 1124.5 2248.9 1124.5 1686.7 2248.9 562.2

Table 5-5. Quality of refined upper and lower-bound solutions: linear travel cost

(Y*, Z*) (y", ZU) (Yl Zl) Rel. cpu
Traffic cpu* (1i cpu' Err Ratio
Intensity Delay (sec) Delay (sec) Delay (sec) ( .) pu cpu*
Low 1337.50 27.42 1385.00 2.57 1392.50 2.38 3.55 10.7 11.5
Medium 1800.00 15.92 1-.., ; 2.66 1815.30 2.90 0.85 6.0 5.5
High 2290.00 95.02 2327.50 4.07 2315.00 1.25 1.09 23.3 76.0

Z = Z") using CONOPT to obtain (Y", Z"). The solution (Y', Z') are obtained in

the same manner. In the two tables, the CPU times for the two approximations are

times for solving both bounding and refinement problems.

For both linear and quadratic travel time functions, the two approximation

schemes provide solutions with relatively small errors using much less CPU time

required to solve DTDTA (see the ratios of the cpu times in Tables 5-5 and 5-6).

For quadratic travel time functions, the approximate solutions are identical to

DTDTA solutions, except for the high traffic intensity case when the approximate

solutions are slightly better (by 0.0 .'.).

Table 5-6. Quality of refined upper and lower-bound solutions: quadratic travel
cost function.

(Y*, Z*) (y", Z) (Y' Z') Rel. cpu
Traffic cpu* < cpu Err Ratio
Intensity Delay (sec) Delay (sec) Delay (sec) ( ) cpu
Low 1054.50 0.88 1054.50 0.09 1054.50 0.08 0.00 9.8 11.0
Medium 1. I ; SO 6.62 1. ; SO 0.14 1'. ; SO 0.34 0.00 47.3 19.5
High 2129.80 501.17 2128.60 0.10 2128.60 0.13 -0.06 5011.7 .-. 2

5.6 Concluding Remarks

This chapter formulates a discrete-time dynamic traffic assignment problem

(DTDTA) in which the planning horizon is treated in a circular fashion and events

occur periodically. Doing so allows positive flows on the network both at the

beginning and at the end of the planning horizon. The structure underlying the

formulation is the time-expansion of the (static) network representation of streets


and highi--,-. The resulting problem is a nonlinear program with binary variables,

a difficult class of problems to solve. Alternatively, two linear integer programs are

constructed to obtain approximate solutions and bounds on the total travel delay.

It is shown that solutions from the latter can be made arbitrarily close to solutions

of DTDTA. Furthermore, numerical results from small test problems si,--l. -1 that

solving the linear integer program is more efficient.


6.1 Introduction to the Chapter

In C'! ipter 5 we have discussed a periodic discrete time dynamic traffic

assignment problem, which is constructed based on two assumptions: (i) all

cars that enter the arc during the same time interval experience the same

travel time and leave the arc during the same time interval, and (ii) the travel

time is a function of the number of cars on the road (see also Nahapetyan and

L i.|- !..ii I-)anich [78]). As we have seen, the initial mathematical formulation

of such model leads to a mixed integer problem with linear constraints and a

nonlinear objective function. By linearizing the objective, one can construct an

upper and a lower bound problems. Although the solution of such problems can

be made arbitrarily close to the solution of the initial problem by decreasing the

discretization parameter 6, observe that the bounding problems belong to the class

of linear mixed integer program, which are computationally expensive to solve. In

the case of the bounding problems, the task becomes more challenging because of

the special structure of the feasible region.

Observe that the model is constructed based on a time-expanded network

and there are binary variables, a(ts), associated with the arcs of the network.

By decreasing the parameter 6, the number of the binary variables increases. For

example, given a traffic network G(N, A) and a set of possible discrete travel

times FP(6), the total number of the binary variables in the DTDTA-U problem

is IA(65) EaEA Ia()l If 6 reduces to 6/2, then IF,(6/2)| = 21|,(6)1|,A(6/2)| =

21A(6)|, and the total number of the binary variables in the new problem is

221A()l aEA I a() 1. Because of resource limitations, MIP solvers cannot solve

large problems.

In this chapter we consider a heuristic algorithm to solve the bounding

problems. Although the same technique can be applied to solve both bounding

problems, we mainly focuss on the upper bound problem DTDTA-U. For

convenience of reference, we restate the problem below.

min qTY

s.t. Byk+ k bIk Vkc C (6-1)

Sgd(k)t h Vk C (6-2)

Ya(t,s) Y (t,s) Vt c A, a E A and s E F, (6 3)

Xa(t) > Ya( ,s) Vt e and a c A (6-4)

Za(t,) 1 Vt A and a c A (6 5)

SOa'-( 6)Za(ts) < Xr(t) "< E a'(s)Za(ts) Vt e A and a e A (6-6)
sEPa sEPa
a(t,s) < ,,, ., Vt e A,a E A and a E Fo (6-7)

k k
Ya (t,s) 9d(k)t X(t) > 0, a(t,s) {0, 1} Vt A, a e A, s E Fa and k C C (6-8)

In the well-known heuristic algorithms such as neighborhood search, greedy

algorithm or tabu search, it is required to move from one feasible solution to

another. However, finding a feasible solution to the DTDTA-U problem is not easy.

To demonstrate, consider a one-arc-network, a = (1, 2), a linear travel time function

o(Ka(t)) = 0.3 + 0.05Xa(t), and a set of possible (discrete) travel times Fa {1, 2, 3}.

In addition, assume that the time horizon [0, 5) is divided into five intervals, and

10 cars enter into the arc at each discrete time t E A {0, 1, 2, 3, 4}. By assigning

those cars to the arcs a(t, 1) of the TE network, one concludes that Xa(t) = 10,

a(xa(t)) = 0.8, and Za(t,1) = 1, Vt A, which satisfies constraints (6-5) and
(6-6) (see the left network in Figure 6-1). It can be shown that this is an optimal

solution to the DTDTA-U problem. However, there is another feasible solution,

where 10 cars are assigned to the arcs a(t, 2) (see the right network in Figure 6-1).

Using those settings, Xa(t) = 20, Oa(xa(t)) = 1.3, and Za(t,2) = 1, Vt A, which again

satisfies constraints (6-5) and (6-6). Assume that the second solution is known

and one decides to improve the solution and move to another feasible solution by

assigning the cars at time t = 0 to arc a(0, 1), i.e., Za(0,1) = 1 and Za(0,2) = 0. As

a result, Xa(l) = 10 and Qa(xa(l)) = 0.8, which violates inequality (6-5). The same

follows from the change of other arcs; thus, we conclude that the second solution

is isolated in the sense that the .,1i ,i:ent solutions, i.e., changing only one arc,

are infeasible. Finding an .,ii ,i:ent feasible solution becomes more complicated

when the static network G(N, A) is larger because, with respect to a given path,

the changes on upstream arcs have an influence on the flows of downstream arcs.

Because heuristic algorithms similar to the neighborhood search, greedy algorithms,

and tabu search require moving from one feasible solution to a neighboring feasible

solution, the difficulties of finding a neighboring solution makes inappropriate the

use of such techniques.

10 o 10 10
10 10 1
10 10

12 u 22 1 2 s22
10 /10 /
13 /23 23
10 10
1424 24

Figure 6-1. Two feasible solutions.

Another approach to solve the DTDTA-U problem is the relaxation of the

integrality of the variable Za(t,s) and constructing an equivalent formulation with

continuous variables. To do so, replace the constraints Za(t,s) E {0, 1} by inequalities

0 < Za(t,s) < 1, i.e. z e [0, 1], n = A(5) EaeA 1a(5)1, and Za(t,s)(1 Za(t,s)) < 0.

The latter can be included into the objective function with a penalty. As a

result, the problem reduces to a continuous concave minimization one and a

global solution of the resulting problem is a solution of the DTDTA-U problem

(see, e.g., Horst et al. [54] or Horst and Tuy [55]). Although it is known that an

optimal vector of the binary variables, z*, represents one of the vertices of the n

dimensional unit cube (each vertex corresponds to an integer solution), because of

constraints (6-1)-(6-8) most of them are infeasible and it is hard to find an optimal


Observe that the LP relaxation of the DTDTA-U problem provides a lower

bound, which is far from an optimal one. To illustrate, consider the DTDTA-U

problem, where the constraints Za(t,s) E {0, 1} are replaced by the inequalities

0 < Za(t,s) < 1. To find optimal values of variables y(t,) and g(k)t in the resulting

problem, it is sufficient to solve the following linear problem.

min qTY

s.t. Byk + gk bk Vk C

t k14 Vk c C

a(t,s) > 0, g(k), > 0, Vt e A, a A, s e F and k C

Because the arcs a(t, 1) have a lower cost in the objective than the arcs a(t, s),

s / 1, one concludes that at optimality only the arcs that corresponds to the free
flow travel time, i.e., the arcs a(t, 1), Va E A and t E A, have positive flows. Using

the optimal values of those variables it is easy to compute values of the variables

X,(t) through equations (6-3) and (6-4). The optimal values of the variables Za(t,s)

can be obtained by solving the following system of equations.

ZsEra Za(t,s) = 1

a )Za(to) < a(t)

e a(s)Z(ts) > x (6 9)

Za(t,s) C [0, 1]
Za(t,s) > Ma()

Notice that the last inequality in (6-9) is satisfied VZa(t,s) E [0,1, s / 1, because

Y(t,s) = 0, Vs c Fa, s / 1. In the case of s 1, a sufficiently large value of .3,
reduces the inequality to Za(t,1) > 0 and makes sure that arcs a(t, 1) are allowed to

have positive flows given any positive values of the variable Za(t,1) (see inequality

(6-7)). Other equations in (6-9) are easy to satisfy and it can be shown that for

any value of x*(t) the set of solutions to the system is not empty and not unique.

However, because of the congestion it is highly unlikely that at optimality of

the DTDTA-U problem all drivers experience the free flow travel time and one

concludes that a solution of the LP relaxation of the problem is not realistic, and

the optimal objective function value of the relaxation problem is far from the

optimal value of the objective function of DTDTA-U.

Despite all complications described above, the DTDTA-U problem has the

following useful property: if at optimality the total inflow into arc a at time t

is zero, i.e., C r Y(ts) = 0, then constraint (6-7) is satisfied for any value of

Za(t,s); therefore, corresponding constraints (6-5)-(6-7) can be removed from the
formulation and the solution of the resulting problem remains the same. Unknown

values of the variables Za(t,s) can be restored by solving the system of equations

(6-5)-(6-6) using the values of xa(t,s).

The above analysis motivates developing a lower bound problem for the

DTDTA-U, which is (i) tighter than the LP relaxation, (ii) easier to solve

than the concave minimization problem discussed above, and (iii) preserves the

above mentioned property. In particular, in this chapter we consider a nonlinear

relaxation of the problem with bilinear constraints. Using the relaxation technique,

we propose a heuristic algorithm to solve the DTDTA-U problem.

For the remainder, Sections 6.2 discusses the nonlinear relaxation of the

DTDTA-U problem. Using the relaxation, in Section 6.3 we propose a heuristic

algorithm to solve the DTDTA-U problem. Numerical experiments on the

algorithm are provided in Section 6.4, and finally, Section 6.5 concludes the


6.2 Nonlinear Relaxation of DTDTA-U Problem

Consider the following continuous nonlinear minimization problem, which we

refer to as DTDTA-R.

min qY
s.t. Byk +gk bk VE C (6-10)

9d(k), = hi Vk cC (6-11)

Ya(t,) = (t,s) Vt e A,a c A and s e (6-12)

a(t) = Ya(r,s Vt c A and a A (6-13)

S-I(s 6y)a(t,s) < Xa(t) Ya(tr) < E .la(S)Y(ts) (6-14)
Vt e A and a e A

y(t,s) > 0, ,, > 0, ,Xa() > 0 Vt e A,a c A,s c Fa and k C (6-15)

Observe that (i) in the DTDTA-R problem constraints (6-10)-(6-13) are the

same as corresponding constraints of the DTDTA-U problem, (ii) the DTDTA-R

problem does not include the binary variables and constraints (6-5) and (6-7), and

(iii) the constraints (6-6) are replaced by bilinear constraints (6-14).

Theorem 6.2.1. The DTDTA-R problem is equivalent to the LP relaxation of the

DTDTA-U problem with additional constraints of the form

Za(t,s) Y. Ya(t,r)= Ya(t,), (6 16)

Proof: Consider the following two case:

Case 1: rEr. Ya(t,r) / 0. From equality (6-16) it follows that Za(t,s)
Ya(t ,) e [0, 1, Vs E F,. The latter satisfies constraints (6-5) and (6-7) given
E-Cra Ya(t,r)
a sufficiently large [,. thus, the constraints can be removed from the formulation.

In addition, observe that after appropriate substitutions of the variables Za(t,s) the

constraint (6-6) transforms into the constraint (6-14), and the variables Za(t,s) can

be removed from the formulation.

Case 2: ErErF Ya(t,r) = 0. Observe that equation (6-16) and constraint (6-7)

are satisfied for any value of the variable Za(t,s), s E Fa. Because constraint (6-7)

is redundant, remove the variables Za(t,s), s E Fa, from the formulation as well as

corresponding constraints (6-5)-(6-7). In addition, notice that constraint (6-14) is

satisfied and can be added to the formulation without changing the feasible region.

Based on the above ,n i i- one concludes that both problems have the same

optimal objective function value and the same optimal values for the variables

Ya(t,s) and Xa(t). In addition, the optimal values of the variables Za(t,s) can be
obtained using the values of y*,s) and x*(). In particular, if rEFr Y*(t,r) / 0 then

z*, *t Otherwise, given the vector x*, to find the optimal value of the
Z (ts) ErEa (t r)
variables Za(t,s) it is sufficient to solve the system of equations (6-5)-(6-6). U

Theorem 6.2.2. The DTDTA-R problem provides a tighter lower bound solution

for the DTDTA-U problem than the LP relaxation.

Proof: Consider a feasible solution, (y, x, z), to the DTDTA-U problem.

Observe that it satisfies equation (6-16) because from constraints (6-5)-(6-7)

it follows that either both sides of (6-16) are zero, i.e., Za(t,s) = Ya(t,s) = 0, or

Za(t,s) = 1 and Zrer, Ya(t,r) Ya(t,s). As a result, adding the equality (6-16) to
the DTDTA-U problem does not change the feasible region. Next, observe that

according to the Theorem 6.2.1 the LP relaxation of the resulting problem is

equivalent to the DTDTA-R problem; therefore, the solution of the DTDTA-R

problem is a lower bound of the DTDTA-U problem.

From Theorem 6.2.1 it also follows that the DTDTA-R problem is equivalent

to the LP relaxation of the DTDTA-U problem with additional constraints of

the form (6-16). Next we show that the constraints are not redundant unless

at optimality all drivers experience the free flow travel time. In particular, the

constraint (6-16) requires sending a portion of the total flow, i.e., Za(t,s) YrEr Ya(t,r),

along the arc a(t, s) if ErEr, Ya(t,r) / 0. On the other hand, recall that in the LP

relaxation of the DTDTA-U problem, at optimality only arcs that correspond to

the free flow travel time have a positive flow, i.e., y,) = 0, Vs E Fa, s / 1, and

the variables Za(t,s) may have positive values for all s E Fa, s / 1, as long as they

solve the system (6-9) (see Section 6.1). It can be shown that the solution satisfies

equation (6-16) only if za(t,i) = 1 solves the system (6-9). The latter holds only

in trivial problems with no congestion. From the above analysis it follows that

the DTDTA-R problem has a tighter feasible region than the LP relaxation of the

DTDTA-U problem. U

The DTDTA-R belongs to the class of global optimization problems because

constraint (6-14) is neither convex nor concave. However, it is computationally

more attractive than the concave minimization problem discussed in Section 6.1.

In particular, the DTDTA-R problem has fewer constraints and variables, and

does not have a penalty term in the objective function. In addition, the DTDTA-R

problem preserves the above mentioned property; if at time t an arc, a, has no

inflow, i.e., Eser, Ya(t,s) = 0, then constraint (6-14) is satisfied for any value of the

variable Xa(t).

6.3 Nonlinear Relaxation Based Heuristic Algorithm

The nonlinear relaxation problem DTDTA-R can be very useful to develop

a heuristic algorithm for solving the DTDTA-U problem because solving the

DTDTA-R problem is a trade off between satisfying constraint (6-14) and solving

the LP relaxation of the DTDTA-U problem. (Notice that the latter is equivalent

to the finding the shortest path in the TE network.) As a result, the optimal values

of the variables Xa(t) better approximate the optimal values of the corresponding

variables of the DTDTA-U problem. However, the solution is unlikely to be feasible

to the original problem; thus it is essential to find an integer solution that has the

objective function value as close as possible to the one provided by the DTDTA-R


In the heuristic algorithm (see Procedure 8), first we solve the LP relaxation of

the DTDTA-U problem, which provides an initial solution for DTDTA-R problem.

Next the procedure solves DTDTA-R problem. In Step 3, finding the values of

Za(t,s) is easy and can be accomplished by performing a simple search technique. By

fixing the binary variables of the DTDTA-U problem to the values of Za(t,s), i.e.,

Za(t,s) Za(t,s) the problem reduces to a linear one. If the resulting LP is feasible
then the algorithm stops and returns the solution. Otherwise, the algorithm runs

the UpSet procedure (see Procedure 9) then goes to Step 2.

Procedure 8 : Heuristic Algorithm for Solving DTDTA-U Problem
Step 1: Solve the LP relaxation of the DTDTA-U problem, and let the solution
be an initial solution for solving the DTDTA-R problem in the next step.
Step 2: Solve the DTDTA-R problem, and let (yR, xR) denote the solution of
the problem
Step 3: Let Xa(t,s) R(t,) and find binary variables Za(t,s) that satisfy
constraints (6-5) and (6-6).
Step 4: In the DTDTA-U problem, fix the binary variables to the values of
Za(t,s) and solve the resulting LP problem.
Step 5: If the LP problem is feasible, stop and return the solution. Otherwise
run the UpSet procedure and go to Step 2.

Procedure 9 : (UpSet) Setting the upper bounds on variables Ya(t,s)
for all a c A, t c A do
Compute values of smax and smin
nmax max{sls E Fa, Ya(ts) / 0} ser Y(t's) 0
m 0 Otherwise
m min{s|s E FC Y(ts) / 0} Eer Y(t) / 0
S 0 Otherwise
if smax smiTn 0 then
relax all bounds on variables Ya(t,s), Vs E Fa
else if max Smin m < 1 and Y("'t"). > a then
2rer Ya(t,r)
for all Vs E F, do
if s > smax + 1 then set the upper bound of Ya(t,s) to zero. Otherwise, relax.
end for
for all Vs c Fa do
if s > smax 1 then set the upper bound of Ya(t,s) to zero. Otherwise, relax.
end for
end if
end for

In the DTDTA-R problem, at optimality variables Ya(t,s) may have positive

values for different indices s C F,. However, in the DTDTA-U problem only one

of them is allowed to be positive. Procedure UpSet restricts the flow on the arcs

of the DTDTA-R problem in order to avoid using a large variety of indices from

the set Fa and, at the same time, remain close to the optimal objective function

value. For example, consider an arc, a, and assume that Fa {2,3,..., 8}. If in

the TE network arcs a(t, 2) and a(t, 8) are used then the procedure sets the upper

bound on the variable Ya(t,s) to zero. As a result, in the next iteration the solution

is required to use indices 2, 3,..., 7 that are more compact in the sense that the

smallest and the largest indices are closer to each other. However, if the set of

used arcs consists of only two i,, _!lhor" indices and the arc with a larger index,

e.g., (, carries a significant portion of the total inflow, e.g., the flow on the arc

a(t,() is greater than a (Ya(t,(c-) + Ya(t,c)), where a c [0, 1], then one may consider

the settings too restrictive and allow a positive flow on the arc a(t, ( + 1). In the

numerical experiments, we found it more useful to take the values for parameter a

from the interval [0.2,0.5]. A similar procedure applies to the case when only one

arc is used. Finally, if the total inflow into the arc is zero, i.e., SEa Y4(t,s) = 0,

then we relax restrictions on the variables Ya(t,s), Vs E F,. Although the DTDTA-R

problem is non-convex and requires finding a global optimum, the UpSet procedure

potentially eliminates the current solution from further consideration by narrowing

the feasible region.

The above described heuristic algorithm may not converge and the performance

of the algorithm is discussed in Section 6.4. However, if the algorithm does not

converge, an alternative objective function (see Table 6-1) can be used to solve

the problem. To construct the alternative objective function, observe that the

cost vector q, consists of discrete travel times s E Fa. On the other hand, the

variable Xa(t) is computed based on the set of indices a,(t) (see equation (6-13)). In

particular, each arc a(t, s), s E Fa, is included into at least one of the sets ao(t), for

some t E A. Notice that the arc may be included into more than one set, and the

total number of such sets is equal to s/6. (Note that s/6 is an integer because s is

a multiple of 6.) As a result,

TY E E -6 a(s) [ Ycjx j 5 5 X 6JT,

aa aEA tEA rEQa() aEA tEA

where e = (1, 1,..., 1). Although the second objective function includes parameter

6, it is not necessarily decreasing with the value of 6. In fact, by decreasing 6,

the number of variables Xa(t,s) and the total number of sets Qa(t) that includes arc

a(t, s) increase, thereby increasing eTx.

Table 6-1. Equivalent objective functions

Objective Function 1 Objective Function 2
min qY min 6eTx
(x,y,z,g) (x,y,z,g)

Full Text


Iwouldliketothankmychairandcochair,Prof.SiriphongLawphongpanichandProf.DonaldW.Hearn,fortheirvaluableadvice,supportandguidanceduringmystudies.Ourmeetingsanddiscussionswerealwaysveryhelpful.AlsoIwouldliketoexpressmysinceregratitudetothecommitteemembersProf.PanosPardalos,Prof.WilliamHager,andProf.RavindraAhujafortheirencouragement.Especially,IamgratefultoProf.PanosPardalosforhisvaluablesuggestionsandadviceonthesupplychainproblemsIhaveworkedon.Thetremendoussupportfrommyparentsisinvaluable,andtherearenowordstoexpressmyappreciationforthat.Finally,Iwouldliketothankallmyfriendsandcollaboratorswhomademystudiesenjoyableandproductive. iv


page ACKNOWLEDGMENTS ............................. iv LISTOFTABLES ................................. viii LISTOFFIGURES ................................ x ABSTRACT .................................... xi CHAPTER 1INTRODUCTION .............................. 1 2ABILINEARREDUCTIONBASEDALGORITHMFORCONCAVEPIECEWISELINEARNETWORKFLOWPROBLEMS ......... 5 2.1IntroductiontotheChapter ...................... 5 2.2ABilinearReductionTechniquefortheConcavePiecewiseLinearNetworkFlowProblem ......................... 7 2.3ConcavePiecewiseLinearProblemswithSeparableObjectiveFunctions ................................ 11 2.4DynamicCostUpdatingProcedure .................. 13 2.5OntheDynamicSlopeScalingProcedure ............... 16 2.6NumericalExperiments ......................... 19 2.7ConcludingRemarks .......................... 21 3ADAPTIVEDYNAMICCOSTUPDATINGPROCEDUREFORSOLVINGFIXEDCHARGENETWORKFLOWPROBLEMS ........... 22 3.1IntroductiontotheChapter ...................... 22 3.2ApproximationoftheFixedChargeNetworkFlowProblembyaTwo-PieceLinearConcaveNetworkFlowProblem .......... 24 3.3AdaptiveDynamicCostUpdatingProcedure ............. 27 3.4OntheDynamicSlopeScalingProcedure ............... 30 3.5NumericalExperiments ......................... 31 3.6ConcludingRemarks .......................... 34 4ABILINEARREDUCTIONBASEDALGORITHMFORSOLVINGCAPACITATEDMULTI-ITEMDYNAMICPRICINGPROBLEMS ... 35 4.1IntroductiontotheChapter ...................... 35 4.2ProblemDescription .......................... 37 v


43 4.4NumericalExperiments ......................... 46 4.5ConcludingRemarks .......................... 49 5DISCRETE-TIMEDYNAMICTRAFFICASSIGNMENTMODELSWITHPERIODICPLANNINGHORIZON:SYSTEMOPTIMUM ... 50 5.1IntroductiontotheChapter ...................... 50 5.2PeriodicPlanningHorizon ....................... 53 5.3Discrete-TimeDynamicTracAssignmentProblemwithPeriodicTimeHorizon .............................. 54 5.4BoundsfortheDTDTAProblem ................... 67 5.5NumericalExperiments ......................... 71 5.6ConcludingRemarks .......................... 76 6ANONLINEARAPPROXIMATIONBASEDHEURISTICALGORITHMFORTHEUPPER-BOUNDPROBLEM .................. 78 6.1IntroductiontotheChapter ...................... 78 6.2NonlinearRelaxationofDTDTA-UProblem ............. 83 6.3NonlinearRelaxationBasedHeuristicAlgorithm ........... 86 6.4NumericalExperiments ......................... 89 6.5ConcludingRemarks .......................... 91 7ADYNAMICTOLLPRICINGFRAMEWORKFORDISCRETE-TIMEDYNAMICTRAFFICASSIGNMENTMODELS ............. 92 7.1IntroductiontotheChapter ...................... 92 7.2TheReducedTime-ExpandedNetworkandUESolution ...... 97 7.3TheDynamicTollSet ......................... 103 7.4DynamicTollPricingProblems .................... 108 7.5IllustrativeExamples .......................... 110 7.6ConcludingRemarks .......................... 113 8DIRECTIONSOFFUTURERESEARCH ................. 115 APPENDIX ACOMPUTATIONALRESULTSFORCHAPTER 2 ............ 117 BCOMPUTATIONALRESULTSFORCHAPTER 3 ............ 121 CCOMPUTATIONALRESULTSFORCHAPTER 4 ............ 125 DCOMPUTATIONALRESULTSFORCHAPTER 6 ............ 127 ECOMPUTATIONALRESULTSFORCHAPTER 7 ............ 129 vi


................................... 131 BIOGRAPHICALSKETCH ............................ 140 vii


Table page 5{1Demandpatterns ............................... 72 5{2Optimalsolutionstothetwo-arcproblem. ................. 73 5{3Solutionsfromthelowerandupper-boundproblems:lineartravelcostfunction. .................................... 75 5{4Solutionsfromthelowerandupper-boundproblems:quadratictravelcostfunction. ................................. 75 5{5Qualityofrenedupperandlower-boundsolutions:lineartravelcostfunction. .................................... 76 5{6Qualityofrenedupperandlower-boundsolutions:quadratictravelcostfunction. .................................... 76 6{1Equivalentobjectivefunctions ........................ 88 6{2Distributionsofparametersofrandomlygeneratedtraveltimefunctions 89 7{1Additionalconstraints ............................ 110 7{2Distributionsofparametersofrandomlygeneratedtraveltimefunctions 111 A{1Setofproblems. ................................ 117 A{2Computationalresultsofsets1-18:qualityofthesolutionandtheCPUtimes. ..................................... 118 A{3Computationalresultsofsets1-18:DSSPvs.DCUP. ........... 119 A{4Computationalresultsforsets19-30. .................... 120 A{5Computationalresultsforthecombinedmode. ............... 120 B{1Setofproblems. ................................ 121 B{2ComputationalresultsofgroupsG1andG2:qualityofthesolutionsandtheCPUtimes. ................................ 122 B{3ComputationalresultsofgroupsG1andG2:thepercentageofproblemwhereoneofthealgorithmsndsabettersolutionthananotherone. .. 123 viii


................ 124 C{1Thequalityofthesolution:Procedure 6 .................. 125 C{2Thequalityofthesolution:Procedure 5 .................. 126 C{3TheCPUtimeoftheprocedures. ...................... 126 D{1Computationalresultsoftheexperiments. ................. 127 D{2Computationalresultsofthecombinedmode. ............... 128 E{1Thetotalcollectedtollandthetotalcostforeachproblemandparameter". ........................................ 129 E{2Thenumberoftollcollectingcentersforeachproblemandparameter". 129 ix


Figure page 3{1Approximationoffunctionfa(xa). ...................... 25 3{2"aa(xa)and"kaa(xa)functions. ........................ 28 4{1Thepriceandtherevenuefunctions. .................... 38 5{1Linearversuscircularintervals. ....................... 53 5{2Eventsoccurringintwoconsecutiveplanninghorizons. .......... 54 5{3Three-nodenetwork. ............................. 55 5{4Timeexpansionofarc(1;2)att=1. .................... 56 5{5Time-expansionofthethree-nodenetwork. ................. 58 5{6a(xa(t))2(s;s]versusxa(t)2(1a(s);1a()]. .......... 68 5{7Two-arcnetwork. ............................... 72 5{8Four-nodenetwork. .............................. 74 6{1Twofeasiblesolutions. ............................ 80 7{14-Nodenetworkandtracdemand. ..................... 92 7{2Userequilibriumowsandtraveltimes. .................. 93 7{3Systemoptimumowsandtraveltimes. .................. 93 7{4Tolleduserequilibriumowsandtraveltimes. ............... 94 7{5Thevalueoft. ................................ 111 D{1TwoNetworks. ................................ 127 E{19-nodenetwork. ................................ 129 E{2Thetollvectorfordierentvaluesof"intheMinRev(")problem. ... 130 E{3Thetollvectorfordierentvaluesof"intheMinCost(")problem. ... 130 x


Inthisdissertationweinvestigatenetworkowproblemswithnonlineararccostfunctions.Therstgroupofproblemsconsistsofconcavepiecewiselinearnetworkow,xedchargenetworkow,anddynamicpricingproblemsthatariseintheareasofsupplychainmanagementandlogistics.BasedontheMIPformulation,weconstructbilinearreductionproblems,inwhichtheglobalsolutionofthelatterisasolutionoftheinitialformulation.Tosolvethereductionproblem,weproposesomeheuristicalgorithms.Intheexperiments,wecomparethesolutionprovidedbyouralgorithmwithanexactsolutionaswellasasolutionprovidedbyotherheuristicalgorithmsintheliterature.Numericalexperimentsonrandomlygeneratedinstancesconrmthequalityofthealgorithms. Thesecondgroupofproblemsisrelatedtothedynamictracassignmentproblem.Inparticular,weconsideraperiodicdiscretetimedynamictracassignmentproblem(DTDTA),inwhichthetraveltimeisafunctionofthenumberofcarsontheroad,andtheplanninghorizoniscircular.Themathematicalformulationbelongstotheclassofnonlinearmixedintegerproblems.Toobtainan xi




Networkowproblemsareminimization/maximizationproblemswithunderlyingnetworkstructure.Althoughtherearedierentrepresentationsofthenetwork,perhapsthemostpopularoneisbasedonowconservationconstraintsviaanode-arcincidencematrix.Apartfromtheowconservationconstraints,manyproblemshaveadditionalrestrictionsonthevariables,e.g.,non-negativityandlower/upperboundaryconstraints.Basedontheobjectivefunctionandotheradditionalconstraintstheproblemscanbeclassiedaslinearornonlinear,wherethelattercanbefurtherdecomposedintoconvex,concave,orotherproblems. Thelinearproblemsassumethattheconstraintsaswellastheobjectivefunctionarelinear.Polynomialtimealgorithmsforsolvingtheproblemsarewellknown.Someclassicalexamplesofthenetworkowproblemswithlinearconstraintsincludeshortestpath,minimumspanningtree,minimumcut,maximumow,minimumcostnetworkow,andotherproblems.Basedontheoptimalityconditionsandotherpropertiesoftheproblem,severalalgorithmshavebeenproposedtosolvetheproblems.Fordetailsonthelinearnetworkowproblems,seeAhujaetal.[ 2 ]. Despitethenicetheoreticalresultsdevelopedforthelinearproblems,mostofthepracticalproblemsarenotlinear,i.e.;theobjectivefunctionand/orsomeoftheconstraintsarenonlinear.Ifitisaconvexminimization(concavemaximization)problemwithaconvexfeasibleregionthenanylocalminimum(maximum)isaglobalsolutionoftheproblem,andappropriatealgorithmssuchthattheFrank-Wolfalgorithm,gradientbasedanddirectionndingmethods,canbeusedtosolvetheproblem.Alargevarietyofalgorithmsforsolvingconvexminimization 1


(concavemaximization)problemscanbefoundinBazaraaetal.[ 5 ].Whentheobjectivefunctionisnotconvex(concave)and/orthefeasibleregionisnotconvex,thesealgorithmsdonotnecessarilyconvergetoaglobaloptimalsolution.Findingaglobalsolutionisahardtask,andglobaloptimizationtechniquesarerequiredtosolvetheproblem(see,e.g.,Horstetal.[ 54 ]andHorstandTuy[ 55 ]). Inthisdissertation,weconsidertwogroupsofnon-convexnetworkowproblems.Therstgroupincludespiecewiselinearnetworkow,xedchargenetworkow,anddynamicpricingproblems(seeChapters 2 3 ,and 4 )thathavealargevarietyofapplicationsintheproductionplanning,scheduling,investmentdecision,networkdesign,locationofplantsanddistributioncenters,pricingpolicy,andmanyotherpracticalproblemsthatariseinsupplychains,logistics,transportationscience,andcomputernetworks.ItiswellknownthattheproblemsintheirgeneralformareNP-Hard;therefore,therearenopolynomialtimealgorithmstosolvetheproblemsunlessP=NP.Althoughthemathematicalformulationbelongstotheclassoflinearmixedintegerproblems,solvinglargeproblemsrequiresalargeamountofCPUtimeandmemory.Ontheotherhand,onecanconsiderapproximationtechniquesthatareabletoprovideagood-qualitysolutionusinglesscomputerresources.Manyofthesetechniquesemployalinearrelaxationoftheproblem.Unlikethoseintheliterature,weproposenonlinearreductiontechniquestosolvetheproblems.Inparticular,inallthreeproblemswedevelopamethodtoreducetheproblemtoabilinearoneandproposeaheuristicalgorithmtosolvetheresultingproblem.Inthenumericalexperimentswecomparetheresultswithanexactsolution(orthebestfeasiblesolution)providedbyMIPsolvers,aswellaswithDynamicSlopeScalingProcedure(DSSP),sinceitisknowntobeoneofthebestheuristicalgorithmstosolvesuchproblems.Numericalexperimentsonrandomlygeneratedproblemsconrmthequalityofthesolutionsprovidedbyouralgorithms.Inparticular,itoutperformstheDSSPin


thequalityofthesolutionaswellasinthecomputationaltime.Inaddition,wetransformtheproblemsintoalternativecontinuousnetworkowproblemswithowdependentcostfunctionandprovethataglobalsolutionoftheresultingproblemisasolutionoftheinitialMIPformulation.Despiteanunusualstructureofthecostfunction,themathematicalformulationoftheproblemsissimilartothesystemoptimumproblemsarisinginthetracassignmentmodelling.Usingthesamecostfunction,wealsoconstructavariationalinequalityproblemsimilartothoseinthetransportationliteratureandprovethattheDSSPconvergestoasolutionoftheresultingproblem;i.e.,itprovidesanequilibriumsolution.However,theproblemrequiresndingasystemoptimumsolution,andthealgorithmsweproposendsanapproximatesolutiontotheproblem. Thesecondgroupofproblemsisrelatedtothedynamictracassignmentproblem.Unlikethestaticcase,wherethetraveltimeisafunctionofthearcow,thedynamicmodelsinvolvethreevariables:inowrate,outowrate,anddensity,andthetraveltimecanbeafunctionofallthreevariables.Intheliteratureseveralcontinuousanddiscretetimemodelshavebeenproposedfordierenttraveltimefunctions.Themodelinthisdissertationassumesthatthetraveltimeisafunctionofthedensity,andallcarsthatenteranarcatthesamepointoftimeexperiencethesametracconditions;therefore,theyleavethearcatthesametime.Inaddition,themodelsintheliteratureassumethatthenetworkisemptyatthebeginningandtheendofaplanninghorizon.Inthecasewhensomecarsarepresentinthenetwork,thetimetoenterthenetworkforthosecarsisunknown,anditishardtomodelthepropagationofthecarsinthenetwork.Unlikeothermodelsintheliterature,weconsideraperiodicplanninghorizonandassumethattheprocessesrepeatthemselvesfromoneperiodtoanother(seeChapter 5 ).Themathematicalformulationoftheproblemminimizesthetotaldelayandbelongstotheclassofnonlinearmixedintegerproblems,ahardproblemtosolve.


Bylinearizingtheobjectivefunctionandtheconstraints,weconstructlinearmixedintegerproblemsthatprovideupperandlowerbounds.Thesolutionoftheboundingproblemscanbemadearbitrarilyclosetoasolutionoftheinitialformulationbydecreasingthediscretizationparameter.However,theboundingproblemsinvolvebinaryvariables,anditishardtosolvelargeproblemsusingMIPsolvers.InChapter 6 wediscussaheuristicalgorithmbasedonanonlinearrelaxationoftheproblem.Inparticular,weconstructacontinuousbilinearproblem,whichprovidesatighterlowerboundthantheLPrelaxation.Usingthebilinearrelaxation,theheuristicalgorithmaimstondanintegersolution,whichhasanobjectivefunctionvalueclosetotheoneprovidedbytherelaxationproblem. Anotherproblemofinterestisthetollpricingframeworkforthedynamictracassignmentproblem(seeChapter 7 ).Similartothestaticcase,weconstructasetofvalidtollvectorssuchthatasystemoptimumsolutionisasolutionofthetolleduserequilibriumproblem.Thelatterisauserequilibriumproblemwherethearccostfunctionsincludetollsinadditiontothetraveltimes.Akeycomponentinthedevelopmentofsuchtechniqueisthereducedtime-expanded(RTE)networkconstructedbasedonafeasiblevector.Usingthenetwork,weshowthatafeasiblevectorisauserequilibriumsolutionifandonlyifitisasolutionofalinearproblemwithanunderlyingRTEnetworkstructure.Thelatterallowstheconstructionofasetofvalidtollsandformulationofatollpricingproblemwithasecondaryobjective,andweprovideseveralexamplesofsuchproblems.


45 ]andGeunesandPardalos[ 41 ]).Weconsidertheconcavepiecewiselinearnetworkowproblem(CPLNF),whichhasdiverseapplicationsinsupplychainmanagement,logistics,transportationscience,andtelecommunicationnetworks.Inaddition,theCPLNFproblemcanbeusedtondanapproximatesolutionfornetworkowproblemswithacontinuousconcavecostfunction.ItiswellknownthattheseproblemsareNP-hard(seeGuisewiteandPardalos[ 45 ]). ThischapterdealswithanonlinearreductiontechniqueforthelinearmixedintegerformulationoftheCPLNFproblem.Inparticular,theproblemisreducedtoacontinuousonewithlinearconstraintsandabilinearobjectivefunction.ThereductionhasaneconomicalinterpretationanditssolutionisproventobethesolutionoftheCPLNFproblem.Basedonthereduction,weproposeanalgorithmforndingalocalminimumoftheproblem,whichwerefertoasthedynamiccostupdatingprocedure(DCUP).Inthechapter,weshowthatDCUPconvergesinanitenumberofiterations. Thetheoreticalresultspresentedinthischaptercanbeextendedtoamoregeneralconcaveminimizationproblemwithaseparablepiecewiselinearobjectivefunctionandlinear/nonlinearconstraints.ItshouldbeemphasizedthatHorst 5


etal.[ 54 ](seealsoHorstandTuy[ 55 ])discussabilinearprogramwithdisjointfeasibleregionsandprovethattheproblemisequivalenttoasubclassofpiecewiselinearconcaveminimizationproblems.Theresultsinthischaptershowthatanyconcaveminimizationproblemwithaseparableconcavepiecewiselinearobjectivefunctionisequivalenttoabilinearprogram.Itiswellknownthatanoptimalsolutionofageneraljointlyconstrainedbilinearprogrambelongstotheboundaryofthefeasibleregionandisnotnecessarilyavertex(seeHorstetal.[ 54 ]).However,thereductiontechniquepresentedinthischapterhasajointlyconstrainedfeasibleregionwithaspecialstructureanditisstillequivalenttoaconcavepiecewiselinearprogram.Fromthelatteritfollowsthattwopartsofasolutionoftheproblemareverticesoftwodierentpolytopesthatare\joined"byasetofconstraints.Inthatsense,thesetypesofproblemsareweaklyjoinedbilinearprograms. TheCPLNFproblemcanbetransformedintoanequivalentnetworkowproblemwithowdependentcostsfunction(NFPwFDCF).UsingNFPwFDCF,itcanbeshownthatthedynamicslopescalingprocedure(DSSP)(seeKimandPardalos[ 61 ]and[ 62 ])convergestoanequilibriumsolutionofNFPwFDCF.AlthoughDSSPprovidesasolution,whichcanbequiteclosetothesystemsolution,itiswellknownthattheequilibriumandthesystemsolutionsingeneralarenotthesame.Ontheotherhand,DCUPconvergestoalocalminimumoftheproblem.Inthenumericalexperiments,wesolvedierentproblemsusingDCUPandDSSPandcomparethequalityofthesolutionaswellastherunningtime.ComputationalresultsshowthatDCUPoftenprovidesabettersolutionthanDSSPandusesfeweriterationsandlessCPUtime.SinceDCUPstartsfromafeasiblevectorandconvergestoalocalminimum,oneconsidersrstsolvingDSSPandthenimprovingthesolutionusingDCUP.Thenumericalexperimentsusingthiscombinedmodeareprovidedaswell.


Fortheremainder,Section 2.2 discussesthenonlinearreductiontechniquefortheCPLNFproblem.Section 2.3 generalizestheresultsfromSection 2.2 foraconcavepiecewiselinearproblemwithaseparableobjectivefunction.Section 2.4 describesDCUPandtheoreticalresultsontheconvergenceandthesolutionoftheprocedure.InSection 2.5 weprovethatthesolutionoftheDSSPisanequilibriumsolutionofanetworkowproblemwithowdependentcostfunctions.TheresultsofnumericalexperimentsonDCUPandDSSPareprovidedinSection 2.6 ,andnally,Section 2.7 concludesthechapter. s.t.Bx=b(2{1) whereBisthenode-arcincidentmatrixofthenetworkG,andfa(xa)arepiecewiselinearconcavefunctions,i.e.,fa(xa)=8>>>><>>>>:c1axa+s1a(=f1a(xa))xa2[0a;1a)cnaaxa+snaa(=fnaa(xa))xa2[na1a;naa];


Usingbinaryvariables,yka,k2Ka,onecanformulatetheCPLNFproblemasthefollowinglinearmixedintegerprogram(CPLNF-IP).minxXa2AXk2Kackaxka+Xa2AXk2Kaskayka whereMisasucientlylargenumber. Intheaboveformulation,equality( 2{6 )makessurethat8a2A,thereisonlyone2Kasuchthatya=1andyka=0,8k2Ka,k6=.Thecorrectchoiceofdependsonthevalueofxaandhastosatisfyconstraint( 2{5 ).Inparticular,ifxa2[1a;a]thenfromconstraints( 2{5 )and( 2{6 ),itfollowsthatya=1.Asfortherestoftheconstraints,inequality( 2{7 )ensuresthatxka=0ifyka=0,andequalities( 2{3 )and( 2{4 )makesurethatthedemandissatisedandthesumofxkaoverallindicesk2Kaisequaltotheowonarca.Inaddition,itiseasytoshowthattheobjectiveoftheproblemisequivalenttotheobjectiveofCPLNFandoneconcludesthattheCPLNFandtheCPLNF-IPproblemsareequivalent. ConsiderarelaxationoftheCPLNF-IPproblemwhereconstraint( 2{7 )andtheintegralityofykaarereplacedby


andyka0,respectively.Observethatintheresultingproblemconstraint( 2{4 )isredundantandfollowsfrom( 2{6 )and( 2{9 );therefore,itcanberemovedfromtheformulation.Inaddition,noticethatonecanremovethevariablexkafromtheformulationaswellbysubstituting( 2{9 )intotheobjectivefunction.ThemathematicaldescriptionoftheresultingproblemisprovidedbelowandwerefertotherelaxationproblemasCPLNF-R.minx;yg(x;y)=Xa2A"Xk2Kackayka#xa+Xa2AXk2Kaskayka=Xa2AXk2Kafka(xa)yka 2{10 )-( 2{13 )arepresentintheCPLNF-IPproblem.Therefore,anyfeasiblevectoroftheCPLNF-IPproblemsatisesconstraints( 2{10 )-( 2{13 ).


thefollowingformminfykajk2Kag"Xk2Kackayka#xa+Xk2Kaskayka=Xk2Ka[ckaxa+ska]yka Letxa2[k1a;ka].Aswehavementionedbefore,fa(xa)=mink2Kaffka(xa)g;thereforefa(xa)=mink2Kaffka(xa)g=mink2Kafckaxa+skag=ckaxa+ska: 2{14 )becausexa2[k1a;ka]and(ii)^ya=argminfPk2Ka[ckaxa+ska]ykajPk2Kayka=1;yka0g.Basedontheabove,oneconcludesthat^yaisanoptimalsolutionoftheproblem.Ifxa2(k1a;ka)then^yistheuniquesolutionoftheproblembecauseckaxa+ska>ckaxa+ska,8k2Ka,k6=k;therefore,y=^y.Ifxa=k1aorxa=ka,thereareexactlytwobinarysolutionsoftheproblem,andbothhavethesameobjectivefunctionvalue.Asaresult,eitheronecanbeusedtoconstructabinarysolution^y.Asimilarresultholdsforallarcsa2A.Regardingvariablexka,given(x;y),theonlyfeasibleoneisxka=xaandxka=0,8k2Ka,k6=k. 2.2.2 ,itfollowsthataglobaloptimumofCPLNF-RiseitherfeasibletoCPLNF-IPorcanbeusedtoconstructafeasiblesolutionwiththesameobjectivefunctionvalue.SinceallfeasiblevectorsofCPLNF-IParefeasibleto


CPLNF-R(seeLemma 2.2.1 ),oneconcludesthataglobalsolutionofCPLNF-RleadstoasolutionofCPLNF-IP. 2.2.1 ,itfollowsthatsolvingtheCPLNF-IPproblemisequivalenttondingaglobaloptimumofthebilinearproblemCPLNF-R.IfthesolutionofCPLNF-RisnotfeasibletoCPLNF-IPthentheproofofLemma 2.2.2 providesaneasywaytoconstructafeasiblesolutionwiththesameobjectivefunctionvalue.OtherpropertiesoflocalminimaoftheCPLNF-RproblemarediscussedinSection 2.4 ItisnoticedthattheCPLNF-Rproblemhasthefollowingeconomicinterpretation.Observethatbecauseofequality( 2{12 ),yka2[0;1],andonecaninterpretthevariablesykaasweights.Underthisassumption,onecanviewtheobjectivefunctionasthesumoftheweightedaveragesofthevariablecostsmultipliedbytheow,Pk2Kackaykaxa,andthexedcosts,Pk2Kaskayka.Inotherwords,theobjectivefunctionconsistsoftheweightedaveragesoffunctionsfka(xa).However,theweightshavetosatisfyconstraint( 2{11 ),wheretheow,xa,isboundedbytheweightedaveragesoftheleftandtherightendsoftheintervals[k1a;ka],k2Ka.AccordingtoLemma 2.2.2 ,alocal(global)optimumleadstoasolutionwheretheweightsareeitherequal0or1. 2{1 )isreplacedbyarequirementx2XRn. wherethefi(xi)arepiecewiselinearconcavefunctions,i.e.,


2{10 )byx2X,i.e., 2.2.1 and 2.2.2 ,andTheorem 2.2.1 ,arestillvalid.Asaresult,oneconcludesthattheCPLPwSOFandtheCPLPwSOF-RproblemsareequivalentinthesensethatasolutionoftheCPLPwSOFproblemcanbeeasilyconstructedfromaglobalsolutionoftheCPLPwSOF-Rproblem. IfthesetXisapolytopethenCPLPwSOF-Risabilinearprogramwithajointlyconstrainedlinearfeasibleregion.LetY=fyjPk2Kiyki=1;yki0gandX+=fxjx2X;xi2[0i;nii]g.DenotebyV(X+)andV(Y)thesetsofverticesofthepolytopesX+andY,respectively.NoticethatthesetsX+andYare\joined"bytheconstraintsPk2Kik1iykixiPk2Kikiyki.Itiswellknownthatanoptimalsolutionofageneralbilinearprogramwithjointly


constrainedfeasibleregionoccursattheboundaryofthefeasibleregionandisnotnecessarilyavertex(seeSection3.2.2,Horstetal.[ 54 ]andtherelatedproblemset).However,CPLPwSOF-RisequivalenttoCPLPwSOF.Inparticular,if(x;y)isaglobalsolutionofCPLPwSOF-RthenfromTheorem 2.2.1 ,itfollowsthatxisasolutionofCPLPwSOF.Thelatterisaconcaveminimizationproblemwherethefeasibleregionisapolytope.Itiswellknownthatthesolutionofsuchaconcaveminimizationproblemisoneoftheverticesofthepolytope;thereforex2V(X+).Inaddition,fromthetheoremitfollowsthatthereexists^y2V(Y)suchthat(x;^y)isaglobalsolutionofCPLPwSOF-Rproblem.Inthatsense,CPLPwSOF-Risaweaklyjoinedbilinearprogram.Theabovediscussionissummarizedinthefollowingtwotheorems. ConsiderthefollowingtwolinearproblemswhichwerefertoasLP(y)andLP(x),wherey(x)denotetheparameteroftheproblemLP(y)(LP(x)),i.e.,xedtoaparticularvalue.


s.t.Bx=bxa2[0;naa] IntheLP(y)problemweassumethatvariablesykaaregiven,andthexaaretheonlydecisionvariables.Similarly,intheLP(x)problem,thexaaregiven,andtheykaarethedecisionvariables.AswehaveshownintheproofofLemma 2.2.2 ,problemLP(x)canbedecomposedintojAjproblemsandthesolutionsofthedecomposedproblemsarebinaryvectors,whichsatisfyconstraint( 2{14 ).Therefore,givenvectorx,asolutionoftheLP(x)problemcanbefoundbyasimplesearchtechniquewhereyka=1ifxa2[k1a;ka]. Weproposeadynamiccostupdatingprocedure(DCUP),whereoneconsiderssolvingtheproblemsLP(x)andLP(y)iteratively,usingthesolutionofoneproblemasaparameterfortheother(seeProcedure 1 ).Althoughintheproceduretheinitialvectory0,issuchthaty10a=1andyk0a=0,8k2Ka,k6=1,onecanchooseanyotherbinaryvector,thatsatisesconstraint( 2{12 ).Itisnoticedthatasimilariterativeprocedurehasbeenusedforsolvingabilinearprogramwithadisjointfeasibleregion(see,e.g.,Horstetal.[ 54 ]andHorstandTuy[ 55 ]).In


theDCUP,LP(y)doesnotincludeconstraint( 2{11 ).Inotherwords,L(x)istheCPLNF-Rproblemwithxedvariableyandrelaxedconstraint( 2{11 ).Thelatterallowsusingtheiterativeproceduretosolvebilinearprogramswithaweaklyjoinedfeasibleregion.LetVrepresentthefeasibleregionofCPLNF-Rand(x;y)bethesolutionoftheDCUP. 54 ]. 2.2.2 wehaveshownthatyisnotuniqueifandonlyifoneofthecomponentsofvectorx,xa,isequaltothevalueofoneofthebreakpointska.However,observethatx=argminfLP(y)g,andthefeasibleregionofproblemLP(y)doesnotinvolvebreakpoints.Asaresult,inpracticeitisunlikelythatxaisequaltooneofthebreakpoints. 2{12 ),DCUPconvergesinanitenumberofiterations.


Observethatineachiterationtheprocedurechangesthebinaryvectory.Ifym=ym1thentheprocedurestopsandg(xm;ym1)=g(xm;ym)=g(xm+1;ym).Ifthereexistm1andm2suchthatm11>m2andym1=ym2,thenxm1+1=argminfLP(ym1)g=argminfLP(ym2)g=xm2+1,i.e.,g(xm2+1;ym2)=g(xm2+1;ym1)=g(xm1+1;ym1).Fromthenon-increasingpropertyofthesequenceitfollowsthatg(xm2+1;ym2)=g(xm2+1;ym2+1);therefore,ym2=ym2+1andthealgorithmmuststoponiterationm2.Fromthelatteritfollowsthatallvectorsym,constructedbytheprocedurebeforeitstops,aredierent.Sincethesetofbinaryvectorsyisnite,oneconcludesthattheprocedureconvergesinanitenumberofiterations. 61 ]and[ 62 ]). AlthoughintheCPLNFproblemtherearenorestrictionsonthevaluesofparameterss1aand0a,bysubtractings1afromfunctionfa(xa)andreplacingthevariablexaby^xa=xa1a,onecantransformtheproblemintoanequivalentone


wheres1a=0and0a=0.Therefore,withoutlossofgenerality,weassumethats1a=0and0a=0. ToinvestigateDSSP,letFa(xa)=8><>:fa(xa) whereMisasucientlylargenumber.ConsiderthefollowingnetworkowproblemwithowdependentcostfunctionsFa(xa)(NFPwFDCF).minxFT(x)x whereF(x)isthevectoroffunctionsFa(xa). 2 ).Thenititerativelyupdates



Thesetofproblemsisdividedintovegroupsthatcorrespondtothenetworkswithdierentsizesandnumbersofsupply/demandnodes.Foreachgroupwerandomlygeneratethreetypesofdemand;U[10;20],U[20;30],orU[30;40],andconsider5or10linearpieces(seeTable A{1 ,Appendix A ).InKimandPardalos[ 62 ]theauthorsconsiderincreasingconcavepiecewiselinearcostfunctionsforexperiments.AlthoughthebilinearreductiontechniqueaswellasDCUParevalidforanyconcavepiecewiselinearfunction,toremainimpartialforcomparisonwegeneratesimilarincreasingcostfunctions.Doingso,rstforeacharcwerandomlygenerateaconcavequadraticfunctionoftheformg(x)=x2+x.Noticethatthemaximumofthefunctionisreachedatthepointx= A Sets1-18havearelativelysmallnetworksize,anditispossibletosolvethemexactlyusingCPLEX(seeTable A{2 ).TherelativeerrorsforthosesetsarecomputedusingthefollowingformulasREDCUP(%)=fDCUPfexact


A{3 ,columnsB,C,andDdescribethepercentageofproblemswhereDCUPisbetterthanDSSP,DSSPisbetterthanDCUP,andtheyarethesame,respectively.ThenumbersincolumnAaretheaverages(maximumvalues)ofthenumbersREDSSPREDCUP,givenREDSSPREDCUP>0.AccordingtothenumericalexperimentsDCUPprovidesabettersolutionthanDSSPinabout41%oftheproblemsandthesamesolutionin36%ofproblems.AlsonoticethatDCUPrequiresfeweriterationstoconvergeandconsumeslessCPUtime.RegardingCPLEX,thecomputationaltimevariesfromseveralsecondsinthesets1-6toseveralthousandsofsecondsinthesets13-18. Inthecaseoftheproblemsets19-30,CPLEXisnotabletondanexactsolutionwithin10,000secondsofCPUtime,andthebestfoundsolutionisnotbetterthantheoneprovidedbytheheuristics;therefore,wecomparetheresultsofDCUPversusDSSP.InTable A{4 ,columnsBandDdescribethepercentageofproblemswhereDCUPisbetterthanDSSP,andDSSPisbetterthanDCUP,respectively.ThenumbersincolumnsAandCarecomputedbasedontheformulafDSSPfDCUP Intheabovenumericalexperiments,wehaveusedthevectory0(8a2A,y10a=1andyk0a=0,8k2Ka,k6=1)asaninitialbinaryvector.However,DCUPcanstartfromanyotherbinaryvectorthatsatisesconstraint( 2{12 ).Inparticular,onecanconsidersthesolutionofDSSPasaninitialvectoranduseDCUPtoimprovethesolution.Table A{5 comparestheresultsofDCUPversus


DSSPwherecolumnAissimilartotheoneinTable A{4 (i.e.thenumbersinthecolumnarecomputedbasedontheformulafDSSPfDCUP Basedonthetheoreticalresults,wehavedevelopedaniteconvergentalgorithmtondalocalminimumofthebilinearrelaxation.ThecomputationalresultsshowthatthedynamiccostupdatingprocedureisabletondanearoptimumoranexactsolutionoftheproblemusinglessofCPUtimethanCPLEX.Inaddition,wecomparethequalityofthesolutionandtherunningtimewiththedynamicslopescalingprocedure.SinceDCUPisfast,onecanaimtondtheglobalminimumbyrandomlygeneratingtheinitialbinaryvectorandrunningDCUP.Inaddition,DCUPcanbeusedincuttingplanealgorithmsforndinganexactsolution.


41 ]). TheFCNFproblemiswellknowntobeNP-Hardandbelongstotheclassofconcaveminimizationproblems.Theproblemcanbemodeledasa0-1mixedintegerlinearprogram(seeHirschandDantzig[ 50 ])andmostsolutionapproachesutilizebranch-and-boundtechniquestondanexactsolution(seeBarretal.[ 4 ],CabotandErenguc[ 12 ],Gray[ 44 ],KenningtonandUnger[ 59 ],andPalekaretal.[ 83 ]).Sincetheconcaveminimizationproblemattainsasolutionatoneoftheverticesofthefeasibleregion,Murty[ 76 ]proposedavertexrankingproceduretosolvetheproblem.However,ndinganexactsolutioniscomputationallyexpensiveanditisnotpracticalforsolvinglargeproblems.SomeheuristicproceduresarediscussedinCooperandDrebes[ 27 ],Diaby[ 31 ],KhangandFujiwara[ 60 ],andKuhnandBaumol[ 63 ].RecentlyKimandPardalos[ 61 ](seealsoKimandPardalos[ 62 ])proposedaheuristicalgorithm,DynamicSlopeScalingProcedure(DSSP),tosolvethexedchargenetworkowproblem.Theproceduresolvesasequenceoflinearproblems,wheretheslopeofthecostfunctionisupdatedbased 22


onthesolutionofthepreviousiteration.ThealgorithmisknowntobeoneofthebestheuristicprocedurestosoleFCNFproblems. Notethatallapproachestosolvetheproblemarebasedonlinearapproximationtechniques.Instead,weapproximateFCNFbyaconcavepiecewiselinearnetworkowproblem(CPLNF),wherethecostfunctionshavetwolinearpieces.AproperchoiceoftheapproximationparameterensurestheequivalencebetweenFCNFandtheresultingCPLNFproblem.However,ndingtheproperparameteriscomputationallyexpensive;therefore,weproposeanalgorithmthatsolvesasequenceofCPLNFproblemsbygraduallydecreasingtheparameteroftheproblem.WeprovethatthestoppingcriteriaofthealgorithmisconsistentinthesensethatasolutionofthelastCPLNFprobleminthesequenceisasolutionoftheFCNFproblem. Despitetheabovementionedtheoreticalresults,thealgorithmrequiresndingexactsolutionsoftheCPLNFproblems,whichareNP-Hard(seeGuisewiteandPardalos[ 45 ]).InChapter 2 (seealsoNahapetyanandPardalos[ 79 ]),wehaveshownthattheCPLNFproblemisequivalenttoabilinearprogram.Inaddition,wehaveproposedaniteconvergentdynamiccostupdatingprocedure(DCUP)tondalocalminimumoftheresultingbilinearprogram.TosolvetheFCNFproblem,inthealgorithmonetransformstheCPLNFproblemsintoequivalentbilinearprogramsandusestheDCUPtosolvetheresultingproblems.Werefertothecombinedalgorithmastheadaptivedynamiccostupdatingprocedure(ADCUP). SimilartotheresultpresentedintheChapter 2 ,weprovethatthesolutionprovidedbyDSSPisasolutiontoavariationalinequalityproblem,whichisformulatedbasedonthefeasibleregionoftheFCNFproblem.Althoughingeneralanequilibriumsolutionandasystemsolutionarenotthesame,thedierencebetweentheobjectivefunctionvaluesofthesolutionscanbefairlysmall.On


theotherhand,ADCUPisaheuristicprocedureforndingasystemoptimumsolution.Tocomparethesetwoprocedures,weconductnumericalexperimentson36problemssetsfordierentnetworksandchoicesofcostfunctions.Thereare30randomlygeneratedproblemsforeachproblemset.Intheexperiments,wecompareADCUPversusDSSPintermsofthequalityofthesolutionaswellasCPUtime.Inaddition,forsmallnetworkswendanexactsolutionoftheproblemsusingMIPsolversofCPLEXandcomputerelativeerrors.ThecomputationalresultsshowthatADCUPprovidesanearoptimumsolutionusinganegligibleamountofCPUtime.Inaddition,theprocedureoutperformsDSSPinthequalityofthesolutionaswellasCPUtime.Thedierencebetweensolutionsismorenoticeableinthecasesofsmallgeneralslopesandlargexedcosts. Fortheremainder,Section 3.2 discussestheapproximationtechniqueandestablishestheequivalencebetweentheFCNFproblemandaCPLNFproblemwithaspecialstructure.AsolutionalgorithmforsolvingtheFCNFproblemisprovidedinSection 3.3 .SomepropertiesoftheDSSPareintroducedinSection 3.4 .TheresultsofnumericalexperimentsonADCUParesummarizedinSection 3.5 ,andnally,Section 3.6 concludesthechapter. ConsiderageneralxedchargenetworkowproblemconstructedonanetworkG(N;A),whereNandAdenotethesetsofnodesandarcs,respectively.Letfa(xa)denotethecostfunctionofarca2A,and


Figure3{1. Approximationoffunctionfa(xa). FCNF:minxf(x)=Xa2Afa(xa)s.t.Bx=b;xa2[0;a];8a2A; Observethatthecostfunctionisdiscontinuousattheoriginandlinearontheinterval(0;a].Althoughweassumethattheowsonthearcsareboundedbya,theboundscanbereplacedbyasucientlylargeM,andtheproblemtransformsintoanunboundedone. Let"a2(0;a],and"aa(xa)=8><>:caxa+saxa2["a;a]c"aaxaxa2[0;"a)



3.2.2 makessurethatbychoosingasucientlysmall">0(e.g.,"a=,8a2A),bothproblemshavethesamesolution;therefore,FCNFisequivalenttoaconcavepiecewiselinearnetworkowproblem. ConsiderProcedure 3 .InStep1,theprocedureassignsinitialvaluesfor"a.Step2solvestheresultingCPLNFproblem.NoticethatCPLNF("1)isindeedalinearproblem,because["1a;a]=fag.If9a2Asuchthatxma2(0;"ma),wedecreasethevalueof"ato"a,whereisaconstantfromtheopeninterval(0;1).Assumethattheprocedurestopsatiterationkandletxk=argminfCPLNF("k)g.


Figure3{2. thedenitionof"-approximationitfollowsthat"aa(xa)"kaa(xa),8a2Aandxa2[0;a](seeFigure 3{2 ),andthesecondinequalityfollows. Observethatbecauseofthestoppingcriteria,xka=0orxka2["ka;a].Since"a<"ka,"aa(xa)="kaa(xa),8a2Aandxa2f0gS["ka;a];therefore,"(xk)="k(xk).Thelattertogetherwith( 3{1 )insuresthat"(x")="k(xk).Sincebothproblems,CPLNF(")andCPLNF("k),havethesameobjectivefunctionvalueatx"andxk,oneconcludesthattheyareequivalent. 3.3.1 itfollowsthat8"suchthat0<"a<"ka,8a2A,theCPLNF(")andtheCPLNF("k)problemshavethesamesetofsolutions.Ontheotherhand,bychoosing0<"a

minimumoftheproblemandcanbeusedinStep2ofProcedure 3 tondasolutionoftheCPLNF("m)problem.TheresultingalgorithmissummarizedinProcedure 4 ,whichwerefertoasadaptivedynamiccostupdatingprocedure(ADCUP).Belowweprovidethemathematicalformulationofthebilinearproblem,whichisequivalenttoCPLNF("m).Fordetailsontheformulationoftheproblem,niteconvergenceandotherpropertiesoftheDCUPwerefertoChapter 2 ProblemCPLNF-R("m)isdenedby:minx;yXa2Ac"maay"maa+cayaxa+Xa2Asayas.t.Bx=b;"mayaxa"may"maa+aya;8a2A;y"maa+ya=1;8a2A;xa0;y"maa0;andya0;8a2A: 3.5 ).


61 ](seealsoKimandPardalos[ 62 ]).Inthepaper,theauthorsdiscussfourvariationsofDSSPbasedonthechoiceoftheinitialvectorandtheslopeupdatingscheme.However,regardlessoftheinitialvectorandtheupdatingscheme,DSSPprovidesanequilibriumtypeofsolution.Toprovethestatement,wersttransformFCNFintoanalternativeproblemthenprovethatthesolutionprovidedbyDSSPisasolutionofavariationalinequalityproblemconstructedbasedonthenewformulation.ThetheoreticalresultsprovidedbelowareverysimilartothoseinChapter 2 ,wherewehaveshownthatthepropertyholdsfortheconcavepiecewiselinearnetworkowproblem. LetFa(xa)=8><>:fa(xa) whereMisasucientlylargenumber.ConsiderthefollowingnetworkowproblemwithowdependentcostfunctionsFa(xa). NFPwFDCF:minxFT(x)x whereF(x)isthevectoroffunctionsFa(xa). 3.4.1 ,Chapter 2 )


61 ],theauthorsproposedierentupdatingschemes,wheretheyreplaceMbyasmallervalue.However,thenexttheoremprovesthatregardlessoftheinitialvectorF0andtheupdatingscheme,thenalsolutionprovidedbyDSSPisasolutionofavariationalinequalityproblem. ndxfeasibleto( 3{2 )and( 3{3 )suchthatFT(x)(xx)0,8xfeasibleto( 3{2 )and( 3{3 ) 3.4.2 itfollowsthatthesolutionofDSSP,x,satisestheinequalityFT(x)xFT(x)x,8xfeasibleto( 3{2 )and( 3{3 ),i.e.,xisanequilibriumsolution.However,sinceNFPwFDCFisequivalenttotheFCNF(seeTheorem 3.4.1 ),oneisinterestedinndingafeasible^xsuchthatFT(x)xFT(^x)^x,8xfeasibleto( 3{2 )and( 3{3 ),i.e.,^xisasystemoptimumsolution.Noticethattheequilibriumandthesystemoptimumsolutionsmaynotbethesame,unlessFa(xa)isconstant.


KimandPardalos[ 61 ]).Tocomparetheresultsoftheheuristicprocedures,inthecaseofsmallproblemswendanexactsolutionusingCPLEXMIPsolverandcomputerelativeerrors.Inthecaseoflargeproblems,CPLEXisnotabletondanexactsolutionwithin5,000secondsofCPUtime;therefore,wecomparetheresultsofDCUPversusDSSP. Intheexperiments,wesolveproblemsusingallfourvariationsofDSSPandchoosethebestsolutiontocomparewiththesolutionprovidedbytheADCUP.Inadditiontothenalsolution(thesolutionthatthealgorithmreturnswhenitstops),duringtheiterativeprocessDSSPaswellasADCUPconstructfeasiblevectorsthatmighthaveabetterobjectivefunctionvalue.Intheprocedures,werecordthosevectorsandchoosethebestone.Thecomparisonbetweenthebestsolutionsofbothalgorithmsisalsoprovided.Withregardtothecomputationaltime,wecomparetheCPUtimeofADCUPversusthebestCPUtimeamongfourvariationsofDSSP. Therearefourgroupsoftestproblemsbasedonthesizeofthenetworkandthenumberofsupply/demandnodes(seeTable B{1 ,Appendix B ).Foreachgroup,weconstructdierenttypesoffunctionsfa(xa),wheretheslopeandthexedcostaregeneratedrandomlyaccordingtothespecieddistributions.Intotal,thereareninesetsofproblems(ninetypesoffunctionfa(xa))foreachgroup,i.e.,onesetofproblemsforeachchoiceofdistributionfortheslopeandthexedcost.Thereare30randomlygeneratedproblemsforeachproblemset.Thecomponentsofthesupply/demandvectoraregenerateduniformlybetween30and50units.ThemodelisconstructedusingtheGAMSenvironmentandsolvedbyCPLEX9.0.ComputationsweremadeonaUnixmachinewithdualPentium43.2Ghzprocessorsand6GBofmemory.AllresultsaretabulatedintheAppendix B Tables B{2 and B{3 illustratethecomputationalresultsforgroupsG1andG2.Sincethesizeofthoseproblemsisnotbig,wehavesolvedtheproblemsexactly


usingtheCPLEXMIPsolver.TherelativeerrorsarecomputedbasedonthefollowingformulasREADCUP(%)=fADCUPfexact InthecaseofgroupsG3andG4,wecompareADCUPversusDSSPusingthefollowingformulaDSSPADCUP(%)=fDSSPfADCUP B{4 .Similartotheprevioustwogroups,oneobservesthatonaverageADCUPprovidesabettersolutionthanDSSP.NoticethatDSSPconsumesmuchmoreCPUtimebeforeterminationthanADCUP.Inaddition,thepercentageofproblemswhereADCUPprovidesabettersolutionthanDSSPishigherthaninthepreviouscases.SimilartogroupsG1andG2,thedierencebetweenthesolutionsprovidedby


bothalgorithmsissmallfortheproblemsetswithalargerslopeandsmallerxedcost.Whentheslopedecreases(orthexedcostincreases),ADCUPprovidesaperceptiblybettersolutionthanDSSP. Inthenumericalexperiments,wehaveshownthattherelativeerrorofthesolutionsofbothproceduresincreasesinthecasesofsmallslopesandlargexedcosts.Toexplainthisphenomena,observethatbydecreasingthevalueoftheslopetheanglebetweenfunctionfa(xa)andtherstlinearpieceoffunction"aa(xa)increases(seeFigure 3{1 ).Asaresult,"aa(xa)doesnotapproximatethefunctionfa(xa)aswellasinthecaseoflargevariablecosts.Thesamediscussionappliestothecaseofalargeslope.


43 ],FlorianandKlein[ 36 ],vanHoeselandWagelmans[ 53 ],Loparicetal.[ 70 ],andLoparicetal.[ 71 ]).Itiswellknownthatuncapacitatedproblemscanbereducedtoashortestpathproblem.FlorianandKlein[ 36 ]studiedcapacitatedsingle-itemproblems,wheretheycharacterizedtheoptimalsolutionandproposedasimpledynamicprogrammingalgorithmforproblemsinwhichthecapacitiesarethesameineveryperiod.Thesingle-itemproblemswithvaryingcapacitiesareknowntobeNP-hard.AclassicationofdierentproblemsandasurveyonexistenceofapolynomialalgorithmforsolvingproblemsfordierentclassescanbefoundinWolsey[ 100 ]andPochetandWolsey[ 88 ].TightformulationsforpolynomiallysolvableproblemsarediscussedinMillerandWolsey[ 75 ]andPochetandWolsey[ 88 ]. Almostallpracticalproblemsinvolvemultipleitems,machinesand/orlevels,andpolynomialresultsforthoseproblemsarelimited.Usingbinaryvariables,onecanconstructamixedintegerlinearprogramming(MIP)formulationoftheproblemwithanimbeddednetworkstructure.Tosolvetheproblem,branch-and-boundandcuttingplanealgorithmshavebeenused(see,e.g.,Barretal.[ 4 ],CabotandErenguc[ 12 ],Gray[ 44 ],KenningtonandUnger[ 59 ],Palekar 35


etal.[ 83 ],Marchandetal.[ 72 ],andWolsey[ 100 ]).Inaddition,severalheuristicalgorithmshavebeenproposed(see,e.g.,CooperandDrebes[ 27 ],Diaby[ 31 ],KhangandFujiwara[ 60 ],KuhnandBaumol[ 63 ],vanHoeselandWagelmans[ 52 ],KimandPardalos[ 61 ]and[ 62 ],NahapetyanandPardalos[ 79 ]and[ 80 ]). Inthischapterwediscussacapacitatedmulti-itemdynamicpricing(CMDP)problemwhereonemaximizestheprotbychoosingaproperproductionlevelaswellaspricingpolicyforeachproduct.Intheproblem,thedemandisadecisionvariable,andinordertosatisfyahigherdemandoneneedstoreducethepriceoftheproduct.Ontheotherhand,reducingthepricecandecreasetherevenue,whichistheproductofthedemandandtheprice.Inaddition,theproblemincludesaninventoryandproductioncostforeachproduct,wherethelatterinvolvesasetupcost.Theobjectiveoftheproblemistondanoptimalproductionstrategy,whichmaximizestheprotsubjecttoproductioncapacitiesthatare\shared"bytheproducts.Dierentvariationsofasingle-itemuncapacitatedproblemwithdeterministicdemandsarediscussedbyGilbert[ 43 ],Loparicetal.[ 71 ],andThomas[ 96 ].Acapacitatedsingle-itemproblemwithtimeinvariantcapacitiesisdiscussedinGeunesetal.[ 42 ].Thepolynomialalgorithmsproposedbytheauthorsarebasedonthecorrespondingresultsforthelot-sizingproblems. InChapters 2 and 3 (seealsoNahapetyanandPardalos[ 79 ]and[ 80 ])wehaveproposedabilinearreductiontechnique,whichcanbeusedtondanapproximatesolutionofconcavepiecewiselinearandxedchargenetworkowproblems.AsimilartechniqueisproposedtosolvetheCMDPproblem.Inparticular,weconsiderabilinearreductiontechniqueoftheproblemandprovethatsolvingtheCMDPproblemisequivalenttondingaglobalmaximumofthebilinearproblem.Thelatterbelongstotheclassofbilinearproblemswithdisjointfeasibleregion,andoneconsidersaheuristicalgorithmtondasolutionoftheproblem.Theheuristicalgorithmemploysawellknowniterativeprocedureforndingalocal


maximumoftheproblem(see,e.g.,Horstetal.[ 54 ]andHorstandTuy[ 55 ]).Numericalexperimentsonrandomlygeneratedproblemsconrmtheeciencyofthealgorithm. Fortheremainder,Section 4.2 providesalinearmixedintegerformulationoftheproblemanddiscussesabilinearreductionoftheproblem.WeprovethatsolvingtheCMDPproblemisequivalenttondingaglobalmaximumofthebilinearreduction.InSection 4.3 weproposeaheuristicalgorithmforsolvingthebilinearproblem.NumericalexperimentsonthealgorithmareprovidedinSection 4.4 ,andnally,Section 4.5 concludesthechapter. LetPandrepresentthesetofproductsanddiscretetimes,respectively.Inaddition,letf(p;j)(d)denotethepriceofproductpattimejasafunctionofthedemandd,andg(p;j)(d)=f(p;j)(d)d,i.e.,g(p;j)(d)representstherevenueobtainedfromsellingdamountofproductpattimej.Intheproblem,weassumethatf(p;j)(d)andg(p;j)(d)arenonincreasingandconcavefunctions,respectively(seeFigures 4{1 ).Iff(p;j)(d)isaconcavefunction,thenitiseasytoshowthatconcavityofg(p;j)(d)follows. Letx(p;i;j)denoteanamountofproductpthatisproducedattimeitosatisfythedemandattimej,andy(p;i)representabinaryvariable,whichequalsoneifproductpisproducedattimeiandzerootherwise.Assumethatinventorycosts,cin(p;i;j),productioncosts,cpr(p;i),andsetupcosts,cst(p;i),aswellasproductioncapacities,Ci,aregiven,wherep,i,andjrepresenttheproduct,producingtime,


Figure4{1. Thepriceandtherevenuefunctions. andsellingtime,respectively.ThefollowingisthemathematicalformulationoftheCMDPproblem. Althoughtheaboveformulationbelongstotheclassofnonlinearmixedintegerprograms,usingstandardtechniquesonecanapproximatetherevenuefunctionbyapiecewiselinearoneandlinearizetheobjectivefunction.Doingso,observethatfromtheconcavityoftherevenuefunctionitfollowsthatthereexistsapoint,~d(p;j),wherethefunctionreachesitsmaximum(seeFigure 4{1 ).Asaresult,producing


andsellingmorethan~d(p;j)isnotprotable,andatoptimalityPi2jijx(p;i;j)~d(p;j).Tolinearizetherevenuefunction,divideh0;~d(p;j)iintoNintervalsofequallength.Letdk(p;j)denotetheendpointsoftheintervals,i.e.,dk(p;j)=k~d(p;j)=N,8k2f1;:::;NgSf0g=KSf0g,andgk(p;j)representsthevalueoftherevenuefunctionatthepointdk(p;j),i.e.,gk(p;j)=g(p;j)(dk(p;j))=f(p;j)(dk(p;j))dk(p;j).Usingthoseparameters,constructthefunction~g(p;j)((p;j))=NXk=0gk(p;j)k(p;j)=NXk=1gk(p;j)k(p;j); 5 ]).Thefollowingisthemathematicalformulationoftheapproximationproblem:maxx;y;Xp2PXj2Xk2Kgk(p;j)k(p;j)Xp2PXi;j2jijhcin(p;i;j)+cpr(p;i)ix(p;i;j)Xp2pXi2cst(p;i)y(p;i);s.t.Xp2PXj2jijx(p;i;j)Ci;8i2;Xj2jijx(p;i;j)Ciy(p;i);8p2Pandi2;Xi2jijx(p;i;j)=Xk2Kdk(p;j)k(p;j);8p2Pandj2;


whereqk(p;i;j)=fk(p;j)cin(p;i;j)cpr(p;i).Observe,thatatoptimalityxk(p;i;j)=0forallindicessuchthatqk(p;i;j)0,andthosevariablescanberemovedfromtheformulation.Therefore,withoutlostofgenerality,intheanalysisbelow,weassumethatqk(p;i;j)>0. DeneX=fxjx0andxk(p;i;j)arefeasibleto( 4{1 )and( 4{3 )g,andY=[0;1]jPjjj.Considerthefollowingbilinearprogram: 4{1 ),


( 4{3 )and( 4{4 ).If(x;^y)violatesconstraint( 4{2 )then9p2Pandi2suchthatPj2jijPk2Kxk(p;i;j)>0and^y(p;i)=0.Fromthelocaloptimalityof(x;^y)itfollowsthatPj2jijPk2Kqk(p;i;j)xk(p;i;j)cst(p;i)0,anditisnotprotabletoproduceproductpattimei.Furthermore,byassigningxk(p;i;j)=0,8j2andk2K,theobjectivefunctionvalueoftheACMDP-Bproblemremainsthesame.Let^xdenotetheresultingvector,i.e., ^xk(p;i;j)=8><>:0ifPj2jijPk2Kqk(p;i;j)xk(p;i;j)cst(p;i)0xk(p;i;j)ifPj2jijPk2Kqk(p;i;j)xk(p;i;j)cst(p;i)>0(4{5) Thevector(^x;^y)isfeasibletotheACMDPaswellastheACMDP-Bproblemandhasthesameobjectivefunctionvalueas(x;y). 4{2 )andrelaxingtheintegralityofthevariabley(p;i).Inotherwords,theACMDP-BproblemisarelaxationoftheACMDPproblem.FromTheorem 4.2.1 itfollowsthataglobalsolutionofACMDP-Bisasolution(orleadstoasolution)oftheACMDPproblem.


solvetheACMDP-BproblemandifthesolutionisnotfeasibletotheACMDPproblem,thenusethemethoddescribedintheproofofTheorem 4.2.1 toconstructafeasibleonewiththesameobjectivefunctionvalue. Observethattheproblembelongstotheclassofbilinearprograms.Byxingvectorxorytoaparticularvalue,theproblemcanbereducedtoalinearone.LetLP(x)andLP(y)denotethecorrespondinglinearprograms,i.e.,LP(x):maxy2YPp2PPi2hPj2jijPk2Kqk(p;i;j)xk(p;i;j)cst(p;i)iy(p;i),andLP(y):maxx2XPp2PPi2Pj2jijPk2Khqk(p;i;j)y(p;i)ixk(p;i;j). NoticethatthesolutionoftheLP(x)iseasytoobtain.Inparticular,y(p;i)=8><>:0ifPj2jijPk2Kqk(p;i;j)xk(p;i;j)cst(p;i)01ifPj2jijPk2Kqk(p;i;j)xk(p;i;j)cst(p;i)>0 isanoptimalsolutionoftheproblem.TheProcedure 5 describesawellknownalgorithm,whichstartsfromaninitialbinaryvectorandconvergestoalocalmaximumoftheACMDP-Bprobleminanitenumberofiterations(seeHorstandTuy[ 55 ]orHorstetal.[ 54 ]). However,theprocedurehasthefollowingdisadvantage.Let(xm;ym)representthesolutionobtainedoniterationm,andassumethat9p2Pandi2suchthatym(p;i)=0.Asaresult,intheLP(ym)problemqk(p;i;j)ym(p;i)=0,8j2,ij,and




5 tondalocalmaximumoftheproblem,whereymisaninitialbinaryvector.Let(xm+1;ym+1)denotethelocalmaximum. Asbefore,thesolutionoftheLP"(x)problemiseasytoobtainbyassigningy(p;i)=8><>:0ifPj2jijPk2Kqk(p;i;j)xk(p;i;j)cst(p;i)"(p;i)1ifPj2jijPk2Kqk(p;i;j)xk(p;i;j)cst(p;i)>"(p;i)=8><>:0if'1(p;i)(x(p;i))'2(p;i)(x(p;i))1if'1(p;i)(x(p;i))>'2(p;i)(x(p;i)): 6 )startswithasucientlylarge"andndsalocalmaximumoftheresulting"-approximationproblem.IfthestoppingcriteriaisnotsatisedinStep3thenitdecreasesthevalueofthevector"to",whereisaconstantfromtheopeninterval(0;1),andtheprocesscontinuesusinganew"-approximationproblem.ObservethatProcedure 6 usesvectorymfromthepreviousiterationasaninitialvector. Theproceduredependsontwoparameters:theinitialvector"andthevalueof.Thevalueof"(p;i)dependsonparametersoftheproblem,andonecanconsiderthemequaltothemaximumprot,whichcanbeobtainedbyproducingonlyproductpattimei.AlthoughsuchmaximizationproblemiseasytosolveusingstandardLPsolvers,forlargejPjandjjonendscomputationallyexpensivesolvingtheproblemforallpairs(p;i)2P.Instead,wepropose


analgorithmforndingthevaluesof"(p;i)(seeProcedure 7 ).Observethatxk(p;i;j)dk(p;j),andthemaximumadditionalprotthatcanbeobtainedusingthevariablexk(p;i;j)isqk(p;i;j)dk(p;j).Usingthisproperty,forallpairs(p;i)theprocedureiterativelyndsthemaximumamongqk(p;i;j)dk(p;j)andassignsthedemand(ortheremanningofthecapacity)tothecorrespondingvariablexk(p;i;j).Thevalueof"(p;i)iscomputedbasedontheformula"(p;i)=Pj2jijPk2Kqk(p;i;j)xk(p;i;j)cst(p;i).Asfortheparameter,itslargervalueincreasesthecomputationaltimeoftheprocedure,anditislikelytoprovideabettersolution. 5 aswellasProcedure 6 usingdierentvaluesfortheparameter.ThelatterprocedureemploysProcedure 7 tondaninitialvalueforthevector".Inaddition,wesolvetheproblemsbytheMIPsolverofCPLEXusingtheACMDPformulation.InthecaseswheretheMIPsolverisnotabletosolvelargeproblemswithinpostedCPUandmemorylimitations,wecomparethesolutionsoftheprocedureswiththebestsolutionsfoundbyCPLEX.Themainpurposeofthecomputationsistheperformanceoftheproceduresfordierentcapacities. Assignxk(p;i;j)=0,8p2P,i;j2,andk2K Assignxkmax(p;i;jmax)=minf^C;dkmax(p;jmax)g,^C=^Cxkmax(p;i;jmax),^qk(p;i;jmax)=0,8k2K,andqmax=maxf^qk(p;i;j)dk(p;j)jk2K;j2;jig


Inthenumericalexperimentsweconsiderproblemsetswithdierentnumbersofproducts,jPj=5,10,or20,andtimehorizons,jj=12or52.Foreachproblemsetwerandomlygeneratecapacitiesforalli2usingtheformulaCi=jPjU,whereUisarandomnumberuniformlygeneratedfrominterval[10;100],[50;150],[100;200],or[150;250].Notethatallintervalsallowgeneratingcapacitiesthataretightatoptimalitywithrespecttotherevenuefunctiondiscussedbelow.Inaddition,usingtermjPjonegeneratescapacitiesthatdependonthenumberofproducts.Thelatterallowscomparingofresultsacrossdierentnumbersofproducts.Asforthecosts,wegeneratetheproductioncostscpr(p;i)andtheinventorycostscin(p;i;j)accordingtotheuniformdistributionsU[20;40]andU[4;8],respectively.Observethatonaveragetheinventorycostisequalto20%oftheproductioncost.Finallythesetupcostcst(p;i)isgenerateduniformlyfrominterval[600;1000]. Intheexperimentswerestrictourselfbyconsideringonlylinearpricefunctionsoftheformf(p;j)(d)=fmax(p;j)fmax(p;j)=dmax(p;j)d.Toavoidgeneratingfunctionsthatatoptimalityresultinunrealisticallylargeprots,weintroduceanindex,where=Pp2PPi2hPj2jijPk2Kfk(p;j)cin(p;i;j)cpr(p;i)xk(p;i;j)cst(p;i)y(p;i)i Pp2PPi2hPj2jijPk2K(cin(p;i;j)+cpr(p;i))xk(p;i;j)+cst(p;i)y(p;i)i: 4{1 ).Inaddition,theproposedpricefunctionanddistributionsofthecostsandcapacitiesallowgeneratingproblemsthathaveanoptimalobjectivefunctionvaluerangingfromhundredsofthousandstoseveral


millions.Finally,intheconstructionofthepiecewiselinearapproximationoftherevenuefunctionweuseN=10. ThemodelisconstructedusingtheGAMSenvironmentandsolvedbyCPLEX9.0withaCPUrestrictionof2000secandamemoryrestrictionof2Gb,wherethelatteristhememorythatisrequiredtostorethetreeinthebranch-and-boundalgorithm.ComputationsaremadeonaUnixmachinewithdualPentium43.2Ghzprocessorsand6GBofmemory.TheresultsaretabulatedintheAppendix C Intheexperimentswesolve10randomlygeneratedproblemsforeachproblemsetandcapacity.Tables C{1 and C{2 comparetheresultsprovidedbyCPLEXwiththesolutionsprovidedbybothprocedures.TherelativeerroriscomputedusingtheformulaRE(%)=ObjCPLEXObjProc: ( 5 ) C{1 ,columnAindicatesthenumberofproblemswheretheheuristicprocedurendsabettersolutionthanCPLEX.NotethatCPLEXisabletoprovideanexactsolutionforallcapacitiesfromtheproblemset5-12.Inallothercases,thesolverstopsafterreachingtheCPUlimitorthememorylimitandreturnsthebestfoundsolution.Althoughtherelativeoptimalitygapofthenalsolutionsofthoseproblemsetsvariesfrom2%to5%,webelievethatthesolutionisanoptimalorclosetoanoptimalone,andthelargeoptimalitygapisduetoimperfectlowerbounds.Thefactthattheheuristicproceduresprovideaslightlybettersolutioninthecasewithjj=52thanjj=12partiallyconrmsourassumptions. TherelativeerrorsintheTable C{1 conrmstheeectivenessoftheheuristicprocedure.Inparticular,inthemajorityoftheproblemstheheuristicalgorithmisabletoprovideasolutionwithin1%fromtheoptimaloneorthebestoneprovidedbyCPLEX.Observethatthelargervalueofprovidesabettersolutionandthenumberofproblemswheretheheuristicprocedurendsabettersolutionthan


CPLEXisincreasingwiththesizeoftheproblem.BycomparingwiththesolutionsprovidedbyProcedure 5 (seeTable C{2 )onenoticesthatProcedure 6 outperformstheProcedure 5 ,anditismorestablewithchangesinthecapacities.AsfortheCPUtime(seeTable C{3 ),theheuristicproceduresrequirefewerresourcesthanCPLEX.Inaddition,unlikeCPLEXtheheuristicproceduresdonotrequiregigabytesofmemorytostorethetree. 4.3 ,intheverybeginningoftheiterativeprocesstheprocedureeliminatessomeproductsfromthefurtherconsideration.Thelatterworsenthequalityofthesolutionreturnedbytheprocedure.Inthesecondprocedureweconstructapproximateproblemsandgraduallydecreaseparametersoftheproblems.Asaresult,duringtheiterativeprocessthecostsoftheeliminatedproductsremainpositiveandtheprocedureconsidersthemagainifneedbe.AlthoughthesecondprocedurerequiresmoreCPUtimetostopthantherstone,itprovidesahigher-qualitysolution.


73 ]and[ 74 ])rstproposedtheirmodelin1978,therehavebeenanumberofpapers(see,e.g.,CareyandSubrahmanian[ 23 ],Carey[ 15 ],Carey[ 13 ],Carey[ 14 ],Friesz[ 37 ],Friesz[ 38 ],Wieetal.[ 98 ],ChenandHsueh[ 25 ],Janson[ 57 ],Ho[ 51 ],Ziliaskopoulos[ 104 ],Drissi-KaitouniandHameda-Benchekroun[ 32 ],Lietal.[ 66 ],Kaufmanetal.[ 58 ],Boyceetal.[ 11 ],RanandBoyce[ 90 ],Ranetal.[ 92 ],andWieetal.[ 99 ])discussingvariationalinequalityormathematicalprogrammingformulationsforthedynamictracassignmentproblemwiththeassumptionthattheplanninghorizonisasetofdiscretepointsinsteadofacontinuousinterval.Manyofthesepapersuseadynamicortime-expandednetwork(see,e.g.,Ahujaetal.[ 2 ])tosimultaneouslycapturethetopologyofthetransportationnetworkandtheevolutionoftracovertime.Implicitlyorotherwise,thesepaperstypicallyassumethatthereisnotracatthebeginningoftheplanninghorizon(orattimezero)andthatalltripsmustexitthenetworkpriortotheend.Whentherearecarsatthetimezero,thetimesatwhichthesecarsenterthenetworkmustbeknowninordertodeterminewhentheywillexitthearcsonwhichtheyweretravelling.Inpractice,datawithsuchdetailsdonotgenerallyexist. Therearetwomainfactorsthatdistinguishthemodelsinpapersreferencedabove.First,some(e.g.,MerchantandNemhauser[ 73 ],CareyandSubrahmanian[ 23 ],Ho[ 51 ],Carey[ 15 ],Ziliaskopoulos[ 104 ],Kaufmanetal.[ 58 ],Garciaetal.[ 40 ])seekasystemoptimalsolutionandothers(e.g.,Janson[ 57 ],Wieet 50


al.[ 98 ],ChenandHsueh[ 25 ],andDrissi-KaitouniandHameda-Benchekroun[ 32 ])computeauserequilibriuminstead.Theotherfactoristhetravelcostfunctionusedbythesemodels.Amongotherparameters(physicalorotherwise),atraveltimeorcostfunctionmaydependonthenumberofcarsonthelinkandtheinputandoutputrates.Many(e.g.,CareyandGe[ 20 ],CareyandMcCartney[ 18 ],Carey[ 16 ],Carey[ 17 ],LinandLo[ 69 ],HanandHeydecker[ 46 ],Daganzo[ 28 ])haveanalyzedtheeectsoftravelcostfunctionsonvariousmodels.Some(e.g.,LinandLo[ 69 ]andHanandHeydecker[ 46 ])haveshownthatsometravelcostfunctionsarenotconsistentwiththemodelsthatusethem. SimilartoCareyandSrinivasan[ 21 ],CareyandSubrahmanian[ 23 ],Carey[ 15 ],ChenandHsueh[ 25 ]andKaufmanetal.[ 58 ],themodelinthischapterisbasedonthetime-expandednetwork.However,insteadofassumingthatthenetworkisemptyatthebeginningorattheend,wetreattheplanninghorizonasacircularintervalinsteadoflinear.Forexample,considertheinterval[0,24],i.e.a24-hourplanninghorizon.Whenviewedinalinearfashion,itistypicallyassumedthatthereisnocarinthenetworkattimes0and24.Inturn,thisimpliesthereisnotraveldemandaftertimek<24.Otherwise,carsthatenterthenetworkaftertimekcannotreachtheirdestinationsbytime24,therebyleavingcarsinthenetworkattheendofthehorizon.Ontheotherhand,ifthereisacarenteringastreetat23:55h(11:55PM)andexitingat24:06h(12:06AM,thenextday)inacircularplanninghorizon,theexittimeofthiscarwouldbetreatedas00:06hinstead.Whenaccountedforinthismanner,itispossibletodeterminetheexittimeforeverycarthatisinthenetworkattimezerowithoutrequiringanyadditionaldata.Additionally,modelsthatviewtheplanninghorizoninacircularfashionaremoregeneralinthattheyincludethosewithalinearplanninghorizon.Bysettingthetraveldemandsandothervariablesduringanappropriatetime


intervaltozero,modelswithacircularplanninghorizoneectivelyreducetooneswithalinearhorizon. ItisoftenarguedthatthenumberofcarsatthebeginningandtheendofthehorizonaresmallandsolutionstoDTAarenotdrasticallyaectedbysettingthemtozero.Whenthepathsthatthesecarsusedonotoverlap,theargumentisvalid.However,whenthesecarshavetotraversethesamearcinreachingtheirdestinations,thenumberofcarsonthearcmaybesignicantandignoringitmayleadtoasolutionsignicantlydierentfromtheonethataccountsforallcars. Thischaptermakestwomainassumptions.Onerequiresthelinktraveltimeattimettobeafunctionofonlythenumberofcarsonthelinkatthattime.CareyandGe[ 20 ]showthatthesolutionsofmodelsusingfunctionsofthistypeconvergetothesolutionoftheLighthill-Whitham-Richardsmodel(seeLighthillandWhitham[ 68 ]andRichards[ 93 ])asthediscretizationoflinksintosmallersegmentsisrened.Becauseminimizingthetotaltraveltimeordelaymitigatesitsoccurrence,modelsdiscussedhereindonotexplicitlyaddressspillback.Ontheotherhand,themodelscanbeextendedtohandlespillbackusingatechniquesimilartotheoneinLieberman[ 67 ]oranalternativetraveltimefunctionthatincludestheeectofspillback(see,e.g.,PerakisandRoels[ 86 ]).However,asindicatedinthereference,usingsuchafunctionmaynotleadtoamodelwithasolution. Fortheremainder,Section 5.2 denestheconceptofperiodicplanninghorizon.Section 5.3 formulatesthesystemversionofthediscrete-timedynamictracassignmentproblemwithperiodicplanninghorizonorDTDTAandprovethatafeasiblesolutionexistsunderarelativelymildcondition.Toourknowledge,thereareonlyfourpapers(Brotcorne[ 10 ],Smith[ 95 ],Wieetal.[ 98 ],andZhuandMarcotte[ 103 ])thataddresstheexistenceissueandsome(see,e.g.,Smith[ 95 ]andZhuandMarcotte[ 103 ])considerthissmallnumbertobelacking.Allfourdeal


Figure5{1. Linearversuscircularintervals. withuserequilibriumproblemsinsteadofsystemoptimal.Section 5.4 describestwolinearintegerprogramsthatprovideboundsforDTDTA.Section 5.5 presentsnumericalresultsforsmalltestproblemsand,nally,Section 5.6 concludesthechapter. 5{1 .Indoingso,time0andTarethesameinstant.Forexample,time0:00hand24:00h(ormidnight)arethesameinstantina24-hourday.Forthisreason,Tisexcludedandtheplanninghorizonishalf-open.Tomakethediscussionhereinmoreintuitive,weoftenrefertotheplanninghorizonasa24-hourday,i.e.,T=24.Intheory,theplanninghorizoncanbeofanylengthaslongaseventsoccurinaperiodicfashion.Ifanevent(e.g.,vecarsenterastreet)occursattimet,thenthesameeventalsooccursattimet+kT,forallintegerk1. Becausetheplanninghorizoniscircular,eventsoccurringtomorrowareassumedtooccurinthesameintervalthatrepresentstoday.Forexample,consideracarthatentersastreetatt1=23:00h(or11PM)todayandtraversesthestreetuntilitleavesatt2=01:00h(or1AM)tomorrow.(SeeFigure 5{2 .)Inacircularplanninghorizon,thesetwoevents,acarenteringandleavingastreet,occurat


Figure5{2. Eventsoccurringintwoconsecutiveplanninghorizons. time23:00hand01:00hinthesameinterval[0,24).Ingeneral,ifacarentersastreetattimet1

Figure5{3. Three-nodenetwork. pairsandthetraveldemandforODpairkduringthetimeinterval[t;t+],t2,ishkt. Thereisalsoatraveltimefunctionassociatedwitheacharcinthenetwork.Intheliterature(see,e.g.,Wuetal.[ 101 ],RanandBoyce[ 89 ]andCareyetal.[ 19 ]),thesefunctionscandependonanumberoffactorssuchasin-owandout-owratesandtracdensities.Weassumeinthisformulationthata,thetraveltimeassociatedwitharca,dependsonlyonthenumberofcarsonthearc.Furthermore,aiscontinuous,non-decreasingandboundedbyT,i.e.,0

Figure5{4. Timeexpansionofarc(1;2)att=1. isa=fs:s=da(w) ToincorporatethetimecomponentintheTEnetwork,everynodeinthestaticnetwork(orstaticnode)is`expanded'orreplicatedonceforeacht2.Forthethree-nodenetwork,staticnode1istransformedintoveTEnodes,oneforeacht2,intheTEnetwork.Forexample,node1isexpandedintonodes10;11;12;13;and14intheTEnetwork.(SeeFigure 5{4 .)Similarly,eacharc(i;j)inthestaticnetwork(orstaticarc)isreplicatedonceforeachpairof(t;s),wheret2ands2(i;j).Considerarc(1;2)inthethree-nodenetwork.Carsthatenterthisarcattime1cantake2,3,or4unitsoftimetotraversedepending(asassumedearlier)onthenumberofcarsonthearcatt=1.Toallowallpossibilities,arc(1;2)isexpandedintothreeTEarcs(11;23),(11;24),and(11;20).Thelatterrepresentsacarthatentersarc(1,2)attime1,takes4unitsoftimetotraverse,andleavesthearcattime5ortime0(ormod(1+4;5))ofthefollowingday.Similarexpansionappliestoeacht2.Ingeneral,eachstaticarc(i;j)expandsintojjj(i;j)jTEarcsoftheform(it;jmod(t+s;T));8t2;s2(i;j).


Figure 5{5 displaysthecompletetimeexpansionofthethree-nodenetwork.Inadditiontothetime-expandednodesandarcs,thegurealsodisplaysthetraveldemandattheoriginTEnodes(i.e.,node1t;8t2)anddecisionvariablesgkd(k)trepresentingnumberofcarsarrivingatthedestinationnoded(k)ofODpairkattimet,i.e.,atnode3t;8t2. ToreferenceowsonTEarcs,letyka(t;s)denotetheamountofowforcommoditykthatentersstaticarcaattimet2,takess2aunitsoftimetotraverseit,andthenexitsthearcattimemodft+s;Tg.Inparticular,ifa=(i;j),thenthesubscripta(t;s)referstoTEarcsoftheform(it;jmod(t+s;T));8t2;s2(i;j).Inaddition,Ya(t;s)=Pk2Cyka(t;s)representsthetotalowonarca(t;s). Tocomputethetimetotraverseastaticarcattimet,let a(t)=f(;s):=[t1]T;[t2]T;;[ts]T;s2ag: Inwords,a(t)containspairsofentrance,,andtraveltimes,s,forstaticarcasuchthat,ifacarentersstaticarcaattimeandtakesstimeunitstotraverseit,thecarwillstillbeonthearcattimet.Forexample,ift=11:00handthetimetotraversearcaisvehoursforthepreviousveconsecutivetimeperiods,thencarsenteringarcaattime=10:00h,9:00h,8:00h,7:00h,and6:00hwillbeonthearcat11:00h.(Weassumeherethatcarsenteringarcaat,e.g.,6:00harestillonthearcat11:00heventhoughitisscheduledorexpectedtoleaveat11:00h.)Whentisrelativelynearthebeginningoftheplanninghorizon,thenotation[]Taccountsforcarsonthearcattimetthatenteritfromthepreviousday.Continuingwiththeforegoingexample,lett=3:00hinstead.Then,carsenteringarcaattime=


Figure5{5. Time-expansionofthethree-nodenetwork.


2:00h,1:00h,0:00h,23:00h,and22:00harestillonthearcat3:00h.Usingtheseta(t),thetotalamountofowonstaticarcaattimetorxa(t)isP(;s)2a(t)Ya(;s). Therearetwoadditionalsetsofdecisionvariables.Onesetconsistsofza(t;s),abinaryvariablethatequalsoneifittakesbetween(s)andsunitsoftimetotraversearcaattimet.Intheformulationbelow,thevalueofza(t;s)dependsonxa(t)and,foreacht,za(t;s)=1foronlyones2a.Theothersetconsistsofgk,avectorwithacomponentforeachnodeintheTEnetwork.ComponentitofgkissettozeroifiisnotthedestinationnodeofODpairk.Otherwise,gkd(k)t,whered(k)denotesthedestinationnodeofODpairk,isadecisionvariablethatrepresentstheamountofowforcommoditykthatreachesitsdestination,d(k),attimet. Belowisamathematicalformulationofthediscrete-timedynamictracassignmentproblemwithperiodicplanninghorizon(DTDTA).min(x;y;z;g)Xt2Xa2A"a(xa(t))Xs2aYa(t;s)#


Intheobjectivefunction,Ps2aYa(t;s)representsthenumberofcarsthatenterarcaattimetand,basedonourassumption,thesecarsexperiencethesametraveltime,a(xa(t)).Thus,thegoalofthisproblemistominimizethetotaltraveltimeordelay.Usingconstraint( 7{10 ),theobjectivefunctioncanbeequivalentlywrittenasmin(x;y;z;g)Xt2Xa2A24a(X(;s)2a(t)Ya(;s))Xs2aYa(t;s)35 (a(P(;s)2a(t)Ya(;s)))whosecomponentsaredenedsothattheirinnerproductisconsistentwiththesummations. Constraint( 7{8 )ensuresthatowsarebalancedateachnodeintheTEnetwork.Inthisconstraint,Bdenotesthenode-arcincidencematrixoftheTEnetworkandbkisaconstantvectorwithacomponentforeachTEnodeanddenedasfollows:bkit=8><>:0ifi6=o(k)hktifi=o(k) whereo(k)denotestheoriginnodeofODpairk.Constraint( 7{9 )guaranteesthatthenumberofcarsarrivingatthedestinationnoded(k)equalsthetotaltraveldemandofODpairkduringtheplanninghorizon.Then,constraint( 7{10 )computesthetotalowoneachTEarcand( 5{4 )determinesthenumberofcarsthatarestillonstaticarcaattimet. Incombination,thenextthreeconstraints,i.e.,constraints( 5{5 )-( 5{7 ),computethetraveltimeforthecarsthatenterarcaattimetandonlyallowowstotraversethecorrespondingarcintheTEnetwork.Inparticular,constraint( 5{5 ),inconjunctionwith( 5{6 ),choosesone(discretized)traveltimes2a


thatbestapproximatesa(xa(t)),i.e.,a(xa(t))2(s;s].Whenarepresentsarc(i;j),constraint( 5{7 )onlyallowsarc(it;imod(t+s;T))tohaveapositiveow.Otherwise,(7)forcesowsonarc(it;imod(t+;T)),for2aand6=s,tobezero.Finally,constraint( 7{11 )makessurethatappropriatedecisionvariablesareeithernonnegativeorbinary. Asformulatedabove,thetraveltimeassociatedwithza(t;s)inequation( 5{6 )canonlytakeondiscretevaluesfromthesetawhilethetraveltimeintheobjectivefunctionvariescontinuously.Althoughitmaybemoreconsistenttousediscretevaluesoftraveltimesintheobjectivefunction,theabovemodelwouldprovideabettersolutionbecausethetruetraveltimeisusedtocalculatethetotaldelay.ThemodelalsohasinterestingpropertiesdiscussedinSection 5.4 .Inaddition,thetreatmentsoftraveltimesinboththeobjectivefunctionandconstraintscanbemadeconsistentbysolvingthe(approximation)renementproblemalsodiscussedinthesamesection. Underarelativelymildsucientcondition,weshowbelowthatDTDTAhasasolutionbyconstructingafeasiblesolution.Infact,thesolutionweconstructbelowisgenerallyfarfrombeingoptimal.However,itsucesforthepurposeofprovingexistence.LetRa(t)beasetofdiscretetimesatwhichacarentersarcaandstillremainsonthearcattimet.Below,werefertoRa(t)astheenter-remainset.Givenxa(t),Ra(t)isaunionoftwosets,i.e.,Ra(t)=fw2:w(t1);w+(xa(w)) 5{4 )-( 5{7 )andrelevantconditionsin( 7{11 )existswhenMaissucientlylarge.


5{4 )to( 5{7 )andtherelevantconditionsin( 7{11 ). Form=1,let Asdenedabove,R1tistheenter-remainsetbasedonthetraveltimes1,avectorofs1!;8!2.Form2,let Sequencesfsmtg,fxma(t)gandfRmtgconstructedabovearemonotonicallynon-decreasing.Considerthesequencefsmtg.Observethats2ts1t;8t2becausex1a(t)0;8t2,andasassumedearliera()isnon-decreasing.Itfollowsthat,foranyt2,!+s2!!+s1!tand!+s2!T!+s1!Tt.Thus,!2R1t


impliesthat!2R2t,i.e.,R1tR2tforallt2.Thelatter,andthefactthatua(t)isnonnegative,implythatx2a(t)=P!2R2tY2a(!;s2!)P!2R1tY1a(!;s1!)=x1a(t);8t2. Assumethattheclaimistrueuptosomexedm.Forallt2,sm+1t=da(xma(t))=eda(xm1a(t))=e=smt;becausexma(t)xm1a(t)anda()isnon-decreasing.Usinganargumentsimilartoabove,RmtRm+1tandxm+1a(t)xma(t).Thus,thethreesequencesaremonotonicallynon-decreasing.Inaddition,allthreesequencesarebounded,i.e.,smt

become(i;j0)and(j;i0).LetpkdenoteapathinG(N;A)connectingtheODpairk,i.e.,pk2Pk.Thesetofthesepaths,=fpk:k2Cg,inducesasubgraphG(bN;bA),wherebNNandbAAdenotethesetsofnodesandarcs,respectively,belongingtothepathsin.Foreachi2bN,dene[i+]=f(i;j):(i;j)2bAgand[i]=f(j;i):(j;i)2bAg.Inwords,[i+]and[i]arethesetsofarcsinG(bN;bA)thatemanatefromandterminateatnodei,respectively.Also,letorder(i)denoteatopologicalorderofnodei(seeAhujaetal.[ 2 ]).If(i;j)2bAandG(bN;bA)canbetopologicallyordered,thenorder(i)

Toconstructthevariablesxa(t),Ya(t;s),za(t;s),andyka(t;s)forarcsemanatingfromnodesofhigherorder,assumethatthedecisionvariablesforarcsemanatingfromnodeswithordermorlesshavebeenconstructedandletnodeibeoforder(m+1). 7{8 )and( 7{9 ). Consideranarca2[i+].Foreachba2[i],deneQ(ba;a)=fk:ba2pk;a2pk;k2Cgand,foreachk2Q(ba;a),letuka(t)denotetheowintoarcaattimetforODpairk.Then,uka(t)=Xfbt:bt+s1ba(bt)=tgykba(bt;s1ba(bt))+Xfbt:bt+s1ba(bt)T=tgykba(bt;s1ba(bt)); 5.3.1 ensuresthatxa(t),Ya(t;s),yka(t;s)andza(t;s)feasibletorelevantconstraintsexist. Thus,whencarriedoutinthetopologicalorderforeveryarcinbA,theaboveprocessmustproduceafeasiblesolutiontoDTDTA.


eachODpairtotraverseoverseveralpathsaslongastheydonotinducecyclesinG(bN;bA).Withmorecumbersomenotation,theaboveargumentcanbeextendedtothecasewithmultiplepathsperODpairaswell. WhenappliedtotheaboveexampleinwhichtheODpairs(i;j)and(j;i)become(i;j0)and(j;i0),theacyclicsubgraphassumptionimpliesthatthepathsfromitoj0andfromjtoi0cannotformacycle.Intuitively,thismeansthattheremustexisttworoutesbetweentheoriginalnodesi(e.g.,home)andj(e.g.,work)withnoroadincommon.Theseroutesneednotbeoptimalandthereisnorequirementinourformulationoralgorithmstousethem.TheyareusedonlytoestablishedtheexistenceinTheorem 5.3.1 TheFirst-In-First-Out(FIFO)conditionrequiresthatcarsenteringanarcattimetmustleavethearcbeforethoseenteringaftertimet.Intheliterature,many(see,e.g.,Ranetal.[ 91 ],ZhuandMarcotte[ 103 ],andParakisandRoels[ 86 ])assumethatthetravelcostfunctionsatisedcertainconditionstoensureFIFO.Toavoidmakingadditionalassumptions,weensureFIFObyaddingthefollowingconstraintstoDTDTAinstead.Doingsomaymaketheproblemhardertosolvebecauseoftheadditionalconstraints.t+Ps2asza(t;s) t2:(t+) t+Ps2asza( t2:(t+)


today)as(t+T)(e.g.,as08:00hofyesterdayplusT)andforcescarsenteringthearcatthistimetodepartafterthosethatenteratyesterday'stime Exceptforconstraint( 5{6 ),theconstraintsforDTDTAarelinear.Todevelopalinearversionof( 5{6 ),assumethatthetraveltimefunction,a,isinvertibleforalla2A.Forexample,ifaisacontinuousandincreasingfunction,then1aexistsontheinterval[a(0);a(Ma)].(SeeFigure 5{6 .)Underthisassumption,a(xa(t))2(s;s]ifandonlyifxa(t)2(1a(s);1a()].Thus,therequirement(s)za(t;s)

Figure5{6. 5{6 ),thefollowingholdforanyfeasiblesolutiontoDTDTA:Xt2Xa2A"Xs2a(s)za(t;s)#"Xs2aYa(t;s)#(Y)TY; 5{7 )impliesthatYa(t;s)>0onlyifza(t;s)=1.Inaddition,constraint( 5{5 )ensuresthat,foreachpair(a;t),za(t; s)=1forsome s)0andYa(t;s)=0;8s2a;s6= s)=Xs2asYa(t;s):


Asimilarresultholdsfortherstsetofsummations.Thus,theaboveinequalitiesreducetothefollowingXt2Xa2AXs2a(s)Ya(t;s)(Y)TYXt2Xa2AXs2asYa(t;s): 7{8 )-( 5{5 ),( 5{9 ),( 5{7 ),and( 7{11 )and,forconvenient,(Y;Z)representsanelementinS().Inaddition,let(Yl;Zl),(Y;Z),and(Yu;Zu)besolutionstothelower-boundproblem(orminfqTlY:(Y;Z)2S()g),theoriginalproblem(orminf(Y)TY:(Y;Z)2S()g),andtheupper-boundproblem(orminfqTuY:(Y;Z)2S()g),respectively.Then,thefollowinglemmaholds. 5.4.1 :qTlYlqTlY(Y)TY: Finally,thelastinequalityholdsbecauseYlisnotnecessarilyoptimaltominfqTuY:(Y;Z)2S()g.


Inviewoftheabovelemma,thesolutionstotheupperandlower-boundproblemsareapproximationsofthesolutiontotheoriginalproblem.Thetheorembelowstatesthattheapproximationcanbemadearbitrarilyclosetotheoriginalproblembychoosingasucientlysmall. Then,thefollowingsequencemusthold:0(quql)TYl=eTYl=Pa2APk2CPt2Ps2ayka(t;s)=Pa2APk2CPt2yka(t;sa(t))Pa2APk2CHk=HPa2A1=HjAj 5.4.2 .Then,theaboverelationshipbetweenquandqlandlettingPk2Cyka(t;s)denoteindividualscomponentsofYlyieldthersttwoequalities.Thethirdequalityfollowsfromthedenitionofsa(t).Followingthis,thesecondinequalityholdsbecausethetotalamountofowon(static)arcaforODpairkduringtheentireplanninghorizoncannotexceedHk.ThesumofthelatterisH,aconstantthatcanbefactoredoutofthesummationoverA.Thisvalidatesthepenultimateequality.Finally,thelast


equalityfollowsfromthefactthatPa2A1simplydenotesthenumberofelementsinthesetA.Choosing=" HjAjguaranteesthatqTuYlqTlYl.WhencombinedwiththeresultsinLemma2,thelatterimpliesthatqTuYu(Y)TYand(Y)TYqTlYl: 5{5 )and( 5{7 )fromtheproblem.InDTDTA,weusexa(t),thenumberofcarsonarcaattimet,tocomputethetraveltimeonarcaand,subsequently,toselectwhichTEarctouseorwhichza(t;s)tosettoone.Thus,whenZisgiven,xa(t)becomesunnecessary.Additionally,lets(t)besuchthatzua(t;s(t))=1foreacht2.Then,constraint( 5{9 ),originally( 5{6 ),reducestorequiringP(;s)2a(t)Ya(;s)tobeintheinterval(s(t);s(t)].Inotherwords,theoriginalproblemwithZ=Zuisanonlinearmulti-commodityowproblemwiththelatterassideconstraints. Let^Yubeanoptimalsolutiontominf(Y)TY:(Y;Zu)2S()g.Then,thefollowingcorollaryshowsthat^YubetterapproximatesY. 5.4.1


Table5{1. Demandpatterns Time TracIntensity 0123456789 Total Low 20253035404035302520 300 Medium 30354045505045403530 400 High 40455055606055504540 500 Inallproblems,theplanninghorizonis[0,10)andthetravelcostfunctionsareeitherlinear,i.e.,(w)=1:5+2:5(w 5{1 .Inallthreepatterns,traveldemandsatdiscretepointsincreasesgraduallyuntiltime4,levelsobriey,andthendecreasesgraduallyaftertime5.Theindividualdemandsinthethreepatternsaredierentandrepresentthreetracintensities:low,medium,andhigh.WeusedGAMS[ 39 ]toimplementandsolveallproblemsusingNEOSServerofOptimization[ 82 ].Inparticular,weusedSBB[ 94 ]tosolveournonlinearintegerprogrammingproblem,i.e.,DTDTA,XPress-XP[ 102 ]tosolveourlinearintegerprograms,i.e.,thelowerandupper-boundproblems,andCONOPT[ 26 ]tosolveourlinearlyconstrainedoptimizationproblems,i.e.,theapproximationrenementproblems.AllCPUtimesreportedhereinarefromtheNEOSserver. ToempiricallyverifythatDTDTAproblemisnotconvex,werstconsiderthetwo-arcnetworkinFigure 5{7 thathasoneODpair.Welet=1.Thus,thediscrete-timeplanninghorizonis=f0;1;;9g.Weusetheabovequadratic Figure5{7. Two-arcnetwork.


Table5{2. Optimalsolutionstothetwo-arcproblem. Solution1 Solution2 Inow Traveltime Inow Traveltime Time 020 1.6001.500 200 1.5001.600 1 250 1.5001.600 025 1.6001.500 2 030 1.6561.500 300 1.5001.656 3 350 1.5001.725 035 1.7251.500 4 040 1.8061.500 400 1.5001.806 5 400 1.5001.900 040 1.9001.500 6 035 1.9001.500 350 1.5001.900 7 300 1.5001.806 030 1.8061.500 8 025 1.7251.500 250 1.5001.725 9 200 1.5001.656 020 1.6561.500 traveltimefunctionforbotharcsandthefunctionyieldstraveltimesintheinterval[1:5;4:0].Because=1,thesetofdiscretetraveltimesis=f2;3;4g.UsingthelowtracintensitydemandpatterninTable 5{1 ,wesolvedDTDTAusingSBBandterminateditwhentherelativeoptimalitygapislessthan0.005(or0.5%).Therearetwooptimalsolutions(seeTable 5{2 )tothetwo-arcproblemwithanoptimaltotaldelayof450. Considertherstsolution,labelled`Solution1',intheTable 5{2 .Attime0,thereare20carstotravelfromnode1tonode2.Atthistime,therearealso20carsalreadyonarca1.Thesecarsenterthearcattime9andhavenotreachedtheirdestinationattime0.BecauseDTDTAassumesthatthetimetotraversearca1dependsonthenumberofcarsonthearcattheentrancetime,thetraveltimeforarca1attime0is1:5+2:5(20 100)2=1:6.Ontheotherhand,thereisnocarona2attime0.Carsthatenterthearcattime8alreadyleftthearcbytime0.Thus,thetraveltimefora2attime0is1.5,thefree-owtraveltime.Tominimizethetraveltime,all20carsenteringthenetworkattime0musttravelona2.Infact,everycarinSolution1travelsatthefree-owtraveltimeof1.5.Thus,therecannotbeanysolutionwithlesstotaldelayandSolution1mustbeoptimal.


Figure5{8. Four-nodenetwork. Becauseofthesymmetryinthedata,switchingtheowsbetweenthetwoarcsinthenetworkyieldsSolution2,anotheroptimalsolution.Furthermore,itiseasytoverifythateveryconvexcombinationofthesetwosolutionsisfeasibletoDTDTAandyields,ontheotherhand,alargertotaldelay,therebyconrmingempiricallythattheobjectivefunctionisnotconvex. Additionally,the\extreme"travelbehaviordisplayedinTable 5{2 maynotbeintuitive.Thisisduetotheassumptionthatthesystemoperatorisextremelysensitivetothedierenceintraveltimesandiswillingtoswitchroutesinordertosaveaminuteamountoftraveltime. Whenthenetworkislarge,itwouldbetootime-consumingtosolveDTDTAoptimallyorotherwise.Inourexperiments,weconsiderfourapproximatesolutionstoDTDTA:(Yl;Zl),(YU;ZU),(^Yl;Zl),and(^YU;ZU),wherethelasttwoarerenementsofthersttwo.Toevaluatethequalityandthecomputationtimesofthesesolutions,weconsiderthefour-nodenetworkinFigure 5{8 withtwoODpairs,(1,4)and(2,4).Inourexperiments,bothODpairshavethesamedemandpatternandallarcshavethesametravelcostfunction,linearorquadratic,asspeciedabove. First,wesolvedthelowerandupper-boundproblemswithusingtwolevelsofdiscretization,=1and=0:5.Asbefore,when=1,thediscrete-timeplanninghorizonis=f0;1;;9g.Ontheotherhand,when=0:5,becomes


Table5{3. Solutionsfromthelowerandupper-boundproblems:lineartravelcostfunction. Trac Intensity low 820.01580.0760.0 1187.51560.0372.5 medium 1200.02230.01030.0 1705.02230.0525.0 high 1500.02875.01375.0 2187.52870.0682.5 5{3 and 5{4 ),weassumethat,when=0:5,thereisnodemandatfractionaltimes(e.g.,at0.5,1.5,2.5,etc.)andthedemandsatintegraltimes(i.e.,1,2,3,etc.)areasshowninTable 5{1 Forbothtypesoftravelcostfunctions,thesizeoftheoptimalitygap(i.e.,qTuYuqTlYl)decreasesbyapproximately50%asdecreasesfrom1to0.5.However,theresultsinTables 5{3 and 5{4 suggestthatthereductioninthegapisduemainlytotheimprovementinthesolution,Yl,ofthelower-boundproblem.TheapproximatetraveldelaysasestimatedbyYuchangerelativelylittleforthetwovaluesof. Tables 5{5 and 5{6 comparethesolutionsfromDTDTA,(Y;Z),againsttwoapproximations,(^Yu;Zu)and(^Yl;Zl).Asinthetwo-nodeproblem,wesolveDTDTAusingSBBtoobtaina(integer)solution(Y;Z)withlessthan0:5%relativeoptimalitygap.Toobtain(^Yu;Zu),werstsolvetheupper-boundproblemusingXPress-MPtoobtain(Yu;Zu),a(integer)solutionwithlessthan0:5%optimalitygap,and,then,solvetheapproximationrenementproblem(with Table5{4. Solutionsfromthelowerandupper-boundproblems:quadratictravelcostfunction. Trac Intensity Low 600.01200.0600.0 900.01200.0300.0 Medium 822.21644.5822.2 1233.31644.5411.1 High 1124.52248.91124.5 1686.72248.9562.2


Table5{5. Qualityofrenedupperandlower-boundsolutions:lineartravelcostfunction. (^Yu;Zu) (^Yl;Zl) Rel. cpu Trac cpu Ratio Intensity Delay(sec) Delay(sec) Delay(sec) (%) cpu 1337.5027.42 1385.002.57 1392.502.38 3.55 10.711.5 Medium 1800.0015.92 1866.302.66 1815.302.90 0.85 6.05.5 High 2290.0095.02 2327.504.07 2315.001.25 1.09 23.376.0 Forbothlinearandquadratictraveltimefunctions,thetwoapproximationschemesprovidesolutionswithrelativelysmallerrorsusingmuchlessCPUtimerequiredtosolveDTDTA(seetheratiosofthecputimesinTables 5{5 and 5{6 ).Forquadratictraveltimefunctions,theapproximatesolutionsareidenticaltoDTDTAsolutions,exceptforthehightracintensitycasewhentheapproximatesolutionsareslightlybetter(by0:06%). Table5{6. Qualityofrenedupperandlower-boundsolutions:quadratictravelcostfunction. (^Yu;Zu) (^Yl;Zl) Rel. cpu Trac cpu Ratio Intensity Delay(sec) Delay(sec) Delay(sec) (%) cpu 1054.500.88 1054.500.09 1054.500.08 0.00 9.811.0 Medium 1543.806.62 1543.800.14 1543.800.34 0.00 47.319.5 High 2129.80501.17 2128.600.10 2128.600.13 -0.06 5011.73855.2




5 wehavediscussedaperiodicdiscretetimedynamictracassignmentproblem,whichisconstructedbasedontwoassumptions:(i)allcarsthatenterthearcduringthesametimeintervalexperiencethesametraveltimeandleavethearcduringthesametimeinterval,and(ii)thetraveltimeisafunctionofthenumberofcarsontheroad(seealsoNahapetyanandLawphongpanich[ 78 ]).Aswehaveseen,theinitialmathematicalformulationofsuchmodelleadstoamixedintegerproblemwithlinearconstraintsandanonlinearobjectivefunction.Bylinearizingtheobjective,onecanconstructanupperandalowerboundproblems.Althoughthesolutionofsuchproblemscanbemadearbitrarilyclosetothesolutionoftheinitialproblembydecreasingthediscretizationparameter,observethattheboundingproblemsbelongtotheclassoflinearmixedintegerprogram,whicharecomputationallyexpensivetosolve.Inthecaseoftheboundingproblems,thetaskbecomesmorechallengingbecauseofthespecialstructureofthefeasibleregion. Observethatthemodelisconstructedbasedonatime-expandednetworkandtherearebinaryvariables,za(t;s),associatedwiththearcsofthenetwork.Bydecreasingtheparameter,thenumberofthebinaryvariablesincreases.Forexample,givenatracnetworkG(N;A)andasetofpossiblediscretetraveltimesa(),thetotalnumberofthebinaryvariablesintheDTDTA-Uproblemisj()jPa2Aja()j.Ifreducesto=2,thenja(=2)j=2ja()j,j(=2)j=2j()j,andthetotalnumberofthebinaryvariablesinthenewproblemis 78


22j()jPa2Aja()j.Becauseofresourcelimitations,MIPsolverscannotsolvelargeproblems. Inthischapterweconsideraheuristicalgorithmtosolvetheboundingproblems.Althoughthesametechniquecanbeappliedtosolvebothboundingproblems,wemainlyfocussontheupperboundproblemDTDTA-U.Forconvenienceofreference,werestatetheproblembelow.min(x;y;z;g)qTuY Inthewell-knownheuristicalgorithmssuchasneighborhoodsearch,greedyalgorithmortabusearch,itisrequiredtomovefromonefeasiblesolutiontoanother.However,ndingafeasiblesolutiontotheDTDTA-Uproblemisnoteasy.Todemonstrate,consideraone-arc-network,a=(1;2),alineartraveltimefunctiona(xa(t))=0:3+0:05xa(t),andasetofpossible(discrete)traveltimesa=f1;2;3g.Inaddition,assumethatthetimehorizon[0;5)isdividedintoveintervals,and10carsenterintothearcateachdiscretetimet2=f0;1;2;3;4g.Byassigningthosecarstothearcsa(t;1)oftheTEnetwork,oneconcludesthatxa(t)=10,


6{5 )and( 6{6 )(seetheleftnetworkinFigure 6{1 ).ItcanbeshownthatthisisanoptimalsolutiontotheDTDTA-Uproblem.However,thereisanotherfeasiblesolution,where10carsareassignedtothearcsa(t;2)(seetherightnetworkinFigure 6{1 ).Usingthosesettings,xa(t)=20,a(xa(t))=1:3,andza(t;2)=1,8t2,whichagainsatisesconstraints( 6{5 )and( 6{6 ).Assumethatthesecondsolutionisknownandonedecidestoimprovethesolutionandmovetoanotherfeasiblesolutionbyassigningthecarsattimet=0toarca(0;1),i.e.,za(0;1)=1andza(0;2)=0.Asaresult,xa(1)=10anda(xa(1))=0:8,whichviolatesinequality( 6{5 ).Thesamefollowsfromthechangeofotherarcs;thus,weconcludethatthesecondsolutionisisolatedinthesensethattheadjacentsolutions,i.e.,changingonlyonearc,areinfeasible.FindinganadjacentfeasiblesolutionbecomesmorecomplicatedwhenthestaticnetworkG(N;A)islargerbecause,withrespecttoagivenpath,thechangesonupstreamarcshaveaninuenceontheowsofdownstreamarcs.Becauseheuristicalgorithmssimilartotheneighborhoodsearch,greedyalgorithms,andtabusearchrequiremovingfromonefeasiblesolutiontoaneighboringfeasiblesolution,thedicultiesofndinganeighboringsolutionmakesinappropriatetheuseofsuchtechniques. Figure6{1. Twofeasiblesolutions.


AnotherapproachtosolvetheDTDTA-Uproblemistherelaxationoftheintegralityofthevariableza(t;s)andconstructinganequivalentformulationwithcontinuousvariables.Todoso,replacetheconstraintsza(t;s)2f0;1gbyinequalities0za(t;s)1,i.e.z2[0;1]n,n=j()jPa2Aja()j,andza(t;s)(1za(t;s))0.Thelattercanbeincludedintotheobjectivefunctionwithapenalty.Asaresult,theproblemreducestoacontinuousconcaveminimizationoneandaglobalsolutionoftheresultingproblemisasolutionoftheDTDTA-Uproblem(see,e.g.,Horstetal.[ 54 ]orHorstandTuy[ 55 ]).Althoughitisknownthatanoptimalvectorofthebinaryvariables,z,representsoneoftheverticesofthendimensionalunitcube(eachvertexcorrespondstoanintegersolution),becauseofconstraints( 6{1 )-( 6{8 )mostofthemareinfeasibleanditishardtondanoptimalone. ObservethattheLPrelaxationoftheDTDTA-Uproblemprovidesalowerbound,whichisfarfromanoptimalone.Toillustrate,considertheDTDTA-Uproblem,wheretheconstraintsza(t;s)2f0;1garereplacedbytheinequalities0za(t;s)1.Tondoptimalvaluesofvariablesyka(t;s)andgkd(k)tintheresultingproblem,itissucienttosolvethefollowinglinearproblem.min(y;g)qTuYs.t.Byk+gk=bk8k2CXt2gkd(k)t=Xt2hkt8k2Cyka(t;s)0;gkd(k)t0;8t2;a2A;s2aandk2C


6{3 )and( 6{4 ).Theoptimalvaluesofthevariablesza(t;s)canbeobtainedbysolvingthefollowingsystemofequations. Noticethatthelastinequalityin( 6{9 )issatised8za(t;s)2[0;1],s6=1,becauseYa(t;s)=0,8s2a,s6=1.Inthecaseofs=1,asucientlylargevalueofMareducestheinequalitytoza(t;1)>0andmakessurethatarcsa(t;1)areallowedtohavepositiveowsgivenanypositivevaluesofthevariableza(t;1)(seeinequality( 6{7 )).Otherequationsin( 6{9 )areeasytosatisfyanditcanbeshownthatforanyvalueofxa(t)thesetofsolutionstothesystemisnotemptyandnotunique.However,becauseofthecongestionitishighlyunlikelythatatoptimalityoftheDTDTA-UproblemalldriversexperiencethefreeowtraveltimeandoneconcludesthatasolutionoftheLPrelaxationoftheproblemisnotrealistic,andtheoptimalobjectivefunctionvalueoftherelaxationproblemisfarfromtheoptimalvalueoftheobjectivefunctionofDTDTA-U. Despiteallcomplicationsdescribedabove,theDTDTA-Uproblemhasthefollowingusefulproperty:ifatoptimalitythetotalinowintoarcaattimetiszero,i.e.,Ps2aYa(t;s)=0,thenconstraint( 6{7 )issatisedforanyvalueofza(t;s);therefore,correspondingconstraints( 6{5 )-( 6{7 )canberemovedfromtheformulationandthesolutionoftheresultingproblemremainsthesame.Unknownvaluesofthevariablesza(t;s)canberestoredbysolvingthesystemofequations( 6{5 )-( 6{6 )usingthevaluesofxa(t;s). TheaboveanalysismotivatesdevelopingalowerboundproblemfortheDTDTA-U,whichis(i)tighterthantheLPrelaxation,(ii)easiertosolve


thantheconcaveminimizationproblemdiscussedabove,and(iii)preservestheabovementionedproperty.Inparticular,inthischapterweconsideranonlinearrelaxationoftheproblemwithbilinearconstraints.Usingtherelaxationtechnique,weproposeaheuristicalgorithmtosolvetheDTDTA-Uproblem. Fortheremainder,Sections 6.2 discussesthenonlinearrelaxationoftheDTDTA-Uproblem.Usingtherelaxation,inSection 6.3 weproposeaheuristicalgorithmtosolvetheDTDTA-Uproblem.NumericalexperimentsonthealgorithmareprovidedinSection 6.4 ,andnally,Section 6.5 concludesthechapter. yka(t;s)0;gkd(k)t0;xa(t)08t2;a2A;s2aandk2C(6{15) Observethat(i)intheDTDTA-Rproblemconstraints( 6{10 )-( 6{13 )arethesameascorrespondingconstraintsoftheDTDTA-Uproblem,(ii)theDTDTA-Rproblemdoesnotincludethebinaryvariablesandconstraints( 6{5 )and( 6{7 ),and(iii)theconstraints( 6{6 )arereplacedbybilinearconstraints( 6{14 ).


Case1:Pr2aYa(t;r)6=0.Fromequality( 6{16 )itfollowsthatza(t;s)=Ya(t;s) 6{5 )and( 6{7 )givenasucientlylargeMa;thus,theconstraintscanberemovedfromtheformulation.Inaddition,observethatafterappropriatesubstitutionsofthevariablesza(t;s)theconstraint( 6{6 )transformsintotheconstraint( 6{14 ),andthevariablesza(t;s)canberemovedfromtheformulation. Case2:Pr2aYa(t;r)=0.Observethatequation( 6{16 )andconstraint( 6{7 )aresatisedforanyvalueofthevariableza(t;s),s2a.Becauseconstraint( 6{7 )isredundant,removethevariablesza(t;s),s2a,fromtheformulationaswellascorrespondingconstraints( 6{5 )-( 6{7 ).Inaddition,noticethatconstraint( 6{14 )issatisedandcanbeaddedtotheformulationwithoutchangingthefeasibleregion. Basedontheaboveanalysis,oneconcludesthatbothproblemshavethesameoptimalobjectivefunctionvalueandthesameoptimalvaluesforthevariablesyka(t;s)andxa(t).Inaddition,theoptimalvaluesofthevariablesza(t;s)canbeobtainedusingthevaluesofyka(t;s)andxa(t).Inparticular,ifPr2aYa(t;r)6=0thenza(t;s)=Ya(t;s) 6{5 )-( 6{6 ). 6{16 )becausefromconstraints( 6{5 )-( 6{7 )itfollowsthateitherbothsidesof( 6{16 )arezero,i.e.,za(t;s)=Ya(t;s)=0,or


6{16 )totheDTDTA-Uproblemdoesnotchangethefeasibleregion.Next,observethataccordingtotheTheorem 6.2.1 FromTheorem 6.2.1 6{16 ).Nextweshowthattheconstraintsarenotredundantunlessatoptimalityalldriversexperiencethefreeowtraveltime.Inparticular,theconstraint( 6{16 )requiressendingaportionofthetotalow,i.e.,za(t;s)Pr2aYa(t;r),alongthearca(t;s)ifPr2aYa(t;r)6=0.Ontheotherhand,recallthatintheLPrelaxationoftheDTDTA-Uproblem,atoptimalityonlyarcsthatcorrespondtothefreeowtraveltimehaveapositiveow,i.e.,yka(t;s)=0,8s2a,s6=1,andthevariablesza(t;s)mayhavepositivevaluesforalls2a,s6=1,aslongastheysolvethesystem( 6{9 )(seeSection 6.1 ).Itcanbeshownthatthesolutionsatisesequation( 6{16 )onlyifza(t;1)=1solvesthesystem( 6{9 ).Thelatterholdsonlyintrivialproblemswithnocongestion.FromtheaboveanalysisitfollowsthattheDTDTA-RproblemhasatighterfeasibleregionthantheLPrelaxationoftheDTDTA-Uproblem. 6{14 )isneitherconvexnorconcave.However,itiscomputationallymoreattractivethantheconcaveminimizationproblemdiscussedinSection 6.1 .Inparticular,theDTDTA-Rproblemhasfewerconstraintsandvariables,anddoesnothaveapenaltytermintheobjectivefunction.Inaddition,theDTDTA-Rproblempreservestheabovementionedproperty;ifattimetanarc,a,hasnoinow,i.e.,Ps2aYa(t;s)=0,thenconstraint( 6{14 )issatisedforanyvalueofthevariablexa(t).


6{14 )andsolvingtheLPrelaxationoftheDTDTA-Uproblem.(NoticethatthelatterisequivalenttothendingtheshortestpathintheTEnetwork.)Asaresult,theoptimalvaluesofthevariablesxa(t)betterapproximatetheoptimalvaluesofthecorrespondingvariablesoftheDTDTA-Uproblem.However,thesolutionisunlikelytobefeasibletotheoriginalproblem;thusitisessentialtondanintegersolutionthathastheobjectivefunctionvalueascloseaspossibletotheoneprovidedbytheDTDTA-Rproblem. Intheheuristicalgorithm(seeProcedure 8 ),rstwesolvetheLPrelaxationoftheDTDTA-Uproblem,whichprovidesaninitialsolutionforDTDTA-Rproblem.NexttheproceduresolvesDTDTA-Rproblem.InStep3,ndingthevaluesof^za(t;s)iseasyandcanbeaccomplishedbyperformingasimplesearchtechnique.ByxingthebinaryvariablesoftheDTDTA-Uproblemtothevaluesof^za(t;s),i.e.,za(t;s)^za(t;s),theproblemreducestoalinearone.IftheresultingLPisfeasiblethenthealgorithmstopsandreturnsthesolution.Otherwise,thealgorithmrunstheUpSetprocedure(seeProcedure 9 )thengoestoStep2. constraints( 6{5 )and( 6{6 ).


forall8s2ado else forall8s2ado endif endfor

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numericalexperiments,wefounditmoreusefultotakethevaluesforparameterfromtheinterval[0:2;0:5].Asimilarprocedureappliestothecasewhenonlyonearcisused.Finally,ifthetotalinowintothearciszero,i.e.,Ps2aYa(t;s)=0,thenwerelaxrestictionsonthevariablesYa(t;s),8s2a.AlthoughtheDTDTA-Rproblemisnon-convexandrequiresndingaglobaloptimum,theUpSetprocedurepotentiallyeliminatesthecurrentsolutionfromfurtherconsiderationbynarrowingthefeasibleregion. TheabovedescribedheuristicalgorithmmaynotconvergeandtheperformanceofthealgorithmisdiscussedinSection 6.4 .However,ifthealgorithmdoesnotconverge,analternativeobjectivefunction(seeTable 6{1 )canbeusedtosolvetheproblem.Toconstructthealternativeobjectivefunction,observethatthecostvectorquconsistsofdiscretetraveltimess2a.Ontheotherhand,thevariablexa(t)iscomputedbasedonthesetofindicesa(t)(seeequation( 6{13 )).Inparticular,eacharca(t;s),s2a,isincludedintoatleastoneofthesetsa(t),forsomet2.Noticethatthearcmaybeincludedintomorethanoneset,andthetotalnumberofsuchsetsisequaltos=.(Notethats=isanintegerbecausesisamultipleof.)Asaresult,qTuY=Xa2AXt2Xs2as Ya(t;s)=Xa2AXt224Xr2a(t)Ya(t;r)35=Xa2AXt2xa(t)=eTx; Table6{1. Equivalentobjectivefunctions ObjectiveFunction1 ObjectiveFunction2 min(x;y;z;g)qTuY

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D{1 ,Appendix D ),wherethe4-nodenetworkhastwoorigin-destinationpairs,(1;4)and(3;4),andthe9-Nodesnetworkhasfourofthem;(1;8),(1;9),(2;8)and(2;9).Inaddition,weconsidertwoplanninghorizons,[0;10)and[0;30),traveltimefunctionsoftheforma(xa(t))=a+a(xa(t)=ca)a,wherea,a,caanda,aregeneratedrandomlyaccordingtothedistributionsspeciedinTable 6{2 .Thediscretizationparameterissettoone. ThedemandforeachODpairateachtimet2isgenerateduniformlybetween20and100cars.Thedistributionofthedemandischosentobeconsistentwiththeparametersofthetraveltimefunction.Inparticular,theaveragedemandisabout1=4oftheaveragecapacityca.Asaresult,onaverageupto4arcswith(average)arcowsof60carscanbeincludedintothecomputationofxa(t).Inotherwords,seta(t)consistsofupto4indices,whichcorrespondtothearcswith(average)owsof60cars.Thelatterisconsistentwithparametersandbecauseonaverage4<5=jaj.Thespecieddistributionsallowgeneratinglargevarietyofeasyandhardsolvableinstances. Foreachchoiceofthenetwork,timehorizonandobjectivefunction(seeTable 6{1 ,Appendix D ),50problemsarerandomlygeneratedandsolvedbytheheuristicalgorithmaswellastheCPLEXMIPsolver.Intheheuristicalgorithm,welimitthenumberofiterationsto500.Ifthealgorithmreachesthelimit,westoptheprocessandreportthattheheuristicalgorithmhasfailedtosolvetheinstance.TheaverageperformanceissummarizedinTable D{1 ,Appendix D .Becausetheheuristicalgorithmcouldnotsolvesomeoftheinstances,theaverageCPU Table6{2. Distributionsofparametersofrandomlygeneratedtraveltimefunctions uniform(2,8) uniform(200,300) random(1or2)

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timeandthequalityofthesolutionarecomputedbasedonthesetofthesolvedproblems.TheratiobetweentheCPUtimesoftheheuristicalgorithmandtheCPLEXsolveriscomputedforeachinstanceandthetablepresentstheaverageofthosenumbers. Onaveragetheheuristicalgorithmisabletosolveabout90%ofthesmallproblems(seethersttworowsinTable D{1 ,Appendix D )usingalmostthesameCPUtimeastheCPLEXsolver.However,inthecaseofthetimehorizon[0;30),theaverageCPUtimeofthelatterincreasesrapidly.Ontheotherhand,theCPUtimeoftheheuristicalgorithmremainsfairlysmallandthealgorithmisabletosolveabout80%oftheproblems.Inthecaseofthe9-nodenetworkandthetimehorizon[0;10),noticethattheperformanceofthealgorithmisdierentfordierentobjectivefunctions.Inparticular,thealgorithmrequiresmuchfewerresourcesandprovidesahigherqualitysolutioninthecaseofthesecondobjectivefunction.Inthelasttwoexperiments,theTEnetworkislarge,andwepostalimitof5000secondsofCPUtimeontheCPLEX.Inmostofthecases,theCPLEXterminatesduetotheCPUrestrictionandreturnsasolution,whichhasalowerqualitythantheoneprovidedbytheheuristicalgorithm.Inparticular,onaveragetheheuristicalgorithmprovidesasolution,whichisabout34%betterthanoneprovidedbytheCPLEXandusesabout1000-1500secondsoftheCPUtime. Althoughtheheuristicalgorithmcouldnotsolvesomeinstancesusingtherstorthesecondobjectivefunctions,onemayconsidersolvingtheproblemsonparallelmachinesusingbothobjectivefunctions,wherethealgorithmstopsifoneofthemsolvestheproblem.Table D{2 inAppendix D providestheresultsforthecombinedmode.ObservethatthepercentageoftheunsolvedproblemsistwicelessthanintheTable D{1 andthequalityofthesolutionremainsthesame.

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7{1 ,whichhastwoorigin-destinationpars,(1;4)and(2;4),andthetracdemandforeachODpairistwocars.Inaddition,assumethatthetraveltimesonarcs(1;3),(1;4),(2;3)and(2;4)areconstantandittakes2,15,4and16unitsof Figure7{1. 4-Nodenetworkandtracdemand. 92

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Figure7{2. Userequilibriumowsandtraveltimes. timetotraversethem,respectively.Ontheotherhand,assumethatthetraveltimeonarc(3;4)isafunctionofow 7{1 ). ItiseasytoverifythatthearcowsinFigure 7{2 satisfytheuserequilibriumcondition.Inparticular,twocarsofODpair(1;4)areassignedtopath1-3-4,whichisshorterthanpath1-4.InthecaseoftheODpair(2;4),paths2-3-4and Figure7{3. Systemoptimumowsandtraveltimes.

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Figure7{4. Tolleduserequilibriumowsandtraveltimes. 2-4havethesametraveltimeandonecarisassignedtoeachpath.Thetotaltraveltimeofallusersinthenetworkis60. NextconsiderthearcowsinFigure 7{3 .Thetotaltraveltimeofallcarsinthenetworkis47:67<60.However,observethatthesolutiondoesnotsatisfytheuserequilibriumcondition.Inparticular,paths1-3-4and2-3-4areshorterthanpaths1-4and2-4,respectively.ItiseasytoverifythattheowvectorinFigure 7{3 isasystemoptimumsolution. Thetollpricingproblemimposesadditionalcostsontheroadstotransformthesystemoptimumsolutionintoasolutiontoauserequilibriumproblemwithtolls(see,e.g.,Beckmann[ 6 ]andDafermosandSparrow[ 30 ]).Intheliteratureatollvectorthatallowsthistransformationisoftencalledavalidtollvector.Intheaboveexample,onemayconsideradding8.10unitsofdelaytothetraveltimeonarc(3;4),i.e.,(3;4)(x(3;4))=x(3;4)+x2(3;4)+8:10.Asaresult,thetraveltimeonpaths1-3-4and1-4arethesameandequalto15unitsoftime(seeFigure 7{4 ).Ontheotherhand,path2-3-4ismoreexpensivethanthealternativeone,i.e.,path2-4;therefore,thereisnoowonit,andweconcludethatthesolutionsatisestheuserequilibriumcondition.Thevalueoftimefordierentuserclassesandits

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inuenceonthesystemoptimumanduserequilibriumproblemsinthepresenceofroadpricingisdiscussedinEngelsonandLindberg[ 33 ]. Inthestaticnetworkowmodels,wherethedemandisknownandtimeinvariant,andthetraveltimeisafunctionofthearcow,thetollpricingframeworkscanbeclassiedasa\rst-best"anda\second-best"problems.Theformerassumesthateveryroadinthenetworkcanbetolled,anditiswellknownthatatollvectorderivedfromthemarginalsocialcostsisavalidtollvector.Bergendoretal.[ 8 ],andHearnandRamana[ 47 ]showthatitisnottheonlyvalidtollvectorandmathematicallydescribethesetofvalidtollvectors.Intheidealcase,whenthetraveltimefunctionisstrictlymonotone,andthesystemoptimumproblemisconvex,thesetofvalidtollvectorsassumesaverysimpleformofaconvexcone.Thetollsetallowsconstructingdierenttollpricingproblemswithasecondaryobjective(e.g.,minimizingthetotalrevenue,maximumtoll,ornumberoftollbooths).YildirimandHearn[ 106 ]extendedtheresultsfortheelasticdemandcase.ComputationalmethodsforsuchmodelsarediscussedinHearnetal.[ 48 ],Baietal.[ 3 ],andYildrimandHearn[ 106 ].Thesecond-bestmodelassumesthatonlyasubsetoftheroadscanbetolled.Thisandothertypesofrestrictionscanyieldtoanemptyvalidtollset.Inotherwords,thereisnoatollvectorthattransformsasystemoptimumsolutionintoasolutionofthetolleduserequilibriumproblem.Mathematicalformulationsofthesecond-bestproblemareeitherbileveloptimizationproblemsormathematicalprogramswithequilibriumconstraints(see,e.g.,Belleietal.[ 7 ],ChenandBernstein[ 24 ],Henderson[ 49 ],Labbeetal.[ 64 ],Ferrari[ 34 ],Brotcorneetal.[ 9 ],YangandLam[ 105 ],PatrikssonandRockafellar[ 84 ],andLawphongpanichandHearn[ 65 ]). Dynamicmodelsassumethatthedemandvariesduringaplanningtimehorizon,andthesetofdecisionvariablesinvolvesinowrates,outowrates,anddensities.Complexrelationshipbetweenthevariablesmakesthemathematical

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descriptionofthesystemoptimumaswellasuserequilibriumproblemsmorecomplicated.Despiteavarietyofmodelsforbothproblem,fewpapersaddressthedynamictollpricingframework.Agnew[ 1 ]presentsadynamicmodelforahighwayandusingoptimalcontroltheoryanalyzesmarginalcostsandassociatedtolls.CareyandSrinivasan[ 22 ]derivesystemmarginalcosts,userperceivedcostsanduserexternalitycosts.Theauthorsconsiderasetoftollsthatdependnotonlyonthelevelofcongestionbutalsotherateofchangeofcongestion.Usingoptimalcontroltheory,HuangandYang[ 56 ]proposeacongestionprisingproblemonanetworkofparallelrouteswithelasticdemand.AnotheroptimalcontrolformulationoftheproblemaswellasamarginalcostbasedtollvectorisdiscussedbyWieandTobin[ 97 ]. Inthischapter,weconsideradynamictollpricingframeworksimilartotheonedevelopedinBergendor[ 8 ],andHearnandRamana[ 47 ]forthestaticcase.Inparticular,theframeworkconsistsofthefollowingsteps:(i)solveasystemoptimumproblem,e.g.,theDTDTAproblemfromChapter 5 ,(ii)derivethesetofvalidtollvectorsforasolutionoftheproblemand(iii)solveatollpricingproblemwithasecondaryobjective.However,theDTDTAproblemaswellasmostofothersystemoptimumproblemswithdynamicsettingsarenon-convexandcanhavemultiplesolutions.Furthermore,ndinganexactsolutionofalargeproblemiscomputationallyexpensive,andheuristicalgorithmsprovideanapproximatesolutionoftheproblem.Inthischapter,weslightlychangethedenitionofthesetofvalidtollvectorsanddenethesetwithrespecttoanapproximateorafeasiblesolutionoftheSOproblem.Inparticular,givenanapproximate(feasible)solutionoftheSOproblematollvectoriscalledvalidwithrespecttothesolutionifittransformsthesolutionintoasolutionofthetolleduserequilibriumproblem.TheabovedenitionallowsdevelopingatollpricingframeworkbasedonanapproximatesolutionoftheSOproblem.

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Thekeycomponentofthetollpricingframeworkinthischapterisareducedtime-expandednetwork(RTE),whichisconstructedbasedonafeasiblesolutiontotheDTDTAproblem(seeChapter 5 ).Usingthenetwork,weprovethatthefeasiblesolutionisauserequilibriumsolutionifandonlyifitisasolutionofalinearminimizationproblemwiththeunderlyingRTEnetwork.ByincludingthetollcomponentintotheobjectiveoftheminimizationproblemandapplyingthedualitytheoryforLP,themathematicalexpressionofthesetofvalidtollvectorsconsistsoflinearequations.Inthecaseofasystemoptimumsolution,weshowthatthereisavalidtollvector,whichhasasimilarstructureasoneobtainedfromthemarginalcostsinthestaticcase. Fortheremainder,Sections 7.2 and 7.3 discussthereducedtime-expandednetworkandthesetofvalidtollvectors,respectively.ThetollpricingproblemswithdierentobjectivesandconstraintsarepresentedinSection 7.4 .SomeillustrativeexamplesareprovidedintheSection 7.5 andnally,Section 7.6 concludesthechapter. Let(^y;^g;^z)denoteafeasiblesolutionoftheDTDTAproblemand^a(t)=a(t)(^y)=a(P(;s)2a(t)Pk2Cyka(;s)).Foreachpair(a;t),a2A,t2,let^sa(t)denotetheelementfromthesetasuchthat^za(t;^sa(t))=1.Observethat^za(t;s)=0,foralls2a,s6=^sa(t).Asaresult,^yka(t;s)=0,8s2a;s6=^sa(t),andintheTEnetworkarcsa(t;s),s6=^sa(t),donothaveows.Byremovingthosearcs,i.e.,all

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PAGE 111;^sa1(t0));:::;an(tn1;^san(tn1))g.Thisshowsthat(pk;t0)correspondstopk(t0).Conversely,itiseasytoseethatpk(t0)alsocorrespondsto(pk;t0).Thus,thereisaone-to-onecorrespondencebetweenpair(pk;t)andpathpk(t).Furthermore,because^ai(ti1)representsthetraveltimeonarcaatthetimetoenterthearcti1,thetraveltimeonpathpkatthetimetoenterthearctisequaltothetraveltimeonpathpk(t)oftheRTE(^z)network. 7.2.1 itfollowsthatgivenafeasiblesolution,(^y;^g;^z),onecanconstructthereducedTEnetworkandanalyzethesolutionusingpathsintheRTE(^z)network.AlthoughRTE(^z)isasubgraphoftheTEnetwork,tosomeextentitcanbeviewedasastaticnetwork.Thismakesiteasiertoprovesomepropertiesofthefeasiblesolution.Becauseoftheone-to-onecorrespondence,belowweusenotationp(t)todescribepathp2Sk2CPkattimet. TheDTDTAproblemdiscussedinChapter 5 isasystemoptimumproblem,wheretheobjectiveistominimizethetotaldelayofalldriversduringthetimehorizon[0;T).Intheuserequilibriumproblem,theobjectiveistondafeasiblevectortotheDTDTAproblem,(^y;^g;^z),suchthatalluserschooseoneoftheshortestpathsamongallalternativepathsavailableatthetimetoenterthenetwork.Givenafeasiblevector(^y;^g;^z),^yka(t;^sa(t))representstheowonthearca(t;^sa(t))oftheRTE(^z)networkthatbelongstotheODpairkaswellasthenumberofcarsthatenterthearcaofthestaticnetworkduringthetimeinterval[t;t+).Let^up(t)denotethenumberofcarsenteringthepathp2Sk2CPkduringthetimeinterval[t;t+).Observethat^up(t)dependsonthevector^y.Similarly,letp(t)(^y)representthetraveltimeonpathpattimet.Usingthesenotations,formallytheconceptoftheuserequilibriumsolutioncanbedenedasfollows:

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7{1 )issatised.Thetheorembelowprovidesanalternativewaytocheckifafeasiblevectorisauserequilibriumsolution.Theprocedureinvolvessolvingthefollowingminimizationproblem,wheretheunderlyingnetworkisRTE(^z). where^anduarethevectorsofp(t)(^y)andup(t),respectively.ObservethattheproblemcanbedecomposedintojCjjjproblems,i.e.,oneproblemforeachpair(k;t)2C,wheretisa(discrete)timetoenterthenetwork.Inaddition,notethateachdecomposedproblemisequivalenttondingashortestpathfromnodeOkttooneofthecopiesofthedestinationnodeofthesameODpairk.AlthoughtheaboveproblemisformulatedbasedontheRTE(^z)network,SP(^y;^z)isanordinaryshortestpathproblemwithpathcostsp(t)(^y). 7{1 )ifandonlyifitisanoptimalsolutionofthecorrespondingSP(^y;^z)problem.

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7.2.1 ).Noticethat^up(t)isfeasibletotheSP(^y;^z)problem.Letup(t)denoteafeasiblesolutionofSP(^y;^z).Observethatup(t)[p(t)(^y)kt]0,8t2,k2Candp(t)2Pk(t).Because^up(t)[p(t)(^y)kt]=0,Xt2Xk2CXp(t)2Pk(t)[up(t)^up(t)][p(t)(^y)kt]0:+Xt2Xk2CXp(t)2Pk(t)p(t)(^y)up(t)^up(t)Xt2Xk2Ckt24Xp(t)2Pk(t)up(t)Xp(t)2Pk(t)^up(t)350+Xt2Xk2CXp(t)2Pk(t)p(t)(^y)up(t)^up(t)0+^Tu^T^u:

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whereB^zisthenode-arcincidentmatrixofRTE(^z),and^andykarethevectorsofa(t)(^y)andyka(t;^sa(t)),respectively.Usingthearcbasedformulation,Theorem 7.2.1 canberestatedasfollows.

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7.2 allowdevelopingatollpricingframeworksimilartothestaticcase(seeHearnandRamana[ 47 ]andBergendoretal.[ 8 ]). Let(^y;^g;^z)and^denoteafeasiblesolutionoftheDTDTAproblemandatollvector,respectively.Werefertothevector^asavalidtollwithrespectto(^y;^g;^z)if(^y;^g;^z)isasolutionofthetolleduserequilibriumproblem.Considerthefollowingproblem:

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wherethefeasibleregionisthesameasintheproblemSP(^y;^z)-Aandtheobjectivefunctionincludesthetollvector^inadditiontothetraveltime.ThefollowingcorollaryisadirectconsequenceofCorollary 7.2.1 ^kd(k)t^k8k2Candt2(7{7) wherekrepresentsthevectorofdualvariablesofcorrespondingnodeconservationconstrains( 7{2 ),andkarethedualvariablesoftheconstraints( 7{3 ).FromCorollary 7.3.1 itfollowsthatvector^isavalidtollifandonlyif(^y;^g)isasolutionoftheSP(^y;^z;^)-Aproblem.Observethat(^y;^g)isfeasibletotheproblem.Usingthedualitytheory,vector(^y;^g)isanoptimalsolutionofSP(^y;^z;^)-Aifandonlyiftherearevectors^kand^feasibletoequations( 7{5 )-( 7{7 ).

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Theorem 7.3.1 providesamathematicaldescriptionofthevalidtollset.Inparticular,givenafeasiblevector(^y;^g;^z),equations( 7{5 )-( 7{7 )describethesetofvalidtollsassociatedwithtriplet(^y;^g;^z).Usingtheset,secondarytollpricingproblemscanbeformulated.Observethatforallfeasiblevectors(^y;^g;^z),^=^0isavalidtollvector,i.e.,thevalidtollsetisnotempty. Let(y;g;z)andRTE(z)denotethesystemoptimumsolution,i.e.,thesolutionoftheDTDTAproblem,andthecorrespondingreducedtime-expandednetwork,respectively.Denea(t)=n(;s):(;s)2a(t);za(;s)=1o.Inwords,a(t)isasubsetofa(t)suchthatarca(;s)remainsintheRTE(z)network.BecauseRTE(z)consistsofarcsoftheforma(t;sa(t)),a(t)=(;sa()):(;sa())2a(t).Inaddition,givenzcomputethelowerandtheupperboundsintheequation( 5{9 ),i.e.,La(t)=Ps2a1a(s)za(t;s)andUa(t)=Ps2a1a(s)za(t;s).ConsiderthefollowingoptimizationproblemwiththeunderlyingRTE(z)networkstructure. whereBzdenotesthenode-arcincidentmatrixoftheRTE(z)networkandthevectorykconsistsofcomponentsyka(t;sa(t)).Inwords,problemDTDTA(z)istheDTDTAproblem,wherethevectorzisxedtothevalueofthevectorz.The

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theorembelowprovestheexistenceofanothervalidtollvectorwithrespecttothesystemoptimumsolution. 7{10 ),anda(t)a(t)=0,8a2Aandt2. wherevectors,,,andarethedualvariablesofcorrespondingconstraints( 7{8 ),( 7{10 ),and( 7{11 ),andiandjdenotethetailandtheheadnodesofthearca(t;sa(t)),respectively.Asaresult,ifyka(t;sa(t))>0thenka(t;sa(t))=0anda(t)(y)+Xr2j(t;sa(t))2a(r)"ryka(t;sa(t))a(r)(y)Xk2Cyka(r;sa(r))+a(r)a(r)#=jkik

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LetMSCa(t;sa(t))=Pr2j(t;sa(t))2a(r)hryka(t;sa(t))a(r)(y)Pk2Cyka(r;sa(r))+a(r)a(r)iandMSCdenotethevectorofMSCa(t;sa(t)).ByviewingthedualmultipliesikaspotentialsofthenodesintheRTE(z)network,itiseasytoshowthatisavalidtollvectorwithrespecttothesystemoptimumsolution(y;g;z);i.e.,(y;g)isanoptimumsolutionofSP(y;z;MSC)-A.Inaddition,ifr2issuchthat(t;sa(t))2a(r)thenryk1a(t;sa(t))a(r)(y)=ryk2a(t;sa(t))a(r)(y)=rxa(r)a(r)xa(t),8k1andk22C,k16=k2. FromtheKKTconditionitalsofollowsthata(t)0@La(t)X(;sa())2a(t)Xk2Cyka(;sa())1A=0; 7.3.1 itfollowsthattherearevectorsandsuchthatthesystemXk2C"(bk)Tk+kXt2hkt#=(+MSC)TXk2CykBTzk+MSC8k2Ckd(k)tk8k2Candt2 issatised.Ontheotherhand,theabovesystemfollowsdirectlyfromtheKKTconditionsoftheDTDTA(z)problem,andcorrespondingdualvariablesand

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ThetollvectorinTheorem 7.3.2 issimilartooneobtainedfromthemarginalsocialcostpriceinthestaticcase.Inparticular,therstcomponentofthetollvector,i.e.,Xr2j(t;sa(t))2a(r)"rxa(r)a(r)(xa(r))Xk2Cyka(r;sa(r))#; 7{10 ),andtheyrepresentthechangesintheobjectivefunctionvalueinthepresenceofminorperturbationsofthevaluesofUa(t)andLa(t),respectively.However,tochangethevaluesofUa(t)orLa(t),thebinaryvariableza(t;s)shouldbechangedaswell,andtheresultingvectormaynotbefeasibleoroptimal.Inthatsense,thesecondcomponentofthetollvector,i.e,Xr2j(t;sa(t))2a(r)a(r)a(r); 7{5 )-( 7{7 ).Intheprevioussection,wehaveshownthat(^;0;0)2=(^y;^g;^z)forallvectors(^y;^g;^z)feasibletotheDTDTA.Inthecaseofthesystemsolution,i.e.,(^y;^g;^z)=(y;g;z),Theorem 7.3.2 providesanothervalidtollvector;therefore,thereareatleasttwovectorsintheset=(y;g;z).Thelattersuggestsdevelopingtollpricingproblemswithasecondaryobjective,wheretheunderlyingfeasibleregionisconstructedbasedontheset=(y;g;z).Forinstance,atracmanagementmightbeinterestedin

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minimizingthetotaloperatingcost,costsofconstructingaproperinfrastructure(e.g.,constructingtollboothes),orminimizing/maximizingthetotalrevenue.Inaddition,thesetofvalidtollscanbenarrowedbyaddingconstraintstotheset=(y;g;z).Thelatterallowsustoconsidermorerealistictollsets.Althoughtheproblemsbelowareconstructedbasedonasystemsolution(y;g;z),asimilarframeworkappliestoanapproximatesolutionorafeasiblevectortotheDTDTAproblem. Letopa(t)andinfadenotebinaryvariablessuchthatopa(t)=8><>:1ifja(t;sa(t))j>00ifja(t;sa(t))j=0;andinfa=8>><>>:1ifPt2ja(t;sa(t))j>00ifPt2ja(t;sa(t))j=0: wherecopandcinfarethevectorsofoperatingandinfrastructurecosts,respectively,andMisasucientlylargenumber.Theoptimizationproblembelowminimizesthetotalrevenue.min(;;)2=(y;g;z)^yT 7{1 .Inparticular,addingthenonnegativityand/ormaximumtoll

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constraintspreventfromcharginganegativeand/oranunrealisticallylargetoll,respectively.Ifthetollsareallowedtobenegative;i.e.,itisallowedtosubsidizethedriversonparticulararcs,thenthemaximumsubsidyconstraintpreventspayingalargeamountofmoneytothedrivers.Theroadandtimerestrictionconstraintspreventtollingcertainroadsatcertaintimesofthetimehorizon,e.g.,from1:00amto5:00am.Finally,thevariabilityconstraintrestrictsthevariationofthetollduringtheplanninghorizon.Itisnoticedthataddingthoseconstraintsmaybetoorestrictiveandviolatethefeasibilityofthetollset.Insuchcasesthesecondbestpricingproblemshouldbeconsidered. Table7{1. Additionalconstraints Constraint 5 .Inthissection,theapproximate

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Table7{2. Distributionsofparametersofrandomlygeneratedtraveltimefunctions U[2,8] U[200,300] random(1or2) solutionisusedtoconstructavalidtollsetandconsiderseveralexamplesofthetollpricingproblemsbasedonthesetofvalidtollsandsomeoftheconstraintsfromTable 7{1 Allexperimentsareconductedonthe9-nodenetwork(seeFigure E{1 ,Appendix E ),whichhasfourODpars:(1,8),(1,9),(2,8),and(2,9).Intheproblems,thetimehorizonis[0;30),and=1,i.e.,=f0;1;:::;29g.Inaddition,thetraveltimefunctionsaregeneratedaccordingtotheformulaa(xa(t))=Aa+Ba(xa(t)=Ca)Da,whereparametersAa,Ba,CaandDa,arerandomnumbers(seeTable 7{2 ). Toimitatethecongestiononthearcs,foreachODpairademandisgeneratedaccordingtotheformulahkt=1:1t,wherethevalueoftdependsontimet(seeFigure 7{5 ),andisanumberuniformlygeneratedfromtheinterval[20;30].Asaresult,thedemandgraduallyincreasesduringthetimeinterval[0;9],remainsonthesamehighlevelduringthenext10unitsoftime,andthendecreasestotheinitiallevelattheendoftheplanninghorizon.Notethataccordingtotheformulathedemandvariesfromto2:36.Asitwasmentionedabove,wendanapproximatesolutionoftheproblembysolvingtheupperboundproblemandthe Figure7{5. Thevalueoft.

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renementproblem.Let(y;g;z)and=(y;g;z)denotetheobtainedsolutionandcorrespondingsetofvalidtolls,respectively. Oneofthetollpricingproblemsminimizesthetotalcollectedtoll,subjecttothesetofvalidtollvectors=(y;g;z)andadditionalnonnegativityandvariabilityconstraints.Belowisthemathematicalformulationoftheresultingproblem,whichwerefertoasMinRev(").min(;;)yTs.t.(;;)2=(y;g;z)"a(a;t;sa(t))(a;t+;sa(t+))"a8a2Aandt2[0;T]0 Inthenumericalexperimentswesolvetheproblemwithoutthevariabilityconstraintsaswellaswithvariabilityconstraints,where"=1,0.5,or0.1. Similarlyconsiderminimizingthetotalcostsubjecttothesamenonnegativityandvariabilityconstraints.Intheexperiments,thecopandcinfcostsarerandomlygeneratedfromtheintervals[5;10]and[100;150],respectively.WerefertotheresultingproblemasMinCost(").min(;;;op;inf)(cop)Top+(cinf)Tinfs.t.(;;)2=(y;g;z)"a(a;t;sa(t))(a;t+;sa(t+))"a8a2Aandt2[0;T]a(t;sa(t))Mopa(t)8a2Aandt2opa(t)infa8a2Aandt20;opa(t);infa2f0;1g;8a2A;andt2

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Inbothproblems,atoptimalitythetotalcollectedtollaswellasthetotalcostiscomputed(seeTable E{1 ,Appendix E ).Observethatinbothproblemstheobjectivefunctionvalueincreaseswiththedecreaseofthevalueof".Inthecaseof"=0:1,thevariabilityconstraintsaretoorestrictive,andthereisnononnegativevalidtollvectorthatsatisestheconstraints.Table E{2 inAppendix E describesthenumberoftollcollectingcentersthatarenecessarytobuild.Figures E{2 and E{3 (seeAppendix E )illustratetheinuenceofthevariabilityconstraintontheoptimaltollvector.Inparticular,theguresillustratethechangesinthetollonthearc(6;9)duringthetimeinterval[19;28].Observethatinthecaseof"a=0:5thetollvectorissmootherthaninothertwocases. IdeallyonewouldliketoconstructthesetofvalidtollsandthecorrespondingtollpricingproblemswithrespecttoanoptimalsolutionoftheDTDTAproblem.However,thelatterbelongstotheclassofnonlinearmixedintegerprograms,anditishardtondanexactsolutionoftheproblem.Section 7.5 discussesseveralexamplesoftollpricingproblemsbasedonanapproximatesolution,whichisobtainedusingthetechniquediscussedinChapter 5 .Forotherapproximatesolutions,thetollpricingframeworkissimilar.Inparticular,givenanapproximatesolution(y;g;z),constructthecorrespondingRTEnetwork,derivethesetof

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validtollsusingequations( 7{5 )-( 7{7 ),i.e.,theset=(y;g;z),andsolveatollpricingproblembasedontheset=(y;g;z).

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Inthisdissertationwehavediscussedbilinearreductionapproachestosolvethepiecewiselinearnetworkow,xedchargenetworkow,anddynamicpricingproblems.InparticularwehaveshownthataglobalsolutionofthebilinearproblemsisasolutionorleadstoasolutiontotheinitialMIPformulation.Theproposedheuristicalgorithmsallowndinganapproximatesolutiontothereductionproblem.Althoughtheproceduresoftenconvergetoaglobalsolution,theycanonlyguaranteetheconvergencetoalocalminimumoftheproblembecausethebilinearreductionproblemisnotconvex.Tondaglobalsolutiontotheproblem,onemayndusefulincorporatingthecuttingplanemethodinwhichthemasterproblemgeneratescuttingplanestoeliminatethecurrentlocalminimumfromthefeasibleregion,andtheheuristicprocedureisusedtondanewlocalminimumoftheresultingproblem.Becauseoftheeectivenessoftheheuristicprocedure,overallperformanceofthecuttingplanealgorithmisexpectedtobeeectiveaswell. Thedynamictracassignmentprobleminthedissertationisconstructedbasedontheassumptionthatthetraveltimeisafunctionofthedensity.Inaddition,weassumethatthedemandisxedandtheeventsoccurinaperiodicfashion.Themathematicalformulationoftheproblembelongstotheclassofnonlinearmixedintegerproblemsandiscomputationallyhardtosolve.However,thetheoreticalresultssuggestsolvingthelinearmixedintegerboundingproblemsinstead.Inparticular,wehaveshownthatbydecreasingthediscretizationparameterasolutionoftheboundingproblemcanbemadearbitrarilyclosetoasolutionoftheinitialproblem.Tosolvetheboundingproblem,wehaveproposed 115

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aheuristicalgorithm,whichisbasedonabilinearrelaxationofthefeasibleregion.Asaresult,ineachiterationitisrequiredtondalocalminimumofthebilinearproblem.AvailablecommercialsolversspendabouthalfofthetotalCPUtimeonndingafeasiblesolutionduringtherstiterationoftheheuristicprocedure;therefore,afastprocedureforndingafeasiblesolutionisdesirable.Inaddition,wewouldliketoexploreotheralgorithmsthatcanbeusedtondalocalminimumoftheproblem. InChapter 7 wehaveusedCPLEXLPandMIPsolverstosolveMinRev(")andMinCost(")tollpricingproblems,respectively.TosolvelargeMinRev(")problemsadecompositiontechniquesimilartheoneinBaietal.[ 3 ]canbeutilized.InthecaseofMinCost("),theprobleminvolvesbinaryvariables,andtosolvelargeproblemsaheuristicalgorithmsshouldbedeveloped.

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2 TableA{1. Setofproblems. SetNo 123522U[10,20]5 2 10 3 U[20,30]5 4 10 5 U[30,40]5 6 10 7 2010033U[10,20]5 8 10 9 U[20,30]5 10 10 11 U[30,40]5 12 10 13 4030044U[10,20]5 14 10 15 U[20,30]5 16 10 17 U[30,40]5 18 10 19 10020002020U[10,20]5 20 10 21 U[20,30]5 22 10 23 U[30,40]5 24 10 25 20050005050U[10,20]5 26 10 27 U[20,30]5 28 10 29 U[30,40]5 30 10

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TableA{2. Computationalresultsofsets1-18:qualityofthesolutionandtheCPUtimes. CPUTime Iterations DCUP DSSP DCUPDSSP DCUPDSSP RE(%) RE(%) Aver.Aver. Aver.Aver. SetNo (min,max) (min,max) (min,max)(min,max) (min,max)(min,max) 1 1.39 1.45 0.020.05 2.235.30 (0.00,8.47) (0.00,9.19) (0.01,0.04)(0.02,0.10) (2,3)(3,9) 2 1.43 1.55 0.050.14 2.205.47 (0.00,9.45) (0.00,9.45) (0.03,0.09)(0.05,0.22) (2,3)(3,9) 3 0.79 0.73 0.020.06 2.175.77 (0.00,3.48) (0.00,5.04) (0.01,0.04)(0.02,0.09) (2,3)(3,8) 4 0.81 0.91 0.050.14 2.205.67 (0.00,3.47) (0.00,10.35) (0.03,0.09)(0.07,0.23) (2,3)(3,8) 5 0.80 0.89 0.070.17 2.376.27 (0.00,4.22) (0.00,6.03) (0.05,0.10)(0.06,0.26) (2,4)(3,9) 6 0.83 0.88 0.060.16 2.506.43 (0.00,4.42) (0.00,6.11) (0.04,0.11)(0.08,0.23) (2,4)(3,9) 7 1.05 1.25 0.070.23 2.478.20 (0.00,6.24) (0.00,5.63) (0.04,0.16)(0.12,0.38) (2,5)(5,13) 8 1.13 1.28 0.060.23 2.478.73 (0.00,6.12) (0.00,5.59) (0.03,0.12)(0.12,0.39) (2,4)(5,14) 9 0.73 1.02 0.080.23 2.738.33 (0.00,3.48) (0.00,5.38) (0.05,0.12)(0.13,0.31) (2,4)(5,11) 10 0.91 1.20 0.030.09 2.477.63 (0.00,5.59) (0.00,5.34) (0.01,0.05)(0.03,0.16) (2,4)(3,14) 11 0.92 0.69 0.070.23 2.508.80 (0.00,4.31) (0.00,4.13) (0.03,0.13)(0.03,0.13) (2,4)(5,13) 12 0.96 0.60 0.070.23 2.879.57 (0.00,4.55) (0.00,3.93) (0.04,0.13)(0.13,0.33) (2,5)(6,14) 13 0.71 1.25 0.070.27 2.5310.00 (0.00,6.27) (0.00,6.40) (0.04,0.10)(0.16,0.41) (2,4)(6,17) 14 0.92 1.46 0.070.30 2.6310.77 (0.00,6.06) (0.00,6.42) (0.03,0.11)(0.13,0.54) (2,3)(6,17) 15 0.99 1.34 0.100.34 3.0010.47 (0.00,3.03) (0.11,3.36) (0.04,0.21)(0.21,0.62) (2,6)(6,18) 16 1.32 1.28 0.070.28 2.8011.30 (0.00,4.85) (0.00,3.43) (0.02,0.14)(0.09,0.56) (2,5)(8,18) 17 1.03 0.96 0.090.36 2.7711.73 (0.00,4.26) (0.00,2.83) (0.06,0.17)(0.06,0.17) (2,6)(8,17) 18 1.03 1.00 0.070.28 3.2712.60 (0.00,4.18) (0.00,2.91) (0.02,0.22)(0.13,0.52) (2,7)(10,17)

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TableA{3. Computationalresultsofsets1-18:DSSPvs.DCUP. SetNo (%)(%)(%)(%) 1 0.2120773 (3.42) 2 0.2523374 (3.43) 3 0.30271756 (4.37) 4 0.52232057 (9.54) 5 0.43403030 (1.80) 6 0.36303060 (1.75) 7 0.71403030 (4.91) 8 0.75333730 (5.14) 9 0.62502030 (5.13) 10 0.80502723 (4.92) 11 0.17473023 (1.09) 12 0.11373726 (0.54) 13 0.67631027 (3.40) 14 0.77501337 (4.33) 15 0.58602020 (2.38) 16 0.58374023 (2.61) 17 0.45504010 (1.28) 18 0.43603010 (1.37) A-aver.(max)improvement;B-DCUPisbetterthanDSSP; C-DSSPisbetterthanDCUP;D-botharethesame.

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TableA{4. Computationalresultsforsets19-30. CPUTime Iterations DCUPvs.DSSP DCUPDSSP DCUPDSSP Set ABCD Aver.Aver. Aver.Aver. No (%)(%)(%)(%) (min,max)(min,max) (min,max)(min,max) 19 0.87570.5843 0.401.69 4.7019.59 (2.02)(1.28) (0.18,0.78)(1.04,2.90) (2,9)(13,34) 20 0.56430.3757 0.492.03 5.1720.90 (1.74)(1.06) (0.27,0.83)(1.11,3.89) (3,10)(13,36) 21 0.63640.4233 0.441.76 5.1020.03 (2.09)(1.26) (0.21,0.91)(1.14,2.19) (3,10)(15,24) 22 0.66700.3030 0.511.96 6.0322.57 (1.74)(0.62) (0.24,1.12)(1.12,2.55) (3,13)(14,32) 23 0.54770.4223 0.462.03 5.3323.13 (1.04)(0.95) (0.81,0.22)(1.32,2.61) (3,9)(16,30) 24 0.50570.4643 0.492.06 5.7023.70 (1.27)(1.52) (0.23,0.88)(1.19,2.94) (3,9)(15,33) 25 0.58330.2767 1.345.31 6.0322.90 (1.38)(0.79) (0.65,2.41)(4.55,6.95) (3,11)(19,29) 26 0.72370.4663 1.716.50 7.8027.83 (1.63)(1.39) (0.60,2.94)(4.63,8.54) (3,14)(21,36) 27 0.56630.2537 1.635.96 6.9025.03 (1.27)(0.71) (0.51,3.59)(4.17,9.95) (3,14)(19,40) 28 0.48700.2830 1.976.83 8.8029.03 (1.14)(1.29) (1.02,3.21)(4.89,9.03) (5,15)(23,42) 29 0.34700.1430 1.836.28 7.9726.27 (0.77)(0.36) (0.84,3.39)(4.78,8.71) (4,14)(21,37) 30 0.39670.2933 2.367.21 10.1330.60 (0.89)(0.69) (1.06,4.11)(5.29,10.75) (4,19)(24,43) A-aver.(max)offDSSPfDCUP C-aver.(max)offDCUPfDSSP Computationalresultsforthecombinedmode. A(%)B(%) A(%)B(%) Aver. Aver. Aver. SetNo (min,max) SetNo (min,max) SetNo (min,max) 1 0.4610 11 0.1947 21 0.17100 (0.02,0.70) (0.01,0.53) (0.01,0.58) 2 0.437 12 0.1643 22 0.1293 (0.01,0.85) (0.01,0.54) (0.01,0.57) 3 0.5920 13 0.6150 23 0.1597 (0.04,1.57) (0.01,3.40) (0.01,0.47) 4 0.6120 14 0.4437 24 0.0997 (0.04,1.87) (0.04,1.40) (0.01,0.39) 5 0.3043 15 0.2057 25 0.17100 (0.07,1.02) (0.03,0.74) (0.01,0.49) 6 0.4030 16 0.1443 26 0.13100 (0.02,1.09) (0.02,0.68) (0.01,0.61) 7 0.3140 17 0.1957 27 0.16100 (0.01,1.14) (0.01,0.90) (0.02,0.44) 8 0.2640 18 0.3060 28 0.14100 (0.02,1.35) (0.01,1.62) (0.02,0.38) 9 0.3347 19 0.2793 29 0.12100 (0.03,0.97) (0.01,1.32) (0.02,0.32) 10 0.5940 20 0.1893 30 0.08100 (0.01,3.48) (0.01,1.08) (0.02,0.29) A-percentageofimprovement;B-percentageofproblemsthatareimproved.

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3 TableB{1. Setofproblems. Set Spp/DemVar.Fixed No No G1 1 20/1003/3U[1,5]U[50,100] 2 U[100,200] 3 U[200,400] 4 U[10,20]U[50,100] 5 U[100,200] 6 U[200,400] 7 U[30,40]U[50,100] 8 U[100,200] 9 U[200,400] G2 1 40/3004/4U[1,5]U[50,100] 2 U[100,200] 3 U[200,400] 4 U[10,20]U[50,100] 5 U[100,200] 6 U[200,400] 7 U[30,40]U[50,100] 8 U[100,200] 9 U[200,400] G3 1 100/100010/10U[1,5]U[50,100] 2 U[100,200] 3 U[200,400] 4 U[10,20]U[50,100] 5 U[100,200] 6 U[200,400] 7 U[30,40]U[50,100] 8 U[100,200] 9 U[200,400] G4 1 150/300015/15U[1,5]U[50,100] 2 U[100,200] 3 U[200,400] 4 U[10,20]U[50,100] 5 U[100,200] 6 U[200,400] 7 U[30,40]U[50,100] 8 U[100,200] 9 U[200,400]

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TableB{2. ComputationalresultsofgroupsG1andG2:qualityofthesolutionsandtheCPUtimes. Average Var.Fixed FinalSol.* BestSol.** CPUTime costcost ADCUPDSSP ADCUPDSSP ADCUPDSSP G1U[1,5]U[50,100] 2.417.5 1.67.8 0.891.47 (0.0,13.3)(0.0,53.0) (0.0,13.3)(0.0,21.8) U[100,200] 3.39.4 2.69.4 0.911.60 (0.0,16.9)(0.0,33.4) (0.0,16.9)(0.0,33.4) U[200,400] 3.513.4 2.710.3 0.901.66 (0.0,14.9)(0.0,34.3) (0.0,14.9)(0.0,34.2) U[10,20]U[50,100] 0.61.1 0.20.2 0.430.45 (0.0,3.7)(0.0,7.3) (0.0,1.5)(0.0,2.3) U[100,200] 0.84.3 0.51.3 0.460.61 (0.0,2.8)(0.0,24.2) (0.0,2.8)(0.0,10.0) U[200,400] 1.612.9 0.95.0 0.450.72 (0.0,8.7)(0.0,43.1) (0.0,4.3)(0.0,16.2) U[30,40]U[50,100] 0.20.5 0.10.1 0.330.20 (0.0,1.7)(0.0,4.8) (0.0,0.9)(0.0,1.5) U[100,200] 0.51.3 0.20.5 0.360.27 (0.0,1.9)(0.0,6.1) (0.0,1.9)(0.0,2.9) U[200,400] 0.91.8 0.40.9 0.390.48 (0.0,5.3)(0.0,10.1) (0.0,2.5)(0.0,6.0) G2U[1,5]U[50,100] 3.417.2 2.511.5 1.113.16 (0.0,15.3)(0.0,41.1) (0.0,11.1)(0.0,34.1) U[100,200] 4.117.0 3.715.1 1.093.32 (0.0,15.3)(0.0,37.8) (0.0,14.4)(0.0,31.2) U[200,400] 5.413.8 5.113.4 1.053.77 (0.0,13.3)(0.0,36.7) (0.0,12.2)(0.0,31.4) U[10,20]U[50,100] 0.52.4 0.11.2 0.541.20 (0.0,2.2)(0.0,17.8) (0.0,4.4)(0.0,4.4) U[100,200] 1.08.9 0.54.1 0.531.46 (0.0,6.1)(0.0,28.1) (0.0,4.3)(0.0,15.2) U[200,400] 2.511.5 1.66.3 0.491.39 (0.0,6.0)(0.0,29.0) (0.0,4.7)(0.0,15.0) U[30,40]U[50,100] 0.20.7 0.10.4 0.560.76 (0.0,1.4)(0.0,2.4) (0.0,1.6)(0.0,1.6) U[100,200] 0.82.0 0.21.1 0.571.15 (0.0,5.1)(0.0,6.5) (0.0,1.8)(0.0,4.9) U[200,400] 1.44.5 0.92.7 0.531.30 (0.0,4.9)(0.0,18.8) (0.0,4.6)(0.0,7.5) A-ADCUPisbetterthanDSSP;B-DSSPisbetterthanADCUP. *Thenalsolutionreturnedbythealgorithmwhenitstops.**Thebestsolutionfoundduringtheiterativeprocess.

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TableB{3. ComputationalresultsofgroupsG1andG2:thepercentageofproblemwhereoneofthealgorithmsndsabettersolutionthananotherone. Var.Fixed FinalSol.* BestSol.** costcost AB AB G1U[1,5]U[50,100] 7710 770 U[100,200] 7013 6010 U[200,400] 870 703 U[10,20]U[50,100] 237 1310 U[100,200] 4320 307 U[200,400] 6310 5313 U[30,40]U[50,100] 2310 1313 U[100,200] 3017 3010 U[200,400] 3020 2020 G2U[1,5]U[50,100] 9010 937 U[100,200] 903 900 U[200,400] 903 873 U[10,20]U[50,100] 5713 6310 U[100,200] 873 807 U[200,400] 807 803 U[30,40]U[50,100] 4017 407 U[100,200] 6713 537 U[200,400] 7310 7010 A-ADCUPisbetterthanDSSP;B-DSSPisbetterthanADCUP. *Thenalsolutionreturnedbythealgorithmwhenitstops.**Thebestsolutionfoundduringtheiterativeprocess.

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TableB{4. ComputationalresultsofgroupsG3andG4. PercentageofProblems Average Var.Fixed Aver.(min,max) FinalSol.* BestSol.** CPUTime costcost FinalSol.* BestSol.** AB AB ADCUPDSSP G3U[1,5]U[50,100] 16.8 13.8 973 1000 2.1012.86 (-0.4,39.5) (0.4,23.6) U[100,200] 17.0 16.9 973 973 1.9214.16 (-2.4,29.6) (-0.7,28.7) U[200,400] 16.0 15.3 1000 1000 1.9614.34 (3.6,29.8) (3.6,29.8) U[10,20]U[50,100] 3.3 1.2 937 903 2.1513.08 (-0.4,7.4) (-0.2,2.9) U[100,200] 7.7 3.9 973 970 2.2213.25 (-0.8,16.4) (0.0,7.6) U[200,400] 15.7 8.1 1000 1000 2.2412.97 (0.7,29.9) (0.2,13.8) U[30,40]U[50,100] 1.3 0.5 907 837 2.1111.38 (-0.3,3.7) (-0.1,1.6) U[100,200] 1.9 1.2 8713 930 2.1613.04 (-1.2,5.5) (0.0,3.5) U[200,400] 4.7 3.2 1000 1000 2.2613.56 (0.5,14.4) (0.3,7.5) G4U[1,5]U[50,100] 21.6 17.1 1000 1000 3.7261.31 (8.2,32.6) (7.8,26.4) U[100,200] 21.2 20.7 1000 1000 3.7267.87 (13.5,32.5) (14.1,31.5) U[200,400] 18.4 18.4 1000 1000 3.5967.57 (10.7,31.8) (10.7,31.5) U[10,20]U[50,100] 5.0 1.1 1000 900 4.4558.87 (0.1,10.6) (0.0,3.6) U[100,200] 12.3 2.9 1000 970 4.3963.07 (4.2,23.0) (0.0,6.6) U[200,400] 18.2 6.0 1000 1000 4.2369.47 (8.9,29.8) (1.1,11.8) U[30,40]U[50,100] 1.8 0.3 1000 900 4.2458.46 (0.0,3.9) (0.0,1.0) U[100,200] 4.0 0.7 1000 870 4.3057.12 (0.6,9.4) (0.0,2.0) U[200,400] 8.8 1.3 1000 7723 4.4859.79 (3.2,15.5) (-1.5,3.8) A-ADCUPisbetterthanDSSP;B-DSSPisbetterthanADCUP. *Thenalsolutionreturnedbythealgorithmwhenitstops.**Thebestsolutionfoundduringtheiterativeprocess.

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4 TableC{1. Thequalityofthesolution:Procedure 6 Procedure 6 usingProcedure 7 tondvector" =1=2 ObjRE%A ObjRE%A ObjRE%A 5-12[10,100] 132,803 131,2061.19131,3121.11131,6120.88[50,150] 216,831 215,5620.58215,9560.40215,9920.39[100,200] 283,494 282,4180.38282,7520.26282,9130.20[150,250] 320,602 319,3250.39319,5010.34319,5180.345-52[10,100] 543,397 537,5881.07538,1820.97539,2640.76[50,150] 910,778 906,2110.50907,3040.381 909,1560.183 [100,200] 1,209,347 1,206,8400.211,208,1550.102 1,208,4610.072 [150,250] 1,378,794 1,375,4310.251,376,0590.201,377,2270.111 10-12[10,100] 246,410 243,2901.29244,2680.87244,7540.70[50,150] 420,000 417,0320.70417,0340.70418,3370.40[100,200] 563,297 562,0490.221 562,2860.18562,6650.112 [150,250] 648,362 646,2400.33646,1870.331 646,8750.232 10-52[10,100] 1,142,132 1,132,2920.871,134,6300.651,136,3690.51[50,150] 1,878,849 1,872,7710.321,873,8330.271,877,1280.091 [100,200] 2,470,661 2,466,9890.152,469,4740.053 2,470,966-0.017 [150,250] 2,799,250 2,796,3960.101 2,798,0490.043 2,798,8080.025 20-12[10,100] 542,549 537,3460.95539,2100.61539,7540.51[50,150] 881,231 875,9630.59877,1760.46878,8360.27[100,200] 1,152,475 1,149,7010.241,150,2680.191,150,9670.13[150,250] 1,305,170 1,301,5870.281,302,1340.241 1,302,9630.173 20-52[10,100] 2,303,214 2,285,5080.772,286,7160.712,292,5450.46[50,150] 3,791,838 3,776,7680.403,780,3100.313,787,1920.121 [100,200] 4,993,056 4,983,5990.194,987,7460.114,992,3680.013 [150,250] 5,671,140 5,661,3380.175,666,8160.085,669,5740.033 A-NumberofproblemswheretheheuristicprocedureprovidesabettersolutionthanCPLEX.

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TableC{2. Thequalityofthesolution:Procedure 5 Procedure 5 ObjRE% 5-12[10,100] 1.02 132,803 123,7476.84 [50,150] 0.95 216,831 208,9763.65 [100,200] 0.86 283,494 278,3421.82 [150,250] 0.76 320,602 315,9691.45 5-52[10,100] 1.03 543,397 498,5038.27 [50,150] 0.96 910,778 873,7494.08 [100,200] 0.88 1,209,347 1,183,3192.16 [150,250] 0.78 1,378,794 1,357,7181.53 10-12[10,100] 1.13 246,410 226,5388.11 [50,150] 0.98 420,000 403,3603.96 [100,200] 0.98 563,297 553,2781.78 [150,250] 0.84 648,362 641,3891.08 10-52[10,100] 1.06 1,142,132 1,059,6957.24 [50,150] 0.98 1,878,849 1,811,7663.58 [100,200] 0.98 2,470,661 2,425,8021.82 [150,250] 0.77 2,799,250 2,765,5601.20 20-12[10,100] 1.08 542,549 504,7716.97 [50,150] 1.00 881,231 850,3083.51 [100,200] 1.00 1,152,475 1,133,5551.64 [150,250] 0.79 1,305,170 1,292,0681.01 20-52[10,100] 1.08 2,303,214 2,139,8377.11 [50,150] 1.00 3,791,838 3,658,2023.53 [100,200] 1.00 4,993,056 4,900,8791.85 [150,250] 0.79 5,671,140 5,601,9321.22 TheCPUtimeoftheprocedures. Procedure 6 Procedure 5 CPUA CPUA CPUA CPUA 5-12[10,100] 335 0.52641 0.78432 1.78188 0.191,811 [50,150] 1,113 0.711,570 0.831,341 2.29486 0.205,621 [100,200] 1,580 0.602,656 0.841,886 2.55617 0.266,148 [150,250] 559 0.74752 0.96583 3.10180 0.212,651 5-52[10,100] 821 10.9675 14.9855 40.9720 3.77218 [50,150] 348 12.7327 18.4219 52.067 4.8272 [100,200] 391 15.8625 19.0620 56.017 5.7967 [150,250] 596 12.7147 28.5121 60.4310 3.23185 10-12[10,100] 820 1.19688 1.64500 4.27192 0.392,096 [50,150] 748 1.24602 1.87401 5.48137 0.421,802 [100,200] 802 1.35594 2.05392 5.77139 0.441,814 [150,250] 660 1.47449 2.16306 5.30124 0.431,542 10-52[10,100] 376 24.5415 35.4211 103.824 8.3045 [50,150] 324 25.6913 39.518 115.943 10.1932 [100,200] 408 27.1915 39.9810 118.083 11.8934 [150,250] 327 26.3412 37.679 107.773 7.1646 20-12[10,100] 857 2.51341 3.69232 10.8279 0.851,005 [50,150] 657 2.72242 4.21156 12.7252 0.92717 [100,200] 633 3.07206 4.00158 12.0553 0.93684 [150,250] 675 2.75246 4.04167 10.7263 0.77873 20-52[10,100] 890 52.0217 76.7412 237.574 17.3851 [50,150] 750 55.2714 86.829 256.993 28.6426 [100,200] 756 58.0413 84.789 248.123 33.3123 [150,250] 785 57.7014 77.9810 236.523 14.1156

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6 FigureD{1. TwoNetworks. TableD{1. Computationalresultsoftheexperiments. HeuristicApproach Cplex Rel. ACPUIter. CPU Err. B (%)(sec.) (sec.) (%) (ratio) 4-Nodes,[0,10),OBJ1 12%0.8110.23 0.84 4.09% 1.74 4-Nodes,[0,10),OBJ2 8%1.0310.35 1.00 4.13% 1.41 4-Nodes,[0,30),OBJ1 20%19.7634.47 376.58 3.92% 22.62 4-Nodes,[0,30),OBJ2 22%27.0538.64 396.51 3.32% 9.55 9-Nodes,[0,10),OBJ1 28%633.0040.58 1,515.44 4.16% 7.80 9-Nodes,[0,10),OBJ2 10%111.929.98 1,758.65 2.09% 38.91 9-Nodes,[0,30),OBJ1 30%1338.377.56 5,000.30 -3.33% 12.65 9-Nodes,[0,30),OBJ2 27%1029.794.53 4,945.14 -4.22% 14.28 A-percentageoftheproblemsnotsolvedbytheheuristic B-CPUCplex/CPUheur:

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TableD{2. Computationalresultsofthecombinedmode. HeuristicApproach Cplex Rel. ACPU CPU Err. B (%)(sec.) (sec.) (%) (ratio) 4-Nodes,[0,10) 4%0.59 0.97 3.63% 1.85 4-Nodes,[0,30) 10%19.91 543.00 3.70% 24.22 9-Nodes,[0,10) 6%166.86 1,359.16 2.50% 30.27 9-Nodes,[0,30) 14%706.03 4,952.60 -3.62% 16.97 A-percentageoftheproblemsnotsolvedbytheheuristic B-CPUCplex/CPUheur:

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7 FigureE{1. 9-nodenetwork. TableE{1. Thetotalcollectedtollandthetotalcostforeachproblemandparameter". TotalCost A"a=1"a=0:5"a=0:1 A"a=1"a=0:5"a=0:1 385440155157NoSol. 267032983870NoSol. 543355636978NoSol. 124916242418NoSol. TableE{2. Thenumberoftollcollectingcentersforeachproblemandparameter". NumberofTollCollectingCenters A"a=1"a=0:5"a=0:1 171617N/A 8912N/A A-NoVariabilityConstraint 129

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FigureE{2. Thetollvectorfordierentvaluesof"intheMinRev(")problem. FigureE{3. Thetollvectorfordierentvaluesof"intheMinCost(")problem.

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ArtyomNahapetyanwasbornonApril15,1974,inGiumri,Armenia.HereceivedhisbachelordegreeinmathematicsfromYerevanStateUniversityin1996,MSdegreeinmathematicsfromArmenianStateEngineeringUniversityin1998,andMSdegreeinindustrialandsystemsengineeringfromAmericanUniversityofArmeniain2001.In2002heenteredthegraduateprograminindustrialandsystemsengineeringattheUniversityofFlorida.HereceivedhisPh.D.degreeinindustrialandsystemsengineeringfromtheUniversityofFloridainAugust2006. 140