Variable Disaggregation in Network Flow ... - Semantic Scholar

Variable Disaggregation in Network Flow Problems with Piecewise Linear Costs Keely L. Croxton

Bernard Gendron

Fisher College of Business

D´epartement d’informatique

The Ohio State University

et de recherche op´erationnelle and Centre de recherche sur les transports Universit´e de Montr´eal

Thomas L. Magnanti School of Engineering and Sloan School of Management Massachusetts Institute of Technology October 31, 2002

Network flow problems with non-convex piecewise linear cost structures arise in many application areas, most notably in freight transportation and supply chain management. In the present paper, we consider mixed-integer programming (MIP) formulations of a generic multi-commodity network flow problem with piecewise linear costs. The formulations we study are based on variable disaggregation techniques, which have been used for a while to derive strong MIP formulations for variants of the fixed-charge network flow problem, a special case of our generic problem. To the best of our knowledge, variable disaggregation techniques have not been studied extensively within the framework of a general non-convex piecewise linear cost function, although a few authors have used them in this context (a recent example is the paper by Croxton, Gendron and Magnanti [2]). Given a directed network G = (V, A), with V , the set of nodes, A, the set of arcs, supplies and demands of multiple commodities at the nodes and capacities at the arcs, we consider the problem of finding the minimum cost multi-commodity flow when the objective is the sum of |A| piecewise linear functions. More specifically, on each arc a of the network, the cost is a function, g a , of the total flow, xa , on the arc, with the unit flow cost and fixed charge varying according to the flow on the arc. The function need not be continuous; it can have positive or negative jumps, though we do assume that the function is lower semi-continuous, that is, ga (xa ) ≤ lim inf

1

x0a →xa

ga (x0a ) for

any sequence x0a that approaches xa . Without loss of generality, we also assume, through a simple translation of the costs if necessary, that ga (0) = 0. Such a piecewise linear function can be fully characterized by its segments. On each arc a, each segment s of the function has a non-negative variable cost, csa (the slope), a non-negative fixed cost, fas (the intercept), and upper and lower bounds, bs−1 and bsa , on the flow of that segment. a Since the total flow on each arc can always be bounded from above by either the arc capacity or the total demand flowing through the network, we assume that there is a finite number of segments on each arc a, which we represent by the set Sa . We further introduce the following notation: K denotes the set of commodities, N is the |V |×|A| node-arc incidence matrix, and dk is the vector of size |V | representing supplies and demands for commodity k. The flow on each arc a, xa , can be decomposed in two ways, by commodity or by segment, with xka and xsa representing either the flow of commodity k, or the flow on segment s. More specifically, we define xsa as the total flow on arc a if that flow lies in segment s. For this definition, when a flow value of x ba lies in segment sb, xsba = x ba and xsa = 0 for all segments

s 6= sb. We also define binary variables, yas , such that yas = 1 if segment s contains a non-zero flow,

and yas = 0 otherwise. With this notation, we can express the piecewise linear cost network flow problem (PLCNF) as a MIP formulation, called the basic model: min

X X

csa xsa + fas yas ,

(1)

a∈A s∈Sa

N xk = dk , k ∈ K, X xa = xka , a ∈ A,

(2) (3)

k∈K

xa =

X

xsa ,

a ∈ A,

(4)

s∈Sa s s s s bs−1 a ∈ A, s ∈ Sa , a ya ≤ xa ≤ b a ya , X yas ≤ 1, a ∈ A,

(5) (6)

s∈Sa

xka ≥ 0, yas ∈ {0, 1},

a ∈ A, k ∈ K

(7)

a ∈ A, s ∈ Sa .

(8)

Constraints (2) are the flow balance constraints typical in a multi-commodity network flow formulation. For each node i and each commodity k, dki > 0 denotes an origin node with supply dki , dki < 0 denotes a destination node with demand −dki , and dki = 0 denotes a transshipment node. Constraints (3) and (4) define the flow by commodity and by segment, respectively. forcing constraints, (5), state that if

yas

= 0, then segment s has no flow, i.e.,

(6) assures that we choose at most one yas to be positive on each arc a.

2

xsa

The basic

= 0. Inequality

Croxton, Gendron, and Magnanti [1] prove that the LP relaxation of the basic model approximates the piecewise linear cost function with its lower convex envelope, which, in general, provides a rather poor approximation. In order to improve it, a classical MIP approach is to add valid inequalities that tighten the LP relaxation. The basic model lends itself naturally to two such simple classes of valid inequalities, which use the fact that the flow of commodity k on each arc a can be bounded from above by some constant Mak , usually set to the total supply for that commodity. The first class of valid inequalities, called the strong forcing constraints, is simply stated as follows: xka ≤ Mak

X

yas ,

a ∈ A, k ∈ K.

(9)

s∈Sa

We use the term strong model to designate the basic model with the strong forcing constraints added. The second class of valid inequalities is obtained through defining additional variables that represent flow for each commodity-segment combination: xks a is the flow of commodity k on arc a if the total flow on the arc lies in segment s, and is equal to 0 otherwise. These new, P extended variables, are related to the previous ones via the simple equations: x sa = k∈K xks a and P k ks xa = s∈Sa xa . Using them, we can now define the extended forcing constraints: k s xks a ≤ M a ya ,

a ∈ A, k ∈ K, s ∈ Sa

(10)

The model obtained by adding the extended forcing constraints to the basic model will be called the extended model. The question that naturally arises is the following: under which conditions the extended model improves upon the strong one? One can also ask a similar question regarding the improvement that both models provide compared to the LP relaxation of the basic formulation. The main objective of the present paper is to qualify, through theoretical results, as well as computational experiments, the improvement obtained by the addition of either the strong, or the extended, forcing constraints to the basic PLCNF model. We show through theoretical investigation that the LP relaxation of the extended model approximates the cost function with its lower convex envelope in multiple dimensions. Together, the theoretical and computational results allow us to make suggestions on when each of the formulations might be the most appropriate.

References [1] Croxton, K.L., B. Gendron, and T.L. Magnanti (2002), A Comparison of Mixed-Integer Programming Models for Non-Convex Piecewise Linear Cost Minimization Problems, working paper, Operations Research Center, Massachusetts Institute of Technology. [2] Croxton, K.L., B. Gendron, and T.L. Magnanti (2001), Models and Methods for Merge-inTransit Operations. Transportation Science, forthcoming.

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