Discrete Optimization
A branch-cut-and-price algorithm for the piecewise linear transportation problem

https://doi.org/10.1016/j.ejor.2015.03.039Get rights and content

Highlights

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    We consider a transportation problem with a piecewise linear cost structure.

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    Two new formulations are proposed based on a Dantzig–Wolfe reformulation.

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    One reformulation is extended to an exact solution method.

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    The proposed method is much faster than standard optimization software.

Abstract

In this paper we present an exact solution method for the transportation problem with piecewise linear costs. This problem is fundamental within supply chain management and is a straightforward extension of the fixed-charge transportation problem. We consider two Dantzig–Wolfe reformulations and investigate their relative strength with respect to the linear programming (LP) relaxation, both theoretical and practical, through tests on a number of instances. Based on one of the proposed formulations we derive an exact method by branching and adding generalized upper bound constraints from violated cover inequalities. The proposed solution method is tested on a set of randomly generated instances and compares favorably to solving the model using a standard formulation solved by a state-of-the-art commercial solver.

Introduction

In this paper we consider the problem of finding a minimum cost flow in a bipartite graph between a set of suppliers and a set of customers. The cost of sending goods on an arc follows a piecewise linear structure (see Section 2) and the problem is thereby a natural generalization of the Fixed-Charge Transportation Problem. This problem is termed the Piecewise Linear Transportation Problem (PLTP), and is a versatile problem that is fundamental within supply chain network design and arises in a number of applications. The general form of the cost functions allows for modeling of different transportation modes such as small packages, less-than-truckloads, truckloads, and air freight (see e.g. Croxton, Gendron, Magnanti, 2003b, Lapierre, Ruiz, Soriano, 2004). Additionally, the cost function can be used to model price discounts such as all-unit or incremental discounts, often found in procurement theory (see Davenport, Kalagnanam, 2001, Kameshwaran, Narahari, 2009) or to linearize an otherwise nonlinear cost function.

Kim and Pardalos (2000) present a heuristic for the PLTP based on a linearization of the cost function and subsequent solution of a (standard) transportation problem. In Croxton, Gendron, and Magnanti (2003a) the authors show that the linear programming relaxations of three textbook formulations of a piecewise linear function are equivalent. One of them is the Multiple-Choice Model (MCM) used in Section 2.1. Other studies (e.g. Keha, de Farias, Jr., Nemhauser, 2004, Vielma, Ahmed, Nemhauser, 2010, Vielma, Keha, Nemhauser, 2008) have extended this result to include a number of other formulations and they also perform tests to find the best formulation in terms of solving the problem to optimality by a standard solver. The most recent of these studies suggests that when the number of different transportation modes is relatively small (as in our tests), the MCM, presented in Section 2.1, is preferable. As the linear programming relaxation of the standard models is often very poor, we propose two stronger formulations for the problem, both based on a Dantzig–Wolfe reformulation of the problem.

In Section 2 we give a formal definition of the problem using the standard multiple-choice formulation and two new formulations. The two new stronger formulations rely on a Dantzig–Wolfe reformulation of the original problem and column generation is required to solve the LP relaxation. The strength of the linear programming relaxation of the formulations is investigated in Section 3, along with the computational experience on a test bed of instances. Based on these results we propose an exact solution method based on one of the formulations, in which we add valid inequalities described in Section 4 and by applying the branching rule described in Section 5. In Section 6 we test the solution method on a number of randomly generated instances and compare the method to solving the MCM by a standard commercial solver. Section 7 summarizes our findings and concludes this paper.

Section snippets

Mathematical formulations

In this section we first define the PLTP and introduce notation. Then we introduce a standard formulation and two new formulations of the PLTP.

Let the set of supply nodes (suppliers) be denoted by the set I = {1, …, n}. The total capacity of each supplier i is denoted by Si. The demand nodes (customers) are denoted by the set J = {1, …, m}, where customer j has demand dj. The cost of transporting goods from supplier i ∈ I to customer j ∈ J follows a piecewise linear cost structure with κij line

Strength of the LP relaxation of the models

In this section we investigate the theoretical strength of the new models compared to MCM which corresponds to a standard textbook way of representing piecewise linear cost functions. In addition we investigate the computational effectiveness of the models by presenting their actual strength on a subset of the instances used in Section 6.

The theoretical strength is easy to establish. The LP relaxations of the CBM and SBM are at least as strong as that of the MCM, as these two models are

Valid inequalities for the CBM

Though the solution to the current LP relaxation of the program (13)–(16) satisfies the capacity constraints, the xijt values of the columns used in this solution might not. To strengthen the formulation, we add cover inequalities, which are valid inequalities based on the knapsack structure of inequalities (15). For more information on cover inequalities see e.g. Nemhauser and Vance (1994), Gu, Nemhauser, Savelsbergh, 1998, Gu, Nemhauser, Savelsbergh, 1999, and Wolsey (1998).

Let us define a

Branching rule

Any solution, β˜j, to the CBM corresponds to a solution for the MCM, given by v˜ijq=∑t:vijqt=1β˜jtandx˜ijq=∑t:vijqt=1xijqtβ˜jt.A solution is integer if and only if no v˜ijq is fractional (as discussed in Section 2.4). This insight is crucial in developing branching rules for the CBM. In preliminary tests a number of different approaches were tested and based on these results we employ the following branching rule. Given the current LP solution, calculate the cost on every arc as C˜ij=∑q∈Qx˜ijqci

Computational tests

In this section we describe how a number of test instances were generated and how the proposed method compares to the standard solver CPLEX (version 12.4).

Conclusion

In this paper we proposed two new formulations for the piecewise linear transportation problem. Both formulations have a possibly stronger LP relaxation bound than the standard models considered so far in the literature. In particular, the formulation termed the customer-based model seems to offer a significant reduction in the gap between the LP relaxation of the root node and the optimal solution. Using this model we propose an exact solution method by adding valid inequalities and branching

Acknowledgments

The work of the second author is supported by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office. The authors would like to thank the area editor and one anonymous referee for their valuable insights and suggestions which significantly improved this paper.

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