A branch-bound algorithm for the capacitated facilities location problem

This paper discusses a mixed integer programming method for solving the Facilities Location Problem with capacities on the facilities. The algorithm uses a Decomposition technique to solve the dual of the associated continuous problem in each branch-bound iteration. The method was designed to produce the global optimum solution for problems with up to 100 facilities and 1,000 customers. Computational experience and a complete example are also presented in the appendix.     

Algorithms for the minimax transportation problem

In this paper, we consider a variant of the classical transportation problem as well as of the bottleneck transportation problem, which we call the minimax transportation problem. The problem considered is to determine a feasible flow x_ij from a set of origins I to a set of destinations J for which max_(i,j)εIxJ{c_ijx_ij} is minimum. In this paper, we develop a parametric algorithm and a primal-dual algorithm to solve this problem. The parametric algorithm solves a transportation problem with parametric upper bounds and the primal-dual algorithm solves a sequence of related maximum flow problems. The primal-dual algorithm is shown to be polynomially bounded. Numerical investigations with both the algorithms are described in detail. The primal-dual algorithm is found to be computationally superior to the parametric algorithm and it can solve problems up to 1000 origins, 1000 destinations and 10,000 arcs in less than 1 minute on a DEC 10 computer system. The optimum solution of the minimax transportation problem may be noninteger. We also suggest a polynomial algorithm to convert this solution into an integer optimum solution.     

An efficient heuristic procedure for the capacitated warehouse location problem

This paper introduces an efficient heuristic procedure for solving a special class of mixed integer programming problem called the capacitated warehouse (plant) location problem. This procedure parallels the work reported earlier in [9] on the uncapacitated warehouse location problem. The procedure can be viewed as tracing a judiciously selected path of the branch and bound tree (from the initial node to the terminal node) to arrive at a candidate solution. A simple backtracking scheme is also incorporated in the procedure to investigate possible improvement in the solution. Computational results on problems found in the literature look quite encouraging.     

A method for solving the transportation problem

A new method has been developed f o r solving the transportation problem. This method is a modification and a generalization of the method for solving the multiple assignment problem developed by Dr. A. J. Hoffman and Dr. H. M. Markowitz. This method is of interest for several reasons. In the first place it is significantly different from all other methods for solving the trans -portation problem known to the author. Secondly, it is moderately simple touse and understand. Thirdly, and perhaps most important, it has proved to be very adaptable tri high-speed computer operations. It is now being used by several branches of the armed services.     

New heuristic methods for the capacitated multi‐facility Weber problem

In this paper we consider the capacitated multi‐facility Weber problem with the Euclidean, squared Euclidean, and ?_p‐distances. This problem is concerned with locating m capacitated facilities in the Euclidean plane to satisfy the demand of n customers with the minimum total transportation cost. The demand and location of each customer are known a priori and the transportation cost between customers and facilities is proportional to the distance between them. We first present a mixed integer linear programming approximation of the problem. We then propose new heuristic solution methods based on this approximation. Computational results on benchmark instances indicate that the new methods are both accurate and efficient. © 2006 Wiley Periodicals, Inc. Naval Research Logistics 2006     

Approximation algorithms for the capacitated traveling salesman problem with pickups and deliveries

We consider the Capacitated Traveling Salesman Problem with Pickups and Deliveries (CTSPPD). This problem is characterized by a set of n pickup points and a set of n delivery points. A single product is available at the pickup points which must be brought to the delivery points. A vehicle of limited capacity is available to perform this task. The problem is to determine the tour the vehicle should follow so that the total distance traveled is minimized, each load at a pickup point is picked up, each delivery point receives its shipment and the vehicle capacity is not violated. We present two polynomial‐time approximation algorithms for this problem and analyze their worst‐case bounds. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 654–670, 1999     

A generalized-indices transportation problem

A generalized-indices transportation problem is formulated and an algorithm is presented for its solution. The algorithm is an extension of the modi-method. A theorem on the number of independent variables in the generalized-indices transportation problem is proved. An example problem is solved for the four-indices transportation problem. A computer program has been written to solve any four-indices problem.     

The bottleneck transportation problem

The bottleneck transportation problem can be stated as follows: A set of supplies and a set of demands are specified such that the total supply is equal to the total demand. There is a transportation time associated between each supply point and each demand point. It is required to find a feasible distribution (of the supplies) which minimizes the maximum transportaton time associated between a supply point and a demand point such that the distribution between the two points is positive. In addition, one may wish to find from among all optimal solutions to the bottleneck transportation problem, a solution which minimizes the total distribution that requires the maximum time Two algorithms are given for solving the above problems. One of them is a primal approach in the sense that improving fcasible solutions are obtained at each iteration. The other is a “threshold” algorithm which is found to be far superior computationally.     

一类带容量限制的运输问题

考虑一类带容量限制的运输问题.采用构造辅助网络的方法,将运输网络中的每个配送中心均拆分成两个节点,构造出新弧,形成新的网络,把此类运输问题转换为最小费用流问题来解决.并在此基础上,考虑运输网络中配送中心的容量扩张问题.     

A transportation problem with competition

The classic transportation problem can be generalized with many carriers and one owner. From the formulation the competition in sense of game theory naturally appears. Here we present and solve this problem using a generalized n-person game. Besides the same composition properties about solutions and regarding zones, related results are considered. Finally, the problem in which there is a modification of the set of destinations assigned to the carriers is also studied.     

A decomposable transshipment algorithm for a multiperiod transportation problem

A dynamic version of the transportation (Hitchcock) problem occurs when there are demands at each of n sinks for T periods which can be fulfilled by shipments from m sources. A requirement in period t₂ can be satisfied by a shipment in the same period (a linear shipping cost is incurred) or by a shipment in period t₁ < t₂ (in addition to the linear shipping cost a linear inventory cost is incurred for every period in which the commodity is stored). A well known method for solving this problem is to transform it into an equivalent single period transportation problem with mT sources and nT sinks. Our approach treats the model as a transshipment problem consisting of T, m source — n sink transportation problems linked together by inventory variables. Storage requirements are proportional to T² for the single period equivalent transportation algorithm, proportional to T, for our algorithm without decomposition, and independent of T for our algorithm with decomposition. This storage saving feature enables much larger problems to be solved than were previously possible. Futhermore, we can easily incorporate upper bounds on inventories. This is not possible in the single period transportation equivalent.     

The pure fixed charge transportation problem

The pure fixed charge transportation problem (PFCTP) is a variation of the fixed charge transportation problem (FCTP) in which there are only fixed costs to be incurred when a route is opened. We present in this paper a direct search procedure using the LIFO decision rule for branching. This procedure is enhanced by the use of 0–1 knapsack problems which determine bounds on partial solutions. Computational results are presented and discussed.     

Strong turnpike policies in the single-item capacitated lot-sizing problem with periodical dynamic parameter

This article considers optimization problems in a discrete capacitated lot sizing model for a single product with limited backlogging. The demand as well as the holding and backlogging costs are assumed to be periodical in time. Nothing is assumed about types of the cost functions. It is shown that there exists an optimal infinite inverse policy and a strong turnpike policy. A forward algorithm for computing optimal policies relative to the class of batch ordering type policies is derived. Some backward procedure is adopted to determine a strong turnpike policy. The algorithm is simple, and it terminates after the a number of steps equal to the turnpike horizon. Some remarks on the existence of rolling horizontal plans and forecast horizons are also given. © 1997 John Wiley & Sons, Inc. Naval Research Logistics 44: 775–790, 1997     

The dynamic transportation problem: A survey

The dynamic transportation problem is a transportation problem over time. That is, a problem of selecting at each instant of time t, the optimal flow of commodities from various sources to various sinks in a given network so as to minimize the total cost of transportation subject to some supply and demand constraints. While the earliest formulation of the problem dates back to 1958 as a problem of finding the maximal flow through a dynamic network in a given time, the problem has received wider attention only in the last ten years. During these years, the problem has been tackled by network techniques, linear programming, dynamic programming, combinational methods, nonlinear programming and finally, the optimal control theory. This paper is an up-to-date survey of the various analyses of the problem along with a critical discussion, comparison, and extensions of various formulations and techniques used. The survey concludes with a number of important suggestions for future work.     

A streamlined simplex approach to the singly constrained transportation problem

A primal simplex procedure is developed to solve transportation problems with an arbitrary additional linear constraint. The approach is a specialization of the Double Reverse Method of Charnes and Cooper. Efficient procedures for pricing-out the basis, determining representations, and implementing the change of basis are presented. These procedures exploit the pure transportation substructure in such a manner that full advantage may be taken of the computational schemes and list structures used to store and update the basis in codifying the MODI method. Furthermore, the pricing-out and change-of-basis procedures are organized in a manner that permits the calculations for one to be utilized in the other. Computational results are presented which indicate that this method is at least 50 times faster than the state-of-the-art LP code, APEX-III. Methods for obtaining basic primal “feasible” starts and “good” feasible integer solutions are also presented.     

A note on a stochastic production-maximizing transportation problem

A stochastic production-maximizing problem with transportation constraints is considered where the production rates, R_ij, of man i — job j combinations are random variables rather than constants. It is shown that for the family of Weibull distributions (of which the Exponential is a special case) with scale parameters λ_ij and shape parameter β, the plan that maximizes the expected rate of the entire line is obtained by solving a deterministic fixed charge transportation problem with no linear costs and with “set-up” cost matrix ‖λ_ij‖.     

The enumeration of all efficient solutions for a linear multiple-objective transportation problem

An algorithm is presented by which the set of all efficient solutions for a linear multiple-objective transportation problem can be enumerated. First the algorithm determines an initial efficient basic solution. In a second step all efficient basic solutions are enumerated. Finally, the set of all efficient solutions is constructed as a union of a minimal number of convex sets of efficient solutions. The algorithm is illustrated by a numerical example.     

Capacity improvement,penalties, and the fixed charge transportation problem

Capacity improvement and conditional penalties are two computational aides for fathoming subproblems in a branch‐and‐bound procedure. In this paper, we apply these techniques to the fixed charge transportation problem (FCTP) and show how relaxations of the FCTP subproblems can be posed as concave minimization problems (rather than LP relaxations). Using the concave relaxations, we propose a new conditional penalty and three new types of capacity improvement techniques for the FCTP. Based on computational experiments using a standard set of FCTP test problems, the new capacity improvement and penalty techniques are responsible for a three‐fold reduction in the CPU time for the branch‐and‐bound algorithm and nearly a tenfold reduction in the number of subproblems that need to be evaluated in the branch‐and‐bound enumeration tree. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 341–355, 1999