Transshipment Problem Research Articles

Operations Research: Applications and Algorithms

Operations Research: Applications and Algorithms

Structure of the submarking-reachability problem and network programming

Using a linear programming formulation, a unified treatment of the submarking-reachability problem for both capacitated and uncapacitated marked graphs is presented. In both cases, the problem reduces to that of testing feasibility of the dual transshipment problem of operations research. An algorithm called REACH is presented for the feasibility testing problem; its worst-case time complexity is O(mn), where m and n are, respectively, the number of edges and the number of nodes in the marked graph. The place of this work in the context of general network programming problems is indicated.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Maximum-weight markings in marked graphs: Algorithms and interpretations based on the simplex method

The problem of determining a maximum-weight marking in a marked graph is mathematically dual to the transshipment problem of operations research. The special structure of the transshipment problem facilitates efficient implementation of the simplex method of linear programming, for solving such problems. In this paper, we first show that the maximum-weight marking problem possesses as much structure as its dual, and then present an implementation of simplex for this problem in terms of marked graph concepts and operations. The pivoting operation in the simplex method is shown to correspond to the subgraph firing operation in marked graphs. A diakoptic reachability theorem is also proved. The formulations presented in this paper cover both live- and nonlive-marked graphs with or without capacity constraints.

A fertilizer transportation problem

We investigate a multiperiod multicommodity transshipment problem which arises in the context of fertilizer distribution in the Republic of Indonesia. Taking the special problem structure into account a model is presented which considers the dynamic aspect of the problem only implicitly and all commodities can be handled by successive solution where at most two commodities have to be considered for each subproblem. Hence, the size of the model makes an implementation on a microcomputer possible.

A complementary pivoting algorithm for linear network problems

In this paper, an algorithm is presented for the transshipment problem that is an adaption of the method used by Jones, Saigal, and Schneider for solving singlecommodity, spatial-equilibrium problems. The approach uses a variable-dimension strategy in which a sequence of subproblems is formed by solving the problem ‘one-node-at-a-time’. The algorithm is tested on uncapacitated transportation problems. Although the computational results are not directly comparable to other methods (since the algorithm is implemented in C under UNIX), the results show that the method is very effective and may be competitive with the best available algorithms for linear network problems.

Performance evaluation of independent superbasic sets on nonlinear replicated networks

In this paper we describe a new type of network flow problems that basically consists of the classical transshipment problem with the following extensions: (1) The replication of a network by producing subnetworks with identical structure, being linked by so-called linking arcs; (2) the objective function terms related to the linking arcs are non-differentiable nonlinear functions. By using a networks-specialized implementation of a linearly constrained nonlinear programming algorithm described elsewhere, we report the computational performance of the new concept of independent superbasic sets that allows to obtain in parallel independent pieces of the solution at each iteration.

An Efficient Optimization Method for Transmission Expansion Planning

Abstract This paper describes a network synthesis model to be applied to the long-range transmission planning expansion. The problem is formulated as a mathematical optimization model that considers simultaneously two objectives: - the minimization of equipment investments and the satis-faction of an electric performance index. A projection strategy is applied to the model - a nonlinear mixed-integer problem - in order to determine the optimal relationship between susceptances and flows. It follows that the objective function of the resulting problem can be approximated by a piecewise linear function. The linearized problem can be thought as a fixed charge piecewise transshipment problem which is solved by a specialized branch and bound search program which allows a great efficiency in running times and core storage. Three case studies are presented and discussed. The quality of the solutions obtained by our exact method is compared with other heuristic methods currently in use.

Empty freight car dispatching model under freight car pool concept

In the last few years, growing attention has been given to the cost savings potential of the specialized freight car pool concept. Under this concept, a fleet of single-purpose freight cars are pooled at many loading points, and the cars emptied at unloading terminals can be sent back to any loading point in order to reduce empty car miles and time. As part of a continuing effort to improve the car dispatching method for this concept, this paper offers an improvement to the traditional linear programming transportation problem approach, incorporating daily variations of empty car supply and demand characteristics in the model. The proposed method, therefore, allocates cars between supply and demand points with the dispatching date specified, enabling the dispatcher to make daily car disposition decisions. It can incorporate both the travel distance cost and inventory cost of empty cars at terminals in the objective. If the holding of empty vehicles at the terminals for future dispatch is allowed, the problem is structured as a transshipment problem in which the overnight storage is the transshipment point.

Multiplant MRP

AbstractMany manufacturing firms have multiple manufacturing plants, located in geographically diverse parts of the world. This situation is becoming more common, as firms establish new plants in foreign countries to take advantage of low labor cost. In such cases, it is not unusual for the firm to retain production capability of certain key parts in a backup plant, with the necessary equipment and trained workforce in place. High volume production could be obtained relatively quickly from the backup plant in case of an emergency at the main supplying plant. In such multiplant settings, the transportation costs are significant. Throughout this paper, we use the term “multisourced parts” to describe parts produced in more than one location.Material Requirements Planning (MRP) is the component of a total manufacturing control system that is designed to manage inventory and plan orders for parts and material with dependent demand (demand derived from the demand of other items). Most of the literature on MRP systems discusses MRP methodology in a single‐plant environment. Most MRP software systems in use today are single‐plant systems.Currently, it is common for firms with multiple plants treated as cost centers to use an independent single‐plant MRP system for each and handle the transshipment problems manually. Because of lack of coordination of production schedules between supplying and demanding plants, those firms hold more inventory and experience longer lead times than necessary to compensate for uncertainties in schedules and supply policies.The purpose of this article is to enhance single‐plant MRP systems for coping with multiplant situations in which: the plants are regarded as cost centers, there exist multisourced parts, and the transportation costs are significant. The multiplant MRP system should recognize that parts are produced in different plants, make offset calculations for in‐transit lead times, and consider transportation costs when establishing production requirements and shipping routes for multisourced parts. The objective is, beginning with the corporate‐determined master schedule for finished products, to communicate in one planning cycle time‐phased planned order release schedules and shipping/delivery schedules to each manufacturing plant producing components for the finished products.We first present a simplified framework for the multiplant MRP system, where a transportation algorithm is incorporated into the MRP logic. Then we refine this simplified framework to handle more complex aspects of a multiplant network. These complexities include the treatment of requirements that are not shipped on time and the regeneration of new MRP schedules. We also observe that the solution to the transportation problem described above is affected by the lot‐sizing rules employed. In addition, we discuss several important issues and decisions that confront a firm when implementing a multiplant MRP system.

Solving Generalized Networks

A complete, unified description is given of the design, implementation and use of a family of very fast and efficient large-scale minimum-cost (primal simplex) network programs. The class of capacitated generalized transshipment problems solved includes the capacitated and uncapacitated generalized transportation problems and the continuous generalized assignment problem, as well as the pure network flow models which are specializations of these problems. These formulations are used for a large number of diverse applications to determine how (or at what rate) flows through the arcs of a network can minimize total shipment costs. A generalized network problem can also be viewed as a linear program with at most two nonzero entries in each column of the constraint matrix; this property is exploited in the mathematical presentation with special emphasis on data structures for basis representation, basis manipulation, and pricing mechanisms. A literature review accompanies computational testing of promising ideas, and extensive experimentation is reported which has produced GENNET, an extremely efficient family of generalized network systems.

Unit commitment in a thermal generation system with multiple pumped-storage power stations

A model is presented of the unit-commitment problem which takes into account generator maximum and minimum outputs, maximum loading and unloading rates, minimum shutdown times, and both on- and offline costs. This is formulated as a mixed-integer problem and solved by using a branch and bound method. The linear-programming problems that arise during the search procedure in the branch and bound algorithm are converted into capacitated transshipment problems which can be solved efficiently by a graphically-based solution procedure. Multiple pumped-storage power stations are then scheduled by a search procedure based on the principle that pumping can only take place if the resulting hydrogeneration reduces total production costs. This is applied iteratively to an initial thermal unit schedule until no further cost improvement occurs. Finally, the computational requirements of this method, both with and without pumped-storage stations, are shown to be less than those of other existing techniques and to be applicable to systems of realistic size.

Complements and substitutes among locations in the two-stage transshipment problem

This paper considers complementarity and substitutability among locations for a two-stage transshipment problem with locations being factories, warehouses, and demand centers. A direct generalization of properties known for the transportation problem would be that any two locations of different types are complements and any two locations of the same type are substitutes. Examples show that these properties need not hold for pairs of locations that include at least one warehouse. An algorithm of Nagelhout and Thompson ( European J. Operational Res. 6 (1981) 149–161) for locating warehouses is based on the incorrect supposition that any two warehouses are substitutes, and an example shows that their algorithm need not generate an optimal solution as claimed. For pairs of locations that do not include a warehouse, complementarity and substitutability properties hold just as in the transportation problem.

Optimal Distribution Substation and Primary Feeder Planning VIA the Fixed Charge Network Formulation

The planning methodology presented in this paper is an application of recent advances in optimization techniques for distribution substation siting and radial feeder planning. The concept of a long range horizon year (target) is used together with a time-phased expansion process beginning with the base year and progressing to the target year. An optimal static horizon year design is first determined using a fixed charge transshipment problem formulation (FCNP). The FCNP solution methodology employs the branch-and- bound algorithm and includes explicit modeling of fixed charge and variable cost components for improved accu- racy. The time-phased expansion process selects substation sites and feeder routings from the horizon (or target) year plan, thus providing an orderly and coherent expansion. This method easily incorporates existing facilities in the decision process for maximum benefit. An example of an actual planning study is presented to illustrate the horizon year target system selection and the corresponding time-phased expansion patterns.

Levelling terrain trees: A transshipment problem

Levelling terrain trees: A transshipment problem

Implementation and analysis of a variant of the dual method for the capacitated transshipment problem

This paper presents a variant of the dual simplex method for the capacitated pure network problem and a computational analysis of this algorithm. This work includes the considerations of different list structures to store the original problem data and the basis and the testing of various procedures to select the leaving basic variable. This study also examines the sensitivity of the implementation to changes in problem parameters. The results show that the algorithm which is presented here is superior to earlier dual implementations.

The cyclic method of solving the transshipment problem with an additional linear constraint

AbstractThe “cyclic” method proposed in this paper provides a more efficient computational scheme than currently known adaptations of the simplex method for solving the transshipment problem with an additional linear constraint. Computer codes for the pure network problem can be easily altered to accommodate the cyclic method. Even more important, the cyclic method produces integral solutions at all iterations (except, possibly, the last one); this property may prove valuable in integer programming.

Primal transportation and transshipment algorithms

Capacitated transshipment problems constitute the most general class of single commodity network flow problems. During the last ten years the emphasis has shifted away from the primal-dual solution methods back to specializations of the primal simplex algorithm as the more efficient approach. In this survey numerous variations in data structures, selection rules and implementations of individual steps are extracted from the relevant literature. The exposition is centered around one basic algorithm which is explained in complete detail with particular attention to the intricacies of the basis exchange part. Specialized solution methods for transportation and assignment problems are taken into account.

Graph Structures Underlying Submatrices of Network Flow Tableaux

This paper analyzes the substructure of linear programming tableaux for capacitated transshipment problems. Specifically, it demonstrates that submatrices of these tableaux correspond in a precise way to certain “contractions” of the transshipment network. The results are couched within the GNET/Depth network data structure of Bradley, Brown and Graves or the predecessor/thread/distance structure of Glover, Klingman, et al. An application to fixed-charge network problems is described.

Solving Goal-Programming Manpower Models Using Advanced Network Codes

This paper shows that a certain class of manpower systems that are often modelled using goal programming can also be modelled as capacitated transshipment problems. The goal programming structure can be preserved in most cases through careful definition of the flow-cost function and through the use of multiple arcs in the network. This latter feature is made practical through the use of advanced network codes, such as GNET, that are now available. Solution times and costs are significantly lower than those obtained with the original goal programming formulation through the use of generalized linear programming codes.

Testing of a large-scale network optimization program

This paper describes the experimental results of testing a “large-scale” program for solving minimum-cost network flow problems. With this program, general structure transshipment problems with over ten thousand nodes and thirty thousand arcs have been easily solved without resorting to auxiliary storage. The algorithm is a variant of the primal revised simplex method; the computer code is called LPNET illustrating the close connection between linear programming and network graphs. This approach substantially improves computer processing timeand core storage, especially for relatively large network problems. The results of these experiments are provided. It is emphasized that an organized experimental design and a detailed series of empirical tests are crucial for an efficient implementation.