Capacitated Transshipment Problem Research Articles

Transshipment and Routing Model for Multi-Product Planning and Distribution at a Refreshment Company in Bogota, Colombia

In this article, a model is established for the planning of the weekly distribution of refreshments in a company that produces 6 types of beverages in 3 Production Centers, with the potential use of 3 Distribution Centers and supplying 22 Consumer Centers located in Bogota. Due to the impact of transportation costs on the company profits, it is necessary to determine the optimal planning of the allocation of sources to destinations in the nodal network of the system and its subsequent routing with two types of vehicles. Therefore, a two-phase model is proposed. In the first, the initial optimal assignments are established by comparing two techniques; one based on the Open Transshipment Problem (OTP) and the other on the Capacitated Transshipment Problem (CTP), which determine the trips to be made from production centers to consumer centers using distribution centers as intermediaries. This comparison allowed to analyze its efficiency to choose the model with the lowest cost and number of trips. In the second phase, routing was performed on routes that do not use the full capacity of the vehicles. The routes were established through the Travelling Salesman Problem (TSP). As a result, a comprehensive model was obtained to carry out the planning of the product distribution, obtaining in Phase II the use of the OTM model, which presented 14% less distance; 2.6% less time and 0.28% less than the total cost compared to the CTP model. In Phase II 5 routes were obtained, of which Distribution Center 2 covers 59% of customers.

Variants of transshipment problem

In a transshipment problem all the sources and destinations can function in any direction thus transshipment is very useful to reduce the transportation cost. Sometimes, because of budget/political constraint, the total flow in transshipment problem is also specified by some external decision maker and the optimal solution of such problems is of realistic interest to the decision maker. This has motivated me to discuss impaired and enhanced flow in a transshipment problem. Algorithms are provided for solving such transshipment problems by transforming the original problem into an equivalent transportation problem by adding an additional row and a column. The optimal solution of the transformed transportation problem gives the optimal solution of the given transshipment problem having the same objective function value. I have considered both balanced as well unbalanced transshipment problems and have also discussed various situations emerging out of unbalanced capacitated transshipment problems in the form of inequalities. The algorithms and transformations are easy to understand and serve the managers by providing the solution to a variety of distribution problems. Numerical examples are solved to illustrate the theory and computational work for various higher dimensional problems is also included.

On a class of capacitated transshipment problems with bounds on rim conditions

The transshipment technique is used to find the shortest route from one point in a network to another point and is useful to truncate the cost of transportation. Inspired by dead mileage problem evaluated in terms of running empty buses from various depots to starting points, in this paper we study capacitated transshipment problems with bounds on total availabilities at sources and total destination requirements. Sometimes because of the budget/political constraints, the total flow of transportation is stipulated by some external decision maker which thereby results in impaired or enhanced flow in the market. This motivated us to explore such problems as a particular case of the original problem. The optimal solution of the specified problem is obtained by transforming it into an equivalent capacitated transportation problem. We have also discussed various situations emerging out of unbalanced capacitated transshipment problems in the form of inequalities and numerical examples are also given to illustrate the theory.

An algorithm for solving time minimizing capacitated transshipment problem

In this paper an algorithm is provided for solving transshipment problems with the objective of minimizing transportation duration when capacity constraints for the flow between origin and destination are given. The algorithm determines the optimal time for transportation from origin to destination. The given problem is solved by transforming the original problem into an equivalent capacitated transportation problem and the original problem’s optimal solution is obtained from the optimal solution of the transformed capacitated transportation problem which is solved by defining a profit matrix. The algorithm involves finite number of iterations and is easy to understand and apply, so can serve the managers by providing a solution to a variety of production-distribution problems where the finite capacity of resources, such as vehicles and docks, have to be taken into account. Numerical examples are provided to illustrate the algorithm.

The Capacitated Transshipment Location Problem Under Uncertainty: A Computational Study

We consider the stochastic capacitated transshipment problem for freight transportation where an optimal location of the transshipment facilities, which minimizes the total cost, must be found. The total cost is given by the sum of the total fixed cost plus the expected minimum total flow cost, when the throughput costs of the facilities are random variables with unknown probability distribution. By applying the asymptotic approximation method derived from the Extreme Value Theory, a deterministic non-linear model, which belongs to a wide class of Entropy maximizing models, is then obtained. In this paper, we present a deep analysis of the impact of different probability distributions of the random costs on the problem optimum, as well as on the number of facilities located. The computational results show a very good performance of the deterministic model when compared to the stochastic one.

An effective heuristic for the CLSP with set-up times

The problem of multi-item, single level, capacitated, dynamic lot-sizing with set-up times (CLSP with set-up times) is considered. The difficulty of the problem compared with its counterpart without set-up times is explained. A lower bound on the value of the objective function is calculated by Lagrangian relaxation with subgradient optimisation. During the process, attempts are made to get good feasible solutions (ie. upper bounds) through a smoothing heuristic, followed by a local search with a variable neighbourhood. Solutions found in this way are further optimised by solving a capacitated transshipment problem. The paper describes the various elements of the solution procedure and presents the results of extensive numerical experimentation.

The Conference Paper‐Reviewer Assignment Problem*

Conference organizers often face the following problem: Given a collection of submitted papers, select a subset to be presented at the conference. The bulk of the work often amounts to assembling a pool of reviewers and then sending each submitted paper to several reviewers. We present in this paper a technique for finding a good assignment of papers to reviewers. An important feature of the solution we find is that each paper is sent to at least one reviewer who is “as expert as possible” for that paper. A major component of the problem is modeled as a bottleneck version of a capacitated transshipment problem.

Efficient Solution of the Single-item, Capacitated Lot-sizing Problem with Start-up and Reservation Costs

A capacitated dynamic lot-sizing model, where the costs incurred are a start-up cost for switching the production facility on and another reservation cost for keeping the facility on, whether or not it is producing, is considered. The resulting scheduling problem is NP-hard. An efficient shortest path model of the uncapacitated version of the problem is developed. This model is then included, via a redefinition of variables, into a tight capacitated model; tight in the sense that sharp lower bounds can be produced from it. The lower bound problems are solved efficiently by recovering the shortest path structure through column generation, and effective upper bounds are generated by solving a small capacitated trans-shipment problem. The results of computational tests to verify the computational efficiency of the resulting solution scheme are presented.

Efficient Solution of the Single-item, Capacitated Lot-sizing Problem with Start-up and Reservation Costs

A capacitated dynamic lot-sizing model, where the costs incurred are a start-up cost for switching the production facility on and another reservation cost for keeping the facility on, whether or not it is producing, is considered. The resulting scheduling problem is NP-hard. An efficient shortest path model of the uncapacitated version of the problem is developed. This model is then included, via a redefinition of variables, into a tight capacitated model; tight in the sense that sharp lower bounds can be produced from it. The lower bound problems are solved efficiently by recovering the shortest path structure through column generation, and effective upper bounds are generated by solving a small capacitated trans-shipment problem. The results of computational tests to verify the computational efficiency of the resulting solution scheme are presented.

Algorithms for Capacitated, Multi-Item Lot-Sizing without Set-Ups

The multi-item lot-sizing problem considered here is concerned with finding the lot sizes over a horizon of discrete time periods to meet known future demand without incurring backlogs, such that the total cost of production and inventory holding is minimized. The capacity constraints arise because the production of each item consumes capacitated production resources at a given rate. Production is assumed to occur without set-ups. The problem is formulated as a capacitated trans-shipment problem. Use of modern, minimum-cost network flow algorithms, coupled with appropriate starting procedures, allows realistically large problem instances to be solved efficiently; thus obviating the need for specialized algorithms based on restrictive assumptions regarding cost structures.

Algorithms for Capacitated, Multi-Item Lot-Sizing Without Set-ups

The multi-item lot-sizing problem considered here is concerned with finding the lot sizes over a horizon of discrete time periods to meet known future demand without incurring backlogs, such that the total cost of production and inventory holding is minimized. The capacity constraints arise because the production of each item consumes capacitated production resources at a given rate. Production is assumed to occur without set-ups. The problem is formulated as a capacitated trans-shipment problem. Use of modern, minimum-cost network flow algorithms, coupled with appropriate starting procedures, allows realistically large problem instances to be solved efficiently; thus obviating the need for specialized algorithms based on restrictive assumptions regarding cost structures.

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.

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.

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.

Exceptional Paper—Design and Implementation of Large Scale Primal Transshipment Algorithms

A complete 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 transshipment problems solved is the most general of the minimum cost network flow models which include the capacitated and uncapacitated transportation problems and the classical assignment problem; these formulations are used for a large number of diverse applications to determine how (or at what rate) a good should flow through the arcs of a network to minimize total shipment costs. The presentation tailors the unified mathematical framework of linear programming to networks with special emphasis on data structures which are not only useful for basis representation, basis manipulation, and pricing mechanisms, but which also seem to be fundamental in general mathematical programming. A review of pertinent optimization literature accompanies computational testing of the most promising ideas. Tuning experiments for the network system, GNET, are reported along with important extensions such as exploitation of special problem structure, element generation techniques, postoptimality analysis, operation with problem generators and external problem files, and a simple noncycling pivot selection procedure which guarantees finiteness for the algorithm.

Solution of a Multi-Product Manufacturing and Distribution Problem

An actual multi-product production and distribution problem is described and an optimizing algorithm is developed. The problem, which is an example of a multi-commodity capacitated trans-shipment problem, is solved by the technique suggested by Ford and Fulkerson [Ford, L. R., D. R. Fulkerson. 1958. A suggested computation for maximal multi-commodity network flows. Management Sci. 3 (1) 97–101.]. Due to the size of the problem and computational constraints, the usefulness of this technique is improved by incorporating the Generalized Upper Bound algorithm of Dantzig and Van Slyke [Dantzig, G. B., R. M. Van Slyke. 1967. Generalized upper bounding technique. J. Comput. System Sci. 1 213–226.]. The modified technique requires manipulation of a relatively small matrix whose order is N by N where N is the number of capacitated links in the network. The traditional node-arc formulation solved by linear programming, the original arc-chain technique and the modified arc-chain algorithm are each compared in terms of computer storage requirements and solution times, resulting in considerable time efficiencies, of the order of fifty to one, for the modified Ford-Fulkerson method.