Classical Transportation Problem Research Articles

Discrete infinite transportation problems

The finite classical transportation problem is extended to an infinite one having a countable number of origins and destinations. The approach taken is essentially discrete and requires no compactness, measure theoretic, or metric properties of any of its constructions. Duality results are presented for the infinite transportation problem extension and its dual, as well as for two of the relaxations. A constructive approximation procedure is given for obtaining program values arbitrarily close to the infinite program values of the extension.

The Capacitated Carrier Transportation Problem: An Application of the Axial Three-Dimensional Transportation Problem with Heuristic

The classic transportation problem assumes that either (1) there is a single carrier who can transport all of the required shipments at the given, per unit O-D costs or (2), there are multiple carriers and each O-D pair can be served by its least cost carrier. These assumptions, however, are unrealistic for logistics problems in which the transportation needs exceed the short-term capacity of any one available carrier. In such cases, it cannot be assumed that every potential O-D pair can be served by its least costly carrier. For example, firms in the paper product and tire industries conduct promotional activities (e.g., specials or truck-load sales). Such activities result in extremely large, short-term peaks in their transportation requirements which cannot be met by their private fleets and/or contracted common carriers within the necessary time constraints. We refer to this problem as the capacitated carrier transportation problem (CCTP). In this paper we formulate the CCTP as an integer version of the axial three-dimensional transportation problem and present a heuristic based upon Lagrangian relaxation for solving it. Test problem results indicate that the heuristic identifies near optimal solutions in reasonable time.

An algorithm for finding the optimum solution of solid fixed-charge transportation problem

The solid fixed-charge transportation problem is an extension of the classical solid transportation problem, in which a fixed cost is incurred for every origin and for every requirement of the product. Several procedures have been developed for solving two-dimensional fixed-charge transportation problem. but the method for finding optimum solution in solid fixed-charge transportation problem has deserved less attention In this paper, an algorithm is presented to find the optimum solution of a solid fixed-charge transportation problem. Anumerical example is worked out in support of the given algorithm

An algorithm for the optimum time-cost trade-off in fixed-charge bi-criterion transportation problem bi-criterion transportation problem

The fixed-charge transportation problem is an extension of the classical transportation problem in which a fixed cost, sometimes called a setup cost, is incurred for every origin. In this paper, an algorithm is presented to find the optimum time-cost trade-off pair in a fixed-charge bi-criterion transportation problem giving the same priority to both time and cost. The algorithm identifies efficient time-cost trade-off pairs and thereby arrives at the conclusion that among the trade-off pairs, the optimum trade-off pair has the minimum D1-distance from the ideal point. A numerical example is given to illustrate the developed algorithm

A Generalized Time–cost Trade-off Transportation Problem

In the classical transportation problem if the unit costs and transportation durations are considered, the time-cost trade-off solutions can be determined by the well-known threshold approach assuming that all the transportations are permitted to be simultaneous in time. If all the unit costs are linear functions of time over a specified interval of time, a parametric technique can be applied for identifying all the time-cost trade-off solutions pertaining to this interval. In this paper, the unit costs considered are piecewise linear non-increasing functions of time and transportations are allowed to be simultaneous. It is shown that a parametric method involving a finite sequence of parametric transportation problems reveals all the time-cost trade-off solutions of this generalized trade-off problem. Computational experience is included. If the transportation problem has considerable degeneracy, the parametric approach may pose some computational difficulty. This difficulty can be reduced by using an alternative method involving the bicriteria optimization approach of Aneja and Nair. Also, a direct method is outlined for the case where a finite set of discrete alternatives of unit cost-time pairs is available.

A Generalized Time–Cost Trade-Off Transportation Problem

In the classical transportation problem if the unit costs and transportation durations are considered, the time-cost trade-off solutions can be determined by the well-known threshold approach assuming that all the transportations are permitted to be simultaneous in time. If all the unit costs are linear functions of time over a specified interval of time, a parametric technique can be applied for identifying all the time-cost trade-off solutions pertaining to this interval. In this paper, the unit costs considered are piecewise linear non-increasing functions of time and transportations are allowed to be simultaneous. It is shown that a parametric method involving a finite sequence of parametric transportation problems reveals all the time-cost trade-off solutions of this generalized trade-off problem. Computational experience is included. If the transportation problem has considerable degeneracy, the parametric approach may pose some computational difficulty. This difficulty can be reduced by using an alternative method involving the bicriteria optimization approach of Aneja and Nair. Also, a direct method is outlined for the case where a finite set of discrete alternatives of unit cost-time pairs is available.

Interval and fuzzy extensions of classical transportation problems

For a given Transportation Problem, several assumptions can be considered on the supply and demand levels. In accordance with the kind of information the decision maker has, s/he may express those values in many ways. Here three situations are considered: the decision maker can exactly define them as point values, s/he can give them as interval values or, if s/he has vague information, express them as fuzzy numbers. For each of these three cases, different models (classical, interval and fuzzy) of the Transportation Problem may be obtained, respectively. This paper deals with these three different models. The links among them are provided, focusing on the case of the Fuzzy Transportation Problem, for which methods of solution are proposed and discussed.

Solutions to a class of transport problems with radially dominant convection

For fluid systems dealing with drops and bubbles, there are many situations in which the flow is dominated by a radial field. An analysis is carried out for a general class of problems, in which the primary flow is a purely radial type in a spherical geometry and the secondary flow is a perturbation on it. In particular, the flow solutions are obtained for a particle in extensional flow, rotating particle, and a particle in a linear shear flow. In addition, the steady state heat/mass flow equations with radial convection are solved in a fairly general form for spherical boundaries. The solutions lead to a new class of polynomials for the radial functions of the separated solutions. Some of the fundamental properties of these polynomials have also been derived.

CHARACTERISTICS METHOD FOR SOLVING THE SPATIALLY INHOMOGENEOUS BOLTZMANN EQUATION

A new algorithm for the inhomogeneous Boltzmann equation in one spatial and velocity dimension, based on the method of characteristics, is presented. Using the analytic solution to the Boltzmann equation along its characteristics, the solution to the classical transport problem in semiconducting structures within the relaxation time approximation for the collision integral is obtained iteratively. An n+nn+‐diode is studied numerically and the effects of ballastic electrons on low‐order moments are investigated.

Transportation problem with nonlinear side constraints a branch and bound approach

In a container terminal management, we are often confronted with the following problem: how to assign a reasonable depositing position for an arriving container, so that the efficiency of searching for and loading of a container later can be increased. In this paper, the problem is modeled as a transportation problem with nonlinear side constraints (TPNSC). The reason of nonlinear side constraints arising is that some kinds of containers cannot be stacked in the same row (the space of storage yard is properly divided into several rows). A branch and bound algorithm is designed to solve this problem. The algorithm is based on the idea of using disjunctive arcs (branches) for resolving conflicts that are created whenever some conflicting kinds of containers are deposited in the same row. During the branch and bound, the candidate problems are transformed into classical transportation problems, so that the efficient transportation algorithm can be applied, at the same time the reoptimization technique is employed during the branch and bound. Further, we design a heuristic to obtain a feasible initial solution for TPNSC in order to prune some candidates as early and/or as much as possible. We report computational results on randomly generated problems.

A heuristic algorithm for a stochastic transportation network

A stochastic version of the classical transportation problem is presented. In this model, each arc of the network connects a supply node to a demand node, and the flow of units travelling along each arc of the network forms a stochastic process. In this paper, a heuristic algorithm is presented to obtain the optimal flow rate along each arc of the network such that the total transportation cost is minimized, the total supply rate is equal to the total demand rate, and the average congestion along each arc does not exceed its capacity

Topics of polyhedral combinatorics in transportation problems with exclusions

Polyhedral combinatorics is a new, large, and important branch of operations research theory which has been developing vigorously during the last 10-15 years. The classical problems and first results of polyhedral combinatorics are associated with the names of Descartes, Euler, and Poincar~. The popularity of polyhedral combinatorics is primarily attributable to its applicability for estimating the complexity of optimization methods (primarily the simplex method) and for potential construction of effective algorithms. Alongside classical problems, such as enumeration and classification of polyhedra and characterization of skeletal complexes, polyheral combinatorics deals with a va r i e tyof new problems, which include construction of convex hulls and analysis of metric properties of graphs of polyhedra. Graphs (line skeletons) are analyzed both for abstract polyhedra and for polyhedra of specific optimization problems: traveling salesman prt~blem, standardization problem, choice problem, transportation problem, linear programming problem, and combinatorial optimization problem. Such studies often require classifying polyhedra by various attributes, in particular, by the number of (maximum-dimension) faces, counting the number of vertices, and estimating the diameter, the radius, and the height. Many authors have considered the feasible region of transportation problems (see, e.g., [1-4] and the references therein). So far, however, only the feasible region of the classical transportation problem has been studied in sufficient detail. Much less studied is the feasible region of the transportation problem with exclusions, i.e., the set MG(a, b) of matrices x = II Xij II m• whose elements satisfy the conditions

A realization of the moment method

The difficulties involved in solving a system of moment equations using two-sided distributions are analyzed. The properties of these distributions do not permit the realization of the moment method (in the case of a collision integral in Boltzmann form) in specific transport boundary-value problems. A method of obtaining analytic solutions of the system of moment equations for linearized transport problems is proposed. The accuracy of the method is analyzed with reference to a classical transport problem.

Use of penalties in a branch and bound procedure for the fixed charge transportation problem

The fixed charge transportation problem (FCTP) is an extension of the classic transportation problem in which a fixed cost is incurred for every supply point that is used in the solution. A branch and bound procedure which requires the use of two subproblems (a classic transportation problem and a knapsack problem) can be used to solve the FCTP. Unlike fixed change problems, with the FCTP, there is a many-to-one correspondence between the continuous variables of the transportation problem and the fixed charge variables of the knapsack problem. This paper incorporates penalties into the branch and bound procedure in order to improve the solution times. Also, an adjacent extreme point heuristic is presented that finds a near optimal solution quickly.

Kostengünstige behandlung stoeliastiseher schwankungen der bedarfsgrößen beim transportproblem durch ein niehtlineares diskretes optlmlerangsproblem

Heuristic considerations result in a nonlinear discrete optimization problem, where classical transportation problems take a leading part for finding out a low cost way to reset the moulds in view of stochastic variations of requirements. After working out the optimization problem is treated by Greedy-algorithm and other methods.

A refined simplex algorithm for the classical transportation problem with application to parametric analysis

The well-known sensitivity analysis (SA) of linear programming (LP) is rarely performed on transportation problems (TP) because it is not directly applicable to TP algorithms such as Stepping Stone or its modified versions. These algorithms do not provide the final LP tableau which is prerequisite to all post-optimality analysis. This prerequisite is automatically fulfilled by simplex or dual simplex. However, they are very tedious because they increase computational complexity through the introduction of additional variables. A further complication arises in that TP always has dependencies require that SA receive the more general treatment of parametric analysis (PA). This paper presents an efficient PA for TP.

A Rigorous Solution to the Classical Space-Dependent Neutron Slowing Down Transport Problem

The isotropic form of the classical space-dependent neutron slowing down problem is solved by applying a Fourier transform with respect to optical distance to the appropriate transport equation and...

A Simple Algorithm for Solving Small, Fixed-Charge Transportation Problems

The solution of the classical transportation problem (as generally presented) can be mastered very quickly. The fixed-charge problem is another matter. The reason is that the introduction of fixed costs in addition to variable costs results in the objective function being a step function. Fixed-charge problems are usually solved, therefore, by using sophisticated computer software. This paper deviates from that approach. It presents a low-tech. algorithm for the solution of small, fixed-charge problems.

A Simple Algorithm for Solving Small, Fixed-Charge Transportation Problems

The solution of the classical transportation problem (as generally presented) can be mastered very quickly. The fixed-charge problem is another matter. The reason is that the introduction of fixed costs in addition to variable costs results in the objective function being a step function. Fixed-charge problems are usually solved, therefore, by using sophisticated computer software. This paper deviates from that approach. It presents a low-tech. algorithm for the solution of small, fixed-charge problems.

Planar three-index transportation problems with dominating index

A class of planar three-index transportation problems is described which can be reduced to classical transportation problems. Moreover an example of a polytope with full dimension having this property is given.