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AbstractA fundamental unsolved problem in the programming area is one in which various activities have fixed charges (e.g., set‐up time charges) if operating at a positive level. Properties of a general solution to this type problem are discussed in this paper. Under special circumstances it is shown that a fixed charge problem can be reduced to an ordinary linear programming problem.

AbstractThe fixed charge problem is a nonlinear programming problem of practical interest in business and industry. Yet, until now no computationally feasible exact method of solution for large problems had been developed.In this paper an exact algorithm is presented which is computationally feasible for large problems. The algorithm is based upon a branch and bound approach, with the additional feature that the amount of computer storage required remains constant throughout (for a problem of any given size).Also presented are three suboptimal heuristic algorithms which are of interest because, although they do not guarantee that the true optimal solution will be found, they usually yield very good solutions and are extremely rapid techniques.Computational results are described for several of the heuristic methods and for the branch and bound algorithm.

AbstractThe 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.

AbstractThis paper describes an approximate solution method for solving the fixed charge problem. This heuristic approach is applied to a set of test problems to explore the margin of error. The results indicate that the proposed fixed charge simplex algorithm is capable of finding optimal or near optimal solutions to moderate sized fixed charge problems. In the absence of an exact method, this heuristic should prove useful in solving this fundamental nonlinear programming problem.

Suggests that aspects of property financing will have to change in
the future. Considers the implications which the Insolvency Act of 1986
will have for fixed charge securities.

AbstractIn many decision‐making situations, each activity that can be undertaken may have associated with it both a fixed and a variable cost. Recently, we have encountered serveral practical problems in which the fixed cost of undertaking an activity depends upon which other activities are also undertaken. To our knowledge, no existing optimization model can accomodate such a fixed cost structure. To do so, we have therefore developed a new model called the interactive fixed charge linear programming problem (IFCLP). In this paper we present and motivate problem (IFCLP), study some of its characteristics, and present a finite branch and bound algorithm for solving it. We also discuss the main properties of this algorithm.

AbstractCalculating the solution for a mixed integer linear programming problem has been problematic for any sizable dimension because of the time required. Accordingly, an improved method for the fixed charge problem is presented here. The method is a modification of a bounding technique first suggested by Balinski, and it exploits fully the ratio of costio‐use, first described by Cooper and Drebes. It exploits those local properties which have global application so that enumeration is confined to those relatively few combinations which cannot be evaluated otherwise.

AbstractIn the absence, to date, of an exact method for solving the linear programming problem with fixed charges, two heuristic methods have been proposed and extensively investigated, computationally, for moderate sized problems. The results indicate that the heuristic methods produce optimal solutions in well over 90 percent of the several hundred problems investigated and very close to optimal (a few percent) in the remaining cases. Hence it should be of practical significance to practitioners in the field.

AbstractA computationally simple method for approximating the optimal solution to transportation problems is described. Also, a method to deal with fixed charges is proposed. The computational algorithms have been designed primarily to handle very large numbers of transportation problems each involving a small number of origins and destinations as is frequently the situation in the Navy supply system. The results of empirical tests of the effectiveness of the methods are summarized.

AbstractA mean‐variance portfolio selection model with limited diversification is formulated in which transaction and management costs are incorporated as the sum of a linear cost and a fixed cost. The problem is a fixed charge integer programming problem solved by hypersurface search using dynamic programming. Fathoming is performed in the forward pass of dynamic programming so that values of the state variable which correspond to infeasible solutions are eliminated from the tables. This logic permits the solution of problems with 20–30 possible investments.

During past few decades, fuzzy decision is an important attention in the areas of science, engineering, economic system, business, etc. To solve day-to-day problem, researchers use fuzzy data in transportation problem for presenting the uncontrollable factors; and most of multi-objective transportation problems are solved using goal programming. However, when the problem contains interval-valued data, then the obtained solution was provided by goal programming may not satisfy by all decision-makers. In such condition, we consider a fixed-charge solid transportation problem in multi-objective environment where all the data are intuitionistic fuzzy numbers with membership and non-membership function. The intuitionistic fuzzy transportation problem transforms into interval-valued problem using (α, β)-cut, and thereafter, it reduces into a deterministic problem using accuracy function. Also the optimum value of alternative corresponds to the optimum value of accuracy function. A numerical example is included to illustrate the usefulness of our proposed model. Finally, conclusions and future works with the study are described. ; Portuguese Foundation for Science and Technology ("FCT-Fundacao para a Ciencia e a Tecnologia"), through the CIDMA-Center for Research and Development in Mathematics and Applications UID/MAT/ 04106/2019 ; Spanish Ministry of Economy and Competitiveness, FEDER funds from the European Union TIN2014-55024-P TIN2017-86647-P