Fixed-charge Problem Research Articles

COAL: A new heuristic approach for solving the fixed charge problem — computational results

Computational results are presented for a new heuristic algorithm for solving large general fixed charge problems based on Lagrangian relaxation and cost allocation heuristics; these results are contrasted with results using adjacent extreme point heuristics and branch and bound mixed integer programming. General facility location and capacitated warehouse location problems as well as production planning problems with fixed and variable work force considerations are used. The many test problems evaluated include problems with a large number of fixed charge and continuous variables. The new algorithm demonstrates superiority over alternative methods for solving these problems.

A branch-and-bound algorithm for solving a fixed charge problem in the profit optimization of sawn timber production

This paper deals with the problem of profit optimization in sawn timber production, utilizing a special type of sawmill. Expected rejects and resetting costs are taken into consideration. The present problem is formulated as a fixed charge linear programming problem involving identical fixed charges, one equality constraint and explicit bounds on the variables. Based on the greedy sorting of the variables we develop a branch-and-bound algorithm working on a special subset of all solutions. Through usage of the problem structure for constructing bounds we arrive at an acceptable CPU-time (on a 80386 personal computer) for practical purposes.

An algorithm for solving fixed-charge problems using surrogate constraints

In recent years, many researchers have shown that surrogate constraints can be used to reduce the computational requirements of integer and mixed-integer programming problems. In this paper, we develop an algorithm that uses surrogate constraints io solve fixed-charge problems. This algorithm generates a candidate solution for the fixed-charge problem by solving a set covering problem. If the candidate solution is infeasible in the fixed-charge problem a surrogate constraint is generated and added to the set covering problem to cut this solution from the feasible region. This process is repeated until a feasible solution to the fixed-charge problem is found. This solution is then used as an incumbent solution in a branch-and-bound procedure, which solves the problem to optimality. In those cases where the problem has only fixed-charges, the optimal solution can be found without the branch-and-bound procedure. Computational results for various types of problems are given. These results indicate that this algorithm tends to perform better than existing branch-and-bound solution procedures.

Long-Term Planning for Solid Waste Management in India

Most of the existing models of solid waste management deal with a single point of time and are based on the fixed charge problem. In developing countries the fixed charge problem approach is not adequate in the modelling of solid waste management systems. Moreover, in fast growing cities the problem of locating disposal sites and allocating collection areas to them becomes difficult in the long-term. This paper deals with the application of a transportation model designed to minimise solid waste handling costs over several developmental phases of an urban area. The model is applied to an actual case to demonstrate its usefulness for long-range planning in cities.

A method for the solution of fixed charge problems by Benders' decomposition

A method of solving fixed charge problems by Benders' Decomposition is described. The sub-problem of the Benders' Decomposition process is solved by solving a smaller equivalent problem. Extremal optimal solutions and extreme rays associated with the dual of the sub-problem are obtained from those associated with the smaller problem.

A computational study of using preprocessing and stronger formulations to solve large general fixed charge problems

The application of traditional branch-and-bound (B&B) procedures to the standard formulation of the general fixed charge problem (GFCP) has not proven to be an acceptable approach to solving this well-known problem. It has been suggested by various authors that preprocessing the data and/or adding constraints to the GFCP might lead to improved B&B performance. In this paper, we present extensive computational results to test these hypotheses using the M PSX/MIP/370 package. The results demonstrate that while preprocessing the data is not always effective, simple constraints may be added to the problem allowing us to solve randomly generated GFCPs that are significantly larger than any previously reported in the literature. Results arc also presented which give insight into the effectiveness of a frequently used heuristic for the GFCP.

Solving the fixed charge problem with Lagrangian relaxation and cost allocation heuristics

A new heuristic algorithm is presented for solving large general fixed charge problems based on Langrangian relaxation and cost allocation heuristics. Limited computational results indicate the new algorithm provides good solutions to large problems with reasonable computational requirements. It demonstrates significant improvement over current methods for solving large general fixed charge problems.

Valid inequalities and facets of the capacitated plant location problem

Recently, several successful applications of strong cutting plane methods to combinatorial optimization problems have renewed interest in cutting plane methods, and polyhedral characterizations, of integer programming problems. In this paper, we investigate the polyhedral structure of the capacitated plant location problem. Our purpose is to identify facets and valid inequalities for a wide range of capacitated fixed charge problems that contain this prototype problem as a substructure.

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.

Reformulation of mathematical programming problems as linear complementarity problems and investigation of their solution methods

A family of complementarity problems is defined as extensions of the well-known linear complementarity problem (LCP). These are: A number of well-known mathematical programming problems [namely, quadratic programming (convex, nonconvex, pseudoconvex, nonconvex), linear variational inequalities, bilinear programming, game theory, zero-one integer programming, fixed-charge problem, absolute value programming, variable separable programming] are reformulated as members of this family of four complementarity problems. A brief discussion of the main algorithms for these four problems is presented, together with some computational experience.

Valid Linear Inequalities for Fixed Charge Problems

Many problems in the Operations Research/Management Science literature can be formulated with both zero-one and continuous variables. However, the exact optimization of such mixed zero-one models remains a computational challenge. In this paper, we propose to study mixed problems from a mathematical point of view that is similar in spirit to recent research on purely combinatorial problems that has investigated systems of defining linear inequalities (or facets of the underlying polytope). At least two numerical studies have validated this line of research computationally, and the advances in the problem-solving capabilities are considerable. We expect that similar gains are possible for the mixed zero-one problem. More precisely, we consider the mixed integer programs whose feasible region X is composed of (i) a simple additive constraint in the continuous variables xj, for j = 1, 2, …, n and (ii) constraints 0 ≤ xj ≤ mjyj defined by binary variables yj for j = 1, 2, …, n. This type of feasible region arises in a variety of mixed integer problems, and particularly in network problems with fixed charges on the arcs. We derive two classes of facet-defining linear inequalities of the convex hull of X, and show that the second of these classes gives a complete description of the convex hull when mj = m for all j. We also develop methods to detect violated inequalities from these classes, so that these facets can be used as cutting planes to strengthen the formulations of certain mixed integer problems.

Subset Coefficient Reduction Cuts for 0/1 Mixed-Integer Programming

We describe a method for generating cuts for mixed-integer 0/1 programs. These cuts are designed to tighten an integer program prior to applying linear programming based branch and bound algorithms. The method involves two basic ideas: subset selection and coefficient reduction. Coefficient reduction is a process of reducing the coefficients of the 0/1 variables. Subset selection is combined with coefficient reduction by applying the coefficient reduction process to the coefficients of a subset of variables from the constraints in the problem formulation. The paper exploits these two simple ideas to derive a broad class of cuts for integer programs with both 0/1 and continuous variables. It also reports on the use of this methodology in solving a variety of fixed charge problems.

Fixed charge problems with identical fixed charges

The general problem considered by this paper is a special case of the fixed-charge problem. The further condition imposed is that all variables have the same associated fixed-charge. The problem is discussed in the context of a known commercial application, that being the cutting stock problem. The situation considered is that of cutting given numbers of small rectangles from large rectangular stock-plates. In many such situations major aims are to have low stock-plate usage and a low number of setups of the cutting equipment. These represent conflicting objectives capable of being combined by the use of fixed charges upon the setups but this paper presents an alternative approach incorporating direct manipulation of the number of setups involved in the solution. This approach is compared to a solution technique for the general fixed-charge problem.

Optimization Methodology for Bus Garage Locations

An optimization model is developed for determining both the optimal bus garage locations and optimal assignment of buses to those garages. The model uses a list of candidate garage sites, the transit system’s service and operational characteristics and the characteristics of the regional highway system as input. As output, it produces the optimal number, sizes and locations of bus garage additions or changes. The model considers garage operating and garage construction cost and the costs of deadheading and driver relief under separate operating schedules for morning, all day, and evening route assignments on weekdays, Sundays and Saturdays. In the methodology development, the problem is first formulated as a discrete solution-space fixed-charge facility location problem. The fixed charge problem is formulated as a mixed integer program. However, because of the difficulty involved in solving a large-scale problem with a general purpose mixed-integer program, the problem is solved by a special purpose branch-and-bound process. This process utilizes an efficient sequence of classical transportation problem (a quick solving, special class of linear program). The methodology is demonstrated with a small-scale sample problem.

A branch‐and‐bound algorithm for solving fixed charge problems

AbstractNumerous procedures have been suggested for solving fixed charge problems. Among these are branch‐and‐bound methods, cutting plane methods, and vertex ranking methods. In all of these previous approaches, the procedure depends heavily on the continuous costs to terminate the search for the optimal solution. In this paper, we present a new branch‐and‐bound algorithm that calculates bounds separately on the sum of fixed costs and on the continuous objective value. Computational experience is shown for various standard test problems as well as for randomly generated problems. These test results are compared to previous procedures as well as to a mixed integer code. These comparisons appear promising.

Design of a Receivable Collection System: Sequential Building Heuristics

This paper extends previous work on the use of sequential building heuristics for the design of a collection system, a linear fixed-charge problem that includes the design of a lock-box system as an important subproblem. The heuristic extensions in this paper first present simple methods for eliminating “premature termination” (stopping the heuristics while there are still worthwhile alternatives not included in the collection system) and unprofitable inclusion (including collection alternatives with net benefits less than the fixed cost). Initiation procedures that increase computational efficiency and reduce the likelihood of significant suboptimization are presented. Finally, the simple one-at-a-time building procedure is extended to include joint exclusion-building (a limited switching plus building) and combination switching-building rules that consider combinations of alternatives rather than one-at-a-time evaluation.

An easy solution for a special class of fixed charge problems

The fixed charge problem is a mixed integer mathematical programming problem which has proved difficult to solve in the past. In this paper we look at a special case of that problem and show that this case can be solved by formulating it as a set-covering problem. We then use a branch-and-bound integer programming code to solve test fixed charge problems using the setcovering formulation. Even without a special purpose set-covering algorithm, the results from this solution procedure are dramatically better than those obtained using other solution procedures.

Timber harvest planning—A combined optimization/simulation model

A special cascading fixed charge structure can be used to characterize a forest management planning problem in which the objectives are to identify the optimal shape of forest harvest cutting units and simultaneously to assign facilities for logging those units. We describe a four-part methodology that has been developed to assist forest managers in analyzing areas proposed for harvesting. This methodology performs an analysis of harvesting feasibility, computes the optimal solution to the cascading fixed charge problem, undertakes a GASP IV simulation to provide additional information about the proposed harvesting operation, and permits the forest manager to perform a time-cost analysis that may lead to a more realistic, and thus improved, solution.

Ground-Water Management with Fixed Charges

The problem of where to drill wells, how many to drill, and how to pump them in order to maintain low water levels in an excavation most economically is formulated as a fixed-charge problem. The cost to be minimized includes fixed installation costs and the variable costs of pumping. The governing ground-water equations in numerical form are included as constraints. The solution includes optimal well locations, costs, steady-state discharge schedules, and hydraulic head distributions.