Concave Cost Research Articles

Optimal Facility Location with Concave Costs

The following problem is considered: select plant sites from a given set of sites and choose their production and distribution levels to meet known demand at discrete points at minimum cost. The construction and operating cost of each plant is assumed to be a concave function of the total production at that plant, and the distribution cost between each plant and demand point is assumed to be a concave function of the amount shipped. There may be capacity restrictions on the plants. A branch-and-bound algorithm for identifying an optimal solution is described; it is equivalent to the solution of a finite sequence of transportation problems. The algorithm is developed as a particular case of a simplified algorithm for minimizing separable concave functions over linear polyhedra. Computational results are cited for a computer code implementing the algorithm.

Warehouse Location With Concave Costs

AbstractThis paper presents an improved solution procedure for the warehouse location problem where continuous economies of scale are present. The improvement is achieved by an effective combination of two heuristic methods presented recently by Drysdale and Sandiford, and Khumawala. The solution procedure first focuses on reducing the potential warehouse set using Khumawala’s method. An iterative technique is then used to optimally allocate customers to these warehouses, taking into account their diminishing marginal costs. As demonstrated in this paper, the procedure converges rapidly to a “good” solution. The efficiency of the procedure is compared with another solution method and the results are quite promising.

Bounded Production and Inventory Models with Piecewise Concave Costs

An n period single-product single-facility model with known requirements and separable piecewise concave production and storage costs is considered. It is shown using network flow concepts that for arbitrary bounds on production and inventory in each period there is an optimal schedule such that if, for any two periods, production does not equal zero or its upper or lower bound, then the inventory level in some intermediate period equals zero or its upper or lower bound. An algorithm for searching such schedules for an optimal one is given where the bounds on production are −∞, 0 or ∞. A more efficient algorithm assumes further that inventory bounds satisfy “exact requirements.”

Construction of efficient tree networks: The pipeline problem

AbstractIt is observed that sample cost functions for the pipeline problem [1] are discretely convex, i.e., their graphs are discrete subsets of graphs of convex functions. As a result, it is shown that the pipeline problem may be attacked by using either a minimum cost flow approach, or a combination of dynamic programming and sorting. Problems with concave cost functions are shown to be relatively easy. Even for problems which are neither convex nor concave, it is shown that in certain instances, only a subset of the data for each cost function is relevant. Error bounds are presented when approximations to the original cost functions are used. Results obtained in [4] for a related problem are summarized.

Erratum: Deterministic Production Planning with Concave Costs and Capacity Constraints

Erratum to authors' paper (Florian M., M. Klein. 1971. Deterministic production planning with concave costs and capacity constraints. Management Sci. 18 (1, September) 12–20.

Structural optimization with piece-wise concave cost functionals

A theorem on a necessary condition for a constrained minimum of piece-wise concave functionals is derived and it is used for obtaining minimum weight solutions for anisotropic cylindrical shells having variable rib depth. Existing methods for plastic optimal design are briefly reviewed.

A Property of Minimum Concave Cost Flows in Capacitated Networks*

AbstractA characterization of extreme flows in capacitated networks is obtained. This propertyis useful to determine optimal policies for specially structured problems that arise in the context of production planning and inventory control.

Deterministic Production Planning with Concave Costs and Capacity Constraints

A multi-period single commodity production planning problem is studied in which known requirements have to be satisfied. The model differs from earlier well-known studies involving concave cost functions in the introduction of production capacity constraints. The structure of an optimal solution is characterized and then used in a simple dynamic programming algorithm for problems in which the capacities are the same in every period. Both the nonbacklog and backlog allowed cases are considered.

Design Commonality to Reduce Multi-Item Inventory: Optimal Depth of a Product Line

The problem of commonality is posed as a problem of economic balance. It involves the disutility of refusing to provide each segment of customers with an item fitting its exact requirements versus economies of scale achieved in producing and inventorying each item. The depth of the product line (the optimal number of items to produce) is determined by finding the minimum concave cost flow through a network. An efficient dynamic program (developed in production smoothing) is used to find this flow. Many products consist of interacting subassemblies; such multistage problems of finding the number of types of each subassembly can also be solved; a computational procedure is developed and an example presented. Along with modularity and cannibalization, multistage commonality is an essential component of a theory of efficient engineering design.

Minimum Concave-Cost Solution of Leontief Substitution Models of Multi-Facility Inventory Systems

The paper shows that a broad class of problems can be formulated as minimizing a concave function over the solution set of a Leontief substitution system. The class includes deterministic single- and multi-facility economic lot size, lot-size smoothing, warehousing, product-assortment, batch-queuing, capacity-expansion, investment consumption, and reservoir-control problems with concave cost functions. Because in such problems an optimum occurs at an extreme point of the solution set, we can utilize the characterization of the extreme points given in a companion paper to obtain most existing qualitative characterizations of optimal policies for inventory models with concave costs in a unified manner. Dynamic programming recursions for searching the extreme points to find an optimal point are given in a number of cases. We only give algorithms whose computational effort increases algebraically (instead of exponentially) with the size of the problem.

Minimum Concave Cost Flows in Certain Networks

The literature is replete with analyses of minimum cost flows in networks for which the cost of shipping from node to node is a linear function. However, the linear cost assumption is often not realistic. Situations in which there is a set-up charge, discounting, or efficiencies of scale give rise to concave functions. Although concave functions can be minimized by an exhaustive search of all the extreme points of the convex feasible region, such an approach is impractical for all but the simplest of problems. In this paper some theorems are developed which explicitly characterize the extreme points for certain single commodity networks. By exploiting this characterization algorithms are developed that determine the minimum concave cost solution for networks with a single source and a single destination, for acyclic single source multiple destination networks, and for acyclic single destination multiple source networks. An interesting theorem then establishes that for either single source or single destination networks the multi-commodity case can be reduced to the single commodity case. Applications to the concave warehouse problem, a single product production and inventory model, a multi-product production and inventory model, and a plant location problem are included.

Production Smoothing of Economic Lot Sizes with Non-Decreasing Requirements

This paper considers the problem of finding a production schedule, in terms of how much to produce in each period, that minimizes the total cost of supplying known market requirements for a single product. The costs include a concave production cost, a concave inventory cost, and a piecewise concave cost of changing the production level from one period to the next. Assuming that there is no backlogging of requirements and that the market requirements are monotone increasing, i.e., not decreasing from period to period, the form of the minimum cost production schedule is obtained. This form is then exploited in a dynamic programming algorithm to provide an efficient means of exactly determining the minimum cost schedule. An interesting calculation reducing theorem is also developed to further enhance the efficiency of the dynamic programming algorithm.

A Deterministic Multi-Period Production Scheduling Model with Backlogging

A deterministic multi-period production and inventory model that has concave production costs and piecewise concave [Zangwill, W. I. 1965. A dynamic multi-product, multi-facility production and inventory model. Technical Report 1, Program in Operations Research, Stanford University, Stanford, California, April 29.] inventory costs is analyzed. An essential feature of the model is that it permits backlogging of unsatisfied demand; otherwise the model is similar to the one considered by Manne [Manne, A. S. 1958. Programming of economic lot sizes. Management Sci. 4(2, Jan.) 115–135.], Wagner and Whitin [Wagner, H. M., T. M. Whitin. 1959. Dynamic version of the economic lot size model. Management Sci. 5(1, October) 89–96.], Wagner [Wagner, H. M. 1960. A postscript to “dynamic problems to the firm.” Naval Res. Logist. Quart. 7(March) 7–12.], and Veinott [Veinott, A. F., Jr. 1963. Unpublished notes. Stanford University, Stanford, California.]. By permitting backlogging, which mathematically means that inventories can be negative, the concavity assumptions of these authors are no longer appropriate. Instead we consider piecewise concave cost functions to find the form of the minimum cost production schedule. An efficient dynamic programming algorithm to calculate the minimum cost schedule is also presented.

A Warehouse-Location Problem

This paper describes a method for determining a more profitable geographic location pattern for the warehouses that are employed by a firm in delivering known quantities of its finished product to its customers, where the number of warehouses is also permitted to vary. It is shown that this can normally be expected to be a concave minimization problem. Unfortunately, there are usually no practical computational methods for determining the values of the variables that correspond to the absolute minimum of a concave cost function. However, a local optimum can be determined by a method described in the article. The method involves a sequence of transportation computations that are shown to converge to the solution. An illustrative small scale computation is included. An appendix explains in intuitive terms the nature of the computational difficulties that arise in a concave-programming problem.