The continuous‐time single‐sourcing problem with capacity expansion opportunities

This paper develops a new model for allocating demand from retailers (or customers) to a set of production/storage facilities. A producer manufactures a product in multiple production facilities, and faces demand from a set of retailers. The objective is to decide which of the production facilities should satisfy each retailer's demand, in order minimize total production, inventory holding, and assignment costs (where the latter may include, for instance, variable production costs and transportation costs). Demand occurs continuously in time at a deterministic rate at each retailer, while each production facility faces fixed‐charge production costs and linear holding costs. We first consider an uncapacitated model, which we generalize to allow for production or storage capacities. We then explore situations with capacity expansion opportunities. Our solution approach employs a column generation procedure, as well as greedy and local improvement heuristic approaches. A broad class of randomly generated test problems demonstrates that these heuristics find high quality solutions for this large‐scale cross‐facility planning problem using a modest amount of computation time. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005.     

An analytical evaluation of optimal solution value estimation procedures

The estimation of optimal solution values for large-scale optimization problems is studied. Optimal solution value estimators provide information about the deviation between the optimal solution and the heuristic solution. Some estimation techniques combine heuristic solutions with randomly generated solutions. In particular, we examine a class of jacknife-based estimators which incorporate any heuristic solution value with the two best randomly generated solution values. The primary contribution of this article is that we provide a framework to analytically evaluate a class of optimal solution value estimators. We present closed-form results on the relationship of heuristic performance, sample size, and the estimation errors for the case where the feasible solutions are uniformly distributed. In addition, we show how to compute the estimation errors for distributions other than uniform given a specific sample size. We use a triangular and an exponential distribution as examples of other distributions. A second major contribution of this article is that, to a large extent, our analytical results confirm previous computational results. In particular, the best estimator depends on how good the heuristic is, but seems to be independent of the underlying distribution of solution values. Furthermore, there is essentially an inverse relationship between the heuristic performance and the performance of any estimator. © 1994 John Wiley & Sons, Inc.     

The one-period,N-location distribution problem

This paper studies the one-period, general network distribution problem with linear costs. The approach is to decompose the problem into a transportation problem that represents a stocking decision, and into decoupled newsboy problems that represent the realization of demand with the usual associated holding and shortage costs. This approach leads to a characterization of optimal policies in terms of the dual of the transportation problem. This method is not directly suitable for the solution for large problems, but the exact solution for small problems can be obtained. For the numerical solutions of large problems, the problem has been formulated as a linear program with column generation. This latter approach is quite robust in the sense that it is easily extended to incorporate capacity constraints and the multiproduct case.     

Algorithms for the multi‐item multi‐vehicles dynamic lot sizing problem

We consider a two‐stage supply chain, in which multi‐items are shipped from a manufacturing facility or a central warehouse to a downstream retailer that faces deterministic external demand for each of the items over a finite planning horizon. The items are shipped through identical capacitated vehicles, each incurring a fixed cost per trip. In addition, there exist item‐dependent variable shipping costs and inventory holding costs at the retailer for items stored at the end of the period; these costs are constant over time. The sum of all costs must be minimized while satisfying the external demand without backlogging. In this paper we develop a search algorithm to solve the problem optimally. Our search algorithm, although exponential in the worst case, is very efficient empirically due to new properties of the optimal solution that we found, which allow us to restrict the number of solutions examined. Second, we perform a computational study that compares the empirical running time of our search methods to other available exact solution methods to the problem. Finally, we characterize the conditions under which each of the solution methods is likely to be faster than the others and suggest efficient heuristic solutions that we recommend using when the problem is large in all dimensions. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2006.     

The fixed charge problem

The 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.     

The single and multi‐item transshipment problem with fixed transshipment costs

This article deals with supply chain systems in which lateral transshipments are allowed. For a system with two retailers facing stochastic demand, we relax the assumption of negligible fixed transshipment costs, thus, extending existing results for the single‐item case and introducing a new model with multiple items. The goal is to determine optimal transshipment and replenishment policies, such that the total centralized expected profit of both retailers is maximized. For the single‐item problem with fixed transshipment costs, we develop optimality conditions, analyze the expected profit function, and identify the optimal solution. We extend our analysis to multiple items with joint fixed transshipment costs, a problem that has not been investigated previously in the literature, and show how the optimality conditions may be extended for any number of items. Due to the complexity involved in solving these conditions, we suggest a simple heuristic based on the single‐item results. Finally, we conduct a numerical study that provides managerial insights on the solutions obtained in various settings and demonstrates that the suggested heuristic performs very well. © 2014 Wiley Periodicals, Inc. Naval Research Logistics, 61: 637–664, 2014     

Lot sizing when yields increase during the production run

We investigate the problem of determining lot sizes for multiple items when the expected percentage of acceptable output increases with the duration of the production run, usually due to adjustments made during the early part of the production run. Such problems arise in metal stamping, textile finishing processes, and a variety of other industries. The goal is to minimize the total cost of production, inventory holding costs, and setup costs (where applicable). We develop a heuristic procedure based on a Lagrangian relaxation that differs from relaxations used in earlier studies. We use various properties of the objective function to guide the adjustment of the initial solution from the relaxation toward feasibility. Computational results indicate that, on the average, the heuristic produces solutions within 4.9% of the lower bound obtained from the Lagrangian relaxation. © 1996 John Wiley & Sons, Inc.     

Dual-ascent methods for large-scale multicommodity flow problems

The capacitated multicommodity network flow problem presents itself in a number of problem contexts including transportation, communication, and production. To solve the large-scale multicommodity flow problems encountered in these fields, we develop dual-ascent heuristics and a primal solution generator. The dual-ascent solutions, in addition to determining lower bounds on the optimal objective function value, provide advanced starting solutions for use with primal-based solution techniques. The primal solution generator uses the dual-ascent solution to obtain heuristically primal solutions to the multicommodity flow problems. Computational experiments performed on three test problem sets show that the dual-ascent and primal heuristic procedures typically determine nearoptimal solutions quickly. In addition, by using the dual-ascent procedure to obtain advanced starting solutions, run times for optimal multicommodity flow procedures are reduced significantly and greatly improved solutions are obtained by the new primal solution generator. © 1993 John Wiley & Sons, Inc.     

Minimizing mean tardiness and earliness in single-machine scheduling problems with unequal due dates

We consider a single-machine scheduling problem with the objective of minimizing the mean (or equivalently, total) tardiness and earliness when due dates may differ among jobs. Some properties of the optimal solution are discussed, and these properties are used to develop both optimal and heuristic algorithms. Results of computational tests indicate that optimal solutions can be found for problems with up to 20 jobs, and that two of the heuristic procedures provide optimal or very near optimal solutions in many instances. © 1994 John Wiley & Sons, Inc.     

A nonidentical parallel processor scheduling problem

The problem considered in this article is a generalization of the familiar makespan problem, in which n jobs are allocated among m parallel processors, so as to minimize the maximum time (or cost) on any processor. Our problem is more general, in that we allow the processors to have (a) different initial costs, (b) different utilization levels before new costs are incurred, and (c) different rates of cost increase. A heuristic adapted from the bin-packing problem is shown to provide solutions which are close to optimal as the number of iterations is allowed to increase. Computational testing, over a large number of randomly generated problem instances, suggests that heuristic errors are, on average, very small.     

A heuristic solution procedure for the multiconstraint zero-one knapsack problem

In this article a new heuristic procedure is proposed. This procedure makes use of surrogate duality in solving multiconstraint knapsack problems. Computational effort involved in the procedure is bounded by a polynomial in the number of variables. Extensive computational testing indicates that the procedure generates good feasible solutions regardless of the problem structure. In 98% of the problems solved, the solution generated by the heuristic was within 1% of the optimal solution. This procedure was also tested against other heuristics and was found to compare favorably.     

Using multiple searchers in constrained-path,moving-target search problems

The search theory open literature has paid little, if any, attention to the multiple-searcher, moving-target search problem. We develop an optimal branch-and-bound procedure and six heuristics for solving constrained-path problems with multiple searchers. Our optimal procedure outperforms existing approaches when used with only a single searcher. For more than one searcher, the time needed to guarantee an optimal solution is prohibitive. Our heuristics represent a wide variety of approaches: One solves partial problems optimally, two use paths based on maximizing the expected number of detections, two are genetic algorithm implementations, and one is local search with random restarts. A heuristic based on the expected number of detections obtains solutions within 2% of the best known for each one-, two-, and three-searcher test problem considered. For one- and two-searcher problems, the same heuristic's solution time is less than that of other heuristics. For three-searcher problems, a genetic algorithm implementation obtains the best-known solution in as little as 20% of other heuristic solution times. © 1996 John Wiley & Sons, Inc.     

Interval estimation of a global optimum for large combinatorial problems

Consider an “intractable” optimization problem for which no efficient solution technique exists. Given a systematic procedure for generating independent heuristic solutions, we seek to obtain interval estimates for the globally optimal solution using statistical inference. In previous work, accurate point estimates have been derived. Determining interval estimates, however, is a considerably more difficult task. In this paper, we develop straightforward procedures which compute confidence intervals efficiently in order to evaluate heuristic solutions and assess deviations from optimality. The strategy presented is applicable to a host of combinatorial optimization problems. The assumptions of our model, along with computational experience, are discussed.     

A heuristic routine for solving large loading problems

The loading problem involves the optimal allocation of n objects, each having a specified weight and value, to m boxes, each of specified capacity. While special cases of these problems can be solved with relative ease, the general problem having variable item weights and box sizes can become very difficult to solve. This paper presents a heuristic procedure for solving large loading problems of the more general type. The procedure uses a surrogate procedure for reducing the original problem to a simpler knapsack problem, the solution of which is then employed in searching for feasible solutions to the original problem. The procedure is easy to apply, and is capable of identifying optimal solutions if they are found.     

Discrete equal‐capacity p‐median problem

This paper examines the discrete equal‐capacity p‐median problem that seeks to locate p new facilities (medians) on a network, each having a given uniform capacity, in order to minimize the sum of distribution costs while satisfying the demand on the network. Such problems arise, for example, in local access and transport area telecommunication network design problems where any number of a set of p facility units can be constructed at the specified candidate sites (hence, the net capacity is an integer multiple of a given unit capacity). We develop various valid inequalities, a separation routine for generating cutting planes that are specific members of such inequalities, as well as an enhanced reformulation that constructs a partial convex hull representation that subsumes an entire class of valid inequalities via its linear programming relaxation. We also propose suitable heuristic schemes for this problem, based on sequentially rounding the continuous relaxation solutions obtained for the various equivalent formulations of the problem. Extensive computational results are provided to demonstrate the effectiveness of the proposed valid inequalities, enhanced formulations, and heuristic schemes. The results indicate that the proposed schemes for tightening the underlying relaxations play a significant role in enhancing the performance of both exact and heuristic solution methods for this class of problems. © 2000 John & Sons, Inc. Naval Research Logistics 47: 166–183, 2000.     

A concave‐cost production planning problem with remanufacturing options

We focus on the concave‐cost version of a production planning problem where a manufacturer can meet demand by either producing new items or by remanufacturing used items. Unprocessed used items are disposed. We show the NP‐hardness of the problem even when all the costs are stationary. Utilizing the special structure of the extreme‐point optimal solutions for the minimum concave‐cost problem with a network flow type feasible region, we develop a polynomial‐time heuristic for the problem. Our computational study indicates that the heuristic is a very efficient way to solve the problem as far as solution speed and quality are concerned. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005     

A branch-and-bound algorithm to minimize total flow time with unequal release dates

This article examines the single-machine scheduling problem to minimize total flow time with unequal release dates. This problem has been proven to be NP-hard. We present a necessary and sufficient condition for local optimality which can also be considered as a priority rule. On the basis of this condition, we then define a class of schedules which contains all optimal solutions. We present some efficient heuristic algorithms using the previous condition to build a schedule belonging to this subset. We also prove some new dominance theorems, discuss the results found in the literature for this problem, and propose a branch-and-bound algorithm in which the heuristics are used to provide good upper bounds. We compare this new algorithm with existing algorithms found in the literature. Computational results on problems with up to 100 jobs indicate that the proposed branch-and-bound algorithm is superior to previously published algorithms. © 1992 John Wiley & Sons. Inc.     

Heuristics for the multi‐resource generalized assignment problem

The well‐known generalized assignment problem (GAP) involves the identification of a minimum‐cost assignment of tasks to agents when each agent is constrained by a resource in limited supply. The multi‐resource generalized assignment problem (MRGAP) is the generalization of the GAP in which there are a number of different potentially constraining resources associated with each agent. This paper explores heuristic procedures for the MRGAP. We first define a three‐phase heuristic which seeks to construct a feasible solution to MRGAP and then systematically attempts to improve the solution. We then propose a modification of the heuristic for the MRGAP defined previously by Gavish and Pirkul. The third procedure is a hybrid heuristic that combines the first two heuristics, thus capturing their relative strengths. We discuss extensive computational experience with the heuristics. The hybrid procedure is seen to be extremely effective in solving MRGAPs, generating feasible solutions to more than 99% of the test problems and consistently producing near‐optimal solutions. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48: 468–483, 2001     

A heuristic for capacity expansion planning with multiple facility types

A capacity expansion model with multiple facility types is examined, where different facility types represent different quality levels. Applications for the model can be found in communications networks and production facilities. The model assumes a finite number of discrete time periods. The facilities are expanded over time. Capacity of a high-quality facility can be converted to satisfy demand for a lower-quality facility. The costs considered include capacity expansion costs and excess capacity holding costs. All cost functions are nondecreasing and concave. An algorithm that finds optimal expansion policies requires extensive computations and is practical only for small scale problems. Here, we develop a heuristic that employs so-called distributed expansion policies. It also attempts to decompose the problem into several smaller problems solved independently. The heuristic is computationally efficient. Further, it has consistently found near-optimal solutions.     

An approximative algorithm for the fixed charge problem

This 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.