Showing papers in "Operations Research in 1994"

Optimal Solution of Vehicle Routing Problems Using Minimum K-Trees

Abstract: We consider the problem of optimally scheduling a fleet of K vehicles to make deliveries to n customers subject to vehicle capacity constraints. Given a graph with n + 1 nodes, a K-tree is defined to be a set of n + K edges that span the graph. We show that the vehicle routing problem can be modeled as the problem of finding a minimum cost K-tree with two K edges incident on the depot and subject to some side constraints that impose vehicle capacity and the requirement that each customer be visited exactly once. The side constraints are dualized to obtain a Lagrangian problem that provides lower bounds in a branch-and-bound algorithm. This algorithm has produced proven optimal solutions for a number of difficult problems, including a well-known problem with 100 customers and several real problems with 25–71 customers.

A Greedy Randomized Adaptive Search Procedure for Maximum Independent Set

Abstract: An efficient randomized heuristic for a maximum independent set is presented. The procedure is tested on randomly generated graphs having from 400 to 3,500 vertices and edge probabilities from 0.2 to 0.9. The heuristic can be implemented trivially in parallel and is tested on an MIMD computer with 1, 2, 4 and 8 processors. Computational results indicate that the heuristic frequently finds the optimal or expected optimal solution in a fraction of the time required by implementations of simulated annealing, tabu search, and an exact partial enumeration method.

Day-To-Day Dynamic Network Disequilibria and Idealized Traveler Information Systems

Abstract: In this paper we present tatonnement models for calculating static Wardropian user equilibria on congested networks with fully general demand and cost structures. We present both a qualitative analysis of stability and numerical studies which show that such an approach provides a reliable means for determining static user equilibria. We also describe circumstances for which these models depict day-to-day adjustments from one realizable disequilibrium state to another and how these adjustment processes differ depending on the “quality” of the information being provided by (abstract) traveler information systems. Specifically, we demonstrate that such dynamic adjustment processes settle down to equilibria both when information is complete and when it is incomplete.

Parallel Branch-and-Branch Algorithms: Survey and Synthesis

Abstract: We present a detailed and up-to-date survey of the literature on parallel branch-and-bound algorithms. We synthesize previous work in this area and propose a new classification of parallel branch-and-bound algorithms. This classification is used to analyze the methods proposed in the literature. To facilitate our analysis, we give a new characterization of branch-and-bound algorithms, which consists of isolating the performed operations without specifying any particular order for their execution.

Note on the Complexity of the Shortest Path Models for Column Generation in VRPTW

Abstract: In this note we prove that the relaxation approach in designing the subproblem of pricing out only the feasible routes for the set partition formulation of the VRPTW is justified on complexity grounds. That is, the first dynamic programming model presented in M. Desrochers, J. Desrosiers and M. Solomon 1992, that is able to price out all feasible routes, is NP-hard in the strong sense.

Quantitative Methods in Credit Management: A Survey

Abstract: Many static and dynamic models have been used to assist decision making in the area of consumer and commercial credit. The decisions of interest include whether to extend credit, how much credit to extend, when collections on delinquent accounts should be initiated, and what action should be taken. We survey the use of discriminant analysis, decision trees, and expert systems for static decisions, and dynamic programming, linear programming, and Markov chains for dynamic decision models. Since these models do not operate in a vacuum, we discuss some important aspects of credit management in practice, e.g., legal considerations, sources of data, and statistical validation of the methodology. We provide our perspective on the state-of-the-art in theory and in practice.

Heuristic and special case algorithms for dispersion problems

Abstract: The dispersion problem arises in selecting facilities to maximize some function of the distances between the facilities The problem also arises in selecting nondominated solutions for multiobjective decision making It is known to be NP-hard under two objectives: maximizing the minimum distance (MAX-MIN) between any pair of facilities and maximizing the average distance (MAX-AVG) We consider the question of obtaining near-optimal solutions for MAX-MIN, we show that if the distances do not satisfy the triangle inequality, there is no polynomial-time relative approximation algorithm unless P = NP When the distances satisfy the triangle inequality, we analyze an efficient heuristic and show that it provides a performance guarantee of two We also prove that obtaining a performance guarantee of less than two is NP-hard for MAX-AVG, we analyze an efficient heuristic and show that it provides a performance guarantee of four when the distances satisfy the triangle inequality We also present a polynomial-ti

The total tardiness problem: review and extensions

Abstract: We provide a unified framework for the total tardiness problem by surveying the related literature in the single-machine, parallel machine, flowshop and jobshop settings. We focus on critically evaluating the heuristic algorithms; we also propose new heuristics for both the single-machine and the parallel-machine tardiness problems. Finally, we identify the areas where further research is needed and we give directions for future research.

Needed: An Empirical Science of Algorithms

Abstract: Deductive algorithmic science has reached a high level of sophistication, but its worst-case and average-case results seldom tell us how well an algorithm is actually going to work in practice. I argue that an empirical science of algorithms is a viable alternative. I respond to misgivings about an empirical approach, including the prevalent notion that only a deductive treatment can be “theoretical” or sophisticated. NP-completeness theory, for instance, is interesting partly because it has significant, if unacknowledged, empirical content. An empirical approach requires not only rigorous experimental design and analysis, but also the invention of empirically-based explanatory theories. I give some examples of recent work that partially achieves this aim.

Markov Decision Processes with Imprecise Transition Probabilities

Abstract: We present new numerical algorithms and bounds for the infinite horizon, discrete stage, finite state and action Markov decision process with imprecise transition probabilities. We assume that the transition probability mass vector for each state and action is described by a finite number of linear inequalities. This model of imprecision appears to be well suited for describing statistically determined confidence limits and/or natural language statements of likelihood. The numerical procedures for calculating an optimal max-min strategy are based on successive approximations, reward revision, and modified policy iteration. The bounds that are determined are at least as tight as currently available bounds for the case where the transition probabilities are precise.

The Multi-Airport Ground-Holding Problem in Air Traffic Control

Abstract: Motivated by the important problem of congestion costs (they were estimated to be $2 billion in 1991) in air transportation and observing that ground delays are more preferable than airborne delays, we have formulated and studied several integer programming models to assign ground-holding delays optimally in a general network of airports, so that the total (ground plus airborne) delay cost of all flights is minimized. All previous research on this problem has been restricted to the single-airport case, which neglects “down-the-road” effects due to transmission of delays between successive flights performed by the same aircraft. We formulate several models, and then propose a heuristic algorithm which finds a feasible solution to the integer program by rounding the optimal solution of the LP relaxation. Finally, we present extensive computational results with the goal of obtaining qualitative insights on the behavior of the problem under various combinations of the input parameters. We demonstrate that the...

A Column Generation Approach to the Multiple-Depot Vehicle Scheduling Problem

Abstract: We give a new formulation to the multiple-depot vehicle scheduling problem as a set partitioning problem with side constraints, whose continuous relaxation is amenable to be solved by column generation. We show that the continuous relaxation of the set partitioning formulation provides a much tighter lower bound than the additive bound procedure previously applied to this problem. We also establish that the additive bound technique cannot provide tighter bounds than those obtained by Lagrangian decomposition, in the framework in which it has been used so far. Computational results that illustrate the robustness of the combined set partitioning-column generation approach are reported for problems four to five times larger than the largest problems that have been exactly solved in the literature. Finally, we show that the gap associated with the additive bound based on the assignment and shortest path relaxations can be arbitrarily bad in the general case, and as bad as 50% in the symmetric case.

A priori optimization of the probabilistic traveling salesman problem

Abstract: The probabilistic traveling salesman problem (PTSP) is defined on a graph G = (V, E), where V is the vertex set and E is the edge set. Each vertex vi has a probability pi of being present. With each edge (vi, vj) is associated a distance or cost cij. In a first stage, an a priori Hamiltonian tour on G is designed. The list of present vertices is then revealed. In a second stage, the a priori tour is followed by skipping the absent vertices. The PTSP consists of determining a first-stage solution that minimizes the expected cost of the second-stage tour. The problem is formulated as an integer linear stochastic program, and solved by means of a branch-and-cut approach which relaxes some of the constraints and uses lower bounding functionals on the objective function. Problems involving up to 50 vertices are solved to optimality.

Minimizing Transportation and Inventory Costs for Several Products on a Single Link

Abstract: This paper deals with the problem of determining the frequencies at which several products have to be shipped on a common link to minimize the sum of transportation and inventory costs. A set of feasible shipping frequencies is given. Transportation costs are supposed to be proportional to the number of journeys performed by vehicles of a given capacity. Vehicles may or may not be supposed to carry out completely all materials available, and products assigned to different frequencies may or may not share the same truck. Integer and mixed integer linear programming models are formulated for each of the resulting four situations, and their properties are investigated. In particular, we show that allowing products to be split among several shipping frequencies makes trucks traveling at high frequencies to be filled up completely. In this situation, trucks may always be loaded with products shipped at the same frequency.

Optimal Control of a Two-Station Tandem Production/Inventory System

Abstract: A manufacturing facility consisting of two stations in tandem operates in a make-to-stock mode: After production, items are placed in a finished goods inventory that services an exogenous Poisson demand. Demand that cannot be met from inventory is backordered. Each station is modeled as a queue with controllable production rate and exponential service times. The problem is to control these rates to minimize inventory holding and backordering costs. Optimal controls are computed using dynamic programming and compared with the kanban, basestock and buffer control mechanisms that have been proposed for manufacturing facilities. Conditions are found under which certain simple controls are optimal using stochastic coupling arguments. Insights are gained into when to hold work-in-process and finished goods inventory, comparable to previous studies of production lines in make-to-order and unlimited demand environments.

Dynamic Capacity Expansion Problem with Multiple Products: Technology Selection and Timing of Capacity Additions

Abstract: This paper examines a multiproduct dynamic investment model for making technology choices and expansion decisions over a finite planning horizon. The motivation for our problem comes from recent developments in the field of flexible technology such as CAD, CAM, and CIM that permit firms to invest in these more expensive, flexible technologies to provide a competitive edge in the form of an ability to respond rapidly to changing product mix. On the other hand, more specialized dedicated equipment may be less costly. The decisions on appropriate mixes of dedicated and flexible capacity involve many complex considerations such as economies of scale, demand patterns, and mix flexibility. We formulate the problem as a mathematical program with the objective of minimizing total investment cost. Since the problem is difficult to solve optimally, we develop a two-phased approach and present heuristics to obtain good expansion schedules. These procedures are based on an easily solvable sequence of subproblems derived from the planning problem. Our computational results suggest that these methods work well and provide acceptable solutions with reasonable effort.

A Branch-and-Bound Algorithm for the Capacitated Vehicle Routing Problem on Directed Graphs

Abstract: We consider the asymmetric capacitated vehicle routing problem CVRP, a particular case of the standard asymmetric vehicle routing problem in which only the vehicle capacity constraints are imposed. CVRP is known to be NP-hard and finds practical applications in distribution and scheduling. We describe two new bounding procedures for CVRP, based on the so-called additive approach. Each procedure computes a sequence of nondecreasing lower bounds, obtained by solving different relaxations of CVRP. Effective implementations of the procedures are also outlined which considerably reduce the computational effort. The two procedures are combined into an overall bounding algorithm. A branch-and-bound exact algorithm is then proposed, whose performance is enhanced by means of reduction procedures, dominance criteria, and feasibility checks. Extensive computational results on both real-world and random test problems are presented, showing that the proposed approach favorably compares with previous algorithms from the literature.

A Procurement Decision Support System in Business Volume Discount Environments

Abstract: Emergence of a new discount pricing schedule called Business Volume Discount becomes a major obstacle for procurement managers in finding the best purchasing strategy. In the context of business volume discount, a supplier offers discounts on total dollar amount of sales volume, not on the quantity or variety of the products purchased from the supplier. This paper describes a Procurement Decision Support System PDSS that has been successfully implemented to improve the purchasing activities of regional Bell telephone companies. In most purchasing operations, a flexible procurement plan is considered a necessity because of the uncertainties associated with product demand and procurement budget. PDSS provides this flexibility by combining two different purchasing strategies-purchasing on an annual commitment basis, and on an as-ordered basis. Although the development of the system is motivated by a particular application procurement of transmission plug-in in one of the regional telephone companies, the system is being used for a variety of procurement decisions involving business volume discount in the telecommunications industry. The results of the implementation indicate that cost savings of up to 15% are achieved. PDSS use is not limited to the companies in the telecommunications industry. The PDSS model is applicable to any organization having a centralized procurement operation with business volume discount.

Using Values in Operations Research

Abstract: Values pervade the field of operations research. Expressed as objectives, goals, criteria, performance measures, and/or objective functions, they are necessary in theoretical operations research models and in applications. Because of their critical role, it is useful to develop these expressions of values from basic principles. This paper outlines how to identify values for a specific decision problem, how to structure these values to facilitate thinking and analysis, and how to quantify values. Since values provide the basis for interest in a problem, these same values should guide all of our effort on that problem. Two important uses of values are to create better alternatives for decision problems and to define decision problems that are more appealing than those that confront us. On another level, the operations researcher's values are crucial in selecting the research and applications that he or she pursues. I illustrate this with a brief summary of a few projects concerning both life-threatening risks and the storage of nuclear waste. The presentation concludes with a challenge to operations researchers to consider devoting some effort and talent to what I think of as the mega-risks facing our country.

The Stability of a Capacitated, Multi-Echelon Production-Inventory System Under a Base-Stock Policy

Abstract: Most models of multilevel production and distribution systems assume unlimited production capacity at each site. When capacity limits are introduced, an ineffective policy may lead to increasingly large order backlogs: The stability of the system becomes an issue. In this paper, we examine the stability of a multi-echelon system in which each node has limited production capacity and operates under a base-stock policy. We show that if the mean demand per period is smaller than the capacity at every node, then inventories and backlogs are stable, having a unique stationary distribution to which they converge from all initial states. Under i.i.d. demands we show that the system is a Harris ergodic Markov chain and is thus wide-sense regenerative. Under a slightly stronger condition, inventories return to their target levels infinitely often, with probability one. We discuss cost implications of these results, and give extensions to systems with random lead times and periodic demands.

Insights on Modeling from a Dozen Experts

Abstract: Twelve selected expert modelers described themselves as modelers, the models they make, the problems they model, and the way they model. They also expressed opinions about the qualities of effective models, modelers, modeling processes, and desirable modeling clients. Finally, they provided stories about their modeling experiences. Their responses provide benchmarks for developing a modeling science and validate the importance of the craft aspects of OR/MS practice. The paper suggests changes in instructors' approaches to OR/MS content and describes a new course that responds to the experts' insights.

On the Convergence Rates of IPA and FDC Derivative Estimators

Abstract: We show that under the (sufficient) conditions usually given for infinitesimal perturbation analysis (IPA) to apply for derivative estimation, a finite-difference scheme with common random numbers (FDC) has the same order of convergence, namely O(n−1/2), provided that the size of the finite-difference interval converges to zero fast enough. This holds for both one- and two-sided FDC. This also holds for different variants of IPA, such as some versions of smoothed perturbation analysis (SPA), which is based on conditional expectation. Finally, this also holds for the estimation of steady-state performance measures by truncated-horizon estimators, under some ergodicity assumptions. Our developments do not involve monotonicity, but are based on continuity and smoothness. We give examples and numerical illustrations which show that the actual difference in mean square error (MSE) between IPA and FDC is typically negligible. We also obtain the order of convergence of that difference, which is faster than the c...

Sample Path Derivatives for (s, S) Inventory Systems

Abstract: for (s, S) inventory systems, we derive sample path derivatives of performance measures with respect to the two parameters s and S. These derivatives yield derivative estimators which can be estimated from a single sample path or simulation of the inventory system, in some cases not even requiring actual knowledge of the underlying demand distribution. Such derivative estimates would be useful in sensitivity analysis or in gradient-based optimization techniques. We consider the nondiscounted periodic review system with general independent and identically distributed (i.i.d.) continuous demands, full backlogging, and general holding and shortage costs. For the infinite horizon model, consistency proofs are given for some special cases, although we argue why the estimators should be correct for the more general case.

Devising a Cost Effective Schedule for a Baseball League

Abstract: In this paper, we discuss the problem of devising a cost effective schedule for a baseball league. Sports scheduling is a notoriously difficult problem. A schedule must satisfy constraints on timing such as the number of games to be played between every pair of teams, the bounds on the number of consecutive home (or away) games for each team, that every pair of teams must have played each other in the first half of the season, and so on. Often, there are additional factors to be considered for a particular league, for example, the availability of venues on specific dates, home-game preferences of teams on specific dates, and balancing of schedules so that games between two teams are evenly-spaced throughout the season. In addition to finding a feasible schedule that meets all the timing restrictions, the problem addressed in this paper has the additional complexity of having the objective of minimizing travel costs. We discuss some structural properties of a schedule that meets the timing constraints and ...

Fenchel Cutting Planes for Integer Programs

Abstract: A technique for generating cutting planes for integer programs is introduced that is based on the ability to optimize a linear function on a polyhedron rather than explicit knowledge of the underlying polyhedral structure of the integer program. The theoretical properties of the cuts and their relationship to Lagrangian relaxation are discussed, the cut generation procedure is described, and computational results are presented.

The Devolution of OR/MS: Implications from a Statistical Content Analysis of Papers in Flagship Journals

Abstract: Ackoff has decried the “devolution” of OR/MS, Corbett and Van Wassenhove have spoken of its “natural drift,” and a sociologist has described its “regression” as typifying that of other learned professions. To shed light on these views, we undertook a detailed survey of a segment of the OR/MS literature, with particular focus on the space in flagship journals devoted to theory on the one hand and applications on the other. While the literature of OR/MS contains many articles and texts with the word application in the title and the word data in the text, the definitions and uses of these terms are not precise. The claimed applications differ in degree and the actual data differ in kind. To encompass these different meanings we used a five-point scale to classify articles in the 1962 and 1992 volumes of Operations Research and Management Science and the 1972 and 1992 volumes of Interfaces. The resulting statistical analyses shed considerable light on the direction that OR/MS is taking and raise questions abo...

Flow Shop Scheduling with Resource Flexibility

Abstract: This paper explores the improvements in manufacturing efficiency that can be achieved by broadening the scope of production scheduling to include both the sequencing of work and the coordination of the resource inputs required to perform work. Recognizing that some resources are inherently flexible and thus can be reassigned dynamically to processing centers as needed, and that job processing times are often a function of the amount of resource dedicated to specific operations, we formulate the flexible-resource scheduling problem with the objective of simultaneously determining the permutation job sequence, resource allocation policy, and operation start times that optimize system performance. Focusing on flexible-resource scheduling in flow shop production systems, we discuss problem complexity, identify properties of and establish lower bounds for optimal schedules, develop optimal and heuristic solution approaches, and report the results of extensive computational experimentation designed to explore t...

Co-Production Processes with Random Yields in the Semiconductor Industry

Abstract: A wide variety of manufacturing operations can be characterized as co-production with substitutable demand. That is, there are many situations in which the availability of two or more items are related, and because of randomness in either supply or demand, it can be advantageous to substitute one of these items for another. Our research was motivated by the semiconductor industry, where chips are produced in large batches. Because of the presence of randomness in the process, individual chips in a given batch can perform differently. Because some customers have stricter specifications than others, chips within the same batch can be classified and sold as different products according to their measurable performance. We model the production and inventory problem as a stochastic dynamic program in which the objective is to minimize the costs of meeting contractual obligations. After developing heuristic methods of solving the problem in practice, we validate them against a lower bound on the cost of an optim...

A Greedy Randomized Adaptive Search Procedure for the Two-Partition Problem

Abstract: We present a greedy randomized adaptive search procedure (GRASP) for the network 2-partition problem. The heuristic is empirically compared with the Kernighan-Lin (K&L) method on a wide range of instances. The GRASP approach dominates K&L with respect to solution value on a large percentage of the instances tested. The ability of GRASP to find optimal solutions is assessed by comparing its performance with a general purpose mixed integer programming package.

Scheduling Multiple Variable-Speed Machines

Abstract: We examine scheduling problems where we control not only the assignment of jobs to machines, but also the time used by the job on the machine. For instance, many tooling machines allow control of the speed at which a job is run. Increasing the speed incurs costs due to machine wear, but also increases throughput. We discuss some fundamental scheduling problems in this environment and give algorithms for some interesting cases. Some cases are inherently difficult so for these we give heuristics. Our approach illustrates the exploitation of underlying network structure in combinatorial optimization problems. We create heuristics that optimally schedule a large portion of the jobs and then attempt to fit in the remainder. This also gives a method for quickly finding valid inequalities violated by the linear relaxation solution. For the problem of minimizing the sum of makespan and production costs, a rounding heuristic is within a constant factor of optimal. Our heuristics are compared to other classical heu...