Showing papers on "Integer programming published in 1988"

Integer and Combinatorial Optimization

Abstract: FOUNDATIONS. The Scope of Integer and Combinatorial Optimization. Linear Programming. Graphs and Networks. Polyhedral Theory. Computational Complexity. Polynomial-Time Algorithms for Linear Programming. Integer Lattices. GENERAL INTEGER PROGRAMMING. The Theory of Valid Inequalities. Strong Valid Inequalities and Facets for Structured Integer Programs. Duality and Relaxation. General Algorithms. Special-Purpose Algorithms. Applications of Special- Purpose Algorithms. COMBINATORIAL OPTIMIZATION. Integral Polyhedra. Matching. Matroid and Submodular Function Optimization. References. Indexes.

Short-term scheduling of thermal-electric generators using Lagrangian relaxation

Abstract: This paper presents an expanded formulation of the unit commitment problem in which hundreds of thermal-electric generators must be scheduled on an hourly basis, for up to 7 days at a time. The underlying model incorporates the full set of costs and constraints including setup, production, ramping, and operational status, and takes the form of a mixed integer nonlinear control problem. Lagrangian relaxation is used to disaggregate the model by generator into separate subproblems which are then solved with a nested dynamic program. The strength of the methodology lies partially in its ability to construct good feasible solutions from information provided by the dual. Thus, the need for branch-and-bound is eliminated. In addition, the inclusion of the ramping constraint provides new insight into the limitations of current techniques. Computational experience with the proposed algorithm indicates that problems containing up to 100 units and 48 time periods can be readily solved in reasonable times. Duality gaps of less than 1% were achieved in all cases.

Optimal VAr planning by approximation method for recursive mixed-integer linear programming

Abstract: The authors propose an algorithm for solving reactive power planning problems. The optimization approach is based on a recursive mixed-integer programming technique using an approximation method. A fundamental feature of this algorithm is that the number of capacitor or reactor units can be treated as a discrete variable in solving large-scale VAr (volt-ampere reactive) planning problems. Numerical results have verified the validity and efficiency of the algorithm. >

An Integer Programming Approach to the Optimal Product Line Selection Problem

Abstract: A zero-one integer mathematical programming formulation is proposed to solve the optimal product line selection problem. Based on individual consumer measurements from conjoint analysis, the formulation seeks to find an optimal subset of products that is drawn from a proposed set of product alternatives with specified product characteristics. In addition to its inherent flexibility in incorporating various realistic constraints, a feature of the proposed approach is its potential determination of verifiable optimal solutions to the product line selection problem. The results of computational experience with an optimization code, known as the X-system, are presented. Using a variety of representative simulated cases, it is shown that the X-system can be used to effectively solve product line selection problems on both mainframe and AT personal computers.

Lot-size models with back-logging: strong reformulations and cutting planes

Abstract: We examine mixed integer programming reformulations of the uncapacitated lot-sizing problem with backlogging. First we consider the effect of using a standard reformulation technique for fixed charge network flow problems which involves the introduction of new variables, leading to a known plant location reformulation and a shortest path reformulation. Each of these reformulations is strong in the sense that its linear programming relaxation solves the lot-sizing problem. Secondly we attempt to treat the problem in the space of the original variables. We give an implicit description of the convex hull of solutions, and show how the problem of finding a violated cutting plane can be solved as a linear program. We also describe a family of strong valid inequalities which can be generated rapidly by a heuristic and which have proved effective in a cut generation algorithm. The efficiency of both the shortest path formulation and the cutting plane algorithm have been tested on a series of multi-item capacitated lot-sizing problems with backlogging. Near optimal solutions have been found to problems with 8 periods and up to 100 times.

A New Algorithm for the 0-1 Knapsack Problem

Abstract: We present a new algorithm for the optimal solution of the 0-1 Knapsack problem, which is particularly effective for large-size problems. The algorithm is based on determination of an appropriate small subset of items and the solution of the corresponding "core problem": from this we derive a heuristic solution for the original problem which, with high probability, can be proved to be optimal. The algorithm incorporates a new method of computation of upper bounds and efficient implementations of reduction procedures. The corresponding Fortran code is available. We report computational experiments on small-size and large-size random problems, comparing the proposed code with all those available in the literature.

Integer linear programming neural networks for job-shop scheduling

Abstract: The authors present an integer linear programming neural network (ILPNN) based on a modified Tank and Hopfield neural network model to solve job-shop scheduling, an NP-complete constraint satisfaction problem. The constraints of the job-shop problem are formulated as a set of integer linear equations. The cost function for minimization is the total starting times of all jobs subject to precedence constraints. In the authors' approach, the set of integer linear equations is solved by an iterative linear programming with integer adjustments (ILPIA) technique, without a branch-and-bound search. In particular, the linear and nonlinear zero-one variables are represented by linear sigmoid and nonlinear high-gain amplifiers with a response of a step function, respectively. Simulations based on solving a linear differential equation show that the ILPNN approach produces optimal or near-optimal solutions, although it does not guarantee optimal solutions. The authors also analyze the hardware implementation of ILPNNs and study the feasibility of this approach for large-scale problems. >

A Heuristic for Minimizing the Number of Tool Switches on a Flexible Machine

Abstract: This paper addresses the problem of scheduling N jobs on a single machine equipped with an automatic tool interchange mechanism. We consider the case where the total number of tools required to process all N jobs is greater than the capacity of the tool magazine, and where processing times and switching times are independent. The underlying problem is to find the job sequence and tool replacement policy that minimizes the total number of switches. This is equivalent to minimizing the makespan. Two industrial applications of the model are cited. The problem is formulated as a nonlinear integer program and solved with a dual-based relaxation heuristic designed to quickly find good local solutions. An example is given to highlight the computations and a series of test cases is examined to gauge the performance of the proposed methodology. The results demonstrate that in almost all cases global optimality is obtained, but in notably less time than current techniques admit. This points up the practica...

Generality versus specificity: an experience with AI and OR techniques

Abstract: This paper contains an in-depth study of a particular problem in order to evaluate several approaches to the solving of discrete combinatorial problems. We take a warehouse location problem as a case study and present solutions to it by using Integer Programming, a specialized program based on A* and the constraint logic programming CHIP. The merits of each approach are discussed and compared in the light of the problem. Finally, we conclude by arguing that CHIP provides a valuable addition to the current set of tools for solving discrete combinatorial problems.

Lagrangian relaxation methods for solving the minimum fleet size multiple traveling salesman problem with time windows

Abstract: We consider the problem of finding the minimum number of vehicles required to visit once a set of nodes subject to time window constraints, for a homogeneous fleet of vehicles located at a common depot. This problem can be formulated as a network flow problem with additional time constraints. The paper presents an optimal solution approach using the augmented Lagrangian method. Two Lagrangian relaxations are studied. In the first one, the time constraints are relaxed producing network subproblems which are easy to solve, but the bound obtained is weak. In the second relaxation, constraints requiring that each node be visited are relaxed producing shortest path subproblems with time window constraints and integrality conditions. The bound produced is always excellent. Numerical results for several actual school busing problems with up to 223 nodes are discussed. Comparisons with a set partitioning formulation solved by column generation are given.

Generalized resolution and cutting planes

Abstract: This paper illustrates how the application of integer programming to logic can reveal parallels between logic and mathematics and lead to new algorithms for inference in knowledge-based systems. If logical clauses (stating that at least one of a set of literals is true) are written as inequalities, then the resolvent of two clauses corresponds to a certain cutting plane in integer programming. By properly enlarging the class of cutting planes to cover clauses that state that at least a specified number of literals are true, we obtain a generalization of resolution that involves both cancellation-type and circulant-type sums. We show its completeness by proving that it generates all prime implications, generalizing an early result by Quine. This leads to a cutting-plane algorithm as well as a generalized resolution algorithm for checking whether a set of propositions, perhaps representing a knowledge base, logically implies a given proposition. The paper is intended to be readable by persons with either an operations research or an artificial intelligence background.

On server allocation in multiple center manufacturing systems

Abstract: We study the problem of allocating a given number of identical servers among the work centers of a manufacturing system. The problem is formulated as a nonlinear integer program of allocating servers in a closed queueing network to maximize throughput. We show that the throughput of the closed queueing network has a monotonicity property, such that any optimal allocation must give more servers to stations with a higher workload. The number of allocations that satisfy this property is much smaller than the total number of feasible allocations. This property and a bounding technique for the throughput of the closed queueing network are combined to develop a search algorithm to obtain an optimal allocation of servers. A greedy heuristic is also developed, and its optimality proven in the special case of a two-center system in the general case, its optimality remains a conjecture.

Determining optimal reorder intervals in capacitated production-distribution systems

Abstract: The problem of determining consistent and realistic reorder intervals in complex production-distribution environments was formulated as a large scale, nonlinear, integer programming problem by Maxwell and Muckstadt (Maxwell, W. L., J. A. Muckstadt. 1985. Establishing consistent and realistic reorder intervals in production-distribution systems. Oper. Res. 33(6, November–December) 1316–1341.). They show how the special structure of the problem permits its solution by a standard network flow algorithm. In this paper, we review the Maxwell-Muckstadt model, provide necessary and sufficient conditions that characterize the solution, and show that the optimal partition of nodes in the production-distribution network is invariant to an arbitrary scaling of the set-up and holding cost parameters. We consider two capacitated versions of the model: one with a single constrained work center, and the other with multiple constrained work centers. For single constraint problems, the invariance corollary provides a simp...

Optimizing Resource Acquisition Decisions by Stochastic Programming

Abstract: This paper reports on the application of stochastic programming with recourse to strategic planning decisions regarding resource acquisition. A resource directed decomposition method, which simultaneously exploits stochastic programming and mixed integer programming model structures, is proposed. Computational experience with the method applied to fuel contract and plant construction decisions faced by an electric utility is presented.

On the Development of a Mixed-Integer Linear Programming Model for the Flowshop Sequencing Problem

Abstract: This paper describes the development of a mixed-integer linear programming (MILP) model for the standard N-job, M-machine flowshop sequencing problem. Based on an earlier all-integer model developed by Wagner, this MILP model has been used to solve optimally problems with as many as 25 jobs and as many as 10 machines. Variants of the standard flowshop model, including a variety of performance measures, are also presented. Computational experience involving the successful solution of over 175 flowshop problems is discussed, and suggestions for future research projects are offered.

Introduction to operations research

Abstract: Using a style of presentation that makes even the more difficult topics easy to understand, this text covers all the important quantitative models of operations research. The formulation of problems in mathematical terms is explained, together with the solution of the resulting models, and the interpretation of results. Coverage is given of linear programming, network flows, integer programming, nonlinear programming, dynamic programming, queueing models, inventory models and discrete-event simulation. The manner in which the topics are developed leads students to discover the underlying concepts for themselves, with a minimum of both notation and unnecessary jargon. An appendix provides notational conventions for matrix algebra.

Mathematical Programming Models for Cutting-Stock Problems in the Clothing Industry

Abstract: The cutting-stock problem in the clothing industry does not conform to the classical representation of such problems. Waste minimization is only a subsidiary objective. The overall objective is to maximize long-run profitability, but operational planning requires limiting consideration to a series of problems covering short planning periods. This requires a more complex objective to incorporate interaction between periods. Production constraints with unique characteristics occur in the laying, cutting and sewing operations. The conventional pattern-design problem is dealt with by use of a commercial computer-aided graphical design system. The question addressed in this paper is how to combine such patterns so as to satisfy the various objectives and constraints. This leads to both integer and quadratic formulations of appropriate mathematical programming models.

Locating concentrators for primary and secondary coverage in a computer communications network

Abstract: A model is developed that assigns primary and secondary (backup) concentrator coverage to each terminal site. The objective is to minimize communications costs as well as costs for setting up and operating the concentrators subject to capacity constraints. A relaxation of the problem is studied, and an effective solution procedure that makes the use of this relaxation is developed. Experimental results over a wide range of problem structures show that this solution procedure is very effective. It is also found to be significantly faster than a state-of-the-art commercial integer programming package. >

Special ordered sets and an application to gas supply operations planning

Abstract: Special Ordered Sets provide a powerful means of modeling nonconvex functions and discrete requirements, though there has been a tendency to think of them only in terms of multiple-choice zero-one programming. This paper emphasizes the origins and generality of the special ordered set concept, and describes an application in which type 2 sets are used in several forms to model both logical conditions and nonlinear functions.

Implicit optimal and heuristic labor staffing in a multiobjective, multilocation environment*

Abstract: This paper presents a tractable set of integer programming models for the days-off scheduling of a mix of full- and part-time employees working α to β days/week (cycle) in a multiple-objective, multiple-location environment. Previous models were formulated to specifically schedule part-time employees working either two or three days per week. These models were intractable because they required complete employee schedule information. The new models are deemed implicit optimal since they are required to supply only essential information. While the number of variables in previous models is an exponential increasing function of β-α, the size of three of the new models is independent of α and β. The first three models developed here (as in [18]) deal with the trade-offs between idle time, the number of employees required to work at multiple “locations,” and the size of the total labor pool. The inherent flexibility of the implicit modeling approach is illustrated by the presentation of various modifications of the basic models. These modifications permit the use of preference weights on the number of employee work days/week (cycle) or the minimization of payroll costs where differential pay rates exist. These latter models may also be formulated such that idle time is ignored, constrained or minimized. The execution time for the implicit models (on a CDC CYBER 730 computer with commercially available software) averaged well under five seconds on 1200 trial problems for the type of application considered in [18]. A solution was obtained in less than 46 seconds of CPU time for a trial problem which would have required over 1.4 million integer variables with previous models. The availability of optimal solutions was invaluable in the development of two heuristics designed to deal with the trade-offs of [16]. In an experimental analysis a previous heuristic produced results which averaged from 74 to 508 percent above optimum across six experimental conditions. The comparable new heuristic produced results which averaged from 3 to 8 percent above optimum for the same experimental conditions. The paper concludes by developing a framework to integrate the results of this research with the tour scheduling problem and by identifying several other areas for related research.