Showing papers on "Integer programming published in 1978"

Integer Programming Approach to the Problem of Optimal Unit Commitment with Probabilistic Reserve Determination

Abstract: A method for determining the unit commitment schedule for hydro-thermal systems using extensions and modifications of the Branch and Bound method for Inteler Programming has been developed. Significant features of the method include its computational practicability for realistic systems and proper representation of reserves associated with different risk levels. Contracts relating to power interchange have also been adequately modelled for such an approach.

An Efficient Integer Programming Algorithm with Network Cuts for Solving Resource-Constrained Scheduling Problems

Abstract: In this paper we describe an integer programming algorithm for allocating limited resources to competing activities jobs, tasks, etc. of a project such that the completion time of the project is minimal among all possible completion times. Typical of such problems is the minimization of the completion time of projects of the PERT/CPM variety where limits on resource availability force the postponement of selected activities during project performance. Also included in this class of problems for which our procedure is applicable is the assignment of jobs to machines such that the elapsed time for completing all jobs makespan is a minimum over all possible job-machine assignments. The procedure developed consists of a systematic evaluation enumeration of all possible job finish times for each task in the project. To limit the number of task assignments which have to be explicitly evaluated, an artifice called a network cut is developed which removes from consideration the evaluation of job finish times which cannot lead to a reduced project completion time. Results reported demonstrate that the procedure developed is a reliable optimization technique for projects consisting of up to 30--50 jobs and three different resource types. The procedure is particularly applicable in those instances in which computer primary storage is limited. Many of the mini-computers available today are capable of implementing our approach without extensive programming being required for writing to auxiliary storage, making the technique available to the project manager in those environments in which extensive computer resources for scheduling are not readily available.

On the Computational Complexity of Integer Programming Problems

Abstract: Recently much effort has been devoted to determining the computational complexity for a variety of integer programming problems. In this paper a general integer programming problem is shown to be NP-complete; the proof given for this result uses only elementary linear algebra. Complexity results are also summarized for several particularizations of this general problem, including knapsack problems, problems which relax integrality or non-negativity restrictions and integral optimization problems with a fixed number of variables.

Minimal Resources for Fixed and Variable Job Schedules

Abstract: We treat the following problem: There are n jobs with given processing times and an interval for each job's starting time. Each job must be processed, without interruption, on any one of an unlimited set of identical machines. A machine may process any job, but no more than one job at any point in time. We want to find the starting time of each job such that the number of machines required to process all jobs is minimal. In addition, the assignment of jobs to each machine must be found. If every job has a fixed starting time the interval is a point, the problem is well-known as a special case of Dilworth's problem. We term it the fixed job schedule problem FSP. When the job starting times are variable, the problem is referred to as the variable job schedule problem VSP, for which no known exact solution procedure exists. We introduce the problems by reviewing previous solution methods to Dilworth's problem. We offer an approximate solution procedure for solving VSP based on the entropy principle of informational smoothing. We then formulate VSP as a pure integer programming problem and provide an exact algorithm. This algorithm examines a sequence of feasibility capacitated transportation problems with job splitting elimination side constraints. Our computational experience demonstrates the utility of the entropy approach.

A Note on Heuristic Methods in Optimal System Reliability

Abstract: Many optimization techniques have been used to solve redundancy allocation problems, most of which result in noninteger solutions. A few, including dynamic programming and integer programming, as well as a host of heuristic methods give integer solutions. This note critically reviews six promising heuristic approaches. The advantages and disadvantages of each of the approaches are discussed. An extended approach is presented which incorporates some of the ideas of the previous methods for solving a general non series-parallel system. The extended approach appears to be quite efficient and is general. The simplicity and efficiency of the approach will lend itself to solving large practical problems.

Optimal scheduling of thermal generating units

Abstract: Scheduling the operation of thermal generating units involves the selection of units to be placed in operation and the allocation of the load among them- These two decisions must be taken so as to minimize the sum of tile startup, banking, and expected riming costs subject to the demand, spinning-reserve, and minimum down-time and up-time constraints and, sometimes, to the following two additional limitations: a unit should not be started up more than once a day; and no more than two units of the same plant should be started up simultaneously. This optimization problem is formulated in this paper as a mixed-integer nonlinear programming problem. Then, by following the Bender's approach, the problem is partitioned into a nonlinear and a pure-integer nonlinear programming problem- The First problem, which represents the economic dispatch problem, is not solved here. The second problem, usually called the unit commitment problem, is solved by a variational method and a branch-and-bound algorithm. Numerical results obtained for a network of ten generating units are presented.

On the Extent to Which Certain Fixed-Charge Depot Location Problems Can be Solved by LP

Abstract: This paper considers a class of feasible set fixed-charge depot location problems which have been formulated as mixed-integer programmes. Computational results are reported to show that linear programming often produces integer solutions to uncapa- citated problems as required. It is suggested that this represents a practical solution approach. Computational evidence suggests this convenient property does not extend to capacitated problems. Discussion of reducing infinite set problems to such feasible set problems is included. This paper considers depot location-demand allocation problems where loca- tions are to be chosen from a finite set of candidate sites. This corresponds to the "feasible set approach", discussed by Rand.' The objective is to locate depots so that all customer demand is allocated among the depots while minimizing the sum of variable and fixed depot costs associated with satisfy- ing that demand. Demand is assumed known in each of a number of customer zones. The modelling intent is to answer the four fundamental questions listed by Rand as: How many depots should there be? Where should they be? Which customers should they serve? How big should they be? A mixed integer programming model-sometimes called the fixed-charge plant loca- tion model is used for which efficient special purpose algorithms exist (see Elshafei2 and Geoffrion and Graves3). However, it is a nontrivial task to develop the necessary computer programs. The purpose of this paper is to provide computational results which indicate that ordinary linear program- ming typically produces integer solutions to uncapacitated problems. It is suggested that this represents a practical solution approach if the problem size is not too large, but computational results show that this approach seems inappropriate for capacitated problems. Discussion of reducing certain infinite set problems to feasible set problems is presented in an Appendix.

Integer Programming Solution of a Classification Problem

Abstract: A classification problem is presented in which it is desired to assign a new individual or observation with k characteristics to one of two distinct populations based upon historical sets of samples from the two populations. The resulting classification problem is formulated as a mixed-integer programming problem. The solution, which can be obtained through use of a partitioning algorithm based on Benders decomposition, provides a nonparametric classification statistic which minimizes the expected total cost of misclassification. Also, an enumeration algorithm is developed for the special case of k = 2. Monte Carlo studies are reported which compare the results of the enumeration algorithm with Anderson's “normal” procedure for different underlying distributions. The performance of the enumeration algorithm is shown to be significantly better than Anderson's normal procedure for distributions with uncorrelated normal populations with unequal covariance matrices and for uncorrelated skewed populations with...

Optimal Reliability Allocation by Branch-and-Bound Technique

Abstract: The paper presents an efficient method for finding the exact optimal solutions of reliability allocation problems that are formulated as an integer nonlinear programming problem generalized to handle nonlinear constraints and nonseparable problems. The method is based on branch-and-bound and developed by considering separation and relaxation techniques.

On a Production Allocation and Distribution Problem

Abstract: We describe a production allocation problem which was worked on at Frito-Lay and an integer programming model formulated for its solution. In connection with model formulation, we discuss estimation and collection of the required cost coefficients. Finally, we report on the use of two different linear programs, representing the same integer problem, which gave dramatically different running times to solve the integer programs. Some reasons for the improved running times are given.

Some Basis Theorems for Integral Monoids

Abstract: We consider sets of integer vectors containing the zero vector and closed under addition, the integral monoids, and provide conditions under which they contain a finite subset of integer vectors which generate the entire monoid as nonnegative integer combinations. The paper concludes with some applications to the theory of integer programming.

The reformulation of two mixed integer programming problems

Abstract: Two practical problems are described, each of which can be formulated in more than one way as a mixed integer programming problem. The computational experience with two formulations of each problem is given. It is pointed out how in each case a reformulation results in the associated linear programming problem being more constrained. As a result the reformulated mixed integer problem is easier to solve. The problems are a multi-period blending problem and a mining investment problem.

A Dynamic Resource Allocation Model for A Machine Requirements Problem

Abstract: This paper is concerned with the problem of determining the optimum number of machines to have in a production process. The problem is analyzed as a resource allocation problem involving the minimum cost allocation of limited floor space, capital budget, and available overtime among various types of machines. The dynamic nature of the problem is also included in the analysis. A deterministic mixed integer programming model is described. A numerical example is presented to illustrate the approach and its utility as an alternative way to address the machine requirements problem.

A Converse for Disjunctive Constraints.

Abstract: A necessary and sufficient condition is given for the disjunctive constraints construction to provide all valid cuts for a system of logical constraints on linear inequalities. This condition is then applied to several commonly occurring situations of integer programming, and the issue of verification of the condition is discussed.

Local Unimodularity in the Matching Polytope

Abstract: Publisher Summary This chapter discusses local unimodularity in the matching polytope. In the first decade of linear programming, it was observed that various extremal combinatorial theorems (Dilworth, Menger, etc.), could be derived as applications of the duality principle of linear programming. The basic idea was that the combinatorial theorem would follow from linear programming duality if optimal vertices of both primal and dual problems were integral. In all the cases treated, the linear programming matrix A was totally unimodular (i.e., every minor of A had absolute value 0 or I), so application of Cramer's rule yielded the integrality of the vertices. In some cases, the primal polyhedron is locally strongly unimoduiar at every vertex, for the arguments establish that A contains a nonsingular submatrix of order q that is not only unimodular, but totally unimodular. (The same arguments establish that the dual polyhedron has at least one optimal vertex locally strongly unimodular.) The chapter provides proof that the matching polytope is locally unimodular, at every vertex, though not locally unimodular, provided one includes certain natural but possibly superfluous inequalities

Parametric branch and bound

Abstract: The paper describes a procedure for mixed integer programming that allows branches to be imposed ‘by degrees’, which can subsequently be revised or weeded out according to their relative influence. It is an adaptive approach in which the branch and bound tree can be manipulated and restructured. The approach also yields measures of the costs of imposing the branches that lead to integer solutions, thus providing a built-in form of sensitivity analysis for evaluating the effect of integer restrictions.

The Single-Plant Mold Allocation Problem with Capacity and Changeover Restrictions

Abstract: This paper presents a formulation and a solution technique for a class of production scheduling problems. Plant capacity limitations and changeover restrictions characterize the problems in this class. A member of this class, called the single-plant mold allocation problem, is formulated as a linear integer programming problem. We develop a transformation that permits the reduction of this problem to a minimum cost flow problem.

An algorithm for the 0/1 Knapsack problem

Abstract: The Knapsack problem (maximize a linear function, subject to a unique constraint, all being in integers), although of thenp-complete type, is a well solved case in combinatorial programming. The reason for this is twofold:(i)an upper bound of the objective function is easy to compute(ii)it is quite simple to construct feasible solutions. They give lower bounds of the optimum. This makes it possible to know rapidly the optimal value of many variables, and therefore to reduce the problem. Several studies have appeared recently on the subject [5, 9, 12, 18]. We present a program by which Knapsacks involving up to 60 000 boolean variables were solved in a matter of seconds, on an I.B.M. 370-168.

Generalized Covering Relaxation for 0-1 Programs.

Abstract: : A general purpose algorithm is constructed for solving polynomial 0-1 programming problems. The algorithm is applied directly to the polynomial problem in its original form. Further, no additional variables are introduced in the solution process. The algorithm was tested on randomly generated modest size problems and the preliminary computational results obtained are very encouraging.

Optimization of pavement rehabilitation and maintenance by use of integer programming

Abstract: An integer programming technique has been used to develop an operating computer program called RAMS, which determine optimal maintenance strategies for pavements by maximizing the overall maintenance effectiveness for all highway segments considered. The program can use numerous maintenance strategies, resources, and feasibility constraints to obtain solutions. An example problem that contains actual field data on 15 highway segments located in one Texas highway district was used to demonstrate typical program input and output. This example revealed that, for 9 of the 15 pavement segments studied, maintenance strategies selected by the computer program were essentially identical to those selected by district personnel of the Texas State Department of Highways and Public Transportation. Both RAMS cases presented are optimal with respect to department selections. /Author/

Planning Time-Phased Regional Treatment Systems

Abstract: An economic optimization model for selecting least-cost regional wastewater treatment facility locations, interceptor routes, and capacity expansion schedules is developed. The model explicitly considers the interaction between facility location and capacity expansion decisions on total system cost. Cost functions for treatment facilities and interceptors and growth patterns for future wastewater flows may be of any form. The model is solved with a three-phase heuristic procedure that makes strong use of dynamic programming. In comparison with a mixed integer programming model on a large-scale problem the procedure demonstrates significant advantages in computational time, accuracy, and user requirements. The model is applied to facilities planning in a 208 study area of Massachusetts to demonstrate how the economic impacts of alternative management objectives and demographic projections can be assessed.

Scheduling Algorithms for the Unbalanced Production Line: An Analysis and Comparison

Abstract: This paper presents an analysis and comparison of scheduling algorithms for the unbalanced production line. A new heuristic algorithm is presented accompanied by an index for classifying the configuration of a production line. A factorial experiment was conducted in order to determine those factors which were significant with respect to the performance measure under consideration, the average cost of holding in-process inventory. Additional analyses were then performed on the line factors and scheduling algorithms with appropriate conclusions drawn. The best performing algorithm was then compared to the performance of Single Period integer programming over a multiple time frame. The heuristic algorithm was found to yield better average performance than the integer programming solution, although the differences were not statistically significant. Recommendations concerning the implementation of the heuristic algorithm are provided.

A Zero-One Integer-Programming Formulation of the Problem of Land-Use Assignment and Transportation-Network Design

Abstract: A zero-one integer-programming formulation of the simultaneous optimization of the problems of land-use assignment and transportation-network design is presented. The problem is modeled through a set-partitioning approach and incorporates a multiple-criteria objective function, appropriate upper- and lower-bound constraints on area assignments, and construction costs. A simple example and a more complicated urban-design case study are included to demonstrate the viability of this approach in solving the simultaneous-optimization problem. As a secondary benefit, this set-partitioning model can be reformulated as an integer, generalized-network, flow problem for which new efficient computer codes, capable of solving networks with thousands of nodes and variables, are available.

Multicommodity Flow Problem with Variable Arcs Capacities

Abstract: The problem of maximizing the sum of the flows of all commodities in a network where the capacities of some arcs can be increased by integer numbers within a fixed budget is solved in this paper. Benders' technique is used to decompose the problem. Then Rosen's primal partitioning and non-linear duality theory are used to solve the subproblems generated by the Benders' decomposition. An application of a multicommodity network to the defence problem is mentioned.

Optimal Allocation of Fault Detectors

Abstract: A Markov model is given for a class of series systems which have fault detectors to find component failures. The optimal allocation of fault detectors is determined. This problem is a nonlinear 0-1 integer programming (0-1 IP) problem. The problem is solved easily because the nonlinearity is of a special type. An illustrative example is given.