Showing papers on "Integer programming published in 1983"
TL;DR: It is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable.
Abstract: It is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable. The proof depends on methods from geometry of numbers.
1,256 citations
TL;DR: In this article, the authors extended the maximum covering location model to account for the chance that when a demand arrives at the system it will not be covered since all facilities capable of covering the demand are engaged serving other demands.
Abstract: The maximum covering location model has been used extensively in analyzing locations for public service facilities. The model is extended to account for the chance that when a demand arrives at the system it will not be covered since all facilities capable of covering the demand are engaged serving other demands. An integer programming formulation of the new problem is presented. Several properties of the formulation are proven. A heuristic solution algorithm is presented and computational results with the algorithm are discussed. Directions for future study are also discussed.
763 citations
TL;DR: The results indicate that cutting-planes related to the facets of the underlying polytope are an indispensable tool for the exact solution of this class of problem.
Abstract: In this paper we report on the solution to optimality of 10 large-scale zero-one linear programming problems. All problem data come from real-world industrial applications and are characterized by sparse constraint matrices with rational data. About half of the sample problems have no apparent special structure; the remainder show structural characteristics that our computational procedures do not exploit directly. By today's standards, our methodology produced impressive computational results, particularly on sparse problems having no apparent special structure. The computational results on problems with up to 2,750 variables strongly confirm our hypothesis that a combination of problem preprocessing, cutting planes, and clever branch-and-bound techniques permit the optimization of sparse large-scale zero-one linear programming problems, even those with no apparent special structure, in reasonable computation times. Our results indicate that cutting-planes related to the facets of the underlying polytope are an indispensable tool for the exact solution of this class of problem. To arrive at these conclusions, we designed an experimental computer system PIPX that uses the IBM linear programming system MPSX/370 and the IBM integer programming system MIP/370 as building blocks. The entire system is automatic and requires no manual intervention.
680 citations
TL;DR: In this paper, the authors consider a family of discrete, deterministic, single-criterion, NP-hard problems, including set packing, set covering, and set partitioning.
552 citations
TL;DR: A superstructure that has embedded many potential configurations of utility systems is proposed, as well as its corresponding mixed-integer programming model, for performing structural and parameter optimization in the synthesis of processing systems.
473 citations
01 Dec 1983
TL;DR: The proposed algorithm first finds a “more orthogonal” basis for a lattice than those of Lenstra (1981) and Lenstra, Lenstra and Lovasz (1982), but in time 0(ndn poly (length of input)).
Abstract: The integer programming problem is: Given m×n and m×l matrices A and b respectively of integers, find whether, there exists an all integer n×l vector x satisfying the m inequalities A×≤b. In settling an important open problem, Lenstra (1981) showed in an elegant way that when n, the number of dimensions is fixed, there is a polynomial-time algorithm to solve this problem. His algorithm achieves a running-time of 0(cn3•p(length of data)) where p is some polynomial and c a constant independent of n. Since such an algorithm has several important applications - cryptography (Shamir (1982)), diophantine approximations (Lagarias (1982)), coding theory (Conway and Sloane (1982), etc. it is important to improve the running time. We present an algorithm here that has a running time of 0(n9nL log L) where L is the length of the input. Whereas Lenstra's algorithm in the worst case reduces an n-dimensional problem to cn2−(n−) dimensional problems, our algorithm effectively reduces an n-dimensional problem to at most polynomially many (n−1) dimensional problems, thus achieving our time bound. The algorithm we propose, first finds a “more orthogonal” basis for a lattice (see the next section for the definition of a lattice) than those of Lenstra (1981) and Lenstra, Lenstra and Lovasz (1982), but in time 0(ndn poly (length of input)). It then uses an enumeration technique to solve integer programming and related problems. While this paper presents mainly the theoretical improvements that can be made in the algorithms, we discuss in section 6 why in practice our estimates of running time may be overly pessimistic. The last part of the paper discusses some complexity issues. It is an interesting open problem as to whether finding the Euclidean shortest non-zero vector of a given lattice is NP-hard. (See Lenstra (1981), Van Emde Boas (1981) and Lagarias (1982)).
466 citations
TL;DR: In this article, a digital filter with discrete coefficient values selected from the powers-of-two coefficient space is designed using the methods of integer programming, and the frequency responses obtained are shown to be superior to those obtained by simply rounding the coefficients.
Abstract: FIR digital filters with discrete coefficient values selected from the powers-of-two coefficient space are designed using the methods of integer programming. The frequency responses obtained are shown to be superior to those obtained by simply rounding the coefficients. Both the weighted minimax and the weighted least square error criteria are considered. Using a weighted least square error criterion, it is shown that it is possible to predict the improvement that can be expected when integer quadratic programming is used instead of simple coefficient rounding.
451 citations
TL;DR: The final topic of this paper, Product Structure Compression, is introduced as a method to reduce the size of the problem without losing optimality.
Abstract: This paper introduces a line of research on capacity-constrained multi-stage production scheduling problems. The first section introduces the problem area as it arises from a failure of MRP systems. Then a review of the literature and an analysis of the type of problems that exist are presented in §2. Section 3 outlines linear and mixed integer-linear programming formulations. These formulations compute the required production lead times according to the demands on available capacity, thereby reducing in-process inventory compared to the usual practice in MRP. A discussion of how to use the LP version is included. However, the size of the problems in practice implies that more efficient solution techniques must be found. The final topic of this paper, Product Structure Compression, is introduced as a method to reduce the size of the problem without losing optimality.
394 citations
TL;DR: The development of the cross decomposition method captures profound relationships between primal and dual decomposition, and shows that the more constraints can be included in the Langrangean relaxation, the fewer the Benders cuts one may expect to need.
Abstract: Many methods for solving mixed integer programming problems are based either on primal or on dual decomposition, which yield, respectively, a Benders decomposition algorithm and an implicit enumeration algorithm with bounds computed via Lagrangean relaxation. These methods exploit either the primal or the dual structure of the problem. We propose a new approach, cross decomposition, which allows exploiting simultaneously both structures. The development of the cross decomposition method captures profound relationships between primal and dual decomposition. It is shown that the more constraints can be included in the Langrangean relaxation (provided the duality gap remains zero), the fewer the Benders cuts one may expect to need. If the linear programming relaxation has no duality gap, only one Benders cut is needed to verify optimality.
179 citations
TL;DR: Computational experience is presented to show that an easily implemented application of linear programming frequently produces optimal solutions to shift and days-off scheduling problems.
Abstract: Shift and days-off scheduling problems have received much attention in the literature of integer programming approaches to workforce scheduling. A typical managerial use would be to schedule full-time employees to minimize the number of labor hours while satisfying variable workforce requirements of a service delivery system. We present computational experience to show that an easily implemented application of linear programming frequently produces optimal solutions to these problems. When the context progresses toward a continuous operating environment (service delivery over 24 hours a day, 7 days a week) we stress the need to shed the myopic views of the shift and days-off scheduling formulations in favor of an integrative tour scheduling formulation. For this problem we observe that a simple heuristic initiated by rounding down the associated LP solution consistently produces near optimal solutions. This observation is based on experiments over varying workforce requirement patterns.
150 citations
TL;DR: In this article, a branch and bound algorithm is used for the solution of the Travelling Salesman Problem, where the main feature of the algorithm lies in the relaxation of the subtour elimination constraints.
Abstract: This paper deals with a generalized version of the Travelling Salesman Problem, which consists of finding the shortest Hamiltonian cycle through n sets of nodes. The problem is formulated as an integer linear program including degree constraints, subtour elimination constraints, and integrality constraints. A branch and bound algorithm is used for the solution of the problem; the main feature of the algorithm lies in the relaxation of the subtour elimination constraints. Computational results for Euclidean and non-Euclidean problems are reported.
TL;DR: In this paper, a multiobjective integer programming formulation for the analysis of the land acquisition problem is presented, which incorporates the discrete and multi-objective nature of land acquisition.
TL;DR: A tight lower bound is obtained on the minimal expected delay as well as sets of feasible solutions for the problem of selecting a set of routes which minimizes the expected network end-to-end queueing and transmission delay.
Abstract: The problem of selecting a single route for each class of service and each pair of communicating nodes in an SNA network is considered. The nodes, links, sets of candidate routes, and traffic characteristics are given. The goal is to select a set of routes which minimizes the expected network end-to-end queueing and transmission delay. Queueing is modeled as a network of M/M/1 queues which leads to a nonlinear combinatorial optimization problem. Using Lagrangean relaxation and subgradient optimization techniques, we obtain a tight lower bound on the minimal expected delay as well as sets of feasible solutions for the problem. An experimental interactive system has been used to evaluate the procedure; very favorable results have been obtained on a variety of networks.
TL;DR: Two examples of parametric cost programming problems—one in network programming and one in NP-hard 0-1 programming—are given; in each case, the number of breakpoints in the optimal cost curve is exponential in the square root of thenumber of variables in the problem.
Abstract: Two examples of parametric cost programming problems—one in network programming and one in NP-hard 0-1 programming—are given; in each case, the number of breakpoints in the optimal cost curve is exponential in the square root of the number of variables in the problem.
TL;DR: An improved technique is developed which is capable of handling more realistic problems and requires too much memory and running time to be able to treat problems of realistic size and complexity.
Abstract: The problem considered is that of finding an optimal (minimum cost) design for a centralized processing network given a set of locations, traffic magnitudes between these locations, and a single common source or destination. Several heuristics, which are efficient (in terms of their execution time and memory requirements on a digital computer) and which produce seemingly good results, have already been developed and are currently accepted techniques. Some work has also been done on finding optimal solutions to this problem both as a design tool and as a means of verifying the effectiveness of proposed heuristics. We focus on this latter area. Currently known techniques for the optimal solution of this problem via integer programming have fallen short of the desired objectives as they require too much memory and running time to be able to treat problems of realistic size and complexity. We develop an improved technique which is capable of handling more realistic problems.
Book•
01 Jan 1983
TL;DR: This paper presents a method for Finding Redundancy in Systems of Linear Constraints of a System of Linear Inequalities using the Algorithm PREDUCE, a simple and intuitive approach based on the principles of reinforcement learning.
Abstract: 1 An Introduction to Redundancy- 11 Redundancy- 12 Causes of Redundancy- 13 Consequences of Redundancy- 14 Dealing with Redundancy- 15 A Survey of the Literature- 16 Objective and Plan of the Study- 2 Mathematical Foundations and Notation- 21 Notation- 22 Terminology- 23 A Categorization of Methods- 24 Some Common Theory- 3 A Method for Identifying Redundant Constraints and Extraneous Variables in Linear Programming- 31 An Intuitive Exposition of the Approach- 32 The Algorithm- 33 Theory- 34 An Example- 35 Conclusion- 4 A Method for Determining Redundant Constraints- 41 An Intuitive Exposition of the Method- 42 The Algorithm- 43 Theoretical Background- 44 An Illustrative Example- 45 Conclusion- 5 Identifying Redundancy in Systems of Linear Constraints- 51 Introduction- 52 Intuitive Exposition of the Approach- 53 Description of the Algorithm- 54 Mathematical Theory- 55 Special Aspects of the Approach- 56 Example- 6 Finding Redundant Constraints in Sets of Linear Inequalities- 61 Introduction- 62 Intuitive Exposition of the Approach- 63 The Algorithm- 64 Mathematical Theory- 65 Special Aspects of the Approach- 66 An Example- 67 Conclusion- 7 A Method for Finding Redundant Constraints of a System of Linear Inequalities- 71 An Intuitive Exposition of the Approach- 72 Description of the Algorithm- 73 Mathematical Theory- 74 Special Aspects of the Approach- 75 An Example- 76 Conclusion- 8 Some Reduction of Linear Programs Using Bounds on Problem Variables- 81 Introduction- 82 An Intuitive Exposition of the Approach- 83 Description of the Algorithm- 84 Mathematical Theory- 85 Special Aspects of the Approach- 86 An Example- 87 Conclusion- 9 A Reduction Procedure for Linear and Integer Programming Models- 91 Introduction- 92 Primal and Dual Observations- 93 The Tests- 94 Applying the Tests- 95 Implementation Considerations- 96 Numerical Examples- 97 Conclusions- 10 Preduce - A Probabilistic Algorithm Identifying Redundancy by a Random Feasible Point Generator (RFPG)- 101 Introduction- 102 An Intuitive Exposition of Algorithm PREDUCE- 103 Description of Algorithm PREDUCE- 104 Mathematical Theory- 105 Special Aspects of PREDUCE- 106 A Numerical Example- 11 The Noncandidate Constraint Method- 111 Introduction- 112 An Intuitive Explanation of the Method- 113 Description of the Algorithm- 114 Special Aspects of the Noncandidate Method- 115 Solution of an Example- 116 Conclusions- 12 Structural Redundancy in Large-Scale Optimization Models- 121 Introduction- 122 Overview of the Analysis- 123 Details of the Analysis- 124 Extensions to Mixed Integer and Nonlinear Models- 125 Conclusion- 126 Acknowledgments- 13 Programming the Methods and Experimental Design- 131 Programming the Methods- 132 Performance Monitoring- 133 Test Problems- 134 Summary- 14 Results of the Sign Test Methods- 141 Results for the Randomly Generated Problems- 142 Problem Differences- 143 Method Efficiencies Versus Time- 144 Efficiency of the Various Tests- 145 Results for the Structured Problems- 15 Results of the Other Methods- 151 Boneh's Method- 152 Mattheiss' Method- 153 Klein and Holm's Method- 154 Williams' Method- 155 The Method of Sethi and Thompson- 156 Summary- 16 Improvements and Extensions- 161 The Extended Sign Test Method- 162 The Hybrid Method- 163 The Reduce Method- 17 Results of the Improvements and Extensions- 171 The Extended Sign Test Method- 172 The Hybrid Method- 173 The Reduce Method- 18 Conclusions- 181 Summary of the Test Results- 182 Other Developments and Conclusions- References
01 Apr 1983
TL;DR: In this paper, a mixed-integer formulation of the mine scheduling problem is discussed and applied for the purpose of optimizing both the mine production sequencing and the mill blending and processing problems simultaneously.
Abstract: A mixed-integer formulation of the mine scheduling problem is discussed and applied for the purpose of optimizing both the mine production sequencing and the mill blending and processing problems simultaneously. Previous attempts at solving these problems separately, although successful, fail to take into account the interrelationships between these problems. Project structure serves as the outline of this paper. Generalized in its organization, this concept is useful for long, intermediate, and short range scheduling. Actual applications that show the concept also is applicable to mining a variety of ore are mentioned.
TL;DR: A class of nonlinear knapsack problems where the objective function is quadratic is discussed, which includes an efficient algorithm to find the continuous (relaxed) solution and a reduction rule which computes tight lower and upper bounds on the integer variables.
TL;DR: It is proved that rather tight upper bounds can be given for the number of non-unique assignments that are achieved after solving the linear programming relaxation of some types of mixed integer linear assignment problems.
01 Jan 1983
TL;DR: It is shown in this work that the model is actually only locally inclusive and that it can be solved as a special case of another general, successively exclusive, model that is formulated and solved.
Abstract: A.B. Calvo and D.H. Marks (1973) formulated the problem of locating various types of health facilities, and allocating different types and levels of demand to these facilities, as a zero-one integer programming model. Although the model has been widely referenced in the literature no solution algorithm has been developed to date. It is shown in this work that the model is actually only locally inclusive and that it can be solved as a special case of another general, successively exclusive, model that is formulated and solved. A second model is presented that corrects for a shortcoming in the A.B. Calvo and D.H. Marks model and solves for the successively inclusive hierarchical location-allocation problem.
TL;DR: The purpose of this paper is to document the results of an ongoing research effort which combines the generalized network concept with the technique of fuzzy programming and the resultant, hybrid approach has been found to provide a computationally efficient approach to multi-objective zero-one (or integer) programming problems.
TL;DR: An algorithm for solving the integer programming problem, which arises in the staggering approach to limited capacity inventory systems, is described.
Abstract: This paper describes an algorithm for solving the integer programming problem, which arises in the staggering approach to limited capacity inventory systems. Numerical examples are given.
01 Dec 1983
TL;DR: A practical method for generating optimal schedules for online train traffic control in disturbed situations and it is shown that each subproblem generated in the second part, which is a linear programming problem, can be easily calculated without using the ordinary simplex method.
Abstract: A practical method for generating optimal schedules for online train traffic control in disturbed situations is proposed. This scheduling problem is formulated to a 0-1 mixed integer programming problem. The method of solution proposed here is mainly divided into two parts: the first part generates a suboptimal solution by a heuristic method based upon "Production System", and the second produces an optimal one by the branch-and-bound method using the above sub-optimal value for the initial bound. It is shown that each subproblem generated in the second part, which is a linear programming problem, can be easily calculated without using the ordinary simplex method. Some examples show that the proposed method has enough efficiency for practical use in both computational time and storage.
TL;DR: It is proved that a certain type of knapsack problem cannot have a polynomialtune approxtmatlon algorithm unless NP = P, and it is shown that a decision problem revolving integer variables is NP-complete.
Abstract: It ts shown that a set of linear Dtophantme equations m nonnegative variables with nonnegative coefficients can be reduced to a single equation with the same solution set in polynomial time. A weaker verston of the above statement ~s shown to be true when the coefficients are allowed to be negative Besides being polynomial-trine bounded, the present aggregation scheme differs from existing ones in that the final equation is m variables that are not exphcltly bounded Three applications of this aggregation technique are presented: (i) ~t Is proved that a certain type of knapsack problem cannot have a polynomialtune approxtmatlon algorithm unless NP = P; 00 an analog of Farkas' lemma for integer programming is proved; and 0ii) ~t is shown that a decision problem revolving integer variables is NP-complete.
TL;DR: An analytical scheme leading to the development of optimal plans for the reuse and redevelopment of marine transport networks in New England by utilizing the port inventory and a mathematical programming model to determine the optimal capacity, placement and railrod and marine interface of coal handling facilities.
Book•
03 Aug 1983
TL;DR: Information is provided on how to construct models for linear and nonlinear programming using the Simplex Method, as well as some of the techniques used in Integer Programming and Dynamic Programming.
Abstract: LINEAR PROGRAMMING. An Introduction to Linear Programming. The Simplex Method. Duality Theory and Sensitivity Analysis. The Transportation and Assignment Problems. Goal Programming. MATHEMATICAL PROGRAMMING. Network Models. PERT/CPM Models. Integer Programming. Nonlinear Programming. Dynamic Programming. Game Theory. PROBABLE PROBABILISTIC MODELS. Waiting Line Models. Markov Processes. Inventory Models. Simulation Models. Decision Analysis. SYNTHESIS. Implementation of Management Science. Appendices. Answers to Even Numbered Questions. Index.
IBM1
TL;DR: In this article, the problem of scheduling personnel and jobs to minimize personnel idle time, by integer programming, is formulated and solved for a set of maintenance jobs to be processed over a fixed time horizon, where each job is divided into finite time intervals in which the skills required are known.
Abstract: Given (i) a set of maintenance jobs to be processed over a fixed time horizon, (ii) the breakdown of each job into finite time intervals in which the skills required are known, and (iii) the pool of available manpower for each skill type over the horizon, we formulate and solve the problem of scheduling personnel and jobs to minimize personnel idle time, by integer programming.
TL;DR: Results of a recent study conducted to compare the performance of Exxon's succesive linear programming and successive quadratic programming techniques with the reduced gradient technology employed in other programs are presented.
IBM1
TL;DR: The problem of scheduling plant maintenance personnel has been recast to give the minimum problem-size formulation as discussed by the authors, where the problem of finding the minimum number of maintenance personnel is solved.
Abstract: The problem of scheduling plant maintenance personnel has been recast to give the minimum problem-size formulation.