Showing papers on "Integer programming published in 1990"
TL;DR: It is proved that no polynomial algorithm can achieve a worst-case ratio less than 3/2 unlessP = NP, and a complexity classification for all special cases with a fixed number of processing times is obtained.
Abstract: We consider the following scheduling problem. There arem parallel machines andn independent jobs. Each job is to be assigned to one of the machines. The processing of jobj on machinei requires timep
ij
. The objective is to find a schedule that minimizes the makespan. Our main result is a polynomial algorithm which constructs a schedule that is guaranteed to be no longer than twice the optimum. We also present a polynomial approximation scheme for the case that the number of machines is fixed. Both approximation results are corollaries of a theorem about the relationship of a class of integer programming problems and their linear programming relaxations. In particular, we give a polynomial method to round the fractional extreme points of the linear program to integral points that nearly satisfy the constraints. In contrast to our main result, we prove that no polynomial algorithm can achieve a worst-case ratio less than 3/2 unlessP = NP. We finally obtain a complexity classification for all special cases with a fixed number of processing times.
953 citations
629 citations
TL;DR: In this article, the authors provide integer linear programming formulations for the selective travelling salesman problem and derive upper and lower bounds for exact enumerative algorithms for the problem, which are then embedded in exact enumeration algorithms.
388 citations
TL;DR: A basic implicit enumeration scheme is developed that finds good feasible solutions within relatively few iterations in the case where each player tries to maximize the individual objective function over a jointly constrained polyhedron.
Abstract: A two-person, noncooperative game in which the players move in sequence can be modeled as a bilevel optimization problem. In this paper, we examine the case where each player tries to maximize the individual objective function over a jointly constrained polyhedron. The decision variables are variously partitioned into continuous and discrete sets. The leader goes first, and through his choice may influence but not control the responses available to the follower. For two reasons the resultant problem is extremely difficult to solve, even by complete enumeration. First, it is not possible to obtain tight upper bounds from the natural relaxation; and second, two of the three standard fathoming rules common to branch and bound cannot be applied fully. In light of these limitations, we develop a basic implicit enumeration scheme that finds good feasible solutions within relatively few iterations. A series of heuristics are then proposed in an effort to strike a balance between accuracy and speed. The computational results suggest that some compromise is needed when the problem contains more than a modest number of integer variables.
349 citations
TL;DR: In this paper, a similarity coefficient matrix is used as the input to the assignment problem and closed loops in the form of subtours are identified after solving the problem and are used as a basis for grouping.
Abstract: SUMMARY The problem of grouping of parts has been addressed in the past using clustering methods and integer programming. This paper presents an assignment model to solve the grouping problem. A similarity coefficient matrix is used as the input to the assignment problem. Closed loops in the form of subtours are identified after solving the problem and are used as the basis for grouping. The method has been applied to a number of examples. Compared with the earlier mathematical programming model, viz., the p-median model, the assignment method emerges as a distinctly superior technique both in terms of quality of solution and computational time.
271 citations
TL;DR: The heuristic, developed to get an initial lower bound, finds an optimal solution for most of the random test problems, and an extension to the basic problem that allows for preselected points, which may correspond to existing facility locations.
267 citations
TL;DR: A finite recursive procedure for generating the mixed integer hull of a polyhedron is obtained, analogous to the process of repeatedly taking Chvátal closures in the integer programming case.
Abstract: Chvatal introduced the idea of viewing cutting planes as a system for proving that every integral solution of a given set of linear inequalities satisfies another given linear inequality. This viewpoint has proven to be very useful in many studies of combinatorial and integer programming problems. The basic ingredient in these cutting-plane proofs is that for a polyhedronP and integral vectorw, if max(wx|x ∈ P, wx integer} =t, thenwx ⩽ t is valid for all integral vectors inP. We consider the variant of this step where the requirement thatwx be integer may be replaced by the requirement that\(\bar wx\) be integer for some other integral vector\(\bar w\). The cutting-plane proofs thus obtained may be seen either as an abstraction of Gomory's mixed integer cutting-plane technique or as a proof version of a simple class of the disjunctive cutting planes studied by Balas and Jeroslow. Our main result is that for a given polyhedronP, the set of vectors that satisfy every cutting plane forP with respect to a specified subset of integer variables is again a polyhedron. This allows us to obtain a finite recursive procedure for generating the mixed integer hull of a polyhedron, analogous to the process of repeatedly taking Chvatal closures in the integer programming case. These results are illustrated with a number of examples from combinatorial optimization. Our work can be seen as a continuation of that of Nemhauser and Wolsey on mixed integer cutting planes.
239 citations
TL;DR: In this paper, the authors study several ways of obtaining valid inequalities for mixed integer programs and show how inequalities obtained from a disjunctive argument can be represented by superadditive functions and how they relate to Gomory's mixed integer cuts.
Abstract: We study several ways of obtaining valid inequalities for mixed integer programs. We show how inequalities obtained from a disjunctive argument can be represented by superadditive functions and we show how the superadditive inequalities relate to Gomory's mixed integer cuts. We also show how all valid inequalities for mixed 0–1 programs can be generated recursively from a simple subclass of the disjunctive inequalities.
210 citations
TL;DR: An experimental analysis demonstrated that, when flexible break assignments were modeled, the implicit formulation was superior to the traditional set-covering formulation with respect to execution time, computer memory requirements, and the ability to produce optimal integer solutions to larger problems incorporating greater flexibility.
Abstract: The labor scheduling literature has demonstrated that the use of flexibility in designing employee schedules can result in a substantial improvement in labor utilization. This paper presents a new implicit integer linear programming formulation for the inclusion of meal/rest-break flexibility. Although the use of flexible break assignments in labor staffing decisions has been of research interest since an early article by Segal 1974, due to problem size, the majority of related research has involved the use of heuristics. An experimental analysis using four different labor requirements patterns and ten shift-length combinations demonstrated that, when flexible break assignments were modeled, the implicit formulation was superior to the traditional set-covering formulation with respect to 1 execution time, 2 computer memory requirements, and 3 the ability to produce optimal integer solutions to larger problems incorporating greater flexibility. Finally, a number of possible extensions of the implicit modeling approach for use in other labor scheduling environments are identified.
196 citations
Patent•
19 Dec 1990
TL;DR: In this article, an airline reservation system implemented using a computer wherein reservations are controlled by a seat inventory control system is described, which produces optimal reservation control using network-wide booking limits while taking into account the probabilistic nature of demand.
Abstract: An airline reservation system implemented using a computer wherein reservations are controlled by a seat inventory control system. The inventory control system produces optimal reservation control using network-wide booking limits while taking into account the probabilistic nature of demand. The inventory control system, based on a concept termed Network-Based Expected Marginal Seat Revenue (EMSR), does not require the large number of variables required by the other network-based approaches, and it incorporates a probabilistic demand model without resorting to computationally intractable integer programming.
167 citations
TL;DR: An efficient algorithm based on the SPT rule is developed to solve the problem of minimizing the mean flow time in a general job shop type machining system with alternative machine tool routeings by decomposing it into subproblems that are easier to solve.
Abstract: SUMMARY In this paper we investigate the problem of minimizing the mean flow time in a general job shop type machining system with alternative machine tool routeings. An analytical formulation of the problem as a mixed integer programming is developed. An efficient algorithm based on this formulation is developed to solve the problem by decomposing it into subproblems that are easier to solve. The algorithm solves large problems in relatively short time. A second algorithm based on the SPT rule is developed and its performance is compared with the first algorithm. A greedy procedure is also developed for the case when a penalty cost is associated with adding alternative machines. Numerical examples are given to demonstrate the use of the above algorithms.
TL;DR: An alternative formulation of the AGV flow path layout (FPL) problem was first formulated by Gaskins and Tanchoco (1987) as a zero-one integer programming problem, and a computationally efficient procedure is proposed which is based on the branch-and-bound technique as discussed by the authors.
Abstract: SUMMARY This paper describes an alternative formulation of the AGV flow path layout (FPL) problem which was first formulated by Gaskins and Tanchoco (1987) as a zero-one integer programming problem. A computationally efficient procedure is proposed which is based on the branch-and-bound technique. An algorithm for satisfying the reachability condition for nodes in the AGV flow path network is also presented. A simple illustrative example is discussed to demonstrate the procedure, and a more complex problem is also given.
TL;DR: A general integer linear programming model is presented for the important practical problem of designing minimum-cost survivable networks, and this model is related to concepts in graph theory and polyhedral combinatorics.
Abstract: In this paper a general integer linear programming model is presented for the important practical problem of designing minimum-cost survivable networks, and this model is related to concepts in graph theory and polyhedral combinatorics. In particular, several interesting special cases of this general model are considered, including the minimum spanning tree problem, the Steiner tree problem, and the minimum cost k-edge connected and k-node connected network design problems. The integer polyhedra associated with these problems are studied, those inequalities from natural ILP-formulations that define facets are identified, the separation problem for these facets is addressed, and how good lower bounds can be obtained from the models studied here is indicated.
TL;DR: In this article, a generalized group technology problem of manufacturing a group of parts in which each part can have alternative process plans and each operation in these plans can be performed on alternative machines is considered.
Abstract: SUMMARY In this paper we consider a generalized group technology problem of manufacturing a group of parts in which each part can have alternative process plans and each operation in these plans can be performed on alternative machines. The objective is to model and analyse how alternative process plans influence the resource utilization when the part families and machine groups are formed simultaneously. Accordingly, we develop three integer programming models to successively study the effect of alternative process plans and simultaneous formation of part families and machine groups. An illustrative example is included.
TL;DR: This linearization scheme provides an equivalent mixed integer linear programming problem which yields a tighter continuous relaxation than that obtainable via the alternative linearization techniques available in the literature.
Abstract: This paper is concerned with a new linearization strategy for a class of zero-one mixed integer programming problems that contains quadratic cross-product terms between continuous and binary variables, and between the binary variables themselves. This linearization scheme provides an equivalent mixed integer linear programming problem which yields a tighter continuous relaxation than that obtainable via the alternative linearization techniques available in the literature. Moreover, the proposed technique provides a unifying framework in the sense that all the alternate methods lead to formulations that are accessible through appropriate surrogates of the constraints of the new linearized formulation. Extensions to various other types of mixed integer nonlinear programming problems are also discussed.
TL;DR: In this paper, an algorithm to obtain an approximate optimal solution to the problem of large-scale radial distribution system planning is proposed, where the distribution planning problem is formulated as a MIP (mixed integer programming) problem.
Abstract: An algorithm to obtain an approximate optimal solution to the problem of large-scale radial distribution system planning is proposed. The distribution planning problem is formulated as a MIP (mixed integer programming) problem. The set of constraints is reduced to a set of continuous variable linear equations by using the fact that the basis of the simplex tableau consists of the power flow variables of radial branch. This linear problem is solved by pivot operations which correspond to a branch-exchange of the radial network. Numerical examples are presented to demonstrate the validity and effectiveness of the algorithm. >
11 Nov 1990
TL;DR: The present approach breaks new ground by simultaneously scheduling and allocating in practical execution times, guaranteeing globally optimal solutions for a specific objective function, and providing a polynomial run-time algorithm for solving some instances of this NP-complete problem.
Abstract: A relaxed linear programming model which simultaneously schedules and allocates functional units and registers is presented for synthesizing cost-constrained globally optimal architectures. This approach is important for industrial applications, because it provides exploration of optimal synthesized architectures and early architectural decisions have the greatest impact on the final design. An integer programming formulation of the architectural synthesis problem is transformed into the mode packing problem. Polyhedral theory is used to formulate constraints that decrease the size of the search space, thus improving solution efficiency. Execution times are an order of magnitude faster than for previous heuristic techniques. The present approach breaks new ground by (1) simultaneously scheduling and allocating in practical execution times, (2) guaranteeing globally optimal solutions for a specific objective function, and (3) providing a polynomial run-time algorithm for solving some instances of this NP-complete problem. >
TL;DR: In this article, a reliability-based optimization model for water-distribution systems is developed, which is aimed at the following goals: (1) design of the pipe network including the number, location, and size of pumps and tanks; (2) designing of the pumping system using a reliabilitybased procedure considering both hydraulic failures of the entire network and mechanical failure of the pump system; and (3) determination of the optimal operation of the pumps.
Abstract: A reliability-based optimization model for water-distribution systems has been developed. The model is aimed at the following goals: (1) Design of the pipe network including the number, location, and size of pumps and tanks; (2) design of the pumping system using a reliability-based procedure considering both hydraulic failures of the entire network and mechanical failure of the pumping system; and (3) determination of the optimal operation of the pumps. The optimization problem is a large mixed-integer, nonlinear programming problem that is solved using a heuristic algorithm consisting of a master problem and a subproblem. The master problem is a pure 0–1 integer programming model, and the subproblem is a large nonlinear programming model solved in an optimal control framework. The conservation of flow and energy constraints are solved implicitly for each iteration of the nonlinear optimization procedure using a hydraulic simulation model, and the reliability constraints are also solved implicitly using a reliability model. The nonlinear programming problem is solved using a generalized reduced gradient code.
TL;DR: It appears that one can achieve a high degree of equity by modestly increasing the total risk and by embarking on different routes to evenly spread the risk among the zones, and it appears that the heuristic procedure is excellent in terms of computational requirements as well as solution quality.
Abstract: In this paper, we develop and analyze a model to generate an equitable set of routes for hazardous material shipments. The objective is to determine a set of routes that will minimize the total risk of travel and spread the risk equitably among the zones of the geographical region in which the transportation network is embedded, when several trips are necessary from origin to destination. An integer programming formulation for the problem is proposed. We develop and test a heuristic that repeatedly solves single-trip problems: a Lagrangian dual approach with a gap-closing procedure is used to optimally solve single-trip problems. We report a sampling of our computational experience, based on a real-life routing scenario in the Albany district of New York State. Our findings indicate that one can achieve a high degree of equity by modestly increasing the total risk and by embarking on different routes to evenly spread the risk among the zones. Furthermore, it appears that our heuristic procedure is excellent in terms of computational requirements as well as solution quality. We also suggest some directions for future research.
TL;DR: An area-based forest plan is formulated and solved by mixed integer programming and a random search algorithm and it is shown that the forest plan can be modified to suit the changing environment.
Abstract: An area-based forest plan is formulated and solved by mixed integer programming and a random search algorithm. This is a computationally difficult problem because operational and environmental cons...
TL;DR: This paper will formulate the mixed integer two-level linear programming problem and develop both the exact and heuristic solution procedures based on the branch-and-bound technique for solving the problem.
TL;DR: Alternative methods for constructing a 0–1 integer programming problem from a propositional calculus problem and the use of the resulting mathematical program to solve the related logic problem are discussed.
TL;DR: The work is based on enumerating the possible cutting patterns, and solving the associated integer program by a combination of cutting planes and branch and bound, and the solutions obtained are usually optimal in trim loss terms.
TL;DR: Expert system and integer programming formulations of an NP-complete constraint satisfaction problem are contrasted in terms of performance, ability to encode complex preferences, control of reasoning, and supporting incremental modification of solutions in response to changing input data.
Abstract: Expert system and integer programming formulations of an NP-complete constraint satisfaction problem are contrasted in terms of performance, ability to encode complex preferences, control of reasoning, and supporting incremental modification of solutions in response to changing input data.
TL;DR: In this article, the authors investigate heuristics based on a stochastic search method for the multi-level lot-sizing problem (MLP) and compare the quality and efficiency of these methods with the single-level heuristic methods which are applied on a level-by-level basis.
TL;DR: This paper presents the mathematical formulation of the map label placement problem and describes a computer program implemented to solve it.
Abstract: The placement of text on a map to maximize map legibility, while avoiding graphic overplotting, is a problem in combinatorial optimization. This particular optimization problem can be formulated as a multiple choice integer programming problem, and the integer programming problem has a structure that can be exploited, using a Lagrangian heuristic, to obtain a cost effective solution, even when tens of thousands of variables and thousands of constraints are involved. This paper presents the mathematical formulation of the map label placement problem and describes a computer program implemented to solve it.
TL;DR: Five sequencing methods are reviewed and it is shown that these methods differ widely in their ability to generate good sequences, and a mixed integer programming (MIP) model is presented as a way to find an optimal sequence.
IBM1
TL;DR: The authors identify the key parameters for capturing the behavior of an access plan and propose a two-step methodology consisting of a query analysis step to estimate the parameters and a binary partitioning step which can be applied recursively.
Abstract: Vertical partitioning can be used to enhance the performance of relational database systems by reducing the number of disk accesses. The authors identify the key parameters for capturing the behavior of an access plan and propose a two-step methodology consisting of a query analysis step to estimate the parameters and a binary partitioning step which can be applied recursively. The partitioning uses an integer linear programming technique to minimize the number of disk accesses. Significant performance benefit would be achieved for join if the partitioned (inner) relation could fit into the memory buffer under the inner-outer loop join method, or if the partitioned relation could fit into the sort buffer under the sort-merge join method, but not the original relation. For cases where a segment scan or a cluster index scan is used, vertical partitioning of the relation with the algorithm described is still often found to lead to substantial performance improvement. >
TL;DR: Computer tests show that polyhedral methods based upon a set of facet inequalities developed in this paper can effectively reduce the gap between the value of an integer programming formulation of the problem and its linear programming relaxation by a factor of 94 to 100%.
Abstract: Changeover costs and times are central to numerous manufacturing operations. These costs arise whenever work centers capable of processing only one product at a time switch from the manufacture of one product to another. Although many researchers have contributed to the solution of scheduling problems that include changeover costs, due to the problem's combinatorial explosiveness, optimization-based methods have met with limited success. In this paper, we develop and apply polyhedral methods from integer programming for a dynamic version of the problem. Computational tests with problems containing one to five products and up to 225 integer variables show that polyhedral methods based upon a set of facet inequalities developed in this paper can effectively reduce the gap between the value of an integer programming formulation of the problem and its linear programming relaxation by a factor of 94 to 100%. These results suggest the use of a combined cutting plane/branch-and-bound procedure as a solution approach. In a test with a five product problem, this procedure, when compared with a standard linear programming-based branch-and-bound approach, reduced computation time by a factor of seven.