Showing papers on "Integer programming published in 1987"

Randomized rounding: a technique for provably good algorithms and algorithmic proofs

Abstract: We study the relation between a class of 0–1 integer linear programs and their rational relaxations. We give a randomized algorithm for transforming an optimal solution of a relaxed problem into a provably good solution for the 0–1 problem. Our technique can be a of extended to provide bounds on the disparity between the rational and 0–1 optima for a given problem instance.

Minkowski's convex body theorem and integer programming

Abstract: The paper presents an algorithm for solving Integer Programming problems whose running time depends on the number n of variables as nOn. This is done by reducing an n variable problem to 2n5i/2 problems in n-i variables for some i greater than zero chosen by the algorithm. The factor of On5/2 “per variable” improves the best previously known factor which is exponential in n. Minkowski's Convex Body theorem and other results from Geometry of Numbers play a crucial role in the algorithm. Several related algorithms for lattice problems are presented. The complexity of these problems with respect to polynomial-time reducibilities is studied.

The generalized group technology concept

Abstract: In this paper two classes of clustering models are considered: (1) matrix, and (2) integer programming. The relationship between the matrix model, the p-median model and the classical group technology concept is discussed. A generalized group technology concept, based on generation for one part of a number of different process plans, is proposed. This new concept improves the quality of process (part) families and machine cells. A corresponding integer programming model is formulated. The models discussed are illustrated with numerical examples.

Solving multi-item capacitated lot-sizing problems using variable redefinition

Abstract: Mixed-integer programming models are typically not used to solve realistic-sized production scheduling problems because they require exorbitant solution times. We impose a useful taxonomy on production scheduling problems and develop alternative formulations for a wide variety of problems within the taxonomy. The linear programming relaxation of the new models is very effective in generating bounds. We show that these bounds are equal to those that could be generated using Lagrangian relaxation or column generation. The linear programming bounds increase in effectiveness as the problems become larger. Perhaps of greatest significance is that practitioners can obtain our results using only standard “off-the-shelf” codes such as LINDO or MPSX/370. We report computational experience in several computing environments (hardware and software) on problems with up to 200 products and 10 time periods (2000 0-1 variables).

Approximation algorithms for scheduling unrelated parallel machines

Abstract: We consider the following scheduling problem. There are m parallel machines and n independent jobs. Each job is to be assigned to one of the machines. The processing of job j on machine i requires time pij. 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 unless P = NP. We finally obtain a complexity classification for all special cases with a fixed number of processing times.

An application of simultaneous Diophantine approximation in combinatorial optimization

Abstract: We present a preprocessing algorithm to make certain polynomial time algorithms strongly polynomial time. The running time of some of the known combinatorial optimization algorithms depends on the size of the objective functionw. Our preprocessing algorithm replacesw by an integral valued-w whose size is polynomially bounded in the size of the combinatorial structure and which yields the same set of optimal solutions asw. As applications we show how existing polynomial time algorithms for finding the maximum weight clique in a perfect graph and for the minimum cost submodular flow problem can be made strongly polynomial. Further we apply the preprocessing technique to make H. W. Lenstra’s and R. Kannan’s Integer Linear Programming algorithms run in polynomial space. This also reduces the number of arithmetic operations used. The method relies on simultaneous Diophantine approximation.

Solving Mixed Integer Programming Problems Using Automatic Reformulation

Abstract: In this paper we describe computational experience in solving mixed 0-1 programming problems using strong valid inequalities as cutting planes. In particular we report on the solution to optimality of 18 medium-to large-size problems, including production planning problems with setup costs and capacity constraints, multilevel distribution planning problems, drainage and heating system design problems, and electricity generator scheduling problems. The solution approach uses the theory of strong valid inequalities that we developed in a series of earlier papers. Here we report specifically on the implementation of an experimental system, MPSARX, which consists of the SCICONIC mathematical programming system and an automatic reformulation executor ARX that use this theory, and on the results obtained with this system.

Some Considerations about Computational Complexity for Multi Objective Combinatorial Problems

Abstract: In the field of vector (or multi objective) optimization there has been a relatively little interest in solving combinatorial or discrete problems. During the 70’s just a few papers have been published on multi objective (m.o.) integer linear programming. However no special emphasis was put on the important aspect of computational complexity. This can be certainly ascribed to the fact that the theory of NP—completeness was developing at a fast pace in those same years.

The node-weighted Steiner tree problem

Abstract: The general Node-Weighted Steiner Tree problem is an extension of the standard Steiner Tree problem by the addition of node-associated weights. This article analyzes a special case of that problem, where the set of nodes, which must be included in the solution tree, consists of a single node, and all node weights are negative. The special case is shown to be NP-Complete, its integer programming formulation is presented, and heuristic procedures are proposed. Using Lagrangian relaxation and subgradient optimization, tight lower bounds were derived and utilized by a branch and bound algorithm. The effectiveness of the developed procedures is demonstrated by a set of computational experiments.

Distribution System Planning through a Quadratic Mixed Integer Programming Approach

Abstract: This Paper presents a new approach for the optimal sizing and siting of substations and network routing problem. The solution approach proposed is a nolinear programming approach. The problem has been formulated as a Quadratic Mixed Integer programming (QMIP) problem in terms of the fixed costs of the substations and lines and the present worth of the energy loss costs of the line segments. The solution to this QMIP problem is obtained in two stages. In the first stage the quadratic programming problem is solved following the procedure developed by Wolfe using simplex method and treating all the variables as continuous variables. In the second stage, a procedure has been suggested to integerize the values of the integer variables. The proposed method is validated using a numerical example.

Competitive Location on a Network

Abstract: In this paper we study the problem of locating facilities on a network in the presence of competition. Customers at each node in the network choose from the available facilities so as to minimize the distance traveled. The problem is to find a set of facilities that is stable in the sense that each facility is economically viable and no competitor can successfully open any facilities. We define several versions of stability and establish certain relationships between them. We then present integer programming formulations that identify stable sets, and describe an enumeration algorithm for constructing a profit-maximizing stable set.

Generating alternative mixed-integer programming models using variable redefinition

Abstract: Dropping the “complicating” constraints in a mixed-integer linear program often yields a “special structure subproblem” that can be reformulated using a different set of decision variables. Once the new variables have been identified, the entire problem can be reformulated in terms of the new variables. We develop a theory of variable redefinition based on relating the two sets of decision variables by a linear transformation, and describe methods for reformulating the special structure problem. The reformulated models have a more accurate linear relaxation than the problems from which they were derived, an important property within the context of linear programming-based branch-and-bound modeling approaches.

Reliability Optimization with the Lagrange-Multiplier and Branch-and-Bound Technique

Abstract: A method has been developed for constrained reliability optimization problems. This method incorporates the Lagrange multiplier method and the branch-and-bound technique. The Lagrange multiplier method treats the number of redundancies as real numbers. Once a real number solution is obtained, the branch-and-bound technique is used to obtain the integer solution. With our method, a 4-stage series system with two linear constraints is illustrated for the redundancy allocation problem, and a 5-stage series system with three nonlinear constraints is illustrated for the reliability-redundancy allocation problem. The results show that our method is better than previous methods for both the redundancy allocation problem and the mixed integer-type reliability-redundancy allocation problem. Our method also provides more reasonable explanations when solving reliability optimization problems.

The median shortest path problem : a multiobjective approach to analyze cost vs. accessibility in the design of transportation networks

Abstract: In this paper the authors introduce the median shortest path problem (MSPP). The MSPP is a bicriterion path problem with the objectives being the minimization of the total path length and the minimization of the total travel time required for demand to reach a node on the path. Potential applications of the MSPP include, among others, the location of new highways, railroad lines and subway lines and the design of airline routes. It is particularly applicable in transportation network design problems where the trade-off between operator costs and user costs is important. An algorithm is presented to identify noninferior solutions to the MSPP. This algorithm incorporates a K shortest path algorithm. The algorithm is demonstrated with a sample problem and the results are compared to those obtained using integer programming.

Optimal inspection policies in a serial production system including scrap rework and repair: an MILP approach

Abstract: The allocation of inspection effort problem for aerial systems is formulated as a 0-1 mixed integer linear programming problem. This formulation permits any combination of scrap, rework, or repair at each station and allows the problem to be solved using standard MILP software packages. Moreover electronic spread-sheets may be used to easily calculate the relevant coefficients. An additional advantage of this approach when compared with the traditional dynamic programming approach is the ease with which the basic model may be modified. For example, it is shown how the model may easily be modified to include both a material and a production constraint and to select between various material suppliers. Sensitivity analysis is also easily performed with this approach. This model is then used to show that the optimal inspection policy is dependent on whether a production or a material requirement is used.

A new formulation of operation allocation problem in flexible manufacturing systems: mathematical modelling and computational experience

Abstract: This paper extends the formulation of the operation allocation problem to include the important planning aspects of refixturing and limited tool availability. A 0–1 integer programming formulation is proposed with two objective functions and a set of realistic constraints. The computational behavior of the solution is discussed and a number of observations prompted by the solution methodology have been made.

A vertical partitioning algorithm for relational databases

Abstract: In a relational database environment, transaction response time is likely to be affected by the time required to read the necessary data from secondary storage (disk). In cases where segment scans are used to a significant extent, vertical partitioning of the relation can result in a decrease in the number of disk accesses. The issue is how to set up the criterion for partitioning. In this paper, an optimal binary partitioning algorithm which can be recursively applied is developed. The algorithm is based on an integer linear programming technique to minimize the number of disk accesses. Performance analysis is provided to study the situation when partitioning can be beneficial and quantify the performance impact. This can also be used to demonstrate the superiority of the proposed algorithm as compared with a previously proposed partitioning scheme.

Optimization of stiffened laminated composite plates with frequency constraints

Abstract: The problem considered in this paper concerns the maximization of frequencies of stiffened laminated composite plates subject to frequency separation constraints and an upper bound on weight. The number of plies of given fiber orientations and the stiffener areas form the two sets of design variables and the problem belongs to the class of nonlinear mixed integer programming ( NMIP) Several efficiency measures are adopted to reduce the computational cost of the optimization process. As a result of these efficiency measures, structural optimization of laminated composite plates using nonlinear mixed integer programming becomes a viable alternative to using the conventional practice of obtaining designs by an ad hoc rounding off of the continuous designs.

Structural properties and recognition of restricted and strongly unimodular matrices

Abstract: A (0, ±1) matrix A is restricted unimodular if every matrix obtained from A by setting to zero any subset of its entries is totally unimodular. Restricted unimodular matrices are also known as matrices without odd cycles. They have been studied by Commoner and recently Yannakakis has given a polynomial algorithm to recognize when a matrix belongs to this class. A matrix A is strongly unimodular if any matrix obtained from A by setting at most one of its entries to zero is totally unimodular. Crama et al. have shown that (0,1) matrix A is strongly unimodular if and only if any basis of (A, 1) is triangular, whereI is an identity matrix of suitable dimensions. In this paper we give a very simple algorithm to test whether a matrix is restricted unimodular and we show that all strongly unimodular matrices can be obtained by composing restricted unimodular matrices with a simple operation.

Quantitative Methods: Applications to Managerial Decision Making

Abstract: Probability Concepts Probability Distribution Decision and Utility Theory Forecasting Introduction to Linear Programming and Model Formulation Graphical Solution of Linear Programming Problems The Simplex Method Postoptimality Analysis Goal Programming Transportation, Transshipment and Assignment Problems Network Models PERT/CPM Integer Programming Models Inventory Analysis: Deterministic Models Inventory Analysis: Probabilistic Models Waiting Line Models Computer Simulation Other Quantitative Models Implementation and Integration of Management Science Techniques in the Decision Framework Appendixes Index.

The production equipment requirements problem

Abstract: In this paper a framework for the equipment requirements problem is presented, It allows one to specify requirements for machine tools and materials handling components. The framework is based on two integer programming models. The models require a set of data which are easily available. A numerical example and some computational results are presented. Alternative solution approaches and directions for future research are suggested.