Showing papers on "Integer programming published in 1972"

Integer programming

Abstract: The principles of integer programming are directed toward finding solutions to problems from the fields of economic planning, engineering design, and combinatorial optimization. This highly respected and much-cited text, a standard of graduate-level courses since 1972, presents a comprehensive treatment of the first two decades of research on integer programming.

Integer Programming Algorithms: A Framework and State-of-the-Art Survey

Abstract: A unifying framework is developed to facilitate the understanding of most known computational approaches to integer programming. A number of currently operational algorithms are related to this framework, and prospects for future progress are assessed.

An Efficient Branch and Bound Algorithm for the Warehouse Location Problem

Abstract: This paper introduces an efficient branch and bound algorithm for a special class of mixed integer programming problems called the warehouse location problem. A set of branching decision rules is proposed for selecting warehouses to be constrained open and closed from any node of the branch and bound tree. These rules are tested for their efficiency in reducing computation times and storage requirements to reach optimal solutions. An improved method of solving the linear programming problems at the nodes which substantially reduces the computations is also introduced in this paper.

Lagrangean Relaxation and Its Uses in Integer Programming

Abstract: : Taking a subset of the constraints of a general mixed integer linear program up into the objective function in a Lagrangean fashion (with fixed multipliers) yields what the author calls a Lagrangean relaxation of the original program. The paper gives a reasonably comprehensive development of the use of this simple device in the context of branch- and-bound. The selective application of these ideas can yield significant improvements in performance for special classes of problems.

The Design or Multipoint Linkages in a Teleprocessing Tree Network

Abstract: The problem of designing a minimum cost network with multipoint linkages which connects several remote terminals to a data processing center is studied. The important aspects of a teleprocessing network are queue behavior at the terminals and the cost and reliability of the entire system. In this paper it is assumed that the rate and manner in which information is requested at the terminals is known and that acceptable line loadings are given. An algorithm that determines (in principle) the optimum minimum cost network subject to reliability constraints is developed. A heuristic based on Vogel's approximation method (VAM) and two other heuristics presented by Martin and Esau-Williams were compared with each other and with the optimal algorithm. The Esau-Williams heuristic seems to be the one that gives the best solution and Martin's requires the least processing time. It is shown experimentally that Martin's and Esau-Williams heuristics are, in fact, near-optimal heurstics in the sense that the solutions provided by these heuristics are generally very near the optimal solution. In this paper we make the assumption that all lines of the network have the same capacity.

On the Set Covering Problem. II. An Algorithm.

Abstract: : In an earlier paper the authors proved that any feasible integer solution to the linear program associated with the equality-constrained set covering problem can be obtained from any other feasible integer solution by a sequence of less than m pivots (where m is the number of equations), such that each solution generated in the sequence is integer. However, degeneracy makes it difficult to find a sequence of pivots leading to an integer optimum. In the paper the authors give a constructive characterization of adjacency relations between integer vertices of the feasible set, which enables them to generate edges (all, if necessary) connecting a given integer vertex to adjacent integer vertices. This helps overcome the difficulties caused by degeneracy and leads to a class of algorithms of which two are discussed. (Author Modified Abstract)

Postoptimality analysis in zero‐one programming by implicit enumeration

Abstract: The procedures for postoptimality analysis that are so much a part of linear programming studies have no simple counterparts in an integer programming context In the case of Balas' Additive Algorithm, however, it would appear that the capacity of the technique to facilitate certain types of postoptimality analysis has not been fully exploited This paper proposes an extension of the additive algorithm that utilizes insights generated while solving the original problem to do subsequent analysis upon it In particular, procedures are developed for doing limited parameter ranging and for seeking new optima in light of parameter changes

Site Location via Mixed-integer Programming

Abstract: It is demonstrated that mixed-integer programming can be applied successfully to the solution of certain practical site location problems. A mixed-integer model of a frequently occurring form of warehouse location problem is presented. Experience with models of this type is described with examples of computational performance. The flexibility of mixed-integer models and the approach to their use as aids to decision-making are discussed.

Integer programming and convex analysis: Intersection cuts from outer polars

Abstract: Recently a new, geometrically motivated approach was proposed [1] for integer programming, based on generating intersection cuts from some convex setS whose interior contains the linear programming optimum $$\bar x$$ but no feasible integer point. Larger sets tend to produce stronger cuts, and in this paper convex analysis is used to construct sets as large as possible within the above requirements. Information is generated from all problem constraints within a unit cubeK containing $$\bar x$$ The key concept is that of outer polars, viewed as maximal convex extensions of the ballB circumscribingK, relative to the problem constraints. The outer polarF * of the feasible setF overB is shown to be a convex set whose boundary contains all feasible vertices ofK, and whose interior contains no feasible integer point. The existence of a dual correspondence betweenF andF *, and the fact that polarity is inclusion-reversing, leads to a dualization of operations onF. As one possible procedure based on this approach, we construct a generalized intersection cut, that can be strengthened whenever some vertex ofF is cut off. This makes it possible to fruitfully combine intersection cuts with implicit enumeration or branch-and-bound. While valid for arbitrary integer programs, the theory developed here is relevant primarily to (pure or mixed-integer) 0–1 problems. Other topics discussed include: generalized polars, intersection cuts from related problems, connections with asymptotic theory.

An integer programming approach to a class of combinatorial problems

Abstract: In this paper, we discuss the solution of a class of modified quadratic assignment problems, with particular reference to an application involving decentralization of a large organization The main emphasis is on the use of a standard branch and bound mathematical programming system (UMPIRE) and the problem manipulations required to carry this out efficiently

Design of Optimal Switching Networks by Integer Programming

Abstract: The design of optimal logic networks is formulated as integer programming (IP) problems. This formulation has the following advantages over other methods of logic design. 1) General feed-forward networks can be dealt with rather than two-level or three-level networks usually treated in conventional switching theory. 2) Network restrictions such as fan-in and fan-out restrictions are easily incorporated. 3) Various gate types such as NOR, NAND, AND-OR combination, NOR-AND combination, and those gates with NOR-OR dual outputs can be treated. 4) Various objectives such as the number of gates and the number of connections are minimized. 5) Incompletely specified functions can be handled without additional difficulty. 6) The formulation can be extended to multiple-output networks. To solve the resulting IP problems, the implicit enumeration method of integer programming is found to be suitable. An IP code ILLIP (Illinois Integer Programming Code) is implemented based on the implicit enumeration by incorporating some new gimmicks such as pseudounderlining. Then the ILLIP is used to solve the IP problems for logical design by making use of the inherent structure of our problems. Various optimal networks are derived by a computer as follows: optimal NOR networks and optimal NOR-AND networks for all functions of up through three variables, one-bit adders with various gate types, and others. These results indicate the computational feasibility of the integer programming approach.

Equivalent knapsack‐type formulations of bounded integer linear programs: An alternative approach

Abstract: In this paper we show that every bounded integer linear program can be transformed into an integer program involving one single linear constraint and upper and lower bounds on the variables, such that the solution space of the original problem coincides with that one of the equivalent knapsack-type problem.

Cut search methods in integer programming

Abstract: Cut search is a new approach for solving integer programs based on extending edges of a cone to probe the solution space for sets of hyperplanes that are “proxies” for solution points in the space. Once all proxy hyperplanes associated with a given point have been intersected by at least one of the extended edges, this point is included in a set of points to be examined for feasibility (algorithmically or by inspection). Thereupon, all edges of the cone are extended an additional distance to create a cut by passing a hyperplane through the endpoints of these extended edges. The flexibility of the cut search procedure permits a variety of strategies for exploring and cutting into the solution space. One useful version arises by taking the proxy hyperplanes to be members of a “positive” or “semipositive” coordinate system. Relative to such a system the procedure can be organized to reduce the set of vectors to be examined for feasibility and also to generate deeper cuts at the end of the edge probe.

Generalized implicit enumeration using bounds on variables for solving linear programs with zero‐one variables

Abstract: In an earlier paper [1] we put forth a framework that helps to tie together a number of approaches for solving integer programming problems. We outlined there how Balas' Additive Algorithm can be explained and generalized in terms of the framework. In the present paper we review Balas' algorithm, and our earlier framework, and present an algorithm generalizing Balas' scheme. In addition, some examples are presented and future research to be done is discussed.

Some related problems from network flows, game theory and integer programming

Abstract: We consider several important problems for which no polynomially time bounded algorithm is known These problems are shown to be related in that a polynomial algorithm for one implies a polynomial algorithm for the others

An implicit enumeration program for zero-one integer programming

Abstract: This paper describes some techniques to improve the speed of the implicit enumeration method for solving zero-one integer programming problems. Among these techniques, the most powerful is the one of using a column vector which works as a tag for each inequality, indicating whether or not the inequality should be checked for the current partial solution. A new condition for underlining a variable and the concept of pseudo-underlining are also proposed. These techniques were implemented in the computer programil lip (ILLinois Integer Programming code). The computational results for different types of problems are discussed.

Programme Selection in Research and Development

Abstract: Mathematical programming as an aid to R & D project portfolio selection has been suggested by many authors, but few practical applications have been reported. This paper discusses some, of the difficulties, associated with the application of the proposed models, and describes procedures for handling these problems. Problem areas considered are: Allowance for future opportunities in multi-period models, Generation of alternative risk-return solutions, Inclusion of projects not completed during the planning period, Comparison of linear and integer programming outputs. In the main, both the problems and methods of solution are clarified by means of numerical examples, and details of a practical case study are given.

Optimal Networks of NOR-OR Gates for Functions of Three Variables

Abstract: Optimal networks consisting of NOR-OR gates (each gate produces the NOR and/or the OR of its inputs) are tabulated for all Boolean functions of three variables. Optimality is defined as minimizing first the number of gates and then the number of interconnections. The optimal networks were synthesized for each Boolean function by using an integer programming synthesis technique.

Enumerative inequalities in integer programming

Abstract: For a linear integer programming problem, thelocal information contained at an optimal solution\(\bar x\) of the continuous linear programming extension stems from the theory of L.P. solutions. This paper proposes the use ofenvironmental information (of a global nature but pertaining to the discrete vicinity of\(\bar x\)), in order to isolate the set of integer solutions which may be considered as true candidates for the optimum. The concept ofenumerative inequalities is introduced and it is shown how it can be obtained in the context of the convex outer-domain theory of Balas, Young, et al.

Large scale optimization and the relaxation method

Abstract: An algorithm is given for the maximization of any function of many variables which may be described as the envelope of a family of linear functions. It is shown how the large-scale problems of linear programming to which various decomposition schemes apply can be posed in this way. Computational experience with some of these problems is reviewed.

An a Priori Bounded Model for Transportation Problems with Stochastic Demand and Integer Solutions

Abstract: A linear approximation model is developed for transportation problems with stochastic demand where integer solutions are required. The technique reduces the stochastic integer programming problem to a deterministic linear approximating problem which can be solved as a capacitated transportation problem. Either the transportation algorithm or the primal-dual algorithm may be used thereby insuring integer solutions.

Sensitivity Analysis in Linear Integer Programming

Abstract: A discussion of post-optimality and sensitivity analysis of linear integer programming problems through the construction of Hermitian bases. These bases are closely related to a Gaussian reduction for solving sets of linear equations. It is shown that from such a basis, the optimal integer solution for discrete changes in the constraint vector may be analyzed and bounds established for which the basis remains feasible. In addition, the effects of changes in the objective function can also be investigated. All of these analyses are direct extensions of linear programming post-optimality analysis applied to these special Hermitian bases. Other near optimal solutions can also be obtained.

A fundamental problem in linear inequalities with applications to the travelling salesman problem

Abstract: We consider a system ofm linearly independent equality constraints inn nonnegative variables:Ax = b, x ≧ 0. The fundamental problem that we discuss is the following: suppose we are given a set ofr linearly independent column vectors ofA, known asthe special column vectors. The problem is to develop an efficient algorithm to determine whether there exists a feasible basis which contains all the special column vectors as basic column vectors and to find such a basis if one exists. Such an algorithm has several applications in the area of mathematical programming. As an illustration, we show that the famous travelling salesman problem can be solved efficiently using this algorithm. Recent published work indicates that this algorithm has applications in integer linear programming. An algorithm for this problem using a set covering approach is described.

Generation of all integer points for given sets of linear inequalities

Abstract: We propose to give a computationally feasible procedure for the generation of all the integer points satisfying a given set of inequalities. Five different systems of inequalities will be considered. In order to generate all of these integer points, one requires a particular set of integer points, called fundamental points, and a set of linearly independent vectors with integer components. The number of these fundamental points is given by a simple formula. We show how to generate the fundamental points and the required vectors. We give an application concerning the localization of the integer optimum of a linear objective function subject to constraints which geometrically define a cone or a parallelotope.

Technical Note—Strengthened Dantzig Cuts for Integer Programming

Abstract: In 1959, Dantzig proposed a particularly simple cut for integer programming. However, in 1963, Gomory and Hoffman showed that, in general, this cut does not provide a finite algorithm. In 1968, Bowman and Nemhauser showed that a slightly modified version of the Dantzig cut does provide a finite procedure. We show how this latter cut can be strengthened through the use of group-theoretic techniques.

Minimal disconnecting sets in directed multi‐commodity networks

Abstract: The problem of finding minimal disconnecting sets for multi-commodity directed networks may be solved using an arc-path formulation and Gomory's all-integer integer programming algorithm. However, the number of network constraints may be astronomical for even moderately sized networks. This paper develops a finite algorithm similar to Gomory's, but requiring no more than m rows in the tableau, where m is the number of arcs in the network.

Mixed integer programming.

Abstract: Introduction Linear programming maximizes (or minimizes) a linear objective function subject to one or more constraints. Mixed integer programming adds one additional condition that at least one of the variables can only take on integer values. The technique finds broad use in operations research. The mathematical representation of the mixed integer programming (MIP) problem is Maximize (or minimize) zz = CCCC subject to AX ≤ b, X ≥ 0, some xi are restricted to integer values. where CC = (xx1,xx2, ... , xxnn)′ CC = (cc1, cc2, ... , ccnn) bb = (bb1,bb2, ... , bbmm)

Management science: a view from nonlinear programming

Abstract: A brief history of integer and continuous nonlinear programming is presented as well as the current obstacles to practical use of these mathematical programming techniques. It is forecast that the useful contributions to nonlinear programming actually made in the next few years are more likely to be consolidations than theoretical breakthroughs. These contributions are likely to be the documentation of standard test problems, construction of user oriented software, and comparisons of currently known algorithms to demonstrate which techniques are best for specific problems.