Showing papers on "Integer programming published in 1979"

Continuous equilibrium network design models

Abstract: It is known that the network design problem with the assumption of user optimal flows can be modeled as a 0–1 mixed integer programming problem. Instead, we formulate the network design problem with continuous investment variables subject to equilibrium assignment as a nonlinear optimization problem. We show that this optimization problem is equivalent to an unconstrained problem which we solve by direct search techniques. For convex investment cost functions, the performance of both Powell's method and the method of Hooke and Jeeves is approximately the same with respect to computational requirements for a 24 node, 76 arc network. For the case of concave investment functions, Hooke and Jeeves was superior. The solution to the concave continuous model was very similar to that of the 0–1 model. Furthermore, the required solution time was far less than that required by the corresponding discrete model of the same network. The advantages and disadvantages of the continuous approach as well as the computational requirements are discussed.

A Survey of Lagrangian Techniques for Discrete Optimization.

Abstract: Publisher Summary This chapter proposes Lagrangean techniques for discrete optimization problems A simple method for trying to solve zero–one integer programming (IP) problems is discussed This method is used as a starting point for discussing many of the developments since then The behavior of Lagrangean techniques in analyzing and solving zero–one IP problems is typical of their use on other discrete optimization problems The chapter discusses a number of questions about this method and its relevance for optimizing the original IP problem The goal of Lagrangean techniques is to try to establish sufficient optimality conditions: Lagrangean techniques are useful in computing zero–one solutions to IP problems with soft constraints or in parametric analysis of an IP problem over a family of right hand sides Parametric analysis of discrete optimization problems is also discussed The use of Lagrangean techniques as a distinct approach to discrete optimization has proven theoretically and computationally important for three reasons First, dual problems derived from more complex discrete optimization problems can be represented as linear programming (LP) problems, but ones of immense size, which cannot be explicitly constructed and then solved by the simplex algorithm Second, reason for considering the application of Lagrangean techniques to dual problems, in addition to the simplex algorithm, is that the simplex algorithm is exact and the dual problems are relaxation approximations Lagrangean techniques as a distinct approach to discrete optimization problems emphasize the need they satisfy for exploiting special structures, which arise in various models

A Dynamic Programming Algorithm for Decision CPM Networks

Abstract: This paper proposes a dynamic programming algorithm for decision CPM (DCPM) networks. DCPM is a natural, powerful, and general way of handling the discrete-time/cost-tradeoff problem. Solution approaches developed to date have not been efficient enough to handle realistically sized problems. The main approaches have been general integer programming algorithms and the specialized branch-and-bound methods for DCPM of Crowston and Wagner. Both of these approaches have many inherent shortcomings solution times grow exponentially with the number of decision nodes, storage requirements quickly become excessive, pre-processing or decomposition of the problem must be undertaken before the algorithms themselves can be called upon to solve the problem, and large variations in solution times have been found based on differences in the structure of the problem. The algorithm presented here overcomes all of these shortcomings. Most significantly, it exhibits only a linear growth in the solution times based on the numb...

Theory and algorithms for linear multiple objective programs with zero–one variables

Abstract: A new algorithm and theoretical results are presented for linear multiple objective programs with zero–one variables. A procedure to identify strong and weak efficient points as well as an extension of the main problem are analyzed. Extensive computational results are given and several topics for further research are discussed.

Branch and Bound Methods for Mathematical Programming Systems

Abstract: Branch and Bound algorithms have been incorporated in many mathematical programming systems, enabling them to solve large nonconvex programming problems. These are usually formulated as linear programming problems with some variables being required to take integer values. But it is sometimes better to formulate problems in terms of Special Ordered Sets of variables of which either only one, or else only an adjacent pair, may take nonzero values. Algorithms for both types of formulation are reviewed. And a new technique, known as Linked Ordered Sets, is introduced to handle sums and products of functions of nonlinear variables in either the coefficients or the right hand sides of an otherwise linear, or integer, programming problem.

Some relationships between lagrangian and surrogate duality in integer programming

Abstract: Lagrangian dual approaches have been employed successfully in a number of integer programming situations to provide bounds for branch-and-bound procedures. This paper investigates some relationship between bounds obtained from lagrangian duals and those derived from the lesser known, but theoretically more powerful surrogate duals. A generalization of Geoffrion's integrality property, some complementary slackness relationships between optimal solutions, and some empirical results are presented and used to argue for the relative value of surrogate duals in integer programming. These and other results are then shown to lead naturally to a two-phase algorithm which optimizes first the computationally easier lagrangian dual and then the surrogate dual.

Computer Codes for Problems of Integer Programming

Abstract: Publisher Summary This chapter presents computer codes for the problems of integer programming. The term “integer programming” covers a wide spectrum of models, which can be characterized by mixed integer programming (MIP) at one end and combinatorial programming at the other end. The interest of those working in commercial organizations is currently focused at the MIP end of the spectrum—indeed on problems, which are basically large linear programming (LP) systems with relatively few integer variables. The chapter presents a “consumer research” report on the different products and also the methods for solving pure integer problems—frequently with special combinatorial structures. Thus, in a consumer report, one has to bear in mind, which consumers are intended for each code. The code should be capable of obtaining a guaranteed optimum solution. A large and complex problem may not be capable of yielding an optimum integer solution within feasible cost and time limits on any code so that the user has in fact to be content with a significant solution obtained by heuristic methods.

Parametric integer programming

Abstract: : A parametric integer linear program (PILP) may be defined as a family of closely related integer linear programs (ILP). Within this definition the author incorporates not only continuous scalar parameterizations but also finite parameterizations. These may include an ILP with a finite number of objective functions or right hand sides or constraint matrices or any combination of these. A general framework for PILP is presented. It begins by outlining the need for PILP algorithms. Basic solution methodologies are explained and two rudimentary approaches for the PILP are stated. Theoretical properties for special parameterizations are proved, and techniques for improving algorithmic efficiency are discussed. The framework concludes with an examination of underlying factors which intimately relate to the scheduling of solution priorities in a PILP algorithm.

Improved Computer-Based Planning Techniques. Part II

Abstract: This is Part II of a two-part series. Part I showed how pine and generalized network models, and advances in methods of solving them have resulted in dramatic cost savings for OR/MS practitioners. This paper focuses on network related formulation NETFORM models, which encompass an even wider variety of applications. We show how NETFORM has enabled the efficient solution of problems in scheduling, production, distribution, and other areas that were too large or difficult lo be handled by previously applied techniques, including mixed integer programming.

Crew Planning at Flying Tiger: A Successful Application of Integer Programming

Abstract: This paper presents a successful application of integer programming to the scheduling of flight crews for a cargo airline. The crew planning process is discussed, the role of the set partitioning model is explained, and representative computational experience is reported. The success of this application is shown to rest upon improved problem conceptualization and decomposition rather than on any advances in solution techniques.

Integer Programming Post-Optimal Analysis with Cutting Planes

Abstract: Sufficient conditions have been developed for testing the optimality of solutions to all-integer and mixed-integer linear programming problems after coefficient changes in the right hand side and the objective function, or after introduction of new variables. The same conditions can be used as necessary conditions for coefficient changes to alter an optimal solution. The tests are based on cutting-plane theory, and the application of the tests requires solution of the original integer problem with a cutting-plane algorithm.

A Survey of Multiple Criteria Integer Programming Methods

Abstract: Several methods have been proposed for solving multiple criteria problems involving integer programming. This paper contains a brief survey as well as a typology of several such methods. Although computational date is scanty to date, an attempt is made to evaluate the methods from a user orientation as well as from the perspective of a researcher trying to develop a workable user-oriented method.

Graph Theory and Integer Programming

Abstract: Publisher Summary A very large part of combinatorics deals or can be formulated as to deal with optimization problems in discrete structures. Generally, the constraints and the objective function are linear forms of certain variables that are restricted to integers or, mostly, to 0 and 1. Thus, the combinatorial problem is translated to a linear integer-programming problem. The value of such a translation depends on whether it provides new insight or new methods for the solution. This chapter discusses several most important results in integer programming that have been successfully applied to graph theory and then discusses those fields of graph theory where an integer-programming approach has been most effective. The chapter also discusses many graph theoretical results that have a linear programming flavor but no explicit treatment.

Binding Constraints and Helly Numbers

Abstract: Summary Starting with axioms for an abstract intersectional system, we define the Helly number., Scarf number, and binding constraint number of such a system. The last concept is based on a definition of a mathematical programming problem in the system. From these definitions, we deduce (1) Bell's theorem that a collection of half spaces contains a point of sn if the intersection of every subset of 2n of the half spaces does, and (2) Scarf's theorem that an integer programming problem on zn has at most 2n— 1 binding constraints. Our arguments use coordinates only at the last moment.

A Method of Decomposition for Integer Programs

Abstract: A method of decomposing integer programs with block angular structure is presented. It is based on the notion of searching for the optimal solution to an integer program among the near-optimal solutions to its Lagrangian relaxation. An optimality theorem is obtained and a generic decomposition algorithm is presented. An application of this approach is discussed and some computational results are reported.

Controlled Experimental Design for Statistical Comparison of Integer Programming Algorithms

Abstract: Testing and comparison of integer programming algorithms is an integral part of the algorithm development process. When test problems are randomly generated, the techniques of statistical experimental design can provide a basis around which to structure computational experiments. This paper formulates the problem of constructing and analyzing controlled integer programming tests in the experimental design context and develops approaches to dealing with a number of issues that arise. Both analytic results and empirical evidence from a large experiment are employed in deriving the suggested techniques.

Implementation of Large-Scale Financial Planning Models: Solution Efficient Transformations

Abstract: It has long been recognized in the literature of finance that the robustness and analytical potential of mathematical programming procedures can be utilized to structure highly complex decision environments and to ascertain quickly and efficiently the dominant set(s) of actions for achieving an explicit objective(s). Although some formulations involve nonlinear relationships (for instance [13] [15]), the vast majority of the models appearing in the finance literature are variants of linear programming, including such identifiable methodologies as linear programming, goal programming, networks, integer programming, mixed integer programming, and chance-constrained programming. The decision processes for capital budgeting ([25] [1] [2] [4] [14] [16] [24]), working capital management ([20] [18] [21] [6]), cash management ([17] [23]), and portfolio selection ([22] [24]), have been structured as linear programs and have contributed significantly to understanding the dynamics of financial systems. Given the potential of these mathematical approaches, the limited industrial use of financial optimization models is disturbing.

On the Group Problem and a Subadditive Approach to Integer Programming

Abstract: The study of Gomory's group problem has led to a subadditive approach to integer programming. In this paper, we trace that development using the cyclic group problem and knapsack problem as prototypes. The asymptotic theorem of Gomory is also discussed. Finally, an algorithm giving a constructive proof of a subadditive dual problem for the knapsack problem is presented.

Cutting Stock, Linear Programming, Knapsacking, Dynamic Programming and Integer Programming, Some Interconnections

Abstract: Cutting stock problems have many connections with Linear, Dynamic and Integer Programming, many of the connections being through the Knapsack Problem This paper relates some of the connections arising from the one dimensional cutting stock Little background in mathematical programming is presumed

Seasonal Production and Sales Planning with Limited Shared Tooling at the Key Operation.

Abstract: : Lagrangian relaxation is applied to a class of very large mixed integer linear programming problems representing seasonal production and sales planning in a situation where limited tooling is available at the key production operation A successful application to the injection molding industry is described (Author)

A o( n log n ) algorithm for LP knapsacks with GUB constraints

Abstract: A specialization of the dual simplex method is developed for solving the linear programming (LP) knapsack problem subject to generalized upper bound (GUB) constraints. The LP/GUB knapsack problem is of interest both for solving more general LP problems by the dual simplex method, and for applying surrogate constraint strategies to the solution of 0–1 “Multiple Choice” integer programming problems. We provide computational bounds for our method of o(n logn), wheren is the total number of problem variables. These bounds reduce the previous best estimate of the order of complexity of the LP/GUB knapsack problem (due to Witzgall) and provide connections to computational bounds for the ordinary knapsack problem. We further provide a variant of our method which has only slightly inferior worst case bounds, yet which is ideally suited to solving integer multiple choice problems due to its ability to post-optimize while retaining variables otherwise weeded out by convex dominance criteria.

An Optimal Algorithm for Sales Representative Time Management

Abstract: This paper addresses the time management problem confronted by sales representatives. The sales representative planning his itinerary must decide the best way to ration time among the accounts comprising his territory. The time management problem is formulated as an integer program whereby each admissible call frequency for each account is represented by a zero-one decision variable. A branch-and-bound integer programming algorithm for this problem is presented. The algorithm is unique in that two integer programming formulations of the problem are used simultaneously in the search procedure and an approximation-cum-relaxation is evaluated at each branch in the search. Computational testing of the algorithm shows that it can solve many realistic time management problems optimally in fractions of a second.

An application and case history of a dynamic R&D portfolio selection model

Abstract: Presents an application of stochastic integer-programming formulation to a portfolio of projects (each of which were planned with the aid of a decision-tree structure) Follow-up studies undertaken one year later are described in an attempt to assess the accuracy of the data and adequacy of the model in practice

Enumerative Methods in Integer Programming

Abstract: Publisher Summary This chapter presents enumerative methods in integer programming. The problem areas that require enumeration are discussed. The chapter focuses on integer vector as an entity (partial solution or state) and using auxiliary and logical inequalities. Associated inequalities derivable from them and from the solution procedure are discussed. Especially for mixed integer problems it is often the associated inequalities, which will determine the enumeration process. A form of logical inequalities, in turn derivable from the initial inequality system and/or the associated inequality system is discussed. A proper utilization of the logical inequalities is the best way for replacing a host of (in) feasibility tests, which are proposed for enumeration. Logical inequalities are the proper tools for a rational direction of the enumerative process. Enumerative techniques are usually confined to 0–1 problems. Alternatively, integer problems can be reduced to 0–1 problems by an expansion of the integer variables into polynomials involving binary variables only, but no extensive testing of such options appears to take place.

New integer programming scheme for nonrecursive digital filter design

Abstract: A new mixed-integer linear programming objective function for optimising an f.i.r. linear-phase digital filter is presented. In comparison with the conventional objective function, the new one has the advantages of reducing the number of delays and/or the coefficient wordlength.

A Study of Alternative Relaxation Approaches for a Manpower Planning Problem.

Abstract: This paper examines a variety of relaxation strategies for zero-one integer programming problems, containing from 54 to 2,683 variables, that arise in manpower planning applications. These strategies are compared by a primal criterion, which emphasizes the ability to obtain high quality feasible solutions. This contrasts with the usual dual criterion for comparing relaxations, which emphasizes objective function bounds obtained from solutions that are generally not feasible. The changed emphasis requires a change in the use of relaxations, which may be viewed from the standpoint of generating trial solutions for heuristic programming or as a fundamental component of branch and bound. Computer tests show that a combined surrogate-Lagrangean strategy is the most effective for the problems examined followed by a pure surrogate relaxation strategy. All other approaches, including generalized Lagrangean relaxation, fared substantially worse, particularly in terms of solution quality.