Showing papers on "Integer programming published in 1975"
TL;DR: In this article, the authors present new formulation techniques for capturing the essential nonlinearities of the problem of interest, while producing a significantly smaller problem size than the standard techniques, which can be used to reduce the number of variables and constraints.
Abstract: A variety of combinatorial problems (e.g., in capital budgeting, scheduling, allocation) can be expressed as a linear integer programming problem. However, the standard devices for doing this often produce an inordinate number of variables and constraints, putting the problem beyond the practical reach of available integer programming methods. This paper presents new formulation techniques for capturing the essential nonlinearities of the problem of interest, while producing a significantly smaller problem size than the standard techniques.
730 citations
TL;DR: An algorithm to detect structure is described and this algorithm identifies sets of variables and the corresponding constraint relationships so that the total number of GUB-type constraints is maximized.
Abstract: Large practical linear and integer programming problems are not always presented in a form which is the most compact representation of the problem. Such problems are likely to posses generalized upper bound(GUB) and related structures which may be exploited by algorithms designed to solve them efficiently. The steps of an algorithm which by repeated application reduces the rows, columns, and bounds in a problem matrix and leads to the freeing of some variables are first presented. The ‘unbounded solution’ and ‘no feasible solution’ conditions may also be detected by this. Computational results of applying this algorithm are presented and discussed. An algorithm to detect structure is then described. This algorithm identifies sets of variables and the corresponding constraint relationships so that the total number of GUB-type constraints is maximized. Comparisons of computational results of applying different heuristics in this algorithm are presented and discussed.
252 citations
TL;DR: This characterization of a cut in a network provides a fundamental equivalence between directed pseudosummetric networks and undirected networks and identifies a class of problems which can be solved as minimum cut problems on a network.
Abstract: : The paper is concerned with an integer programming characterization of a cut in a network. This characterization provides a fundamental equivalence between directed pseudosummetric networks and undirected networks. It also identifies a class of problems which can be solved as minimum cut problems on a network. (Author)
244 citations
TL;DR: An algorithm of this type is developed which is guaranteed to find a schedule satisfying problem constraints, if one exists, and which will accept any of an important class of optimality criteria, not just levelness of reserve.
Abstract: The generator maintenance scheduling problem is formulated as a 0-1 integer linear program. Although previous papers have considered a rigorous integer programming approach intractable, an algorithm of this type is developed which (1) is guaranteed to find a schedule satisfying problem constraints, if one exists; (2) is guaranteed to find the optimal feasible schedules; and (3) will accept any of an important class of optimality criteria, not just levelness of reserve. Particular attention is directed to a new criterion incorporating dollar costs/benefits incurred by delaying or advancing maintenance on a unit.
203 citations
TL;DR: This paper gives a precise definition for sets of vectors, called test sets, which will include those sets described above arising in the simplex and flow algorithms, and proves that any “improvement process” which searches through a test set at each stage converges to an optimal point in a finite number of steps.
Abstract: In this paper we consider the question: how does the flow algorithm and the simplex algorithm work? The usual answer has two parts: first a description of the "improvement process", and second a proof that if no further improvement can be made by this process, an optimal vector has been found. This second part is usually based on duality a technique not available in the case of an arbitrary integer programming problem. We wish to give a general description of "improvement processes" which will include both the simplex and flow algorithms, which will be applicable to arbitrary integer programming problems, and which will in themselves assure convergence to a solution. Geometrically both the simplex algorithm and the flow algorithm may be described as follows. At the i th stage, we have a vertex (or feasible flow) to which is associated a finite set of vectors, namely the set of edges leaving that vertex (or the set of unsaturated paths). The algorithm proceeds by searching among this special set for a vector along which the gain function is increasing. If such a vector is found, the algorithm continues by moving along this vector as far as is possible while still remaining feasible. The search is then repeated at this new feasible point. We give a precise definition for sets of vectors, called test sets, which will include those sets described above arising in the simplex and flow algorithms. We will then prove that any "improvement process" which searches through a test set at each stage converges to an optimal point in a finite nmnber of steps. We also construct specific test sets which are the natural extensions of the test sets employed by the flow algorithm to arbitrary linear and integer linear programming problems.
154 citations
TL;DR: In this article, a link performance function is developed to express the loss incurred by platoons traveling through a signal-controlled intersection us a function of link offset, and the optimization problem is formulated as a mixed-integer linear program and a test network is solved by branch and bound techniques using IBM's MPSX pac...
Abstract: Setting traffic signals in a signal-controlled street network involves the determination of cycle time, splits of green time, and offsets. Part I of this paper considers the network coordination problem, i.e., given a common cycle lime and green splits at each intersection, determine offsets for all signals. Part II considers the more general synchronization problem, i.e., determine simultaneously all the control variables for the network including offsets, splits, and cycle time. In Part I, a link performance function is developed to express the loss incurred by platoons traveling through a signal-controlled intersection us a function of link offset. Integer variables enter the formulation because of the periodicity of the traffic lights: The algebraic sum of the offsets around any closed loop of the network must equal an integral number of cycle limes. The optimization problem is formulated as a mixed-integer linear program and a test network is solved by branch-and-bound techniques using IBM's MPSX pac...
154 citations
TL;DR: Various uses, including those in group theory and in other integer programming algorithms, as well as applications from the literature, are discussed, and Dynamic programming, branch and bound, search enumeration, heuristic methods, and other solution techniques are presented.
Abstract: A unifying survey of the literature related to the knapsack problem; that is, maximize , subject to and xi ⩾ 0, integer; where vi, wi and W are known integers, and wi (i = 1, 2, …, N) and W are positive. Various uses, including those in group theory and in other integer programming algorithms, as well as applications from the literature, are discussed. Dynamic programming, branch and bound, search enumeration, heuristic methods, and other solution techniques are presented. Computational experience, and extensions of the knapsack problem, such as to the multi-dimensional case, are also considered.
143 citations
01 Jan 1975
TL;DR: A general class of discrete optimization problem is given for which dual problems of this type may be derived and the use of dual problems for obtaining strong bounds, feasible solutions, and for guiding the search in enumeration schemes for this class of problems is discussed.
Abstract: Meaningful dual problems have recently been identified for the integer programming problem, the resource constrained network scheduling problem and the traveling salesman problem. In this paper a general class of discrete optimization problem is given for which dual problems of this type may be derived. We discuss the use of dual problems for obtaining strong bounds, feasible solutions, and for guiding the search in enumeration schemes for this class of problems. Properties of dual problems and three algorithms are discussed, a primal-dual ascent algorithm, a simplicial approximation algorithm and an algorithm based on the relaxation method for solving systems of inequalities. Finally, computational experience is given for integer programming and resource constrained network scheduling dual problems.
119 citations
TL;DR: Using queuing theory and integer linear programming, a method for scheduling patrol cars so that specified service standards are met at each hour of the day is presented.
Abstract: In any city the arrival rate of calls for police patrol-car service varies considerably through the day. Using queuing theory and integer linear programming, we present a method for scheduling patrol cars so that specified service standards are met at each hour of the day. Two models are used. The first is an M/M/n queuing model with time-dependent parameters that is solved numerically. The second is an integer linear program in which the decision variables are the number of patrol cars working each tour and the times at which they go out of service for meals. The program's constraints are determined by the output of the queuing model. Use of the method with data from the New York City Police Department indicates that it can lead to substantial improvements in police service.
115 citations
TL;DR: The effectiveness of the method is shown with a 15-variable problem, which requires about 1 day's FORTRAN programming effort and 8 seconds of computer time for its solution on an IBM 370/165 digital computer.
Abstract: This paper presents a useful procedure of solving nonlinear integer programming problems. It finds, first, a pseudo-solution to the problem, as if the variables were continuous. Then it uses direct search in the neighbourhood of the pseudo-solution to find the optimum. The effectiveness of the method is shown with a 15-variable problem, which requires about 1 day's FORTRAN programming effort and 8 seconds of computer time for its solution on an IBM 370/165 digital computer.
TL;DR: In this paper, a constructive characterization of adjacency relations between integer vertices of the feasible set that enables us to generate edges (all, if necessary) connecting a given integer vertex to adjacent integer vertex is given.
Abstract: In an earlier paper [Opns Res 20 1153–1161 (1972)] we 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 this paper we give a constructive characterization of adjacency relations between integer vertices of the feasible set that enables us 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 we discuss two
TL;DR: In this paper, the authors formulate a rigorous definition of a mixed-integer model of a given function and study the properties of the functions that can be modelled by such a model.
Abstract: It is well known that mixed-integer formulations can be used tomodel important classes of nonconvex functions, such as fixed-charge functions and linear economy-of-scale cost functions. The purpose of this paper is to formulate a rigorous definition of a mixed-integer model of a given function and to study the properties of the functions that can be so modelled. An interesting byproduct of this approach is the identification of a simple class of functions that cannot be modelled by computer-representable mixed-integer formulations, even though mixed-integer models based on the use of a single arbitrary irrational constant are available for this class.
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TL;DR: In this paper, a family of integer programs whose right-hand-sides lie on a given line segment is considered, called a parametric integer program (PIP), and a simple generalization of the conventional branch-and-bound approach to integer programming is presented.
Abstract: A family of integer programs is considered whose right-hand-sides lie on a given line segment L. This family is called a parametric integer program (PIP). Solving a (PIP) means finding an optimal solution for every program in the family. It is shown how a simple generalization of the conventional branch-and-bound approach to integer programming makes it possible to solve such a (PIP). The usual bounding test is extended from a comparison of two point values to a comparison of two functions defined on the line segment L. The method is illustrated on a small example and computational results for some larger problems are reported.
TL;DR: It has been shown by G. Roodman that useful postoptimization capabilities for the 0-1 integer programming problem can be obtained from an implicit enumeration algorithm modified to classify and collect all fathomed partial solutions.
Abstract: : It has been shown by G. Roodman that useful postoptimization capabilities for the 0-1 integer programming problem can be obtained from an implicit enumeration algorithm modified to classify and collect all fathomed partial solutions. The paper extends the approach as follows: (1) Improved parameter ranging formulae are obtained by higher resolution classification criteria. (2) Parameters may be changed so as to tighten the original problem, in adddition to relaxing it. (3) An efficient storage structure is presented to cope with difficult data collection task implicit in this approach. (4) Finally, computer implementation is facilitated by the elaboration of a unified set of algorithms. (Author)
TL;DR: In this article, the NIP problem is reformulated into a 0-1 linear programming (ZOLP) problem and a one-to-one correspondence is shown between this NIP and the ZOLP problem.
Abstract: Mathematical models for reliability of a redundant system with two classes of failure modes are usually formulated as a nonlinear integer programming (NIP) problem. This paper reformulates the NIP problem into a 0-1 linear programming (ZOLP) problem and a one-to-one correspondence is shown between this NIP problem and the ZOLP problem. A NIP example treated by Tillman is formulated into a ZOLP problem and optimal solutions, identical to Tillman's are obtained by an implicit enumeration method. Calculating the new coefficients of the objective function and the constraints in the ZOLP are straight forward. There are not many constraints or variables in the proposed ZOLP. Consequently, the computation (CPU) time is less.
TL;DR: Polyhedral annexation is a new approach for generating all valid inequalities in mixed integer and combinatorial programming, including the facets of the convex hull of feasible integer solutions, without resorting to specialized notions of group theory, convex analysis or projective geometry.
Abstract: Polyhedral annexation is a new approach for generating all valid inequalities in mixed integer and combinatorial programming. These include the facets of the convex hull of feasible integer solutions. The approach is capable of exploiting the characteristics of the feasible solution space in regions both “adjacent to” and “distant from” the linear programming vertex without resorting to specialized notions of group theory, convex analysis or projective geometry. The approach also provides new ways for exploiting the “branching inequalities” of branch and bound.
TL;DR: In this paper, the authors consider the relevance of various planning methods and decision criteria to multiobjective investment planning under uncertainty and demonstrate how the technique of dynamic programming can be extended to take account of multiple objectives and use dynamic programming as a framework in which they analyze the robustness of an initial decision in the face of various types of uncertainty.
Abstract: In this paper we consider the relevance of various planning methods and decision criteria to multiobjective investment planning under uncertainty. Assuming that a natural reaction to uncertainty is to operate so as to leave open as many good options as possible (as opposed to maximizing subjective expected utility) we argue that the planning process should concentrate on analyzing the effects of the initial decision, and that for this exercise the classical methods of mixed integer programming are inappropriate. We demonstrate how the technique of dynamic programming can be extended to take account of multiple objectives and use dynamic programming as a framework in which we analyze the robustness of an initial decision in the face of various types of uncertainty. In so doing we also analyze the risks involved in both the planning and decision making functions.
TL;DR: In this article, it was shown that for problems involving n integer variables, the examination of at most n + 2 surrogates associated to any pair of constraints in the problem produces all the conclusions derivable from any linear combination of the given constraint pair.
Abstract: Single linear constraints can be used in a straightforward way for deriving bounds on the variables of discrete optimization problems. More powerful conclusions can be obtained from the examination of all the surrogate constraints associated to pairs of constraints. The paper shows that for problems involving n integer variables, the examination of at most n + 2 surrogates (constructively) associated to any pair of constraints in the problem produces all the conclusions derivable from any linear combination of the given constraint pair. The bounds obtained are shown to be the same as those obtained by maximizing or minimizing the individual variables under the pair of constraints. If m denotes the number of constraints in the problem, then 0(m2n2) elementary operations will be performed for testing all constraint-pairs of the problem in order to improve the bounds on the variables and/or to derive binary relations among them.
01 Aug 1975
TL;DR: In this paper, the authors present various vehicle routing problems and provide a unified framework for these very difficult combinatorial programming problems, in conjunction with several widely-used heuristic solution techniques.
Abstract: : An essential element of the newspaper logistics system is the allocation and routing of vehicles for the purpose of delivering newspapers on a daily basis. In this paper, the author presents various vehicle routing problems. Formulations defining the mathematical models are discussed in conjunction with several widely-used heuristic solution techniques. The focus is on providing a unified framework for these very difficult combinatorial programming problems.
TL;DR: Results focus on easily specified coefficient conditions rather than on the use of bounds that must be calculated “externally” in order to produce the desired aggregation.
Abstract: Methods are given for replacing a system of equations in nonnegative integer variables by a single equation with the same solution set. These results focus on easily specified coefficient conditions rather than on the use of bounds that must be calculated “externally” in order to produce the desired aggregation.
TL;DR: This paper reports an empirical discovery in integer programming, where a version of the branch-and-bound approach is used as a control and tested against the same algorithm augmented by the use of Hillier's linear search performed at every node of the search tree.
Abstract: This paper reports an empirical discovery in integer programming. A version of the branch-and-bound approach is used as a control and tested against the same algorithm augmented by the use of Hillier's linear search performed at every node of the search tree. It is shown that the imbedded linear search locates solutions within 1%, and solutions within 5% of the theoretical optimum, which in fact can be seen to have this proximity to the theoretical optimum at the time of termination of computation, over ten times faster than the control.
TL;DR: In this article, an integer programming formulation is developed having identical structure to a warehouse (factory) location problem which has been the focus of a number of investigations, and a model for allocation of inspection resources to points within a discrete manufacturing process at minimum cost.
Abstract: This note presents a model for allocation of inspection resources to points within a discrete manufacturing process at minimum cost. An integer programming formulation is developed having identical structure to a warehouse (factory) location problem which has been the focus of a number of investigations.
TL;DR: An exact method for solving all-integer non-linear programming problems with a separable nondecreasing objective function is presented in this article, which is used to efficiently search candidate hypersurfaces for the optimal feasible integer solution.
Abstract: An exact method for solving all-integer non-linear programming problems with a separable non-decreasing objective function is presented. Dynamic programming methodology is used to efficiently search candidate hypersurfaces for the optimal feasible integer solution. An efficient computational and storage scheme exists and initial calculations give very promising results.
01 Jan 1975
TL;DR: In this article, the authors present a foundation of linear programming with emphasis on the topics that contribute to the development of integer programming theory, including the topics including the Benders approach for partitioning programming problems having two different types of variables into two sub-problems, each having homogeneous variables.
Abstract: Publisher Summary
This chapter presents a foundation of linear programming with emphasis on the topics that contribute to the development of integer programming theory. It also presents the topics including the Benders approach for partitioning programming problems having two different types of variables into two sub-problems, each having homogeneous variables. The Benders approach is useful in solving mixed integer problems. The chapter discusses simplex method, revised simplex method, dual problem, and bounded variables. The selective search of the simplex method, among all the basic solutions, depends on two fundamental conditions: (1) the feasibility condition, given an initial basic feasible solution to the problem, guarantees generating new basic solutions that are always feasible. This is accomplished by replacing one basic vector by an associated non-basic vector. (2) The optimality condition ensures that the non-basic vector that is selected by the feasibility condition would improve the objective value or otherwise indicate that no further improvements could be achieved by generating new basic solutions.
TL;DR: There are three main reasons why a purely linear programming model may not represent a constrained optimization problem adequately: economies of scale, other non-linearities that do not invalidate a local optimum, random data, and integer programming, non- linear programming and stochastic programming.
Abstract: There are three main reasons why a purely linear programming model may not represent a constrained optimization problem adequately: economies of scale, other non-linearities that do not invalidate a local optimum, random data. These reasons lead respectively to integer programming, non-linear programming and stochastic programming. Examples of each type of model are discussed. These have all been solved using a standard mathematical programming system to exploit sparseness efficiently. Economies of scale arise when selecting a set of new pipelines to expand the capacity of a given network. This problem involves non-linear functions, but is essentially an integer programming problem because we must use branch and bound methods to find the best combinations of pipelines, and pipeline diameters. An unsuccessful and a subsequent successful formulation for this problem are discussed. A non-linear programming model for allocating resources in health care is outlined. A model for multi-time-period production scheduling with stochastic demands is also outlined. The model requires data defining the uncertainties in demand forecasts, and the extent to which these are correlated with each other and with past sales. The existence of software for this model may encourage more people to quantify these data.
TL;DR: Criteria for selecting a new LP basis for which the associated relaxation is stronger is state, which may be successively applied to obtain either an optimal IP solution or a lower bound on the cost of such a solution.
Abstract: Consider the relaxation of an integer programming (IP) problem in which the feasible region is replaced by the intersection of the linear programming (LP) feasible region and the corner polyhedron for a particular LP basis. Recently a primal-dual ascent algorithm has been given for solving this relaxation. Given an optimal solution of this relaxation, we state criteria for selecting a new LP basis for which the associated relaxation is stronger. These criteria may be successively applied to obtain either an optimal IP solution or a lower bound on the cost of such a solution. Conditions are given for equality of the convex hull of feasible IP solutions and the intersection of all corner polyhedra.
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TL;DR: In this article, a family of integer programs whose right-hand-sides lie on a given line segment is considered, called a parametric integer program (PIP), and a simple generalization of the conventional branch-and-bound approach to integer programming is presented.
Abstract: A family of integer programs is considered whose right-hand-sides lie on a given line segment L. This family is called a parametric integer program (PIP). Solving a (PIP) means finding an optimal solution for every program in the family. It is shown how a simple generalization of the conventional branch-and-bound approach to integer programming makes it possible to solve such a (PIP). The usual bounding test is extended from a comparison of two point values to a comparison of two functions defined on the line segment L. The method is illustrated on a small example and computational results for some larger problems are reported.
TL;DR: This paper presents a new implicit enumeration algorithm for solving the pure integer linear programming problem and includes a scanning procedure and a method for identifying constraints that become redundant during the course of the algorithm.
Abstract: This paper presents a new implicit enumeration algorithm for solving the pure integer linear programming problem. The theory of equivalent integer programming problems is first used to reformulate the problem. A technique originally used with particular success in the bound-and-scan algorithm to deal with only a subset of the variables is extended to all of the variables in the restructured problem. In addition to the resulting basic enumeration scheme, the algorithm includes a scanning procedure and a method for identifying constraints that become redundant during the course of the algorithm. Computational experience on standard test problems is reported.
TL;DR: A problem that is equivalent to the generalized lattice point problem of nonconvex programming is considered, which can be expressed as the maximization of a certain convex function over a convex polyhedral set.
Abstract: In this paper we consider a problem that is equivalent to the generalized lattice point problem of nonconvex programming. This problem can be expressed as the maximization of a certain convex function over a convex polyhedral set. This formulation can be used with the cutting plane algorithm of Glover and Klingman sometimes to strengthen the cuts their method generates for the generalized lattice point problem. A cut for zero-one integer programming can also be derived.