A branch and bound algorithm for the two-level uncapacitated facility location problem with some side constraints

This paper deals with the problem of locating at minimum total costs both plants and warehouses in a two-level distribution system where commodities are delivered from plants to customers either directly or indirectly via warehouses. Some side-constraints are imposed on the problem, which represent the adjunct relationship of some warehouses to a certain plant. Proposed is an efficient branch and bound solution procedure, employing a set of new devices for lower bounds and simplifications which are obtained by exploiting the submodularity of the objective function and the special structure of the side-constraints. Computational experiments with fifteen test problems are provided.     

A dual-based algorithm for solving lexicographic multiple objective programs

In this paper, we propose an algorithm for solving lexicographic multiple objective programs based upon duality theorem. In the existing algorithm, we should solve several linear programming problems (LPPs); therefore if, in particular, there are several objective functions, this method is not worthwhile from the viewpoint of computation. But in our new algorithm we just solve one LPP.     

A branch-bound algorithm for a wiring problem

A simple computational method for solving a specific wiring problem related to the construction of the RC 4000 computer is described. The intimate relationship between the wiring problem and the traveling salesman problem is established, and the algorithm is based upon the branch and bound technique as employed by J.D.C. Little et al. [1] for solving the latter problem.Adapted from the thesis Fixed-cost and Other Network Flow Problems presented to The Institute of Mathematical Statistics and Operations Research, The Technical University of Denmark in partial fulfilment of the requirements for the licentiate degree, April 1967.     

A polynomial algorithm for the two-connections variant of the tree p-median problem

We consider the variant of the tree $p$-median problem where each node must be connected to the two closest centers. This problem is polynomially solved through a dynamic programming formulation that extends the solution given by A. Tamir for the classical$p$-median problem on a tree.     

A lift-and-project cutting plane algorithm for mixed 0–1 programs

We propose a cutting plane algorithm for mixed 0–1 programs based on a family of polyhedra which strengthen the usual LP relaxation. We show how to generate a facet of a polyhedron in this family which is most violated by the current fractional point. This cut is found through the solution of a linear program that has about twice the size of the usual LP relaxation. A lifting step is used to reduce the size of the LP's needed to generate the cuts. An additional strengthening step suggested by Balas and Jeroslow is then applied. We report our computational experience with a preliminary version of the algorithm. This approach is related to the work of Balas on disjunctive programming, the matrix cone relaxations of Lovász and Schrijver and the hierarchy of relaxations of Sherali and Adams.The research underlying this report was supported by National Science Foundation Grant #DDM-8901495 and Office of Naval Research Contract N00014-85-K-0198.     

A tree search algorithm for the multi-commodity location problem

The multi-commodity location problem is an extension of the simple plant location problem. The problem is to decide on locations of facilities to meet customer demands for several commodities in such a way that total fixed plus variable costs are minimized. Only one commodity may be supplied from any location.In this paper a primal and a dual heuristic for producing good bounds are presented. A method of improving these bounds by using a new Lagrangean relaxation for the problem is also presented. Computational results with problems taken from the literature are provided.     

Open questions concerning Weiszfeld's algorithm for the Fermat-Weber location problem

The Fermat—Weber location problem is to find a point in ⁿ that minimizes the sum of the weighted Euclidean distances fromm given points in ⁿ . A popular iterative solution method for this problem was first introduced by Weiszfeld in 1937. In 1973 Kuhn claimed that if them given points are not collinear then for all but a denumerable number of starting points the sequence of iterates generated by Weiszfeld's scheme converges to the unique optimal solution. We demonstrate that Kuhn's convergence theorem is not always correct. We then conjecture that if this algorithm is initiated at the affine subspace spanned by them given points, the convergence is ensured for all but a denumerable number of starting points.     

Comparison of two algorithms for solving a two‐stage bilinear stochastic programming problem with quantile criterion

The paper is devoted to solving the two‐stage problem of stochastic programming with quantile criterion. It is assumed that the loss function is bilinear in random parameters and strategies, and the random vector has a normal distribution. Two algorithms are suggested to solve the problem, and they are compared. The first algorithm is based on the reduction of the original stochastic problem to a mixed integer linear programming problem. The second algorithm is based on the reduction of the problem to a sequence of convex programming problems. Performance characteristics of both the algorithms are illustrated by an example. A modification of both the algorithms is suggested to reduce the computing time. The new algorithm uses the solution obtained by the second algorithm as a starting point for the first algorithm. Copyright © 2015 John Wiley & Sons, Ltd.     

NP-Completeness of the linear complementarity problem

We consider the linear complementarity problem (q, M) for which the data are the integer column vectorq εR ⁿ and the integer square matrixM of ordern. GLCP is the decision problem: Does (q, M) have a solution? We show that GLCP is NP-complete in the strong sense.     

A graph theoretical bound for the p-median problem

The p-median problem is a minisium network location problem that seeks to find the optimal location of p centres in a network. In the present paper a graph-theoretical bound is developed for the problem. This bound is based on shortest spanning trees and arborescences and other graphical properties of the problem. It is shown that the graph-theoretical bound dominates a bound based on shortest distances.     

A polynomial time algorithm for solving the fermat-weber location problem with mixed norms

This paper discusses the Fermat-Weber location problem, manages to apply the ellipsoid method to this problem and proves the ellipsoid method can be terminated at an approximately optimal location in polynomial time, verifies the ellipsoid method is robust for the lower dimensional location problem     

An inexact NE/SQP method for solving the nonlinear complementarity problem

In this paper, we present an extension to the NE/SQP method; the latter is a robust algorithm that we proposed for solving the nonlinear complementarity problem in an earlier article. In this extended version of NE/SQP, instead of exactly solving the quadratic program subproblems, approximate solutions are generated via an inexact rule.Under a proper choice for this rule, this inexact method is shown to inherit the same convergence properties of the original NE/SQP method. In addition to developing the convergence theory for the inexact method, we also present numerical results of the algorithm tested on two problems of varying size.     

A global optimization approach for solving the convex multiplicative programming problem

We consider a convex multiplicative programming problem of the form% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9qq-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaGG7bGaam% OzamaaBaaaleaacaaIXaaabeaakiaacIcacaWG4bGaaiykaiabgwSi% xlaadAgadaWgaaWcbaGaaGOmaaqabaGccaGGOaGaamiEaiaacMcaca% GG6aGaamiEaiabgIGiolaadIfacaGG9baaaa!4A08!\[\{ f_1 (x) \cdot f_2 (x):x \in X\} \]where X is a compact convex set of ⁿ and f ₁, f ₂ are convex functions which have nonnegative values over X.Using two additional variables we transform this problem into a problem with a special structure in which the objective function depends only on two of the (n+2) variables. Following a decomposition concept in global optimization we then reduce this problem to a master problem of minimizing a quasi-concave function over a convex set in ² ₂. This master problem can be solved by an outer approximation method which requires performing a sequence of simplex tableau pivoting operations. The proposed algorithm is finite when the functions f _i, (i=1, 2) are affine-linear and X is a polytope and it is convergent for the general convex case.Partly supported by the Deutsche Forschungsgemeinschaft Project CONMIN.     

A new linearization technique for multi-quadratic 0-1 programming problems

We consider the reduction of multi-quadratic 0-1 programming problems to linear mixed 0-1 programming problems. In this reduction, the number of additional continuous variables is O(kn) (n is the number of initial 0-1 variables and k is the number of quadratic constraints). The number of 0-1 variables remains the same.     

A new algorithm for solving the general quadratic programming problem

For the general quadratic programming problem (including an equivalent form of the linear complementarity problem) a new solution method of branch and bound type is proposed. The branching procedure uses a well-known simplicial subdivision and the bound estimation is performed by solving certain linear programs.     

A comparison of heuristics and relaxations for the capacitated plant location problem

Approaches proposed in the literature for the Capacitated Plant Location Problem are compared. The comparison is based on new theoretical and computational results. The main emphasis is on relaxations. In particular, dominance relations among the various relaxations found in the literature are identified. In the computational study, the relaxations are compared as a function of various characteristics of the test problems. Several of these relaxations can be used to generate heuristic feasible solutions that are better than the classical greedy or interchange heuristics, both in computing time and in the quality of the solutions found.     

On solving the multi-period single-sourcing problem under uncertainty

We present a framework for solving the strategic problem of assigning retailers to facilities in a multi-period single-sourcing product environment under uncertainty in the demand from the retailers and the cost of production, inventory holding, backlogging and distribution of the product. By considering a splitting variable mathematical representation of the Deterministic Equivalent Model, we specialize the so-called Branch-and-Fix Coordination algorithmic framework. It exploits the structure of the model and, specifically, the non-anticipativity constraints for the assignment variables. The algorithm uses the Twin Node Family (TNF) concept. Our procedure is specifically designed for coordinating the selection of the branching TNF and the branching S3 set, such that the non-anticipativity constraints are satisfied. Some computational experience is reported. D. Romero Morales: The work of this author was supported in part by the National Science Foundation under Grant No. DMI-0355533 The work of the first three authors has been partially supported by the grants TIC2003-05982-C05-05 and SEC2002-00112 from MCyT, Spain     

A problem expanding parametric programming method for solving the job shop scheduling problem

A new zero-one integer programming model for the job shop scheduling problem with minimum makespan criterion is presented. The algorithm consists of two parts: (a) a branch and bound parametric linear programming code for solving the job shop problem with fixed completion time; (b) a problem expanding algorithm for finding the optimal completion time. Computational experience for problems having up to thirty-six operations is presented. The largest problem solved was limited by memory space, not computation time. Efforts are under way to improve the efficiency of the algorithm and to reduce its memory requirements.This report was prepared as part of the activities of the Management Sciences Research Group, Carnegie-Mellon University, under Contract No. N00014-82-K-0329 NR 047-048 with the U.S. Office of Naval Research. Reproduction in whole or in part is permitted for any purpose of the U.S. Government.     

A two-stage successive overrelaxation algorithm for solving the symmetric linear complementarity problem

We propose a two-stage successive overrelaxation method (TSOR) algorithm for solving the symmetric linear complementarity problem. After the first SOR preprocessing stage this algorithm concentrates on updating a certain prescribed subset of variables which is determined by exploiting the complementarity property. We demonstrate that this algorithm successfully solves problems with up to ten thousand variables.This material is based on research supported by National Science Foundation Grants DCR-8420963 and DCR-8521228 and Air Force Office of Scientific Research Grants AFSOR-86-0172 and AFSOR-86-0255 while the author was at the Computer Sciences Department at the University of Wisconsin-Madison, USA.     

Links Between Linear Bilevel and Mixed 0–1 Programming Problems

We study links between the linear bilevel and linear mixed 0–1 programming problems. A new reformulation of the linear mixed 0–1 programming problem into a linear bilevel programming one, which does not require the introduction of a large finite constant, is presented. We show that solving a linear mixed 0–1 problem by a classical branch-and-bound algorithm is equivalent in a strong sense to solving its bilevel reformulation by a bilevel branch-and-bound algorithm. The mixed 0–1 algorithm is embedded in the bilevel algorithm through the aforementioned reformulation; i.e., when applied to any mixed 0–1 instance and its bilevel reformulation, they generate sequences of subproblems which are identical via the reformulation.