Properties of a multifacility location problem involving euclidian distances

This paper considers a problem of locating new facilities in the plane with respect to existing facilities, the locations of which are known. The problem consists of finding locations of new facilities which will minimize a total cost function which consists of a sum of costs directly proportional to the Euclidian distances among the new facilities, and costs directly proportional to the Euclidian distances between new and existing facilities. It is established that the total cost function has a minimum; necessary conditions for a mimumum are obtained; necessary and sufficient conditions are obtained for the function to be strictly convex (it is always convex); when the problem is “well structured,” it is established that for a minimum cost solution the locations of the new facilities will lie in the convex hull of the locations of the existing facilities. Also, a dual to the problem is obtained and interpreted; necessary and sufficient conditions for optimum solutions to the problem, and to its dual, are developed, as well as complementary slackness conditions. Many of the properties to be presented are motivated by, based on, and extend the results of Kuhn's study of the location problem known as the General Fermat Problem.     

反舰导弹仿真中的目标命中判断问题

阐述了模型校核的意义和作用。对属于模型校核范畴的仿真中的系统状态不连贯问题的基本概念通过乒乓球的下落和反舰导弹攻击目标舰艇的例子进行了说明。介绍了已有的解决系统状态不连贯问题的三种方法,并进行了优、缺点分析。给出了反舰导弹仿真中的目标命中判断模型。指出,反舰导弹仿真中的目标命中判断问题是一个系统状态不连贯问题。为解决该问题,提出并应用了一种新方法预测法。利用预测法,最多进行两步最小步长仿真,就能够以要求的精确度检测到任何一个系统状态不连贯。相对于以前的三种方法,在提高仿真效率的同时,预测法还能够避免对系统状态不连贯问题的漏检。     

Machine scheduling with an availability constraint and job delivery coordination

In this paper we study the scheduling problem that considers both production and job delivery at the same time with machine availability considerations. Only one vehicle is available to deliver jobs in a fixed transportation time to a distribution center. The vehicle can load at most K jobs as a delivery batch in one shipment due to the vehicle capacity constraint. The objective is to minimize the arrival time of the last delivery batch to the distribution center. Since machines may not always be available over the production period in real life due to preventive maintenance, we incorporate machine availability into the models. Three scenarios of the problem are studied. For the problem in which the jobs are processed on a single machine and the jobs interrupted by the unavailable machine interval are resumable, we provide a polynomial algorithm to solve the problem optimally. For the problem in which the jobs are processed on a single machine and the interrupted jobs are nonresumable, we first show that the problem is NP‐hard. We then propose a heuristic with a worst‐case error bound of 1/2 and show that the bound is tight. For the problem in which the jobs are processed on either one of two parallel machines, where only one machine has an unavailable interval and the interrupted jobs are resumable, we propose a heuristic with a worst‐case error bound of 2/3. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Structured markovian decision problems

Structured finite action-finite state space discounted Markovian decision problems are analyzed. Any problem of a general class is shown to be equivalent to a “separated” problem with decomposable problem structure. A modified policy iteration approach is developed for this decomposable reformulation. Both analytic and computer evaluations of the decomposition algorithm's effectiveness are presented.     

Linear programming and the reliability of multicomponent systems

Several problems in the assignment of parallel redundant components to systems composed of elements subject to failure are considered. In each case the problem is to make an assignment which maximizes the system reliability subject to system constraints. Three distinct problems; are treated. The first is the classical problem of maximizing system reliability under total cost or weight constraints when components are subject to a single type of failure. The second problem deals with components which are subject to two types of failure and minimizes the probability of one mode of system failure subject to a constraint on the probability of the other mode of system failure. The third problem deals with components which may either fail to operate or may operate prematurely. System reliability is maximized subject to a constraint ori system safety. In each case the problem is formulated as an integer linear program. This has an advantage over alternative dynamic programming formulations in that standard algorithms may be employed to obtain numerical results.     

Sensitivity analysis as a means of reducing the dimensionality of a certain class of transportation problems

Sensitivity analysis of the transportation problem is developed in a way which enables reducing the dimensionality of the associated tableau. This technique is used to reduce the dimensionality of a transportation problem whose origin requirements are relatively small at the majority of origins. A long transportation problem, for which efficient solution procedures exist, results. A second application relates to the location-allocation problem. Reducing the dimensionality of such a problem, accompanied by the partial determination of the optimal solution, should prove helpful in the quest for an analytic solution to the aforementioned problem. In the meantime, reducing dimensionality greatly decreases the effort involved in solution by trial and error. Examples of the two applications are provided.     

AN ALGORITHM FOR THE SEGREGATED STORAGE PROBLEM

The segregated storage problem involves the optimal distribution of products among compartments with the restriction that only one product may be stored in each compartment. The storage capacity of each compartment, the storage demand for each product, and the linear cost of storing one unit of a product in a given compartment are specified. The problem is reformulated as a large set-packing problem, and a column generation scheme is devised to solve the associated linear programming problem. In case of fractional solutions, a branch and bound procedure is utilized. Computational results are presented.     

Concave minimization over a convex polyhedron

A general algorithm is developed for minimizing a well defined concave function over a convex polyhedron. The algorithm is basically a branch and bound technique which utilizes a special cutting plane procedure to' identify the global minimum extreme point of the convex polyhedron. The indicated cutting plane method is based on Glover's general theory for constructing legitimate cuts to identify certain points in a given convex polyhedron. It is shown that the crux of the algorithm is the development of a linear undrestimator for the constrained concave objective function. Applications of the algorithm to the fixed-charge problem, the separable concave programming problem, the quadratic problem, and the 0-1 mixed integer problem are discussed. Computer results for the fixed-charge problem are also presented.     

Computing bounds for the optimal value in linear programming

Consider a standard linear programming problem and suppose that there are bounds available for the decision variables such that those bounds are not violated at an optimal solution of the problem (but they may be violated at some other feasible solutions of the problem). Thus, these bounds may not appear explicitly in the problem, but rather they may have been derived from some prior knowledge about an optimal solution or from the explicit constraints of the problem. In this paper, the bounds on variables are used to compute bounds on the optimal value when the problem is being solved by the simplex method. The latter bounds may then be used as a termination criteria for the simples iterations for the purpose of finding a “sufficiently good” near optimal solution. The bounds proposed are such that the computational effort in evaluating them is insignificant compared to that involved in the simplex iterations. A numerical example is given to demonstrate their performance.     

The fixed charge problem

A fundamental unsolved problem in the programming area is one in which various activities have fixed charges (e.g., set-up time charges) if operating at a positive level. Properties of a general solution to this type problem are discussed in this paper. Under special circumstances it is shown that a fixed charge problem can be reduced to an ordinary linear programming problem.     

Optimal flowshop schedules with no intermediate storage space

In this paper the problem of finding an optimal schedule for the n-job, M-machine flowshop scheduling problem is considered when there is no intermediate space to hold partially completed jobs and the objective function is to minimize the weighted sum of idle times on all machines. By assuming that jobs are processed as early as possible, the problem is modeled as a traveling salesman problem and solved by known solution techniques for the traveling salesman problem. A sample problem is solved and a special case, one involving only two machines, is discussed.     

Benders' cuts guided large neighborhood search for the traveling umpire problem

This article introduces the use of Benders' cuts to guide a large neighborhood search to solve the traveling umpire problem, a sports scheduling problem inspired by the real‐life needs of the officials of a sports league. At each time slot, a greedy matching heuristic is used to construct a schedule. When an infeasibility is recognized first a single step backtracking is tried to resolve the infeasibility. If unsuccessful, Benders' cuts are generated to guide a large neighborhood search to ensure feasibility and to improve the solution. Realizing the inherent symmetry present in the problem, a large family of cuts are generated and their effectiveness is tested. The resulting approach is able to find better solutions to many instances of this problem. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Lagrangian relaxation approach to the targeting problem

In this paper, we consider a new weapon–target allocation problem with the objective of minimizing the overall firing cost. The problem is formulated as a nonlinear integer programming model. We applied Lagrangian relaxation and a branch‐and‐bound method to the problem after transforming the nonlinear constraints into linear ones. An efficient primal heuristic is developed to find a feasible solution to the problem to facilitate the procedure. In the branch‐and‐bound method, three different branching rules are considered and the performances are evaluated. Computational results using randomly generated data are presented. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 640–653, 1999     

The formulation of some allocation and connection problems as integer programs

In this paper a component placement problem and a digital computer backboard wiring problem are formulated as integer linear programs. The component placement problem consists of making a unique assignment of components to column positions such that wireability is maximized. The backboard wiring problem consists of three interrelated subproblems, namely, the placement, the connection, and the routing problems. The placement and connection problems are combined and solved as one, thereby giving the optimal circuit connections as well as minimizing the total lead length. It is shown that under certain assumptions, the number of inequalities and variables in the problem can be greatly reduced. Further simplifying assumptions lead to a near optimal solution. Examples of other allocation problems to which the models presented here are applicable are given. The following concepts are formulated as linear inequalities: (1) the absolute magnitude of the difference between two variables; (2) minimize the minimum function of a set of functions; and (3) counting the number of (0, 1) adjacent component pairs in a vector.     

Postoptimality analysis in integer programming by implicit enumeration: The mixed integer case

This paper develops a method for doing postoptimality analysis on the mixed integer programming problem. The proposed procedures form a natural adjunct to enumerative I.P. algorithms that are linear programming based, and they are designed, in effect, to capitalize on insights generated as the problem is initially solved to do subsequent analysis upon it. In particular, limited ranging analysis is possible on selected parameters, as is the efficient resolving of the problem following parameter changes.     

An algorithm for 0-1 multiple-knapsack problems

The 0-1 multiple-knapsack problem is an extension of the well-known 0-1 knapsack problem. It is a problem of assigning m objects, each having a value and a weight, to n knapsacks in such a way that the total weight in each knapsack is less than its capacity limit and the total value in the knapsacks is maximized. A branch-and-bound algorithm for solving the problem is developed and tested. Branching rules that avoid the search of redundant partial solutions are used in the algorithm. Various bounding techniques, including Lagrangean and surrogate relaxations, are investigated and compared.     

Sensitivity analysis of the optimal location of a facility

We perform a sensitivity analysis of the Euclidean, single-facility minisum problem, which is also known as the Weber problem. We find the sensitivity of the optimal site of the new facility to changes in the locations and weights of the demand points. We apply these results to get the optimal site if some of the parameters in the problem are changed. We also get approximate formulas for the set of all possible optimal sites if demand points are restricted to given areas, and weights must be within given ranges, which is a location problem under conditions of uncertainty.     

A note on duality in homogeneous fractional programming

For a linear fractional programming problem, Sharma and Swarup have constructed a dual problem, also a linear fractional program, in which the objective functions of both primal and dual problems are the same. Craven and Mond have extended this result to a nonlinear fractional programming problem with linear constraints, and a dual problem for which the objective function is the same as that of the primal. This theorem is now further extended from linear to differentiable convex constraints.     

Mathematical control theory solution of an interactive accounting flows model

In this paper we have applied the mathematical control theory to the accounting network flows, where the flow rates are constrained by linear inequalities. The optimal control policy is of the “generalized bang-bang” variety which is obtained by solving at each instant in time a linear programming problem whose objective function parameters are determined by the “switching function” which is derived from the Hamiltonian function. The interpretation of the adjoint variables of the control problem and the dual evaluators of the linear programming problem demonstrates an interesting interaction of the cross section phase of the problem, which is characterized by linear programming, and the dynamic phase of the problem, which is characterized by control theory.     

Minimax and maximin facility location problems on a sphere

The problem dealt with in this article is as follows. There are n “demand points” on a sphere. Each demand point has a weight which is a positive constant. A facility must be located so that the maximum of the weighted distances (distances are the shortest arcs on the surface of the sphere) is minimized; this is called the minimax problem. Alternatively, in the maximin problem, the minimum weighted distance is maximized. A setup cost associated with each demand point may be added for generality. It is shown that any maximin problem can be reparametrized into a minimax problem. A method for finding local minimax points is described and conditions under which these are global are derived. Finally, an efficient algorithm for finding the global minimax point is constructed.