Locating a 1-Center on a Manhattan Plane with “Arbitrarily” Shaped Barriers

Barriers commonly occur in practical location and layout problems and are regions where neither travel through nor location of the new facility is permitted. Along the lines of (Larson and Sadiq, 1983) we divide the feasible location region into cells. To overcome the additional complications introduced by the center objective, we develop new analysis and classify cells based on number of cell corners. A procedure to determine the optimal location is established for each class of cells. The overall complexity of the approach is shown to be polynomially bounded. Also, an analogy is drawn to the center problem on a network and generalizations of the model are discussed.     

Contour line construction for a new rectangular facility in an existing layout with rectangular departments

In a recent paper, Savas et al. [S. Savas, R. Batta, R. Nagi, Finite-size facility placement in the presence of barriers to rectilinear travel, Operations Research 50 (6) (2002) 1018–1031] consider the optimal placement of a finite-sized facility in the presence of arbitrarily shaped barriers under rectilinear travel. Their model applies to a layout context, since barriers can be thought to be existing departments and the finite-sized facility can be viewed as the new department to be placed. In a layout situation, the existing and new departments are typically rectangular in shape. This is a special case of the Savas et al. paper. However the resultant optimal placement may be infeasible due to practical constraints like aisle locations, electrical connections, etc. Hence there is a need for the development of contour lines, i.e. lines of equal objective function value. With these contour lines constructed, one can place the new facility in the best manner. This paper deals with the problem of constructing contour lines in this context. This contribution can also be viewed as the finite-size extension of the contour line result of Francis [R.L. Francis, Note on the optimum location of new machines in existing plant layouts, Journal of Industrial Engineering 14 (2) (1963) 57–59].     

Planar Location Problems with Block Distance and Barriers

This paper considers one facility planar location problems using block distance and assuming barriers to travel. Barriers are defined as generalized convex sets relative to the block distance. The objective function is any convex, nondecreasing function of distance. Such problems have a non-convex feasible region and a non-convex objective function. The problem is solved by modifying the barriers to obtain an equivalent problem and by decomposing the feasible region into a polynomial number of convex subsets on which the objective function is convex. It is shown that solving a planar location problem with block distance and barriers requires at most a polynomial amount of additional time over solving the same problem without barriers.     

Facility placement with sub-aisle design in an existing layout

In this paper, we consider the integration of facility placement in an existing layout and the configuration of one or two connecting sub-aisles. This is relevant, for example, when placing a new machine/department on a shop floor with existing machines/departments and an existing aisle structure. Our work is motivated by the work of Savas et al. [Savas, S., Batta, R., Nagi, R., 2002. Finite-size facility placement in the presence of barriers to rectilinear travel. Operations Research 50 (6), 1018–1031], that considered the optimal planar placement of a finite-size facility in the presence of existing facilities. Our work differs from theirs in that we consider material handling to be restricted to the aisle structure. We do not allow the newly placed facility to overlap with existing facilities or with the aisle structure. Facilities are rectangular and travel is limited to new or existing aisles. We show that there are a finite number of candidate placements for the new facility. Algorithms are developed to find the optimal placement and the corresponding configurations for the sub-aisles. Complexity of the solution method is analyzed. Also, a numerical example is provided to explore the impact of the number of sub-aisles added.     

带惩罚的容错设施布局问题的近似算法

在带惩罚的容错设施布局问题中, 给定顾客集合、地址集合、以及每个顾客和各个地址之间的连接费用, 这里假设连接费用是可度量的. 每位顾客有各自的服务需求, 每个地址可以开设任意多个设施, 顾客可以被安排连接到某些地址的一些开设的设施上以满足其需求, 也可以被拒绝, 但这时要支付拒绝该顾客所带来的惩罚费用. 目标是确定哪些顾客的服务需求被拒绝并开设一些设施, 将未被拒绝的顾客连接到不同的开设设施上, 使得开设费用、连接费用和惩罚费用总和最小. 给出了带惩罚的容错设施布局问题的线性整数规划及其对偶规划, 进一步, 给出了基于其线性规划和对偶规划舍入的4-近似算法.     

Combinatorial approximation algorithms for the robust facility location problem with penalties

In this paper, we consider the robust facility location problem with penalties, aiming to serve only a specified fraction of the clients. We formulate this problem as an integer linear program to identify which clients must be served. Based on the corresponding LP relaxation and dual program, we propose a primal–dual (combinatorial) 3-approximation algorithm. Combining the greedy augmentation procedure, we further improve the above approximation ratio to 2.     

Computing Approximate Solutions of the Maximum Covering Problem with GRASP

We consider the maximum covering problem, a combinatorial optimization problem that arises in many facility location problems. In this problem, a potential facility site covers a set of demand points. With each demand point, we associate a nonnegative weight. The task is to select a subset of p > 0 sites from the set of potential facility sites, such that the sum of weights of the covered demand points is maximized. We describe a greedy randomized adaptive search procedure (GRASP) for the maximum covering problem that finds good, though not necessarily optimum, placement configurations. We describe a well-known upper bound on the maximum coverage which can be computed by solving a linear program and show that on large instances, the GRASP can produce facility placements that are nearly optimal.     

Location of Multiple-Server Congestible Facilities for Maximizing Expected Demand, when Services are Non-Essential

We formulate a model for locating multiple-server, congestible facilities. Locations of these facilities maximize total expected demand attended over the region. The effective demand at each node is elastic to the travel time to the facility, and to the congestion at that facility. The facilities to be located are fixed, so customers travel to them in order to receive service or goods, and the demand curves at each demand node (which depend on the travel time and the queue length at the facility), are known. We propose a heuristic for the resulting integer, nonlinear formulation, and provide computational experience.     

A mixed integer programming formulation for the 1-maximin problem

In this paper, I present a mixed integer programming (MIP) formulation for the 1-maximin problem with rectilinear distance. The problem mainly appears in facility location while trying to locate an undesirable facility. The rectilinear distance is quite commonly used in the location literature. Our numerical experiments show that one can solve reasonably large location problems using a standard MIP solver. We also provide a linear programming formulation that helps find an upper bound on the objective function value of the 1-maximin problem with any norm when extreme points of the feasible region are known. We discuss various extension alternatives for the MIP formulation.     

Solvability of a multivalued filtering problem in a heterogeneous environment with a distributed source

In this paper we formulate a generalized filtering problem in a heterogeneous environment in the presence of a source distributed along a line. Incompressible fluids obey a multivalued law with a linear growth at infinity. In this study we use the additive singularity extraction in the right-hand side of the problem constraint. We represent the pressure field as the sum of a known solution to a certain linear problem and an unknown “additive term”. We reduce the problem under consideration to a variational inequality of the second kind in a Hilbert space (with respect to the mentioned “additive term”) and prove its solvability.     

随机容错设施布局问题的近似算法

在确定性的容错设施布局问题中, 给定顾客的集合和地址的集合. 在每个地址上可以开设任意数目的不同设施. 每个顾客j有连接需求r_j. 允许将顾客j连到同一地址的不同设施上. 目标是开设一些设施并将每个顾客j连到r_j个不同的设施上, 使得总开设费用和连接费用最小. 研究两阶段随机容错设施布局问题(SFTFP), 顾客的集合事先不知道, 但是具有有限多个场景并知道其概率分布. 每个场景指定需要服务的顾客的子集. 并且每个设施有两种类型的开设费用. 在第一阶段根据顾客的随机信息确定性地开设一些设施, 在第二阶段根据顾客的真实信息再增加开设一些设施．给出随机容错布局问题的线性整数规划和基于线性规划舍入的5-近似算法.     

An efficient heuristic approach for a multi-period logistics network redesign problem

In this paper, a multi-period logistics network redesign problem arising in the context of strategic supply chain planning is studied. Several aspects of practical relevance are captured, namely, multiple echelons with different types of facilities, product flows between facilities in the same echelon, direct shipments to customers, and facility relocation. A two-phase heuristic approach is proposed to obtain high-quality feasible solutions to the problem, which is initially modeled as a large-scale mixed-integer linear program. In the first phase of the heuristic, a linear programming rounding strategy is applied to find initial values for the binary location variables. The second phase of the heuristic uses local search to correct the initial variable choices when a feasible solution is not identified, or to improve the initial feasible solution when its quality does not meet given criteria. The results of a computational study are reported for randomly generated instances comprising a variety of logistics networks.     

Solving Rectilinear Planar Location Problems with Barriers by a Polynomial Partitioning

This paper considers planar location problems with rectilinear distance and barriers where the objective function is any convex, nondecreasing function of distance. Such problems have a non-convex feasible region and a nonconvex objective function. Based on an equivalent problem with modified barriers, derived in a companion paper [3], the non convex feasible set is partitioned into a network and rectangular cells. The rectangular cells are further partitioned into a polynomial number of convex subcells, called convex domains, on which the distance function, and hence the objective function, is convex. Then the problem is solved over the network and convex domains for an optimal solution. Bounds are given that reduce the number of convex domains to be examined. The number of convex domains is bounded above by a polynomial in the size of the problem.     

Optimal supplier choice with discounting

This paper investigates a model for pricing the demand for a set of goods when suppliers operate discount schedules based on total business value. We formulate the buyers's decision problem as a mixed binary integer program, which is a generalization of the capacitated facility location problem (CFLP). A branch and bound (BnB) procedure using Lagrangean relaxation and subgradient optimization is developed for solving large-scale problems that can arise when suppliers’ discount schedules contain multiple price breaks. Results of computer trials on specially adapted large benchmark instances of the CFLP confirm that a sub-gradient optimization procedure based on Shor and Zhurbenko's r-algorithm, which employs a space dilation in the direction of the difference between two successive subgradients, can be used efficiently for solving the dual problem at any node of the BnB tree.     

On measuring the inefficiency with the inner-product norm in data envelopment analysis

A technique for assessing the sensitivity of efficiency classifications in Data Envelopment Analysis (DEA) is presented. It extends the technique proposed by Charnes et al. (A. Charnes, J.J. Rousseau, J.H. Semple, Journal of Productivity Analysis 7 (1996) 5–18). An organization's input–output vector serves as the center for a cell within which the organization's classification remains unchanged under perturbations of the data. The maximal radius among such cells can be interpreted as a stability measure of the classification. Our approach adopts the inner-product norm for the radius, while the previous work does the polyhedral norms. For an efficient organization, the maximal-radius problem is a convex program. On the other hand, for an inefficient organization, it is reduced to a nonconvex program whose feasible region is the complement of a convex polyhedral set. We show that the latter nonconvex problem can be transformed into a linear reverse convex program. Our formulations and algorithms are valid not only in the CCR model but in its variants.     

Location-allocation on a line with demand-dependent costs

This paper extends the location-allocation formulation by making the cost charged to users by a facility a function of the total number of users patronizing the facility. Users select their facility based on facility charges and transportation costs. We explore equilibria where each customer selects the least expensive facility (cost and transportation) and where the facility is at a point that minimizes travel costs for its customers. The problem in its general form is quite complex. An interesting special case is studied: facilities and customers are located on a finite line segment and demand is distributed on the line by a given density function.     

Optimal Positioning of a Mobile Service Unit on a Line

In this paper we address the problem of locating a mobile response unit when demand is distributed according to a random variable on a line. Properties are proven which reduce the problem to locating a non-mobile facility, transforming the original optimization problem into an one-dimensional convex program.In the special case of a discrete demand (a simple probability measure), an algorithm which runs in expected linear time is proposed.     

An efficient algorithm for facility location in the presence of forbidden regions

This paper investigates a constrained form of the classical Weber problem. Specifically, we consider the problem of locating a new facility in the presence of convex polygonal forbidden regions such that the sum of the weighted distances from the new facility to n existing facilities is minimized. It is assumed that a forbidden region is an area in the plane where travel and facility location are not permitted and that distance is measured using the Euclidean-distance metric. A solution procedure for this nonconvex programming problem is presented. It is shown that by iteratively solving a series of unconstrained problems, this procedure terminates at a local optimum to the original constrained problem. Numerical examples are presented.     

Successive computation of some efficient locations of the Weber problem with barriers

We transform a continuous Weber problem with barriers to a discrete problem by computing some efficient locations, which are considered as promising locations for a new facility. A theorem determining conditions for a location to be efficient is introduced. A method for constructing the weighted coefficients and travel costs modeled by some distance metrics, is derived. This method provides successive computation of some efficient locations. Also, we propose an algorithm for the verification of location’s efficiency. Numerical examples are provided using squared Euclidean and Manhattan distance metrics, as well as the implementation details in programming package MATHEMATICA.     

Cutting and packing problems for irregular objects with continuous rotations: mathematical modelling and non-linear optimization

We further improve our methodology for solving irregular packing and cutting problems. We deal with an accurate representation of objects bounded by circular arcs and line segments and allow their continuous rotations and translations within rectangular and circular containers. We formulate a basic irregular placement problem which covers a wide spectrum of packing and cutting problems. We provide an exact non-linear programming (NLP) model of the problem, employing ready-to-use phi-functions. We develop an efficient solution algorithm to search for local optimal solutions for the problem in a reasonable time. The algorithm reduces our problem to a sequence of NLP subproblems and employs optimization procedures to generate starting feasible points and feasible subregions. Our algorithm allows us to considerably reduce the number of inequalities in NLP subproblems. To show the benefits of our methodology we give computational results for a number of new challenger and the best known benchmark instances.