Optimal model for warehouse location and retailer allocation

Warehouse location and retailer allocation is a high‐level strategic decision problem that is commonly encountered by logisticians and supply chain managers, especially during the supply chain design phase. Considering the product distribution cost and warehouse capital cost trade‐offs, this paper models the warehouse location and retailer allocation problem as a 0–1 integer programming problem and provides an efficient two‐stage set covering heuristic algorithm to solve large‐sized problems. Finally, concluding remarks and some recommendations for further research are also presented. Copyright © 2007 John Wiley & Sons, Ltd.     

Heuristics for the capacitated plant location model

The paper generalizes heuristics for the uncapacitated plant location model to the capacitated case. The heuristics are ADD, DROP, SHIFT, ALA (Alternate Location Allocation) and VSM (Vertex Substitution Method). The generalizations take place within an unifying framework based on elementary ADD and DROP operations.Heuristics from the literature are discussed and compared to the procedures developed in this paper.     

Analysis of collaborative savings and cost allocation techniques for the cooperative carrier facility location problem

Transport companies may cooperate to increase their efficiency levels by, for example, the exchange of orders or vehicle capacity. In this paper a new approach to horizontal carrier collaboration is presented: the sharing of distribution centres (DCs) with partnering organisations. This problem can be classified as a cooperative facility location problem and formulated as an innovative mixed integer linear programme. To ensure cooperation sustainability, collaborative costs need to be allocated fairly to the different participants. To analyse the benefits of cooperative facility location and the effects of different cost allocation techniques, numerical experiments based on experimental design are carried out on a UK case study. Sharing DCs may lead to significant cost savings up to 21.6%. In contrast to the case of sharing orders or vehicles, there are diseconomies of scale in terms of the number of partners and more collaborative benefit can be expected when partners are unequal in size. Moreover, results indicate that horizontal collaboration at the level of DCs works well with a limited number of partners and can be based on intuitively appealing cost sharing techniques, which may reduce alliance complexity and enforce the strength of mutual partner relationships.     

A Lagrangian heuristic for concave cost facility location problems: the plant location and technology acquisition problem

We propose a Lagrangian heuristic for facility location problems with concave cost functions and apply it to solve the plant location and technology acquisition problem. The problem is decomposed into a mixed integer subproblem and a set of trivial single-variable concave minimization subproblems. We are able to give a closed-form expression for the optimal Lagrangian multipliers such that the Lagrangian bound is obtained in a single iteration. Since the solution of the first subproblem is feasible to the original problem, a feasible solution and an upper bound are readily available. The Lagrangian heuristic can be embedded in a branch-and-bound scheme to close the optimality gap. Computational results show that the approach is capable of reaching high quality solutions efficiently. The proposed approach can be tailored to solve many concave-cost facility location problems.     

The plant location problem with demand-dependent setup costs and centralized allocation

Suppose that customers are situated at the nodes of a transportation network, and a service company plans to locate a number of facilities that will serve the customers. The objective is to minimize the sum of the total setup cost and the total transportation cost. The setup cost of a facility is demand-dependent, that is, it depends on the number of customers that are served by the facility. Centralized allocation of customers to facilities is assumed, that is, the service company makes a decision about allocation of customers to facilities. In the case of a general network, the model can be formulated as a mixed integer programming problem. For the case of a tree network, we develop a polynomial-time dynamic programming algorithm.     

A hierarchical model for the location of perinatal facilities in the municipality of Rio de Janeiro

We present a 3-level hierarchical model for the location of maternal and perinatal health care facilities in Rio de Janeiro. Relaxations and heuristics are developed for this model and computational results are given for problems available in the literature, for networks ranging from 10 to 400 vertices. The quality of the solutions produced by the procedures we developed do not differ significantly among themselves. The model is also applied to a case study corresponding to real 1995 data of the municipality of Rio de Janeiro.     

A fuzzy algorithm for continuous capacitated location allocation model with risk consideration

This paper presents a continuous capacitated location-allocation model with fixed cost as a risk management model. In the presented model, the fixed cost consists of production and installation costs. The model considers risk as percent of unsatisfied demands. The fixed cost is assigned to a zone with a predetermined radius from its center. Because of uncertain environment, demand in each zone is investigated as a fuzzy number. The model is solved by a fuzzy algorithm based on α-cut method. After solving the model based on different α-values, the zones with the largest possibilities are determined for locating new facilities and the best locations are calculated based on the obtained possibilities. Then, the model is solved based on different α-values to determine best allocation values. Also, this paper proposes a Cross Entropy (CE) algorithm considering multivariate normal and multinomial density functions for solving large scale instances and is compared with GAMS. Finally, a numerical example is expressed to illustrate the proposed model.     

Reimbursement systems and regional inpatient allocation: a non-linear optimisation model

^** Email: marion.rauner{at}univie.ac.at^*** Email: georg.schneider{at}univie.ac.at^**** Email: kurt.heidenberger{at}univie.ac.at This study presents a non-linear optimisation model for investigatingthe optimal allocation of both budgets and inpatients with differenttreatments among hospitals within a geographic region such asVienna. The objective function maximises the overall qualityof treatment provided by regional hospitals. We compare theeffects of two different reimbursement systems—fixed versusvariable budgets—on optimal allocation strategies. Thecombination of modelling ideas from hospital location-allocationmodels and economic models to solve such a problem is new accordingto the literature. We found that fixed budgets outperformedvariable budgets as fewer Euros had to be invested for an incrementalunit of quality of care provided in most of the policy scenariosanalysed. Regional demand and supply patterns for differenttreatments affect the decision makers' choice of the most suitablereimbursement system. In our illustrative example, two hospitalsappeared inefficient regardless of the reimbursement system.Vienna policy makers are currently considering restructuringthese hospitals. They plan to merge one with nearby hospitalsand transform the other into a nursing home.     

A global supply chain model with transfer pricing and transportation cost allocation

We present a model for the optimization of a global supply that maximizes the after tax profits of a multinational corporation and that includes transfer prices and the allocation of transportation costs as explicit decision variables. The resulting mathematical formulation is a non-convex optimization problem with a linear objective function, a set of linear constraints, and a set of bilinear constraints. We develop a heuristic solution algorithm that applies successive linear programming based on the reformulation and the relaxation of the original problem. Our computational experiments investigate the impact of using different starting points. The algorithm produces feasible solutions with very small gaps between the solutions and their upper bound (UB).     

A quadratic integer program for the location of interacting hub facilities

This paper reports a new formulation of a general hub location model as a quadratic integer program. Non-convexity of the objective function makes the problem difficult. A variety of alternative solution strategies are discussed. Computational results from two simple heuristics are presented for the task of siting 2, 3 or 4 hubs to serve interactions between sets of 10, 15, 20 and 25 U.S. cities. The effects of different computational shortcuts are examined.     

Preprocessing and cutting for multiple allocation hub location problems

In this paper we consider formulations and solution approaches for multiple allocation hub location problems. We present a number of results, which enable us to develop preprocessing procedures and tightening constraints for existing mixed integer linear programming formulations. We employ flow cover constraints for capacitated problems to improve computation times. We present the results of our computational experience, which show that all of these steps can effectively reduce the computational effort required to obtain optimal solutions.     

New formulations for the uncapacitated multiple allocation hub location problem

In this paper we review the integer linear formulations of the uncapacitated multiple allocation hub location problem, we study the scope of validity of these formulations and give new ones that generalize the older formulations. Our formulations allow one or two visits to hubs and include a more general cost structure that needs not satisfy the triangle inequality. We prove that the constraints defined by cliques of a related (intersection) graph are tighter constraints than the classical ones. We also discuss a pre-processing of the problem, which is very useful for reducing its size. Finally, we check the strength of the new formulations and compare them with others in the literature by solving instances of two commonly used data sets: the CAB (Civil Aeronautics Board) and AP (Australian Post), and also randomly generated instances. Our computational results clearly show that our formulations outperform all others previously used for small and medium problems.     

A multicriteria approach to the location of public facilities

We argue that practical problems involving the location of public facilities are really multicriteria problems, and ought to be modeled as much. The general criteria are those of cost and service, but there exist several distinct criteria in each of those two categories. For the first category, fixed investment cost, fixed operating cost, variable operating cost, total operating cost, and total discounted cost are all reasonable criteria to consider. In terms of service, both demand served and response time (or distance traveled) are appropriate criteria, either agglomerated or considered on the basis of the individual clients. In this paper we treat such multicriteria questions in the framework of a model for selecting a subset of M sites at which to establish public facilities in order to serve client groups located at N distinct points. We show that for some combinations of specific criteria, parametric solutions of a generalized assignment problem (GAP) will yield all efficient solution. In most other cases the efficient solutions can be found through parametric solution of a GAP with additional constraints of a type which can be incorporated into an existing algorithm for the GAP. Rather than attempting to find all efficient solutions, however, we advocate an interactive approach to the resolution of multicriteria location problems and elaborate on a specific interactive algorithm for multicriteria optimization which for the present model solves a finite sequence of GAP's or GAP-type problems. Finally, some similar aspects of private sector location problems are discussed.     

Low cost method for location service in the WCDMA system

A new and low cost method for a location service (LCS) in the Wideband Code Division Multiple Access (WCDMA) system is outlined. This method, which is called TDOA+RTT, enables calculation of the geographical position of a mobile station (MS) without knowledge of relative time differences (RTDs) between base stations (BSs). The TDOA+RTT method is based on the measurement of round trip times (RTTs) between the MS and the serving BS, and the solution of a nonlinear system of equations with six variables. The elimination of the RTD parameters significantly simplifies the localisation process in 3G (3rd Generation) real-life cellular networks. The paper concentrates on solving the nonlinear system of equations with six variables. In this context, a new algorithm for solving nonlinear equations is proposed.     

Sequential location of two facilities: comparing random to optimal location of the first facility

We review our recent results in the development of optimal algorithms for the minimization of a strictly convex quadratic function subject to separable convex inequality constraints and/or linear equality constraints. A unique feature of our algorithms is the theoretically supported bound on the rate of convergence in terms of the bounds on the spectrum of the Hessian of the cost function, independent of representation of the constraints. When applied to the class of convex QP or QPQC problems with the spectrum in a given positive interval and a sparse Hessian matrix, the algorithms enjoy optimal complexity, i.e., they can find an approximate solution at the cost that is proportional to the number of unknowns. The algorithms do not assume representation of the linear equality constraints by full rank matrices. The efficiency of our algorithms is demonstrated by the evaluation of the projection of a point to the intersection of the unit cube and unit sphere with hyperplanes.     

Optimizing pricing and location decisions for competitive service facilities charging uniform price

In this paper, we present the problem of optimizing the location and pricing for a set of new service facilities entering a competitive marketplace. We assume that the new facilities must charge the same (uniform) price and the objective is to optimize the overall profit for the new facilities. Demand for service is assumed to be concentrated at discrete demand points (customer markets); customers in each market patronize the facility providing the highest utility. Customer demand function is assumed to be elastic; the demand is affected by the price, facility attractiveness, and the travel cost for the highest-utility facility. We provide both structural and algorithmic results, as well as some managerial insights for this problem. We show that the optimal price can be selected from a certain finite set of values that can be computed in advance; this fact is used to develop an efficient mathematical programming formulation for our model.     

A facility location model with safety stock costs: analysis of the cost of single-sourcing requirements

We consider a supply chain setting where multiple uncapacitated facilities serve a set of customers with a single product. The majority of literature on such problems requires assigning all of any given customer??s demand to a single facility. While this single-sourcing strategy is optimal under linear (or concave) cost structures, it will often be suboptimal under the nonlinear costs that arise in the presence of safety stock costs. Our primary goal is to characterize the incremental costs that result from a single-sourcing strategy. We propose a general model that uses a cardinality constraint on the number of supply facilities that may serve a customer. The result is a complex mixed-integer nonlinear programming problem. We provide a generalized Benders decomposition algorithm for the case in which a customer??s demand may be split among an arbitrary number of supply facilities. The Benders subproblem takes the form of an uncapacitated, nonlinear transportation problem, a relevant and interesting problem in its own right. We provide analysis and insight on this subproblem, which allows us to devise a hybrid algorithm based on an outer approximation of this subproblem to accelerate the generalized Benders decomposition algorithm. We also provide computational results for the general model that permit characterizing the costs that arise from a single-sourcing strategy.     

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.