A capacitated competitive facility location problem

We consider a mathematical model similar in a sense to competitive location problems. There are two competing parties that sequentially open their facilities aiming to “capture” customers and maximize profit. In our model, we assume that facilities’ capacities are bounded. The model is formulated as a bilevel integer mathematical program, and we study the problem of obtaining its optimal (cooperative) solution. It is shown that the problem can be reformulated as that of maximization of a pseudo-Boolean function with the number of arguments equal to the number of places available for facility opening. We propose an algorithm for calculating an upper bound for values that the function takes on subsets which are specified by partial (0, 1)-vectors.     

An upper bound for the competitive location and capacity choice problem with multiple demand scenarios

A new mathematical model is considered related to competitive location problems where two competing parties, the Leader and the Follower, successively open their facilities and try to win customers. In the model, we consider a situation of several alternative demand scenarios which differ by the composition of customers and their preferences.We assume that the costs of opening a facility depend on its capacity; therefore, the Leader, making decisions on the placement of facilities, must determine their capacities taking into account all possible demand scenarios and the response of the Follower. For the bilevel model suggested, a problem of finding an optimistic optimal solution is formulated. We show that this problem can be represented as a problem of maximizing a pseudo- Boolean function with the number of variables equal to the number of possible locations of the Leader’s facilities.We propose a novel systemof estimating the subsets that allows us to supplement the estimating problems, used to calculate the upper bounds for the constructed pseudo-Boolean function, with additional constraints which improve the upper bounds.     

Optimal location of public health centres which provide free and paid services

A model and a heuristic are presented for finding the most effective location of public health centres providing non-vital services in competition with existing private health centres. While private centres provide only services to customers who can pay for them, public centres provide both paid services to affluent customers, and subsidised services to customers belonging to low-income groups (a hierarchical structure). While low-income customers are assigned to fixed public centres, high-income customers can choose which centre to patronise. To find the solution of this problem, the equilibrium between maximum coverage of low-income population (within a pre-specified distance), and an adequate capture of high-income population must be found. Thus, in the public service, the revenues obtained from paid services are used to partly cover the costs of the subsidised services, and the number of centres that can be located depends on how many high-income clients can be captured. Capture of a high-income client happens when a public centre is located closer to the client than any of the existing private centres. Computational experience with optimal, as well as special heuristic, methods for solving this problem is described.     

The multi-facility min-max Weber problem

An algorithm is presented to solve the problem of the locating a given number of facilities on the plane amongst given customers so that the maximum weighted distance from any facility to the customers it services is minimised. The algorithm successfully overcomes the allocation aspects of this problem by generating partitions of customers using a method originally designed for graph colouring embedded within a modified bisection search. Problems of 50 customers and three facilities can be solved in entirely acceptable computer times.     

On the credibility of insurance claim frequency: Generalized count models and parametric estimators

We analyze the concept of credibility in claim frequency in two generalized count models–Mittag-Leffler and Weibull count models–which can handle both underdispersion and overdispersion in count data and nest the commonly used Poisson model as a special case. We find evidence, using data from a Danish insurance company, that the simple Poisson model can set the credibility weight to one even when there are only three years of individual experience data resulting from large heterogeneity among policyholders, and in doing so, it can thus break down the credibility model. The generalized count models, on the other hand, allow the weight to adjust according to the number of years of experience available. We propose parametric estimators for the structural parameters in the credibility formula using the mean and variance of the assumed distributions and a maximum likelihood estimation over a collective data. As an example, we show that the proposed parameters from Mittag-Leffler provide weights that are consistent with the idea of credibility. A simulation study is carried out investigating the stability of the maximum likelihood estimates from the Weibull count model. Finally, we extend the analyses to multidimensional lines and explain how our approach can be used in selecting profitable customers in cross-selling; customers can now be selected by estimating a function of their unknown risk profiles, which is the mean of the assumed distribution on their number of claims.     

A location–allocation heuristic for the capacitated multi-facility Weber problem with probabilistic customer locations

The capacitated multi-facility Weber problem is concerned with locating m facilities in the Euclidean plane, and allocating their capacities to n customers at minimum total cost. The deterministic version of the problem, which assumes that customer locations and demands are known with certainty, is a non-convex optimization problem and difficult to solve. In this work, we focus on a probabilistic extension and consider the situation where the customer locations are randomly distributed according to a bivariate distribution. We first present a mathematical programming formulation, which is even more difficult than its deterministic version. We then propose an alternate location–allocation local search heuristic generalizing the ideas used originally for the deterministic problem. In its original form, the applicability of the heuristic depends on the calculation of the expected distances between the facilities and customers, which can be done for only very few distance and probability density function combinations. We therefore propose approximation methods which make the method applicable for any distance function and bivariate location distribution.     

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 model for rational abandonments from invisible queues

We propose a model for abandonments from a queue, due to excessive wait, assuming that waiting customers act rationally but without being able to observe the queue length. Customers are allowed to be heterogeneous in their preferences and consequent behavior. Our goal is to characterize customers' patience via more basic primitives, specifically waiting costs and service benefits: these two are optimally balanced by waiting customers, based on their individual cost parameters and anticipated waiting time. The waiting time distribution and patience profile then emerge as an equilibrium point of the system. The problem formulation is motivated by teleservices, prevalently telephone- and Internet-based. In such services, customers and servers are remote and queues are typically associated with the servers, hence queues are invisible to waiting customers. Our base model is the M/M/m queue, where it is shown that a unique equilibrium exists, in which rational abandonments can occur only upon arrival (zero or infinite patience for each customer). As such a behavior fails to capture the essence of abandonments, the base model is modified to account for unusual congestion or failure conditions. This indeed facilitates abandonments in finite time, leading to a nontrivial, customer dependent patience profile. Our analysis shows, quite surprisingly, that the equilibrium is unique in this case as well, and amenable to explicit calculation. This revised version was published online in June 2006 with corrections to the Cover Date.     

Facility location for market capture when users rank facilities by shorter travel and waiting times

A firm wants to locate several multi-server facilities in a region where there is already a competitor operating. We propose a model for locating these facilities in such a way as to maximize market capture by the entering firm, when customers choose the facilities they patronize, by the travel time to the facility and the waiting time at the facility. Each customer can obtain the service or goods from several (rather than only one) facilities, according to a probabilistic distribution. We show that in these conditions, there is demand equilibrium, and we design an ad hoc heuristic to solve the problem, since finding the solution to the model involves finding the demand equilibrium given by a nonlinear equation. We show that by using our heuristic, the locations are better than those obtained by utilizing several other methods, including MAXCAP, p-median and location on the nodes with the largest demand.     

The Single Period Coverage Facility Location Problem: Lagrangean heuristic and column generation approaches

In this paper we introduce the Single Period Coverage Facility Location Problem. It is a multi-period discrete location problem in which each customer is serviced in exactly one period of the planning horizon. The locational decisions are made independently for each period, so that the facilities that are open need not be the same in different time periods. It is also assumed that at each period there is a minimum number of customers that can be assigned to the facilities that are open. The decisions to be made include not only the facilities to open at each time period and the time period in which each customer will be served, but also the allocation of customers to open facilities in their service period.     

The gravity p-median model

In this paper we propose a new model for the p-median problem. In the standard p-median problem it is assumed that each demand point is served by the closest facility. In many situations (for example, when demand points are communities of customers and each customer makes his own selection of the facility) demand is divided among the facilities. Each customer selects a facility which is not necessarily the closest one. In the gravity p-median problem it is assumed that customers divide their patronage among the facilities with the probability that a customer patronizes a facility being proportional to the attractiveness of that facility and to a decreasing utility function of the distance to the facility.     

A discrete competitive facility location model with variable attractiveness

We consider the discrete version of the competitive facility location problem in which new facilities have to be located by a new market entrant firm to compete against already existing facilities that may belong to one or more competitors. The demand is assumed to be aggregated at certain points in the plane and the new facilities can be located at predetermined candidate sites. We employ Huff's gravity-based rule in modelling the behaviour of the customers where the probability that customers at a demand point patronize a certain facility is proportional to the facility attractiveness and inversely proportional to the distance between the facility site and demand point. The objective of the firm is to determine the locations of the new facilities and their attractiveness levels so as to maximize the profit, which is calculated as the revenue from the customers less the fixed cost of opening the facilities and variable cost of setting their attractiveness levels. We formulate a mixed-integer nonlinear programming model for this problem and propose three methods for its solution: a Lagrangean heuristic, a branch-and-bound method with Lagrangean relaxation, and another branch-and-bound method with nonlinear programming relaxation. Computational results obtained on a set of randomly generated instances show that the last method outperforms the others in terms of accuracy and efficiency and can provide an optimal solution in a reasonable amount of time.     

An Iterated Local Search for the Budget Constrained Generalized Maximal Covering Location Problem

Capacitated covering models aim at covering the maximum amount of customers’ demand using a set of capacitated facilities. Based on the assumptions made in such models, there is a unique scenario to open a facility in which each facility has a pre-specified capacity and an operating budget. In this paper, we propose a generalization of the maximal covering location problem, in which facilities have different scenarios for being constructed. Essentially, based on the budget invested to construct a given facility, it can provide different service levels to the surrounded customers. Having a limited budget to open the facilities, the goal is locating a subset of facilities with the optimal opening scenario, in order to maximize the total covered demand and subject to the service level constraint. Integer linear programming formulations are proposed and tested using ILOG CPLEX. An iterated local search algorithm is also developed to solve the introduced problem.     

Competitive facility location problem with attractiveness adjustment of the follower: A bilevel programming model and its solution

We are concerned with a problem in which a firm or franchise enters a market by locating new facilities where there are existing facilities belonging to a competitor. The firm aims at finding the location and attractiveness of each facility to be opened so as to maximize its profit. The competitor, on the other hand, can react by adjusting the attractiveness of its existing facilities with the objective of maximizing its own profit. The demand is assumed to be aggregated at certain points in the plane and the facilities of the firm can be located at predetermined candidate sites. We employ Huff’s gravity-based rule in modeling the behavior of the customers where the fraction of customers at a demand point that visit a certain facility is proportional to the facility attractiveness and inversely proportional to the distance between the facility site and demand point. We formulate a bilevel mixed-integer nonlinear programming model where the firm entering the market is the leader and the competitor is the follower. In order to find the optimal solution of this model, we convert it into an equivalent one-level mixed-integer nonlinear program so that it can be solved by global optimization methods. Apart from reporting computational results obtained on a set of randomly generated instances, we also compute the benefit the leader firm derives from anticipating the competitor’s reaction of adjusting the attractiveness levels of its facilities. The results on the test instances indicate that the benefit is 58.33% on the average.     

Location and sizing of facilities on a line

The location-allocation problem in its basic form assumes that the number of new facilities to be located is known and the capacities are unlimited. When the locations of the facilities and demand points (or customers) are restricted to the real line, the basic model may be solved efficiently by dynamic programming. In this note, we show that when the number of facilities and their capacities are included in the decision process, the problem may actually be easier to solve.     

Solving the uncapacitated multi-facility Weber problem by vector quantization and self-organizing maps

The uncapacitated multi-facility Weber problem is concerned with locating m facilities in the Euclidean plane and allocating the demands of n customers to these facilities with the minimum total transportation cost. This is a non-convex optimization problem and difficult to solve exactly. As a consequence, efficient and accurate heuristic solution procedures are needed. The problem has different types based on the distance function used to model the distance between the facilities and customers. We concentrate on the rectilinear and Euclidean problems and propose new vector quantization and self-organizing map algorithms. They incorporate the properties of the distance function to their update rules, which makes them different from the existing two neural network methods that use rather ad hoc squared Euclidean metric in their updates even though the problem is originally stated in terms of the rectilinear and Euclidean distances. Computational results on benchmark instances indicate that the new methods are better than the existing ones, both in terms of the solution quality and computation time.     

Interactive fuzzy goal programming for a multi-objective closed-loop logistics network

Over the last decade, there has been increased attention to closed-loop logistics networks. Environmental legislation requires companies to be more responsible by collecting used products from customers. Companies can also benefit from savings that are related to recovering and recycling used products. Unlike previous studies, which only consider single products or a single period of time in multi-objective problems, this paper considers a multi-product multi-period closed-loop logistics network with different types of facilities. A?multi-objective mixed-integer nonlinear programming formulation is developed to minimize the total cost, the delivery time of new products, and the collection time of used products. Thus, this model better approximates real-life applications of closed-loop logistics problems. Interactive fuzzy goal programming (IFGP) is applied to solve the model for handling multiple objective problems with conflicting objectives and to address the imprecise nature of decision-makers?? aspiration levels for goals. The results from computational experiments performed here show that by changing the upper or lower bound of each objective function, one can obtain a better final solution of the problem and also can provide more options for decision makers to choose from based on their situation. Finally, the utilization rate of facilities is shown to be an important indicator when designing a logistics network.     

Channel Coordination with Cooperative Advertising Considering Effect of Advertising on Willingness to Pay

In this paper, cooperative advertising in a manufacturer–retailer supply chain is studied. Advertising can enhance willingness to pay (WTP) of customers. This trade-off between the benefits of increasing WTP of customers and the advertising expenditure is a key to understanding the retailers optimal advertising decision. On the other hand, it is interesting to understand in which condition supporting the retailer for his advertising expenditure is beneficial for the manufacturer. In this study, in order to capture pricing and advertising strategies of the channel member, three non-cooperative games including Nash, Stackelberg retailer and Stackelberg manufacturer game-theoretic models are established. In spite of the related studies which restrict price in order to prevent negative demand, the proposed model allows channel members to increase their prices by enhancing WTP of customers. In this study, contrary to similar additive form demand functions applied in the co-op ad literature which limits their studies for cases that profit function is concave with respect to variables, optimal prices and advertising strategies are obtained for all the solution space. Surprisingly for the very high values of the advertising effect coefficient, a finite optimal advertising expenditure is achieved.     

Capacitated location allocation problem with stochastic location and fuzzy demand: A hybrid algorithm

In this article, a capacitated location allocation problem is considered in which the demands and the locations of the customers are uncertain. The demands are assumed fuzzy, the locations follow a normal probability distribution, and the distances between the locations and the customers are taken Euclidean and squared Euclidean. The fuzzy expected cost programming, the fuzzy β-cost minimization model, and the credibility maximization model are three types of fuzzy programming that are developed to model the problem. Moreover, two closed-form Euclidean and squared Euclidean expressions are used to evaluate the expected distance between customers and facilities. In order to solve the problem at hand, a hybrid intelligent algorithm is applied in which the simplex algorithm, fuzzy simulation, and a modified genetic algorithm are integrated. Finally, in order to illustrate the efficiency of the proposed hybrid algorithm, some numerical examples are presented.     

Optimal pricing for service facilities with self-optimizing customers

Many service industries (e.g., walk-in clinics, vehicle inspection facilities, and data-processing centers) have customers who choose among congested facilities, and select the facility with the lowest combination of travel cost plus congestion cost at the facility. In general, customers over-utilize attractive facilities, causing higher costs than if customers were assigned to facilities to minimize total costs. Optimal facility prices induce customers to select facilities that minimize total cost. We find optimal facility prices and show they equal charging customers for the impact (net costs and benefits) they cause for others. We explore a rich flexibility that allows a range of optimal prices, useful when negotiating the implementation of facility fees. Facility prices can be positive or negative (price discounts), and can be adjusted to be all positive, or to provide net subsidy or net revenue. We contribute to unifying and generalizing several disparate streams of research.