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.     

Strategic facility location: A review

Facility location decisions are a critical element in strategic planning for a wide range of private and public firms. The ramifications of siting facilities are broadly based and long-lasting, impacting numerous operational and logistical decisions. High costs associated with property acquisition and facility construction make facility location or relocation projects long-term investments. To make such undertakings profitable, firms plan for new facilities to remain in place and in operation for an extended time period. Thus, decision makers must select sites that will not simply perform well according to the current system state, but that will continue to be profitable for the facility's lifetime, even as environmental factors change, populations shift, and market trends evolve. Finding robust facility locations is thus a difficult task, demanding that decision makers account for uncertain future events. The complexity of this problem has limited much of the facility location literature to simplified static and deterministic models. Although a few researchers initiated the study of stochastic and dynamic aspects of facility location many years ago, most of the research dedicated to these issues has been published in recent years. In this review, we report on literature which explicitly addresses the strategic nature of facility location problems by considering either stochastic or dynamic problem characteristics. Dynamic formulations focus on the difficult timing issues involved in locating a facility (or facilities) over an extended horizon. Stochastic formulations attempt to capture the uncertainty in problem input parameters such as forecast demand or distance values. The stochastic literature is divided into two classes: that which explicitly considers the probability distribution of uncertain parameters, and that which captures uncertainty through scenario planning. A wide range of model formulations and solution approaches are discussed, with applications ranging across numerous industries.     

A competitive hub location and pricing problem

We formulate and solve a new hub location and pricing problem, describing a situation in which an existing transportation company operates a hub and spoke network, and a new company wants to enter into the same market, using an incomplete hub and spoke network. The entrant maximizes its profit by choosing the best hub locations and network topology and applying optimal pricing, considering that the existing company applies mill pricing. Customers’ behavior is modeled using a logit discrete choice model. We solve instances derived from the CAB dataset using a genetic algorithm and a closed expression for the optimal pricing. Our model confirms that, in competitive settings, seeking the largest market share is dominated by profit maximization. We also describe some conditions under which it is not convenient for the entrant to enter the market.     

Optimal location and design of a competitive facility

A single facility has to be located in competition with fixed existing facilities of similar type. Demand is supposed to be concentrated at a finite number of points, and consumers patronise the facility to which they are attracted most. Attraction is expressed by some function of the quality of the facility and its distance to demand. For existing facilities quality is fixed, while quality of the new facility may be freely chosen at known costs. The total demand captured by the new facility generates income. The question is to find that location and quality for the new facility which maximises the resulting profits.It is shown that this problem is well posed as soon as consumers are novelty oriented, i.e. attraction ties are resolved in favour of the new facility. Solution of the problem then may be reduced to a bicriterion maxcovering-minquantile problem for which solution methods are known. In the planar case with Euclidean distances and a variety of attraction functions this leads to a finite algorithm polynomial in the number of consumers, whereas, for more general instances, the search of a maximal profit solution is reduced to solving a series of small-scale nonlinear optimisation problems. Alternative tie-resolution rules are finally shown to result in problems in which optimal solutions might not exist.Mathematics Subject Classification (2000):90B85, 90C30, 90C29, 91B42Partially supported by Grant PB96-1416-C02-02 of the D.G.E.S. and Grant BFM2002-04525-C02-02 of Ministerio de Ciencia y Tecnología, Spain     

Perception and information in a competitive location model

This paper considers a situation in which two competitors locate one facility each at the vertices of a tree. Their decisions are based on the capture of demands at the vertices of the tree. As opposed to the usual models, it is assumed that the true demands are unknown and that each competitor has its own perception of them. The paper investigates a facility planner's advantage that results from knowledge concerning his competitor's perception. Stackelberg solutions are analyzed as well as possible advantages to the competitor who locates first.     

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.     

Static competitive facility location: An overview of optimisation approaches

We give an overview of the research, models and literature about optimisation approaches to the problem of optimally locating one or more new facilities in an environment where competing facilities are already established.     

Developments in network location with mobile and congested facilities

We review four facility location problems which are motivated by urban service applications and which can be thought of as extensions of the classic Q-median problem on networks. In problems P1 and P2 it is assumed that travel times on network links change over time in a probabilistic way. In P2 it is further assumed that the facilities (servers) are movable so that they can be relocated in response to new network travel times. Problems P3 and P4 examine the Q-median problem for the case when the service capacity of the facilities is finite and, consequently, some or all of the facilities can be unavailable part of the time. In P3 the facilities have stationary home locations but in P4 they have movable locations and thus can be relocated to compensate for the unavailability of the busy facilities. We summarize our main results to date on these problems.     

Recent insights in Huff-like competitive facility location and design

Recently, several articles appeared on the location–design problem that firms face when entering a competing market. All use a Huff-like attraction model. We discuss the formulation of the base model, the different settings studied in the papers and summarise their findings.     

A robust and efficient algorithm for planar competitive location problems

In this paper we empirically analyze several algorithms for solving a Huff-like competitive location and design model for profit maximization in the plane. In particular, an exact interval branch-and-bound method and a multistart heuristic already proposed in the literature are compared with uego (Universal Evolutionary Global Optimizer), a recent evolutionary algorithm. Both the multistart heuristic and uego use a Weiszfeld-like algorithm as local search procedure. The computational study shows that uego is superior to the multistart heuristic, and that by properly fine-tuning its parameters it usually (in the computational study, always) find the global optimal solution, and this in much less time than the interval branch-and-bound method. Furthermore, uego can solve much larger problems than the interval method.     

Relaxed voting and competitive location under monotonous gain functions on trees

We examine competitive location problems where two competitors serve a good to users located in a network. Users decide for one of the competitors based on the distance induced by an underlying tree graph. The competitors place their server sequentially into the network. The goal of each competitor is to maximize his benefit which depends on the total user demand served. Typical competitive location problems include the (1,X₁)-medianoid, the (1,1)-centroid, and the Stackelberg location problem.An additional relaxation parameter introduces a robustness of the model against small changes in distance. We introduce monotonous gain functions as a general framework to describe the above competitive location problems as well as several problems from the area of voting location such as Simpson, Condorcet, security, and plurality.In this paper we provide a linear running time algorithm for determining an absolute solution in a tree where competitors are allowed to place on nodes or on inner points. Furthermore we discuss the application of our approach to the discrete case.     

Sensitivity analysis of a continuous multifacility competitive location and design problem

A continuous location problem in which a firm wants to set up two or more new facilities in a competitive environment is considered. Other facilities offering the same product or service already exist in the area. Both the locations and the qualities of the new facilities are to be found so as to maximize the profit obtained by the firm. This is a global optimization problem, with many parameters to be estimated, and whose behavior is not really well understood. Using random problems and a robust evolutionary algorithm recently proposed for solving this problem, the behavior of optimal solutions in various environments and changes in the basic model parameters are researched. These comprise the quality of existing and new facilities, cost function and presence of the chain. Some economic implications are derived.     

Capacitated dynamic location problems with opening, closure and reopening of facilities

^** Email: joana{at}fe.uc.pt In this paper a capacitated dynamic location problem with opening,closure and reopening of facilities is formulated and a primal–dualheuristic that can solve this problem is described. In thisproblem both maximum and minimum capacity restrictions are considered.The problem formulated is NP-hard. Computational results arepresented and discussed.     

Multi-objective competitive location problem with distance-based attractiveness and its best non-dominated solution

The multi-objective competitive location problem (MOCLP) with distance-based attractiveness is introduced. There are m potential competitive facilities and n demand points on the same plane. All potential facilities can provide attractiveness to the demand point which the facility attractiveness is represented as distance-based coverage of a facility, which is “full coverage” within the maximum full coverage radius, “no coverage” outside the maximum partial coverage radius, and “partial coverage” between those two radii. Each demand point covered by one of m potential facilities is determined by the greatest accumulated attractiveness provided the selected facilities and least accumulated distances between each demand point and selected facility, simultaneously. The tradeoff of maximum accumulated attractiveness and minimum accumulated distances is represented as a multi-objective optimization model. A proposed solution procedure to find the best non-dominated solution set for MOCLP is introduced. Several numerical examples and instances comparing with introduced and exhaustive method demonstrates the good performance and efficiency for the proposed solution procedure.     

Discretization results for the Huff and Pareto-Huff competitive location models on networks

In this paper we prove that there always exists a finite set that includes an optimal solution for the Huff and the Pareto-Huff competitive models on networks with the assumption of a concave function of the distance. In the Huff model, there is always a vertex of the network that belongs to the solution set. For the Pareto-Huff model, we prove that there is always an optimal solution at, or an ε-optimal solution close to, a vertex or an isodistant point, a new concept introduced in this paper.     

Retail store location and pricing within a competitive environment using constrained multinomial logit

The purpose of this paper is to report a pricing and retail location model using the constrained multinomial logit (CMNL), which takes into account customers’ utility and maximum willingness to pay via cut-off soft-constraints. The proposed model is probabilistic and non-linear, therefore a PSO metaheuristic approach was designed to determine the most suitable price, store locations and demand segmentation. The results obtained in test-cases showed a close relationship between price and location decisions. In addition, the results suggest that not only price, but also location decisions are affected when the consumers’ maximum willingness to pay is considered.     

Dynamic facility location when the total number of facilities is uncertain: A decision analysis approach

Models developed to analyze facility location decisions have typically optimized one or more objectives, subject to physical, structural, and policy constraints, in a static or deterministic setting. Because of the large capital outlays that are involved, however, facility location decisions are frequently long-term in nature. Consequently, there may be considerable uncertainty regarding the way in which relevant parameters in the location decision will change over time. In this paper, we propose two approaches for analyzing these types of dynamic location problems, focussing on situations where the total number of facilities to be located in uncertain. We term this type of location problem NOFUN (Number Of Facilities Uncertain). We analyze the NOFUN problem using two well-established decision criteria: the minimization of expected opportunity loss (EOL), and the minimization of maximum regret. In general, these criteria assume that there are a finite number of decision options and a finite number of possible states of nature. The minisum EOL criterion assumes that one can assign probabilities for the occurrence of the various states of nature and, therefore, find the initial set of facility locations that minimize the sum of expected losses across all future states. The minimax regret criteria finds the pattern of initial facility locations whose maximum loss is minimized over all possible future states.     

Adaptation of the plant location model for regional environmental facilities and cost allocation strategy

This paper is divided into two parts. In the first part of the paper, the plant location model is adapted for the special case of siting wastewater treatment facilities when the wastewater sources and treatment facilities are arranged in a chain or linear configuration. In this problem, flows or shipments may be merged in common pipes that provide economies of scale in transport. In order to apply the plant location model, an appropriate definition of the additional cost incurred when a waste source joins a regional facility is required. In addition, sequential priority constraints are developed in the siting model in order to make possible proper accounting of transport costs. The new siting model can be conveniently solved by linear programming.In the second part of the paper, a dual of the plant location model is explored as a cost allocation method for the fixed charge facility siting problem. The constraint sets of the dual model can be shown to imply the core conditions of the related cost game; hence, a set of the dual variables from the dual problem can be regarded as rational cost allocations. The analysis places both facility siting and cost allocation in a common framework.     

Solving a Huff-like competitive location and design model for profit maximization in the plane

A chain wants to set up a single new facility in a planar market where similar facilities of competitors, and possibly of its own chain, are already present. Fixed demand points split their demand probabilistically over all facilities in the market proportionally with their attraction to each facility, determined by the different perceived qualities of the facilities and the distances to them, through a gravitational or logit type model. Both the location and the quality (design) of the new facility are to be found so as to maximise the profit obtained for the chain. Several types of constraints and costs are considered.     

A decomposition approach for facility location and relocation problem with uncertain number of future facilities

In this paper, we discuss two challenges of long term facility location problem that occur simultaneously; future demand change and uncertain number of future facilities. We introduce a mathematical model that minimizes the initial and expected future weighted travel distance of customers. Our model allows relocation for the future instances by closing some of the facilities that were located initially and opening new ones, without exceeding a given budget. We present an integer programming formulation of the problem and develop a decomposition algorithm that can produce near optimal solutions in a fast manner. We compare the performance of our mathematical model against another method adapted from the literature and perform sensitivity analysis. We present numerical results that compare the performance of the proposed decomposition algorithm against the exact algorithm for the problem.