درباره موقعیت امکانات جدید برای گسترش زنجیره تحت قیمت گذاری تحویل

We study the problem of locating new facilities for one expanding chain which competes for demand in spatially separated markets where all competing chains use delivered pricing. A new network location model is formulated for profit maximization of the expanding chain assuming that equilibrium prices are set in each market. The cannibalization effect caused by the entrance of the new facilities is integrated in the objective function as a cost to be paid by the expanding chain to the cannibalized facilities. It is shown that the profit of the chain is maximized by locating the new facilities in a set of points which are nodes or iso-marginal delivered cost points (points on the network from which the marginal delivered cost equals the minimum marginal delivered cost from the existing facilities owned by the expanding chain). Then the location problem is reduced to a discrete optimization problem which is formulated as a mixed integer linear program. A sensitivity analysis respect to both the number of new facilities and the cannibalization cost is shown by using an illustrative example with data of the region of Murcia (Spain). Some conclusions are presented.

The location of facilities is a major decision for a chain that competes for customers demand with other chains offering the same type of product. A variety of location models have been proposed to cope with this kind of problem (see for instance [1], [2] and [3]). When the competing chains use delivered pricing, profit is strongly affected by both the location of their facilities and the price they set in each market area. For spatially separated markets and homogeneous products, the main feature of many of these location problems is that each chain monopolizes a group of markets once the locations of the facilities are fixed. Therefore, each chain sets the optimal price in each one of its monopolized markets. Delivered prices are frequently used when the ratio of transportation cost to the total price paid by the customers is high, which has been observed in many markets [4]. With this price policy, once the facility locations are fixed, a Nash equilibria in price for the competing chains can be found under quite general conditions. Hoover [5] analyzed this price policy for the first time, considering each chain locates one facility, and concluded that the equilibrium market price of a chain with the lowest delivery cost is equal to the next lowest delivery cost. This result has been extended to a spatial duopoly on a compact subset of the plane [6] and to a network [7]. The existence of Nash equilibrium under delivered pricing has been recently reconsidered in [8]. As result of price competition, the competing chains will set the equilibrium prices, if they exist, once they know the location of their facilities. In such a case, the location-price decision problem for a chain under competitive delivered prices reduces to a location problem. This kind of location problem has been mainly studied on a network location space within two frameworks, one considering that all competing chains decide on location, another considering that there is one entering chain which decides on location and competes with other chains which have their facility locations already fixed. The first is seen as a non-cooperative game, for which a node-optimality property and some location equilibrium results have been given when each chain opens one facility [7] and [9]. A procedure to find location equilibria when each chain opens more than one facility is shown in [10]. The second is seen as an optimization problem, which has been solved in discrete location space by integer linear programming formulations for fixed demand in [11] and variable demand in [12]. In network location space, a node-optimality property is shown and the problem is solved for variable demand by a mixed integer linear programming formulation in [13]. The aim of this paper is to study the location problem in a new framework in which the existence of fixed facilities owned by some competing chains operating in the network is considered and one of such chains wants to expand by locating new facilities on the network. The new facilities will compete with each other, as well as with any existing facility, owned by the expanding chain or by any of its competitors. Therefore, the pre-existing facilities owned by the expanding chain can lose profit as a consequence of the expansion. This effect is known as cannibalization and it was first considered in franchise distribution systems (see [14] and [15]), but it has been almost ignored in the recent location literature. To our knowledge, cannibalization has been taken into account mostly in Huff-like location models, but it has not been studied under delivered pricing. In discrete location, it has been considered in a single-objective location model with variable expenditure functions in [16]. In that model the effect is not explicitly present, but the cannibalized facilities may increase their demand due to market expansion as result of the entrance of new facilities. In [17], a model to simultaneously optimize the locations and designs of a set of new facilities is studied in which the cannibalization effect is captured. In planar location, it has been considered as a secondary objective in multi-objective location of a single facility in [18] and [19], where lexicographic approaches are used. In this paper, a sharing profit model is formulated where the cannibalization effect is integrated in the objective function as a cost instead of as a secondary objective. Our contribution is the formulation and study of a new location model under delivered pricing in which the profit lost by the existing facilities owned by the chain as result of the expansion is compensated by side payments. The income of the entire chain is a portion of the profit of its facilities which is determined under the assumption that the competing chains will set the equilibrium prices in each market once the new facilities are fixed. Minimum profit constraints are considered to make the new facilities economical to operate. We formulate this sharing profit model on a network location space where nodes and any point on the edges are location candidates. First we show that the profit of the chain (income minus side payments) is maximized by locating the new facilities in a finite set of points given by the nodes and the points in the network from which the marginal delivered cost equals the minimum marginal delivered cost from the existing facilities owned by the chain. Then the location model is reduced to a discrete optimization problem which is formulated as a mixed integer linear programming problem. Finally, the cannibalization effect caused by the new facilities is analyzed by using an illustrative example with real data. A sensitivity analysis with respect to the number of new facilities and the cost of side payment is also carried out. In Section 2, basic hypothesis and notation are given, the equilibrium prices are determined, and the location model is formulated. In Section 3, the results concerning the maximization of the net profit are shown. In Section 4, the location model is reduced to a discrete optimization problem which is formulated as a mixed integer linear programming problem that can be solved by standard optimizers. In Section 5, an illustrative example with data of the region of Murcia (Spain) is analyzed. Finally, some conclusions are presented in Section 6.

A new location model for an expanding chain, which takes into account the cannibalization effect caused by the new facilities, has been proposed under the assumption that its competitors have their locations fixed and all competing chains use delivered pricing. It is shown that the optimal facility locations on a network can be found in a finite set of points and a mixed integer linear programming formulation (MILP problem) to find the optimal locations has been proposed. This formulation allows to make a sensitivity analysis with respect to the number of new facilities to be located and the cannibalization cost. This is illustrated by solving 560 test problems with data from the Region of Murcia (Spain). The optimal location candidates are nodes and points on the network from which the marginal delivered cost equals the minimum marginal delivered cost from the pre-existing facilities owned by the expanding chain. This set is the same for any demand function at any population node. Furthermore, the set of location candidates does not depend on the location of the competitors of the expanding chain, but it only depends on the delivered costs of the pre-existing facilities of the expanding chain. For the test problems, 45% of the optimal locations were population nodes, 24% linking nodes and 31% iso-mdc points. The average number of optimal solutions which are population nodes increases in the number of new facilities, but it is nearly the same for each value of the side payment. With respect to profit and cannibalization cost, when the number of new facilities varies, it was obtained that the profit of the entire chain is strongly increasing with the number of new facilities, but the profit per new facility decreases with the number of new facilities. However, both the cannibalization cost and the cannibalization cost per new facility smoothly increase with the number of new facilities. When the side payment varies, it was obtained that both the profit of the entire chain and the cannibalization cost smoothly decrease as the side payment increases. Therefore, the cannibalization effect caused by chain expansion can be reduced by increasing side payments without a strong decrease in profit. Thus, side payment is a good way to cope with cannibalization.