Introduction: The Voronoi game is a simple geometric model for Competitive Facility Location problem which is played by two players, White and Black, in a continuous space (one-dimensional or two-dimensional). In the one-round game, White places all his n points. After that, Black places the same number of points on the game space. The players Cannot change or reuse a point that was placed before. At the end of the game, the Voronoi diagram of all 2n points is constructed and a player who obtains the larger total area is the winner...

In this paper, a nonlinear model for locating service facilities is introduced in a Competitive ‎region. In the proposed model the factors of travel time, quality of service facilities and price ‎are considered that are the most important effective factors in attracting and maintaining the ‎customers in the Competitive environments. For identifying the customers’ probabilistic ‎behavior in selecting the competitor’s and entering firm’s facilities, the spatial interaction ‎model and Logit function are used. Since the decision making for locating is not only based on ‎the sale quantity and market demand captured and ignoring other important organizational ‎goals such as minimizing the cost may provide problems in long term for firms, therefore in ‎addition to the capturing demand, the fixed cost of locating is considered in the model that ‎will provide better results in the real world problems. Due to the computational complexity, a ‎genetic algorithm is developed for solving the model. The numerical results verify the ‎performance of the proposed model and effectiveness of‏ ‏‎ the genetic algorithm in solving it‏ ‏and show that the study of costumers’‎‏ ‏probabilistic demand considering the factors of price, ‎travel time and quality of service facilities has the major role in the increasing of sale and profit ‎of the entering firm, so that, in spite of selecting the most appropriate Locations by the ‎competitor, the‏ ‏sale and profit quantities of entering firm are more than the competitor’s, in the ‎equal situation of price, quality and number of facilities.‎

In this paper, there are two competitors in a planar market. The first competitor, called, the leader, opens p new facilities. After that, the second competitor, the follower, reacts to the leader's action and opens r new facilities. The leader and the follower have got some facilities in advance in this market. The optimal Locations for leader and follower are chosen among predefined candidate Locations. In this paper, it is assumed that the demand is inelastic. Considering the huff model, customers share their demand among all facilities, probabilistically. The leader's objective function is the maximization of its market share after the follower's new facilities entry. The leader and follower problems are solved via an exact solution method.

In the Competitive Location problems, the matter of the optimal Location of single or multiple facilities are in a condition in which competitors exist as well. This paper deals with a type of a Competitive Location on which a leader possibly uses the investment of other investors through concession and receives a percentage of their income. He also can place his own facilities on potential Locations that are available. In fact there are three decision-makers, one as a leader, others as followers who get in the game of Facility Location for placing their facilities. The Location of facilities is a simultaneous game in which decisions are made in two levels, in the first level they are asynchronous and sequential, in the second level decisions are synchronous, namely the leader selects places for his own facilities, where other decision-makers (investor) play the simultaneous and non-cooperative game, and according to Nash equilibrium, select the optimal Location. The leader uses new Locations to optimize his objective function. Eventually, a numerical example is presented to test the model and a comprehensive sensitivity analysis is carried out to extract some managerial insights.

Facility Location along with taking account the competition is a critical component of strategic planning. The Location affects the Competitiveness and market share of new entrants, in long-term. Because of the complications of Location problems, most of the Location models are faced whit many different limitations. In this paper a Competitive Facility Location model with three competitors which produce two commodities is developed. The problem is modeled in the form of a Location-price game. A bi-level model is developed to investigate the Location and price. In Location-pricing models, the Location decision should come before the pricing game. From the second level and the use of Nash equilibrium, equilibrium will be obtained. In this paper, probabilistic costumer behavior with the help of exponential function is modeled. A metaheuristic based on tabu search is proposed to search the optimal Location-price solution of the model. Findings show that changing the equilibrium price affect the profitability of new entry firms.

The Facility Location problem is a strategic decision-making for a supply chain which determines the profitability and sustainability of its components. This paper deals with a scenario where two supply chains, consisting of a producer, a number of distribution centers and several retailers provided with similar products, compete to maintain their market shares by opening new distribution centers because of increasing demand. The competition problem is formulated as a non-linear integer bi-level mathematical model, where the upper level represents the decisions of the leader producer and the lower level administrates the decisions of the follower producer. It has been shown that even in small-scale problems, bi-level mathematical programming problems are strongly NP-hard, so an adapted bi-level ant colony algorithm with inter-level information sharing is developed to solve the problem. To evaluate the performance of the developed ant colony algorithm, the upper bound of the Competitive Facility Location problem is determined by solving the upper-level problem as an integer linear programming model without considering the follower’ s decision. Comparing the computational results of the developed ant colony algorithm with those of the determined upper bounds shows the satisfactory capability of the proposed approach for solving even medium-and large-scale problems.

This paper presents a bi-objective Competitive Facility Location model for congested systems in which the entering facilities compete with the competitors’ facilities for capturing the market share. In the proposed model, customers can choose which Facility to patronize based on the gravity function that depends on both the quality of service provider and the travel time to facilities. The proposed model attempts to simultaneously maximize the captured demand from each Facility and minimize the total waiting time in the system. Two multi-objective evolutionary algorithms, involving a multi-objective harmony search algorithm (MOHS) and a non-dominated sorting genetic algorithm-II (NSGA-II), are designed to solve the proposed model. The performance of solution procedures are compared on the basis of different performance metrics including generational distance, spacing metric, diversification metric, and number of non-dominated solution. The computational results based on different test problems show that in general MOHS outperforms NSGA-II.

In this article, authors propose a mixed integer bicriterion linear programming model for Facility Locations network design problem under uncertainty in the Competitive environment. This model takes into consideration the possibility of competition and interruption in servicing availability. The objective functions are of cost minimization and Facility attraction maximization. The cost components are Facility construction cost, the routes' enhancement, improvement and or development expenses, and transportation costs. Due to the facts that customers' demands as well as transportation expenses are uncertain authors have employed robust type modeling of the problem taking scenario approach into consideration. We also have taken into considered the concept of reliable network design and attraction function for Facility Location in the competition environment. In order to show the application of the proposed model, a real case study discussing the Facility Location design and Facility implementation for a new CT-Scan system in Fars province was studied. The result of this study indicates that Fars providence has capacity for four CT-Scans that can be positioned in the cities of Fasa, Marvedasht, Estahban and Shiraz. For sensitivity analysis purposes, two parameters are considered separately and simultaneously. Parameter can have impacts on the cost objective function and hence on the final solution of the problem. The aim of this research is obtaining an answer that is less sensitive to the changes in data and meantime increases the system's reliability. A trade-off between the solution robustness and model robustness for various values of parameter is demonstrated by the figure in the body of the article. This tradeoff can help the decision makers in determining suitable weight for Due to the fact that when the number of scenarios increases the use of exact methods of solutions are impossible for the reason of being an Np-hard problem one needs to employ a meta-heuristic approach to solve this problem. This can be considered as a future research for those who want to work on the extension of this problem.