Efficient locations to optimize market share and maximum travel time in a duopoly competition model

We consider the problem of locating a new facility on a given path in a tree network with regard to two criteria: maximization of market share and minimization of maximum distance between the facility and each of the customers that it serves. An existing facility is already located on the tree and customers are located on the nodes. Once a new facility is located each customer at node i selects its closest facility to be served, ties are broken by assigning a proportion p(i) (such that 0 < p(i) <1), of the demand at node i to the new facility. We show that this problem can be reduced to a problem on linear market by a slight modification of the two objective functions. The problem is analyzed on that linear market and an algorithm is given to generate the set of efficient locations for the new facility.