$$\varepsilon $$ε-Constraint method for bi-objective competitive facility location problem with uncertain demand scenario

We consider a model of two parties’ competition organized as a Stackelberg game. The parties open their facilities intending to maximize profit from serving the customers that behave following a binary rule. The set of customers is unknown to the party which opens its facilities first and is called the Leader. Instead, a finite list of possible scenarios specifying this set is provided to the Leader. One of the scenarios is to be realized in the future before the second party, called the Follower, would make their own decision. The scenarios are supplied with known probabilities of realization, and the Leader aims to maximize both the probability to get a profit not less than a specific value, called a guaranteed profit, and the value of a guaranteed profit itself. We formulate the Leader’s problem as a bi-objective bi-level mathematical program. To approximate the set of efficient solutions of this problem, we develop an $$\varepsilon $$ε-constraint method where a branch-and-bound algorithm solves a sequence of bi-level problems with a single objective. Based on the properties of feasible solutions of a bi-level program and mathematical programming techniques, we developed three upper bound procedures for the branch-and-bound method mentioned. In numerical experiments, we compare these procedures with each other. Besides that, we discuss relations of the model under investigation and the stochastic competitive location model with uncertain profit values.