1 Mathematical model

Assume that the set I of potential facility locations and the set J of clients are finite. For each facility i ∈ I we have the set R_{i} of design scenarios and this set is finite as well. For each pair i ∈ I, r ∈ R_{i} we have the fixed costs f_{ir} and g_{ir} of opening facility i with design scenario r by the leader and by the follower, respectively. Moreover, we know the attractiveness a_{ir} of the leader facility and the similar parameter b_{ir} of the follower facility. The last two features are important for describing the client behavior. Each client j splits own demand w_{j} probabilistically over all facilities directly proportional with attraction to each facility and inversely proportional to the distance d_{ij} between client j and facility i. Following [1], we consider the utility function u_{ijr} of leader facility i with design scenario r for client j and the similar function v_{ijr} for follower facility:

u_{ijr} = a_{ir}/(d_{ij} + 1)^{β}, v_{ijr} = b_{ir}/(d_{ij} + 1)^{β}, i ∈ I, r ∈ R_{i}, j ∈ J,

where β is a distance sensitivity parameter. Now we introduce the decision variables for the players: x_{ir} is equal to 1 if facility i is open by the leader with design scenario r and 0 otherwise; y_{ir} is equal to 1 if facility i is open by the follower with design scenario r and 0 otherwise. For client j, the total utility U_{j} from the leader facilities and the total utility V_{j} from the follower facilities are defined as:

U_{j} = ∑_{i∈I} ∑_{r∈R_{i}} u_{ijr}x_{ir}, V_{j} = ∑_{i∈I} ∑_{r∈R_{i}} v_{ijr}y_{ir}, j ∈ J.

The total market share of the leader is given by ∑_{j∈J} w_{j}U_{j}/(U_{j} + V_{j}). The leader wishes to maximize own market share, anticipating that the follower will react to the decision by opening own facilities. The market share of the follower is given by ∑_{j∈J} w_{j}V_{j}/(U_{j} + V_{j}). The follower maximizes own market share. In opposite [2], we assume that the players can open facilities at the same site. This Stackelberg game can be presented as the following nonlinear 0–1 bilevel optimization problem [3]:

max_{x} ∑_{j∈J} w_{j} \frac{∑_{i∈I} ∑_{r∈R_{i}} u_{ijr}x_{ir}}{∑_{i∈I} ∑_{r∈R_{i}} u_{ijr}x_{ir} + ∑_{i∈I} ∑_{r∈R_{i}} v_{ijr}y_{ir}^{∗}}  (1)

subject to

∑_{i∈I} ∑_{r∈R_{i}} f_{ir}x_{ir} ≤ B_{l};  (2)
∑_{r∈R_{i}} x_{ir} ≤ 1,  i ∈ I;  (3)
x_{ir} ∈ {0, 1},  r ∈ R_{i}, i ∈ I;  (4)

where y_{ir}^{∗} is the optimal solution for the follower problem:

max_{y} ∑_{j∈J} w_{j} \frac{∑_{i∈I} ∑_{r∈R_{i}} v_{ijr}y_{ir}}{∑_{i∈I} ∑_{r∈R_{i}} u_{ijr}x_{ir} + ∑_{i∈I} ∑_{r∈R_{i}} v_{ijr}y_{ir}}  (5)

subject to

∑_{i∈I} ∑_{r∈R_{i}} g_{ir}y_{ir} ≤ B_{f};  (6)
∑_{r∈R_{i}} y_{ir} ≤ 1,  i ∈ I;  (7)
y_{ir} ∈ {0, 1},  r ∈ R_{i}, i ∈ I.  (8)

Objective functions (1) and (5) are market shares of the players. Inequalities (2) and (6) are the budget constraints: B_{l} is the budget of the leader, B_{f} is the budget of the follower. Inequalities (3) and (7) ensure the only design scenario for each open facility.

References

[1] Aboolian R., Berman O., Krass D. Competitive facility location and design problem. European J. Oper. Res, 2007. Vol. 182. P. 40–62.

[2] E. Alekseeva, Yu. Kochetov, N. Kochetova, and A. Plyasunov. Heuristic and exact methods for the discrete (r|p)–centroid problem. LNCS, 2010. Vol. 6022. P. 11–22.

[3] Yu. Kochetov, N. Kochetova, and A. Plyasunov. A matheuristic for the leader-follower facility location and design problem. 10th international Metaheuristics Conference (MIC 2013). Singapure 5-8 August, 2013.