An Approximation Algorithm for the Facility Location Problem with Lexicographic Minimax Objective

Figure 1

Schematic illustrating the definition of customers. The set of aggregate customers 𝐽 is composed of two elements, that is, 𝐽={1,2}, and the set of individual customers is 𝐽={1,2,3,4,5}. The aggregate customer 1 includes individual customers 1 and 2, and the aggregate customer 2 stands for individual customers 3, 4, and 5. Thus, in this case the function 𝑗(𝑘), mapping the individual customer 𝑘∈𝐽 to the aggregate customer, returns these values 𝑗(1)=1, 𝑗(2)=1, 𝑗(3)=2, 𝑗(4)=2, and 𝑗(5)=2, and the weights (multiplicities) assigned to aggregate customers are 𝑏₁=2 and 𝑏₂=3.