Optimizing Facility Placement: A Mathematical Model for Customer Allocation

Problem Background

Many companies across industries must at some point make strategic decisions about where to build facilities that support their operations, such as designing supply chains for manufacturers (factories, distribution centers, warehouses, retail stores) or determining hospital locations for healthcare providers to maximize population coverage.

Problem Description

The customer allocation problem is closely related to the facility location problem, which seeks the optimal placement of facilities (chosen from a set of candidate sites) to minimize the distance between facilities and customers. When facilities have unlimited capacity, each customer is assumed to be served by the nearest facility.

If the number of customers is too large, they can be grouped into clusters, and cluster centers can replace individual customer locations. This preprocessing assumes that all customers in a given cluster will be served by the facility assigned to that cluster. The clustering can be performed using the k-means algorithm, which partitions objects into k non-overlapping clusters.

Mathematical Model

Sets and Indices

I: index of customer clusters

J: index of potential facility locations

K: index of allowable customer‑facility pairings

Parameters

d_ij: maximum allowable distance between customer cluster i and facility j

P: maximum number of facilities to be opened

n_i: number of customers in cluster i

c_ij: distance from customer cluster i to facility j

Decision Variables

x_j: equals 1 if facility j is selected, otherwise 0

y_ij: equals 1 if customer cluster i is assigned to facility j, otherwise 0

Objective Function

Total Distance : Minimize the total distance from customer clusters to the selected facilities.

Constraints

Facility Limit : The number of opened facilities cannot exceed the maximum P.

Facility Activation : An opened facility can be matched to at most one customer cluster.

Nearest Facility Assignment : Each customer cluster can be matched to only one facility.