Facility Location

Geographic Representation

• Location problems are based on how geography is modeled.
• In 2D, location can be modeled as x-y coordinate plane, or in a network.
• Two ways of measuring distances between two points with coordinates ($x_i$, $y_i$) and ($x_j$, $y_j$)

  • Euclidean distance

    $$ d_{ij} = \sqrt{(x_i - x_j)^2 + (y_i - y_j)^2} $$

  • Metropolitan metric distance

    $$ d_{ij} = |x_i - x_j|+|y_i - y_j| $$

Number of Facilties

• Single facility location problem can be solved analytically with relative ease.
• Multi-facility location problems are difficult because of demand assignment at nodes. Also for some services, the type of facilities may vary.

Optimization Criteria

Objective function in the location problem is different for different services. Depending on the owner of the facility, the objectives could be:

• Maximization of utility, profit, social benefit
• Minimization of travel times, cost.

Single Dimension Single Facility Location Problem

$$ \text{Maximize } Z = \sum_{i=1}^m w_i (s-x_i) + \sum_{i=m+1}^n w_i (s-x_i) $$

where,

$w_i$ = relative demand at the location $i$,

$x_i$ = the location $i$ of the demand on the beach,

$s$ = the location where the vendor should set up the shop

$$ \frac{\partial Z}{\partial s} = 0 => \sum_{i=1} ^m w_i = \sum_{i=m+1}^n w_i $$