The Set Covering / Packing / Partitioning Problem

Prerequisites

• Please look at Getting Started first for the most basic functions and the setup of OPTANO.Modeling

The mathematical Model

Sets: $$U = \text{the set containing }n\text{ elements}$$

Parameters: $$c_j = \text{cost of subset } j$$

$$a_{ij} = \begin{cases} 1, \text{if element i is contained in set j} \newline 0, \text{else} \end{cases} $$

Variables: $$x_j = \begin{cases} 1, \text{ if you pick set } j \newline 0, \text{ else} \end{cases} $$

Objective: $$ min \sum\limits_{j=1}^n c_j x_j $$

Restrictions: \begin{array}{l} \sum\limits_{j=1}^n a_{ij} x_j \ge 1 & \qquad \forall i=1,...,m & \qquad \text{(Set Covering)}\newline \sum\limits_{j=1}^n a_{ij} x_j = 1 & \qquad \forall i=1,...,m & \qquad \text{(Set Partitioning)}\newline \sum\limits_{j=1}^n a_{ij} x_j \le 1 & \qquad \forall i=1,...,m & \qquad \text{(Set Packing)} \end{array}

The Set Covering / Packing / Partitioning Problem

• Step 1: Create Business objects for your Model
• Step 2: Create your Model Class
• Step 3: Retrieve the Solution of your Model