Market Sharing: Assigning Retailers to Company Divisions

Objectives

The first objective is to minimize the following (nonlinear, nondifferentiable) $L_1$ norm function:

\[ \Variable{Objective1} = \sum _{\substack{\text {category} \in \text {CATEGORIES},\\ \text {division} \in \text {DIVISIONS}}} \left| \Variable{MarketShare[category,division]} - \Argument{target[division]} \right| \]

The second objective is to minimize the following (nonlinear, nondifferentiable) $L_\infty $ norm function:

\[ \Variable{Objective2} = \max _{\substack{\text {category} \in \text {CATEGORIES},\\ \text {division} \in \text {DIVISIONS}}} \left| \Variable{MarketShare[category,division]} - \Argument{target[division]} \right| \]