Decomposition of Multi-Player Linear Programs

Abstract

We consider a set of n players, each of which produces and consumes a set of q commodities. Commodities produced by player i are consumed by that same player, and/or exported to players j = 1,… ‚ n possibly with some loss factor along arc ij. We set for player i the following mathematical program

$$ \left( {L{P_i}} \right)\left\{ {{A_i}{x_i} - \sum\limits_j {E_{ij}^{ - 1}{s_{ij}} + } \mathop {\mathop {\sum\limits_j^{\mathop {\min {c_i}{x_i}}\limits_{_{{x_i}}} } {{s_{ji}} \ge {b_i}} }\limits_{{B_i}{x_i} \ge {d_i}} }\limits_{{s_{ij}} \ge 0\,\,\,\,all\,j,} } \right. $$

(a)

where x _i is the vector of activities of player i C _i is the vector of unit costs of the activities S _ij is the q dimensional vector of exportations from i to j (delivered to j) E _ij is a q × q diagonal matrix (eiℓj) with eiℓjequal to the fraction of flow ℓ along ij that arrives at destination j A _i defines the production constraints of player i b _i is the vector of demands for the q commodities by player i B _i , d _i define linear constraints involving only player i. These constraints will also be denoted x _i ∈ L _i in the sequel.