Lexicographically Minimum and Maximum Load Linear Programming Problems

In this paper we introduce the lexicographically minimum load linear programming problem and we provide a polynomial approach followed by the proof of correctness. This problem has applications in numerous areas where it is desirable to achieve an equitable distribution or sharing of resources. We consider the application of our technique in the problem of lexicographically minimum load in capacitated multi-commodity networks and discuss a special non-linear case, the so-called Kleinrock load function. We next define the lexicographically maximum load linear programming problem and deduce a similar approach. An application in the lexicographically maximum concurrent flow problem is depicted followed by a discussion on the minimum balance problem as a special case of the lexicographically maximum load problem.