A generalized bi-criteria fuzzy integer flow sharing problem

The flow sharing problem is a class of techniques that can be used to find the optimal flow in a capacitated network, which realizes an equitable distribution of flows. This paper extends the integer flow sharing problem by considering fuzzy capacities and fuzzy weights such that the flux received at each sink node and the flow value through each arc are restricted to be multiples of some block unit. Fuzzy capacity describes the flexibility of the upper limit of flow value through each arc. Fuzzy weight represents the degree of satisfaction of the flux to a sink node. Our model has the two following criteria: to maximize the minimal degree of satisfaction among all of the fuzzy capacity constraints and to maximize the minimal degree of satisfaction among the fluxes to all of the sink nodes. Because an optimal flow pattern that simultaneously maximizes the two objectives is usually not feasible, we define non-domination in this setting and propose a pseudo-polynomial algorithm that finds some non-dominated flow patterns. Finally, a numerical example is presented to demonstrate how our algorithm works.