Shared Investment for Network Flow under Fuzzy Membership Scenarios

Abstract

Capacity improvement is often planned from one network assessment even though edge suitability varies across operating conditions. We study a common investment vector that supports a required terminal flow in every member of a finite collection of fuzzy membership scenarios. Routing can change with the scenario, while the installed capacity increments are shared. The model is a linear program with explicit investment caps. We prove equivalence with scenario dependent cut inequalities, derive a dual cost bound and characterize the largest feasible service target. The optimal investment cost is a nondecreasing piecewise linear convex function of that target. A six vertex example shows that a design requiring no investment under the nominal assessment can miss the requested service in other scenarios; a shared investment of 10 normalized units meets all three requirements. Forty generated perturbations compare nominal and shared designs using identical cost data. The formulation combines a clear membership to capacity conversion with independently checkable flow and cut conditions. The worked optimum also admits a direct lower bound from three disjoint sets of cut arcs, so its cost can be verified by hand. The analysis distinguishes common installation from adaptive routing and explains when a componentwise smallest scenario makes nominal design sufficient. Scenario coverage is conditional on the chosen assessments.