A Variational Inequality-Based Optimization Model for Flexible Manufacturing Resource Allocation

Abstract

Flexible manufacturing systems need allocation schemes that can take care of alternative resource eligibility, capacity coupling, nonlinear processing effects, congestion, and workload imbalance altogether. The conventional single objective optimization approach cannot be able to capture adequately the directionality aspect of interaction among allocation variables with asymmetric and non-conservative marginal effects. This paper introduces a variational inequality (VI)-based mathematical model for flexible manufacturing resource allocation where the equilibrium can be characterized directly via an allocation operator. Quadratic cost allocations, nonlinear processing time effects, capacity-dependent congestion, workload imbalance, and asymmetric resource interaction effects are taken into account in the formulation. An artificial restriction on the feasible set is imposed to preserve the positivity of the distance from the case of total utilization of resources so as to ensure boundedness of the derivative of the congestion function and local Lipschitz continuity of the operator. Existence follows from the continuity and convexity of the feasible set, while uniqueness follows from the strong monotonicity of the operator. With respect to the asymmetric Jacobian matrix, strong monotonicity is shown via the symmetric part of the matrix without assuming symmetry of the Jacobian matrix. Capacity multipliers are used to express endogenously resource-clearing prices. A projected VI method is presented, and its convergence is proved under explicit strong monotonicity and Lipschitz conditions. Numerical examples of allocation instances of sizes 10×10, 50×50, and 100×100 illustrate feasibility, linear residual convergence, sensitivity of the step sizes, flexibility effects, and sensitivity of capacity factors.