A duality‐based relaxation and decomposition approach for inventory distribution systems

Abstract

We propose a new method for making the inventory replenishment decisions in distribution systems. In particular, we consider distribution systems consisting of multiple retailers that face random demand and a warehouse that supplies the retailers. The method that we propose is based on formulating the distribution problem as a dynamic program and relaxing the constraints that ensure the nonnegativity of the shipments to the retailers by associating Lagrange multipliers with them. We show that our method provides lower bounds on the value functions and a good set of values for the Lagrange multipliers can be obtained by maximizing a concave function in a relatively straightforward manner. Computational experiments indicate that our method can provide significant improvements over the traditional approaches for making the inventory replenishment decisions, in terms of both the tightness of the lower bounds on the value functions and the performance of the policies.

Linear programming based decomposition methods for inventory distribution systems

“…However, these papers either do not use Lagrange multipliers of any kind to penalize the violations of the relaxed constraints or do not give general purpose methods to compute Lagrange multipliers for any demand distribution. Kunnumkal and Topaloglu (2008) build on the work described in the paragraph above by relaxing the constraints that ensure the nonnegativity of the shipments to the retailers, but they use Lagrange multipliers to explicitly penalize the violations of the relaxed constraints. Their contribution is to give a general purpose method that chooses a good set of Lagrange multipliers by solving a convex optimization problem.…”

“…Their contribution is to give a general purpose method that chooses a good set of Lagrange multipliers by solving a convex optimization problem. The computational experiments in Kunnumkal and Topaloglu (2008) indicate that the quality of the lower bounds and the performance of the inventory replenishment policies can be improved by associating Lagrange multipliers with the relaxed constraints. However, the unfortunate aspect of the work in Kunnumkal and Topaloglu (2008) is that solving the aforementioned convex optimization problem to find a good set of Lagrange multipliers is a computationally intensive and complex process.…”

“…The computational experiments in Kunnumkal and Topaloglu (2008) indicate that the quality of the lower bounds and the performance of the inventory replenishment policies can be improved by associating Lagrange multipliers with the relaxed constraints. However, the unfortunate aspect of the work in Kunnumkal and Topaloglu (2008) is that solving the aforementioned convex optimization problem to find a good set of Lagrange multipliers is a computationally intensive and complex process. Solving their convex optimization problem through subgradient optimization requires calibrating the step size parameter and devising a stopping criterion, for which there are no hard and fast rules.…”

“…The computational overhead associated with the large amount of matrix algebra, combined with a large number of iterations required for subgradient optimization, results in extensive computation times. Our interactions with supply chain practitioners indicate that the approach proposed by Kunnumkal and Topaloglu (2008) is theoretically interesting, but may not be practically attractive. In particular, extensive computation times may not only prevent dealing with industrial size systems, but may also prevent doing quick what if analyses.…”

“…One of our goals is to address the shortcomings of Kunnumkal and Topaloglu (2008). We propose efficient methods to obtain a good set of Lagrange multipliers.…”