Lagrangian relaxation bounds for a production-inventory-routing problem

We consider a single item Production-Inventory-Routing problem with a single producer/supplier and multiple retailers. Inventory management constraints are considered both at the producer and at the retailers, following a vendor managed inventory approach, where the supplier monitors the inventory at retailers and decides on the replenishment policy for each retailer. We assume a constant production capacity. Based on the mathematical formulation we discuss a classical Lagrangian relaxation which allows to decompose the problem into four subproblems, and a new Lagrangian decomposition which decomposes the problem into just a production-inventory subproblem and a routing subproblem. The new decomposition is enhanced with valid inequalities. A computational study is reported to compare the bounds from the two approaches

Operational fixed job scheduling problem under spread time constraints: a branch-and-price algorithm

This study addresses the operational fixed job scheduling problem under spread time constraints. The problem is to select a subset of jobs having fixed ready times and deadlines for processing on identical parallel machines such that total weight of the selected jobs is maximised. We first give a mathematical formulation of the problem and then reformulate it using Dantzig-Wolfe decomposition. We propose a branch-and-price algorithm that works on the reformulation of the problem. Computational results show that our algorithm is far superior to its competitor in the literature. It solves instances that could not be solved in one hour CPU time in less than a second and is able to solve large-scale instances in reasonable times which make it a computationally viable tool for decision-making