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Using valid inequalities to solve the integrated production-inventory-distribution-routing problem

The production-inventory-distribution-routing problem is an integrated supply chain management problem that combines decisions on several functions. The objective is to minimise the total costs without violating demand fulfilment policy. A production-inventory-distribution-routing problem of medium size is a combinatorial optimisation problem mostly intractable to solve using exact methods. The main contribution of this work is to introduce valid inequalities for a problem with a single plant, multiple products and multiple heterogeneous vehicles to improve the quality of lower bounds, obtain a good approximation of the convex hull of the polyhedron of the problem and reduce its hypervolume, so that the computation time can be reduced without a significant effect on the quality of the solutions found. The results showed that adding the valid inequalities to the model can improve the percentage gaps for all the tested instances with a significant improvement in the lower bounds from the poor bounds obtained from the linear programming relaxation (up to 98.8% for the dataset of 50 customers and up to 79.7% for the dataset of 100 customers).