A Branch-Price-and-Cut Algorithm for Single-Product Maritime Inventory Routing

Abstract

A branch-price-and-cut algorithm is developed for a complex maritime inventory-routing problem with varying storage capacities and production/consumption rates at facilities. The resulting mixed-integer pricing problem is solved exactly and efficiently using a dynamic program that exploits certain “extremal” characteristics of the pricing problem. The formulation is tightened by using the problem's boundary conditions in preprocessing and to restrict the set of columns that are produced by the pricing problem. Branching schemes and cuts are introduced that can be implemented efficiently and that preserve the structure of the pricing problem. Some of the cuts are inspired by the capacity cuts known for the vehicle-routing problem, whereas others specifically target fractional solutions brought about by individual vessels “competing” for limited inventory at load ports and limited storage capacity at discharge ports. The branch-price-and-cut approach solves practically sized problems motivated by the operations of an oil company to optimality, and it provides reasonable bounds for larger instances.

Discrete time and continuous time formulations for a short sea inventory routing problem

“…The inequalities used in this paper impose either a minimum number of visits to ports or a minimum number of (un)loads. Similar valid inequalities have been used in related papers for constant rate case and for the non constant consumption rates case, see for the last case, [1,2,18,20,28]. When consumptions are not constant new inequalities based on lot-sizing relaxations have been used, see [1,2,18].…”

“…These problems are often called Maritime Inventory Routing Problems (MIRPs). Most of the published MIRP contributions are based on real cases from the industry, see for the single product case [7,9,18,16,17,20,29] and for the multiple products case [5,11,24,25,27,28].…”

“…In [26] valid inequalities are included in order to enhance the proposed formulations to an oil products transportation problem, and in [23] valid inequalities are developed within a column generation approach for a maritime inventory routing problem. Also, in [18] valid inequalities are derived for a single-product maritime inventory routing, which are used within a branch-pricecut algorithm. In [20] valid inequalities are included to improve the formulation presented for the liquefied natural gas inventory routing problem.…”

“…Mathematical formulations, and related discussion, for maritime inventory routing problems have received some attention during the last decades, see for instance, [1,2,3,5,8,18,28]. However, to the best of our knowledge, comparison of different formulations for a given MIRP have been considered only in [1] and [2].…”

“…For each approach, following [2] (see also [1] for a completely discrete model), we develop an arc-load formulation and two extended formulations. Arc-load formulations are the most used formulations in MIRPs, see [5,9,18,28]. The extended formulations use new sets of variables that provide additional information about the solution.…”