## Abstract

The Inventory Routing Problem (IRP) aims at determining the best distribution plan of a certain commodity from a supplier to a set of customers over a given planning horizon. Customers face a per period consumption and the plan has to be such that they can always satisfy it. The aim is to minimize the total cost of the distribution plan, which is given by the sum of routing and inventory holding cost. A replenishment strategy that has become common in the recent literature is the Maximum Level (ML) strategy, where no restriction is imposed on when the customers are visited and what the quantity delivered is, provided that a maximum inventory level at each customer (corresponding to the customer′s warehouse capacity) is not exceeded. This strategy is highly flexible and this clearly provides cost advantages. However, it might also create inconveniences to customers, who might receive visits when their inventory level is still relatively high. Towards addressing this issue, we study in this paper the Zero-Inventory-Ordering (ZIO) replenishment strategy for the IRP, where customers can be served only when their inventory level is zero. The aim is to analyze the impact of a policy which is more customer-orientedthan the ML. The analysis is made in terms of advantages from the customers′ perspective and disadvantages from the system perspective, i.e., increase of the total cost. We propose a formulation based on two-commodity flow variables, and strengthen it through new valid inequalities tailored for our problem as well as with valid inequalities inherited from the literature on the IRP. We develop branch-and-cut algorithms and experimentally compare the ZIO and ML policies on benchmark IRP instances. The results show that the application of the ZIO policy allows to reduce the inventory costs at the customers without substantially increasing, or even decreasing, the number of visits to them. This comes at the expense of an increase of the total costs. Results also show that the increase in total cost is inversely proportional to the relative weight of the inventory costs at the customers. More precisely, inventory costs at customers are decreased by more than 20% on average while total cost increases, on average, by 17.6% when the inventory costs are high and of 9.2% when inventory costs are low. In addition, we generated new sets of instances in which inventory costs are more aligned with practical applications with respect to what happens in the benchmark IRP instances. Results show how the ZIO policy consistently provides solutions where the inventory cost to customers is lower than the one associated with the ML policy. Also, the gap in total cost reduces when the inventory cost at customers increases.

Original language | English (US) |
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Pages (from-to) | 1024-1038 |

Number of pages | 15 |

Journal | European Journal of Operational Research |

Volume | 312 |

Issue number | 3 |

DOIs | |

State | Published - Feb 1 2024 |

## Keywords

- Branch-and-cut
- Inventory routing problem
- Routing
- Zero-inventory-ordering

## ASJC Scopus subject areas

- Computer Science(all)
- Modeling and Simulation
- Management Science and Operations Research
- Information Systems and Management