The inventory routing problem under uncertainty with perishable products: an application in the agri-food supply chain

Abstract

In this paper, we propose a dynamic and stochastic approach for an inventory routing problem in which products with a high perishability must be delivered from a supplier to a set of customers. This problem falls within the agri-food supply chain (${\mathcal {ASC}}$) management field, which includes all the activities from production to distribution. The need for high-quality products that are subject to perishability is a critical issue to consider in the ${\mathcal {ASC}}$ optimization. Moreover, the demand uncertainty makes the problem very challenging. In order to effectively manage all these features, a rolling horizon approach based on a multistage stochastic linear program is proposed. Computational experiments over medium-size instances designed on the basis of the real data provided by an agri-food company operating in Southern Italy show the effectiveness of the proposed approach.

Route generation procedure

This section illustrates the algorithm designed for generating the subset of feasible routes, for each instance and for each scenario tree. The procedure creates clusters of retailers by considering their aggregate demand for a relevant subset of scenarios, and generates one or more feasible routes for each cluster. For each scenario and for each time level, a subset of retailers is identified according to the following rule: a retailer is included in the subset if the service is necessary to avoid stock-out at that time, in the scenario. Then, for each subset of retailers a Capacitated Vehicle Routing Problem (${\mathcal {CVRP}}$) is solved. The quantity to deliver to each retailer in the subset is equal to the expected demand in the specific scenario and for the given time level. The routes obtained by the solution of the ${\mathcal {CVRP}}$ are evaluated to be included in the set of all the routes that are considered in the mathematical formulation presented in Sect. 3.1: only the routes that occur in almost a given number of solutions are added to the subset of all feasible routes. In the following, a brief description of the algorithm is proposed.

For a generic scenario s, let $N_s$ be the set of nodes which belong to s. Let $C_{s,n}$ be the cluster of retailers for node $n \in N$, and let $D_{s,n}$ be the overall demand of the retailers in s at level n. Moreover, let $\alpha _{k}$ be the number of times route k occurs in any solution. Finally, let R be the set of routes that are generated. Initially $R = \{ \overline{r}, r_1, \ldots , r_k, \ldots , r_{|K|} \}$, where $\overline{r}$ is the giant tour and $r_k, \ \forall k = 1, \ldots , |K|,$ is the direct delivery for retailer $k \in K$. The overall procedure is outlined in Algorithm 1

More specifically, step 1 builds clusters for each scenario $s \in S$ and for each node $n \in N_s$ are constructed. The details of the clustering algorithm are reported in Algorithm 2

Step 2 performs a check of the vehicle capacity with aim to build clusters such that the total amount of product to deliver is less or equal to the total capacity of the fleet. Details are provided by Algorithm 3

Step 3 outlines how routes are constructed and selected. Algorithm 4 provides a whole overview of the routes generation procedure.