Two-phase Matheuristic for the vehicle routing problem with reverse cross-docking

Abstract

Cross-dockingis a useful concept used by many companies to control the product flow. It enables the transshipment process of products from suppliers to customers. This research thus extends the benefit of cross-docking with reverse logistics, since return process management has become an important field in various businesses. The vehicle routing problem in a distribution network is considered to be an integrated model, namely the vehicle routing problem with reverse cross-docking (VRP-RCD). This study develops a mathematical model to minimize the costs of moving products in a four-level supply chain network that involves suppliers, cross-dock, customers, and outlets. A matheuristic based on an adaptive large neighborhood search (ALNS) algorithm and a set partitioning formulation is introduced to solve benchmark instances. We compare the results against those obtained by optimization software, as well as other algorithms such as ALNS, a hybrid algorithm based on large neighborhood search and simulated annealing (LNS-SA), and ALNS-SA. Experimental results show the competitiveness of the matheuristic that is able to obtain all optimal solutions for small instances within shorter computational times. For larger instances, the matheuristic outperforms the other algorithms using the same computational times. Finally, we analyze the importance of the set partitioning formulation and the different operators.

Appendix A: Supplier delivery process constraints

The constraints occurring in the supplier delivery process are formulated as follows.

$$ L \sum\limits_{v \in V} \sum\limits_{i \in S \cup 0, i \neq j} x_{ij}^{\prime\prime\prime v} \geq \sum\limits_{i \in O} r_{ij}^{\prime\prime} + \sum\limits_{i \in C} r_{ij}^{\prime} - \sum\limits_{v \in V} \sum\limits_{i \in O} A_{ij}^{\prime\prime v} \forall j \in S $$

(75)

$$ \sum\limits_{i \in O} r_{ij}^{\prime\prime} + \sum\limits_{i \in C} r_{ij}^{\prime} - \sum\limits_{v \in V} \sum\limits_{i \in O} A_{ij}^{\prime\prime v} \geq \epsilon - L \Bigg(1-\sum\limits_{v \in V} \sum\limits_{i \in S \cup 0, i \neq j} x_{ij}^{\prime\prime\prime v}\Bigg) \forall j \in S $$

(76)

$$ \sum\limits_{i \in S} \sum\limits_{j \in S, j \neq i} x_{ij}^{\prime\prime\prime v} \leq L \sum\limits_{j \in S} x_{0j}^{\prime\prime\prime v} \forall v \in V \\ $$

(77)

$$ \sum\limits_{i \in S \cup 0, i \neq l} x_{il}^{\prime\prime\prime v} = \sum\limits_{j \in S \cup 0, j \neq l} x_{lj}^{\prime\prime\prime v} \forall l \in S, \forall v \in V \\ $$

(78)

$$ \sum\limits_{v \in V} \sum\limits_{i \in S \cup 0} x_{ij}^{\prime\prime\prime v} \leq 1 \forall j \in S \\ $$

(79)

$$ \sum\limits_{v \in V} A_{j}^{\prime\prime\prime v} \geq \Bigg(\sum\limits_{i \in O} r_{ij}^{\prime\prime} + \sum\limits_{i \in C} r_{ij}^{\prime} - \sum\limits_{v \in V} \sum\limits_{i \in O} A_{ij}^{\prime\prime v}\Bigg) - L \Bigg(1 - \sum\limits_{v \in V} \sum\limits_{i \in S \cup 0} x_{ij}^{\prime\prime\prime v}\Bigg) \forall j \in S \\ $$

(80)

$$ \sum\limits_{v \in V} A_{j}^{\prime\prime\prime v} \leq \Bigg(\sum\limits_{i \in O} r_{ij}^{\prime\prime} + \sum\limits_{i \in C} r_{ij}^{\prime} - \sum\limits_{v \in V} \sum\limits_{i \in O} A_{ij}^{\prime\prime v}\Bigg) + L \Bigg(1 - \sum\limits_{v \in V} \sum\limits_{i \in S \cup 0} x_{ij}^{\prime\prime\prime v}\Bigg) \forall j \in S \\ $$

(81)

$$ L \sum\limits_{i \in S \cup 0, i \neq j} x_{ij}^{\prime\prime\prime v} \geq A_{j}^{\prime\prime\prime v} \forall j \in S, \forall v \in V \\ $$

(82)

$$ q_{0}^{\prime\prime\prime v} = \sum\limits_{j \in S} A_{j}^{\prime\prime\prime v} \forall v \in V \\ $$

(83)

$$ q_{i}^{\prime\prime\prime} \geq q_{0}^{\prime\prime\prime v} - A_{i}^{\prime\prime\prime v} - L (1-x_{0i}^{\prime\prime\prime v}) \forall i \in S, \forall v \in V \\ $$

(84)

$$ q_{i}^{\prime\prime\prime} \leq q_{0}^{\prime\prime\prime v} - A_{i}^{\prime\prime\prime v} + L (1-x_{0i}^{\prime\prime\prime v}) \forall i \in S, \forall v \in V \\ $$

(85)

$$ q_{j}^{\prime\prime\prime} \geq q_{i}^{\prime\prime\prime} - A_{j}^{\prime\prime\prime v} - L (1 - x_{ij}^{\prime\prime\prime v}) \forall i,j \in S, \forall v \in V \\ $$

(86)

$$ q_{j}^{\prime\prime\prime} \leq q_{i}^{\prime\prime\prime} - A_{j}^{\prime\prime\prime v} + L (1 - x_{ij}^{\prime\prime\prime v}) \forall i,j \in S, \forall v \in V \\ $$

(87)

$$ q_{0}^{\prime\prime\prime v} \leq q \forall v \in V \\ $$

(88)

$$ u_{j}^{\prime\prime\prime} \geq u_{i}^{\prime\prime\prime} + 1 - |S|\Bigg(1-\sum\limits_{v \in V} x_{ij}^{\prime\prime\prime v}\Bigg) \forall i,j \in S \\ $$

(89)

Constraints (75) and (76) ensure that if either there exist outlets’ returned products or there are some returned products from customers that are not sent to any outlet (including the defective products), then the supplier that supplies the product will be visited. Constraint (77) ensures that for every vehicle used in this process, it always starts its trip from the cross-dock. Constraint (78) ensures the outflow and inflow of a vehicle in each supplier node. Constraint (79) ensures that each supplier is visited at most once. The amount of products delivered to each supplier is calculated in constraints (80) and (81), while ensuring no split delivery occurs as in constraint (82). Constraints (83) to (87) track the total load inside a vehicle. Constraint (88) ensures that the total amount of products delivered to all suppliers in a vehicle does not exceed vehicle capacity. Constraint (89) is the sub-tour elimination constraint.