Multi-Product Multi-Depot Periodic Distribution Problem

In this paper, we present a methodology for solving a multi-product, multi-depot periodic vehicle routing problem. The problem focuses on allocating products and assigning appropriate delivery days to a set of customers, and designing a set of delivery routes for every day of the planning period. A tabu-search based heuristic is proposed to find the best allocation and routing plan that would give a minimum total cost over the planning period. A simple numerical example is solved to demonstrate the steps of the proposed algorithm. Keywords Periodic Vehicle Routing Problem; Tabu-Search Heuristic; Distribution 1. Introduction The periodic vehicle routing problem is a problem of designing a set of delivery routes for each day of a planning period where customers may not receive deliveries on every day in the period. This problem typically arises in distribution industries such as grocery industry and beverage industry. In general, most customers of such industries will have low-volume daily product usages. It is therefore not economical for the distributors to deliver the products to their customers on a daily basis. Instead, the customers’ daily demands will be aggregated into larger quantities and delivered periodically (e.g. once a week, twice a week, etc.). How frequent the customers will receive the deliveries will depend on their product usages, their storage spaces and/or their preferences. In this paper, a multiple-product periodic vehicle routing problem under the multiple-depot environment is considered. Typically, if a distributor owns many distribution centers in a region, it will specify the territory for each depot and let each depot operate independently from the others. However, some customers may be located approximately equi-distant from two or three distribution centers. In such cases, it may be beneficial to incorporate the choice of the distribution center as a suitable decision variable in the development of the model and have the model select the distribution center to make deliveries from, than it is to assign each customer to a particular distribution center. In the classical vehicle routing problem and periodic vehicle routing problem, every vehicle is allowed to make at most one delivery trip per day. However, in real-world applications, if a vehicle finishes its first trip in a short time, the vehicle dispatcher may consider assigning it for another trip to fully utilize the resources. Thus, multiple trips for a vehicle on a single day are also considered. 2. Mathematical Model We first define the notations, parameter and variables used in the formulation of our model. Then, we present a mathematical model for the multi-product, multi-depot periodic distribution problem. 2.1 Notations P = 1, 2,…, p products; V = 1, 2,…, n nodes; V = { V1 ∪ V2}; V1 = set of depot nodes; V2 = set of customer nodes; T = 1,2,…, t days; K = 1,2,…, k vehicles; K = { K1 ∪ K2 ∪ ...∪ Kl }; Kl = set of vehicles k which belong to depot l; k S = set of all nodes visited by vehicle k; Ri = set of delivery patterns r allowable to customer i 2.2 Parameters zrt = binary variable equal to1 if day t corresponds to delivery pattern r; 0 otherwise ; ir γ = penalty cost for assigning pattern r to customer i; M’ = maximum number of trips that a vehicle could make per day; Vp = volume per piece of product p ; Qk = capacity of the vehicle k; pit d = demand of store i for product p on day t ; Bpi = backorder cost of product p at customer i; spl0 = initial inventory of product p at depot l ; Srplt = scheduled receipts of product p at depot l on day t; cij = travel time from store i to store j; [ai ,bi] = delivery time window of store i; mijk = per-minute travel cost of vehicle k; wk = driver wage per minute on vehicle k; f1 = average loading setup time; f2 = average unloading setup time at a customer; g1 = average loading time per piece; g2 = average unloading time per piece 2.3 Variables ijkmt x = binary variable equal to 1 if customer j is visited immediately after customer i by vehicle k on its m th trip on day t (i ≠ j); 0 otherwise; yir = binary variable equal to 1 if delivery pattern r is assigned to customer i; 0 otherwise; qpit = quantity of product p delivered to customer i on day t pieces); pit I = inventory of product p at customer i on day t; pit I = backorder of product p at customer i on day t; αikmt = arrival time at customer i (or at depot i) of vehicle k on its m trip on day t; eit = waiting time at customer i on day t 2.4 Model Minimize [ ] ijkmt V i i j V j K k M m T t it k k ijk ij x f e w w m c ∑ ∑ ∑ ∑ ∑ ∈ ≠ ∈ ∈ ∈ ∈ + + + ' 2 ) ( ) ( + ∑ ∑ ∑ ∈ ∈ ∈ − 2 V i T t P p pit pi I B + ∑ ∑ ∈ ∈ 2 V i R r ir ir i y γ