3.3
Problem description
This chapter studies a supply chain problem with multiple suppliers and customers, which is formulated as a rich MDVRP. It is assumed that the distribution process is carried out by a homogeneous fleet of vehicles. The problem consists of opti- mizing the distribution routing plan considering different sustainability dimensions. The three components of sustainability (economic, environmental, and social im- pacts) are represented by travel distances and times, carbon emissions, and risk of accidents. Several studies have addressed the economic impacts as a variable mainly influenced by travel distances. Therefore, most existing models seek to min- imize travel distances. However, achieving this goal does not guarantee a minimum economic impact, since many time-related factors are not being considered –e.g., congestion, speed limits, traffic signs, vehicle crashes, etc. (Wang et al., 2016). In fact, the shortest paths in urban zones are sometimes the slowest ones too. Ac- cordingly, the tackled problem also considers travel times to represent these urban attributes.
Formally, the MDVRP can be defined on a complete undirected graph G=(N,A), where N = {Nd, Nc} is a set of nodes, Nd and Nc represent the subsets of depots and customers respectively, and A = {(i, j) : i, j ∈ N, i 6= j} is the set of edges connecting all nodes in N. Each depot i (∀ i ∈ Nd) has a capacity si (si > 0), and each customer j (∀ j ∈ Nc) has a demand rj (rj > 0). The vehicle fleet K is composed of o identical vehicles (K = {1, 2, ..., o}). Finally, Q, D, and Qd denote, respectively, the capacity and the maximum-distance-allowed associated with each vehicle, and the capacity of each depot. Each edge (i, j) ∈ A has an associated travel time (tij), and travel distance (dij). Typically, the aim is to design a set of routes minimizing the total cost, which depends on travel distances or times. Besides satisfying the constraints related to the capacities and the maximum-distance for each vehicle, a feasible solution has to ensure that each customer is visited only once. In addition, each route must start and end at the same depot. The binary variable xijk is employed to represent the solution (i.e., the set of routes): xijk = 1 if the edge (i, j) is traversed by vehicle k, and xijk= 0 otherwise.
The rich MDVRP variant presented in this chapter aims to find a sustainable solution by assessing and minimizing the negative impacts associated. The following sustainability dimensions are considered:
• Economic dimension. This dimension is composed by total travel time and fuel consumption (fij), which are monetized based on the driver wage (DW ), the vehicle fixed cost (FC ), and the oil price (Cf). The costs of the route associated with vehicle k are computed as follows:
X
(i,j)∈A
42 The sustainable multi-depot vehicle routing problem X
(i,j)∈A
Cf · fij · xijk (3.2)
• Environmental dimension. CO2 emissions estimates assume that the internal combustion process of vehicles burns the carbon of the fuel and it is released as carbon dioxide. Thus, emissions are assumed to depend on fuel consumption. Expression (3.3) computes the cost of environmental impacts for the route associated with vehicle k, considering a factor for carbon emissions (Ce).
X
(i,j)∈A
Ce· fij · xijk (3.3)
• Social dimension. Accidents are an externality caused by speed variations on roads, among other factors. These variations represent the state and stability of the roads, and are associated with an accident risk for pedestrians and vehicles (Wang et al., 2016). Expression (3.4) represents the social cost, and depends on a given coefficient (aij), vehicle loading (yijk), and travel distance. In particular, flow variables yijk represent the load in the route associated with vehicle k servicing customer j after visiting customer i.
X
(i,j)∈A
aij · dij · yijk (3.4)
The fuel consumption is estimated as suggested in Kuo (2010) andZhang et al.
(2016). Thus, in Equation (3.5), lphij represents the fuel consumption per unit of time, and p is a penalty for each additional load (M). This value is determined by the average miles per fuel liter (kplij) and velocity (vij) (Equation (3.6)):
fijk = lphij · dij vij · 1 + p · yijk M ∀ (i, j) ∈ A, k ∈ K (3.5) lphij = vij kplij ∀ (i, j) ∈ A (3.6)
Without loss of generality, it will be assume that p is equal to 0. Thus, fijk can be represented by fij. All in all, the objective is defined on a multi-criteria function to minimize, which considers the total travel time, the total travel distance, the environmental cost, and the social cost:
X
k∈K X
(i,j)∈A
(DW · tij + F C + Cf · fij + Ce· fij) · xijk+ aij · dij · yijk (3.7) Based on the model presented by Mirabi et al. (2010), the constraints are as
3.3 Problem description 43 follows: X i∈N X k∈K xijk= 1 ∀ j ∈ Nc (3.8) X (i,j)∈A|j∈Nc rj· xijk ≤ Q ∀ k ∈ K (3.9) X (i,j)∈A dij · xijk ≤ D ∀ k ∈ K (3.10) Ulk− Ujk + |N | · xljk ≤ |N | − 1 ∀ l, j ∈ Nc, k ∈ K (3.11) X j∈N xijk=X j∈N xjik ∀ i ∈ Nc, k ∈ K (3.12) X j∈Nc X k∈K xijk ≤ p ∀ i ∈ Nd (3.13) X i∈Nd X j∈Nc xijk ≤ 1 ∀ k ∈ K (3.14) X j∈Nc xijk = X j∈Nc xjik ∀ i ∈ Nd, k ∈ K (3.15) X k∈K X j∈Nc yijk ≤ Qd ∀ i ∈ Nd (3.16) X i∈N yijk−X i∈N yjik = rj·X i∈N xijk ∀ j ∈ Nc, k ∈ K (3.17)
rj· xijk ≤ yijk≤ (Q − ri) · xijk ∀ (i, j) ∈ A, k ∈ K (3.18)
xijk = 0 ∀ i, j ∈ Nd, k ∈ K (3.19)
yijk ≥ 0 ∀ (i, j) ∈ A, k ∈ K (3.20)
xijk ∈ {0, 1} ∀ (i, j) ∈ A, k ∈ K (3.21)
Ulk≥ 0 ∀ l ∈ Nc, k ∈ K (3.22)
Equation (5.6) assigns each customer to exactly one route. Equation (5.7) limits the total demand that may be served by a vehicle. Equation (5.8) defines the max- imum distance allowed per vehicle. Equation (5.9) eliminates sub-tours. The flow conservation is introduced by Equation (3.12). Equation (3.13) limits the number of routes to the same number of vehicles available in each depot (p). Equation (3.14) imposes that each vehicle relates to a single route or zero. Equation (5.10) ensures that each route starts and ends at the same depot. Equation (5.11) makes sure that the total demand of the customers allocated to a depot is not greater than its capacity. Equation (5.12) states that the load in the vehicle arriving at customer j
44 The sustainable multi-depot vehicle routing problem minus the demand of that customer equals the load in the vehicle leaving it after the service. Equation (5.13) sets lower and upper bounds for the loads. Equation (5.14) avoids the creation of routes among depots. Finally, Equations (5.15) and (5.17) define variable domains.