# A Hyper Heuristic Algorithm to Solve the Low-Carbon Location Routing Problem
## 2 Review
### Mathematical Model  
RLCLRRPTWSPDHF can be defined as a complete graph \(G=(V,E)\), where \(V\) is composed of candidate depots \(M\) and \(N\) customers. Each customer has demand and time windows, with known geographical locations. The capacity, location, and open cost of each candidate depot are known. \(E\) is the edge set. Different types of vehicles depart from the depot to serve a series of customers and return to the original starting point. The goal of the model was to minimize total costs, including the fixed open costs of depots, vehicles costs, fuel and carbon emissions costs, and penalty costs. The model was based on the following assumptions: (1) every customer is served exactly once; (2) the vehicle must return to the original depot after serving all customers; (3) the vehicle capacity is not exceeded; (4) the depot capacity is not exceeded; and (5) each depot has enough vehicles to serve the corresponding customers. The meanings of the symbols and variables that appear in the model are as follows: customer nodes \(I=\{1,2,\ldots,N\}\), potential depot nodes \(J=\{N+1,N+2,\ldots,N+M\}\), heterogeneous fleet \(H=\{L1,L2,M\}\), the set of nodes \(V=I\cup J\), and the arc set \(E=\{(i,j):i,j\in V,i\
eq j\}\backslash\{(i,j):i\in J,j\in J\}\). \(FD_{j}\) and \(CD_{j}\) are the open costs and capacities of the potential depots, respectively. \(p_{i}\) and \(q_{i}\) are the pickup and delivery demand of customer \(i\). \([e_{i},l_{i}]\) is the soft time window of customer \(i\). \(ST_{i}\) is the service time for customer \(i\). \(CV_{h}\) and \(FV_{h}\) are the capacity and fixed dispatch cost, respectively, of vehicle \(h\). \(AT_{ih}\) is the time when the vehicle arrives to the customer \(i\). \(TT_{ijh}\) is the travel time of vehicle \(h\) in arc\((i,j)\). \(Q_{ijh}\) is the dynamic load of vehicle \(h\) on arc\((i,j)\). \(F_{ijh}\) is the fuel consumption of vehicle \(h\) on arc\((i,j)\). \(C_{f}\) is the fuel consumption cost (yuan/liter). \(C_{\omega 2}\) is the cost of carbon emissions (yuan/kg), and \(C_{\omega w}/C_{\omega w}\) is the early or late penalty coefficient of the vehicle (yuan/min). The decision variables are as follows: \(x_{ijh}\) indicates whether arc \((i,j)\) is served by vehicle \(h\). If it is, \(x_{ijh}\) takes the value of 1; otherwise, \(x_{ijh}\) is 0. \(z_{ij}\) indicates whether client \(i\) is served by candidate depot \(j\). If it is, then \(z_{ij}\) takes the value of 1; otherwise, it is 0. \(y_{j}\) indicates whether candidate depot \(j\) is leased. If it is, \(y_{j}\) takes the value of 1; otherwise, it is 0. Based on the above assumptions and definitions, the model is constructed as follows:  
\[ \begin{split}\text{min}C=\sum\limits_{j\in J}FD_{j}y_{j}+\sum \limits_{i\in J}\sum\limits_{j\in h\in H}FV_{h}x_{ijh}+\left(C_{f}+C_{\omega 2}\eta_{j}\right)\sum\limits_{i\in V}\sum\limits_{j\in H }F_{ijh}x_{ijh}+C_{\omega w}\sum\limits_{i\in J}\sum\limits_{j\in V}x_{ijh} \text{max}[e_{i}-AT_{ih},\ 0]+\\ C_{\omega}\sum\limits_{i\in J}\sum\limits_{j\in V}\sum\limits_{i\in H }x_{ijh}\text{max}\{\text{max}\{AT_{ih},e_{i}\}+ST_{i}-l_{i},\ 0];\end{split}  \tag{6}\]  
\[\sum\limits_{i\in V}\sum\limits_{h\in H}x_{ijh}=1,\ \forall j\in I; \tag{7}\]  
\[\sum\limits_{h\in H}\sum\limits_{i\in V}x_{ijh}=\sum\limits_{h\in H}\sum \limits_{i\in V}x_{ijh},\forall j\in V; \tag{8}\]  
\[\sum\limits_{j\in J}z_{ij}=1,\ \forall i\in I; \tag{9}\]  
Figure 2: Illustration of speed zones.