# A Proactive Approach to Extended Vehicle Routing Problem with Drones (EVRPD)
## 3 Problem Statement and Mathematical Model
### Mathematical Model  
The model for the EVRPD was formulated as a CSP (constraint satisfaction problem) with a set of questions. Ultimately, depending on the question, the model took the form of a CSP or BIP (binary integer programming). Table 1A presents questions Q1-8 for the presented version of the model. Both the set of questions and the set of constraints may be modified in subsequent versions of the model. \begin{table}
\begin{tabular}{p{128.0 pt} p{128.0 pt} p{128.0 pt}} \hline (A) & & \\ \hline Symbol & Description & P/R \\ Q1 & Is it possible to make deliveries? & P/R \\ Q2 & What is the minimum cost of deliveries? & P/R \\ Q3 & Is it possible to make deliveries within the given time T? & P/R \\ Q4 & What is the minimum cost of deliveries in the given time T? & P/R \\ Q5 & What is the minimum number of mobile points (mobile hubs—MHs) required to make deliveries? & P \\ Q6 & What is the minimum number of drones needed to make deliveries? & P \\ Q7 & Can deliveries be made if the drone \(dx\) is damaged? & P \\ \hline Q8 & Is it possible to make deliveries if it is not possible to use the mobile point \(mx\)? & P \\ \hline \multicolumn{3}{p{128.0 pt}}{**P/R—proactive/reactive**} \\ \hline (B) & & \\ \hline Question & Set of constraints & Problem type & Model type & Decision Variable \\ \hline Q1 & (2)–(8) and (10)–(15) & feasible & CSP & R\({}_{8}\) \\ \hline Q2 & (1a) and (2)–(8) and (10)–(15) & optimal & BIP & X\({}_{h,d,c,j},\) R\({}_{8}\) \\ \hline Q3 & (2)–(15) & feasible & CSP & R\({}_{8}\) \\ \hline Q4 & (1a) and (2)–(15) & optimal & BIP & X\({}_{h,d,c,j},\) R\({}_{8}\) \\ \hline Q5 & (2)–(8) and (10)–(15) and min MS & optimal & BIP & Z\({}_{h}\) \\ \hline Q6 & (1b) and (2)–(8) and (10)–(15) & optimal & BIP & Y\({}_{h,d,c,j,s}\) \\ \hline Q7 & (2)–(8) and (10)–(15) and VC\({}_{di}\) = 0 & feasible & CSP & R\({}_{8}\) Y\({}_{h,d,c,j,s}\) \\ \hline Q8 & (2)–(8) and (10)–(15) and VK\({}_{mx}\) = 0 & feasible & CSP & Z\({}_{h}\), X\({}_{h,d,c,j}\) \\ \hline \end{tabular}
\end{table} 
Table 1: (A) Set of questions for the EVRPD. (B) Description of the modeled problems with the use of the EVRPD model. The way the model was formulated as questions and constraints, along with the nature of the proposed questions, enables a proactive way of using the model in the decision support process. Models for the VRPD known from the literature are usually reactive in nature. For the proposed model, questions Q1-4 may be of both a proactive and reactive nature, depending on the data set and the time the question is asked. On the other hand, questions Q5-8 are typically proactive (Table 1A). In the current version, the proposed EVRPD model made it possible, depending on the choice of question (Q1-8), subset of constraints, and corresponding decision variables, to model eight different problems, the characteristics of which are presented in Table 1B. The sets of the model indices, parameters, and decision variables, along with their descriptions, are presented in Table 2. The descriptions and meanings of the model constraints formulated in (1a)-(15) are presented in Table 3.  
\[\min\sum_{d\in D}\sum_{c\in C\cup H}\sum_{j\in C\cup H}\sum_{h\in H}\text{FA}_{ cj}\cdot\text{X}_{h,d,cj} \tag{1a}\] \[\min\sum_{h\in H}\sum_{d\in D}\sum_{c\in M}\sum_{j\in C}\text{X}_{h,d,cj}\] (1b) \[\sum_{j\in c\cup H}\text{X}_{h,d,cj}=\sum_{j\in c\cup H}\text{X}_{h,d,cj} \forall h\in H,d\in D,c\in C\cup M\] (2) \[\text{X}_{h,d,cj}\leq\sum_{k\in K}\text{Y}_{h,d,cj,s}\forall h\in H,\ d \in D,\ c\in C,\ j\in C\] (3) \[\text{Y}_{h,d,cj,s}\leq\text{X}_{h,d,cj}\forall h\in H,\ d\in D,\ c\in C \cup M,\ j\in C\cup M,\ s\in S\] (4) \[\sum_{s\in S}\text{VS}_{k}\cdot\text{Y}_{h,d,cj,s}\leq\text{VC}_{d} \forall d\in D,\ c\in C\cup M,\ j\in C\cup M\] (5) \[\sum_{h\in H}\sum_{d\in D}\sum_{c\in C\cup m}\text{Y}_{h,d,cj,s}= \text{RA}_{s,c}-\text{R}_{s}\forall j\in O,\ s\in S:\text{RA}_{s,c}=1,\ \text{UA}_{s}=1\] (6) Table 2.0 Table 2.1 (7) Table 2.2 (8) Table 2.3 Table 2.4 (9) Table 2.5 (10) Table 2.6 Table 2.7 (11) Table 2.8 (12) Table 2.9 (13) Table 3.0\[