## 3 Overview of the Egg Distribution Problem
### Indices, Parameters and Decision Variables  
This section defines the indices, parameters and variables used in the proposed mixed-integer linear programming model. The variables and parameters for the model are summarized in Table 2.  
Objective function  
\[ \begin{array}{ll}&min\sum\limits_{i=1}^{N+M}\sum\limits_{j=1}^{N+M}\sum \limits_{k=1}^{K}C_{ij}x_{ijk}+\sum\limits_{k=1}^{K}f_{k}u_{k}\\ &+(\sum\limits_{i=1}^{N+M}\sum\limits_{p=1}^{P}(v_{ip}H_{ip}g_{i})-\sum\limits _{i=1}^{N+M}\sum\limits_{j=1}^{M}\sum\limits_{k=1}^{K}(z_{ij}d_{jk}H_{ip}))\end{array}  \tag{1}\]  
Subject to  
\[ \begin{array}{ll}\sum\limits_{i=N+1}^{N+M}\sum\limits_{j=1}^{N}x_{ijk}\ \leq u_{k}&\forall k=1,\ldots,K\end{array}  \tag{2}\] \begin{table}
\begin{tabular}{c l} \hline \hline
**Symbol** & **Description** \\ \hline Indices & \\ \(i,j\) & Indices of all customers and depots; \(i,j=1,2,\ldots,N+M\) \\ \(k\) & Index of all vehicles; \(k=1,2,\ldots,K\) \\ \(p\) & Index of all egg sizes; \(p=1,2,\ldots,P\) \\ Parameters & \\ \(N\) & Number of customers \\ \(M\) & Number of depots \\ \(K\) & Number of vehicles \\ \(P\) & Number of products \\ \(C_{ij}\) & Transportation cost between customer \(i\) and customer \(j\) \\ \(D_{ij}\) & Distance between customer \(i\) and customer \(j\) \\ \(t_{ij}\) & Travel time between customer \(i\) and customer \(j\) \\ \(f_{k}\) & Labor cost of vehicles \(k\) \\ \(Q_{kp}\) & Maximum capacity of vehicle \(k\) for egg size \(p\) \\ \(v_{ip}\) & Inventory of egg size \(p\) at depot \(i\) \\ \(I_{ip}\) & Holding cost of egg size \(p\) at depot \(i\) \\ \(d_{ip}\) & Demand of customer \(i\) for egg size \(p\) \\ \(L\) & Positive Large number \\ \(D^{\prime}\) & Maximum distance a vehicle can travel \\ \(K^{\prime}_{i}\) & Maximum number of vehicles departing from depot \(i\) \\ \(s_{i}\) & Service time of the delivery at customer \(i\) \\ \(e_{i}\) & Earliest arrival/delivery time at customer \(i\) \\ \(l_{i}\) & Latest time at customer \(i\) \\ \(T_{k}\) & Maximum duration time of vehicle \(k\) \\ \(I_{i}\) & Maximum inventory at depot \(i\) \\ Decision variable & \\ \(x_{ijk}\) & \(\left\{\begin{array}{l}1\ if\ vehicle\ k\ travels\ from\ customer\ i\ to\ customer\ j\\ 0\ otherwise\\ \end{array}\right.\) \\ \(u_{k}\) & \(\left\{\begin{array}{l}1\ if\ vehicle\ k\ is\ used\\ 0\ otherwise\\ \end{array}\right.\) \\ \(g_{i}\) & \(\left\{\begin{array}{l}1\ if\ depot\ i\ is\ operated\\ 0\ otherwise\\ \end{array}\right.\) \\ \(y_{ipk}\) & \(\left\{\begin{array}{l}1\ if\ the\ demand\ of\ customer\ i\ for\ egg\ size\ p\ is\ fulfilled\ by\ vehicle\ k\\ 0\ otherwise\\ \end{array}\right.\) \\ \(z_{ij}\) & \(\left\{\begin{array}{l}1\ if\ the\ customer\ j\ is\ allocated\ to\ depot\ i\\ 0\ otherwise\\ \end{array}\right.\) \\ \(a_{i}\) & Arrival time of the delivery vehicle at customer \(i\) \\ \(w_{i}\) & Waiting time of the delivery vehicle at customer \(i\) \\ \(A_{ik}\) & A non-negative time that indicates when vehicle \(k\) departs from customer \(i\) \\ \(T_{ik}^{\mu}\) & Variable used for elimination of sub-tours. \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Indices Parameters and Decision variable of egg distribution problem.