# Optimizing Perishable Product Supply Chain Network Using Hybrid Metaheuristic Algorithms
## 2 Problem Description
### Notation  
Sets and index:  
\(i\)Manufacturer index \(i\in I\)  
\(j\)DC index \(j\in J\)  
\(k\)Retailer index \(k\in K\)  
\(v\)Vehicle index \(v\in V\)  
\(p\)Perishable product index \(p\in P\)  
\(t\)Period index \(t\in T\)  
Parameters  
\(f_{i}^{1}\)The fixed operating \(\cos\)ts for manufacturer \(i\)  
\(pc_{ip}\)The unit production \(\cos\)t of perishable product \(p\) at manufacturer \(i\)  
\(tc_{ip}^{1}\)The unit transportation \(\cos\)t of perishable product \(p\) from manufacturer \(i\) to DC \(j\)  
\(m_{ipt}^{dep}\)The capacity of manufacturer \(i\)  
\(s_{i}^{1}\)The distance from manufacturer \(i\) to DC \(j\)  
\(f_{j}^{2}\)The fixed operating \(\cos\)t for DC \(j\)  
\(hc_{ip}\)The inventory \(\cos\)t of perishable product \(p\) at DC \(j\)  
\(tc_{lmcp}^{2}\)The transportation \(\cos\)t of perishable product \(p\) from DC \(j\)to retailer \(k\)  
\(\alpha\)The waste rate during the transportation process in all systems  
\(s_{ln}^{2}\)The distance from node \(l\) to node \(n\)  
\(rc_{p}\)The waste \(\cos\)t for perishable product \(p\)  
\(d_{kpt}\)The demand of perishable product \(p\)at retailer \(k\)during period \(t\)  
\(c_{cp}^{dep}\)The capacity of vehicle \(v\)  
\(\tau_{p}\)The shelf life of perishable product \(p\)  
M  
A sufficiently large number  
Decision variables  
\(x_{j}\)  
\(1\), if DC \(j\) is operated; \(0\), otherwise  
\(y_{ijt}^{1}\)  
\(1\), if DC \(j\) is supplied by manufacturer \(i\) for perishable product \(p\) during period \(t\); \(0\), otherwise  
\(y_{jkpt}^{2}\)  
\(1\), if retailer \(k\) is supplied by DC \(j\) for product \(p\) during period \(t\); \(0\), otherwise  
\(y_{jkpt}^{3}\)  
\(1\), if the order quantity from DC \(j\) to retailer \(k\) is delivered by vehicle \(v\) during period \(t\); \(0\), otherwise  
\(1\), if link \(l-\)  
\(z_{lmpt}\)\(n\)is a part of the vehicle route for delivering perishable product \(p\) by vehicle \(v\)  
during period \(t\); \(0\), otherwise  
\(u_{jkmt}\)  
\(q_{ipt}^{1}\)Production quantity of perishable product \(p\) at manufacturer \(i\) during period \(t\)  
\(w_{ijt}^{1}\)Purchasing quantity of perishable product \(p\) by DC \(j\) from manufacturer \(i\) during  
\(w_{ipt}^{2}\)Inventory quantity of perishable product \(p\) at DC \(j\) during period \(t\)  
\(w_{ipt}^{2}\)Waste quantity of perishable product \(p\) at DC \(j\) during period \(t\)  
\(w_{ipt}^{4}\)Total delivered quantity of perishable product \(p\) by DC \(j\) during period \(t\)  
Quantity of perishable product \(p\) procured during period \(m\) and delivered out from  
\(w_{ipmt}^{5}\)DC \(j\) during period \(t\)  
Quantity of perishable product picked up from DC \(j\) by vehicle \(v\) at period \(t\)  
Load quantity of perishable product \(p\) transported by vehicle \(v\) from node \(l\) to node  
\(w_{lnpt}^{7}\)\(n\)during period \(t\)  
Allocation quantity of perishable product \(p\) from DC \(j\) to retailer \(k\) during period \(t\)  
Delivered quantity of perishable product \(p\) by vehicle \(v\) from DC \(j\) to retailer \(k\)  
during period \(t\)  
Auxiliary non-negative variable used for subtour elimination