# Optimizing Perishable Product Supply Chain Network Using Hybrid Metaheuristic Algorithms
## 2 Problem Description
### Mathematical Model
\begin{array}{c}w_{jp(m+1)t}^{5}\Big{(}1-u_{jp(m+1)t}\Big{)}\ \geq\ \sum\limits_{h=1}^{m}\left(\sum\limits_{i}w_{ijph}^{1}y_{ijph^{\prime}}^{1}- \sum\limits_{g=h}^{t}w_{jRg}^{8}y_{jkRg}^{2}-w_{jp(h+\tau_{p})}^{3}\right)\\ -\sum\limits_{k}w_{jRg}^{8}y_{jkRg}^{2}+1\quad\forall j,p,m,t,m+\tau_{p}\geq| T|\\ w_{jp(m+1)t}^{5}\Big{(}1-u_{jp(m+1)t}\Big{)}\ \geq\ \sum\limits_{h=1}^{m}\left(\sum\limits_{i}w_{ ijph}^{1}y_{ijph^{\prime}}^{1}-\sum\limits_{g=h}^{t}w_{jRg}^{8}y_{jkRg}^{2} \right)\\ -\sum\limits_{k}w_{jkRg}^{8}y_{jkRg}^{2}+1\quad\forall j,p,m,t,m+\tau_{p}\geq| T|\end{array} \right., \tag{19}\]  
\[(1-\alpha)w_{j_{k}pt}^{6}=\sum\limits_{k\in K}w_{jkrypt}^{9}y_{jkopt}^{3}\quad \forall j,v,p,t, \tag{20}\]  
\[w_{j_{k}pt}^{6}\leq r_{cp}^{cap}\quad\forall j,v,p,t, \tag{21}\]  
\[w_{j_{k}pt}^{6}=w_{j_{k}pt}^{7}\quad\forall j,k,v,p,t, \tag{22}\]  
\[\sum\limits_{l\in(\bigcup K)}z_{lkypt}*\left(\sum\limits_{l\in(\bigcup K)}w _{lkypt}^{7}-w_{jkypt}^{9}-\sum\limits_{l\in(\bigcup K)}w_{lkypt}^{7}\right)= 0\quad\forall k,v,p,t, \tag{23}\]\[w^{7}_{kjypt}=0\quad\forall j,k,v,p,t, \tag{24}\]  
\[w^{8}_{kpt}y^{2}_{jkpt}=\sum_{v\in V}w^{9}_{jkpt}y^{3}_{jkypt}\quad\forall j,k,p,t, \tag{25}\]  
\[\sum_{j\in I}\sum_{v\in V}w^{9}_{jkpt}y^{3}_{jkypt}=d_{kpt}\quad\forall k,p,t, \tag{26}\]  
\[\sum_{m=t}^{T}w^{5}_{jptm}u_{jptm}\leq\sum_{i\in I}w^{1}_{ijpt}y^{1}_{ijpt}\ \forall j,p,t, \tag{27}\]  
\[\sum_{i\in I}\sum_{t\in T}w^{1}_{ijpt}y^{1}_{ijpt}=\sum_{k\in K}\sum_{t\in T}w ^{8}_{jkpt}y^{2}_{jkpt}+\sum_{t=T}^{T}w^{2}_{jpt}+\sum_{t\in T}w^{3}_{jpt}\quad \forall j,p, \tag{28}\]  
\[\left\{ \begin{array}{c}\sum\limits_{i\in I}w^{1}_{ijpt}y^{1}_{ijpt}\geq\sum _{k\in K}w^{8}_{jkpt}y^{2}_{kpt}\quad\forall j,p,t=1\\ \sum\limits_{i\in I}w^{1}_{ijpt}y^{1}_{ijpt}\geq w^{2}_{jpt}-w^{3}_{jpt}-\sum _{k\in K}w^{8}_{jkpt}y^{2}_{jkpt}\quad\forall j,p,t\geq 2^{{}^{\prime}} \end{array} \right. \tag{29}\]  
\[x_{j},y^{1}_{ijpt},y^{2}_{jkpt},y^{3}_{jkypt},u_{jbmt},z_{lnypt}\in\{0,1\}, \tag{30}\]  
\[q^{1}_{ijpt},w^{1}_{ijpt},w^{2}_{jpt},w^{3}_{jpt},w^{4}_{jpt},w^{5}_{jpmt},w^{6 }_{jpmt},w^{7}_{jkpt},w^{8}_{jkpt},w^{9}_{jkpt}\geq 0. \tag{31}\]  
The objective Function (1) aims to minimize the total cost of the supply chain. Constraint (2) ensures that an operated DC is served by at least one manufacturer. Constraints (3) and (4) require that each retailer is served by an operated DC. Constraint (5) reveals the relationship between service and delivery vehicles from DCs to retailers. Constraints (6) and (7) ensure that when a vehicle leaves the DCs for a retailer, it eventually returns to the starting point. Constraint (8) guarantees that each retailer is served by only one vehicle for each product in each period. Constraint (9) implies that whenever a vehicle arrives at a retailer location, it must leave again. Constraint (10) is adopted in the proposed model for subtour elimination. Constraint (11) specifies that all arcs must comply with the service relationship between the DC and the retailer. Constraint (12) calculates the production quantity of each product for the manufacturer in each period. Constraint (13) specifies the capacity level of the manufacturer. Constraints (14) and (15) indicate the inventory and waste quantity at the DCs. Constraint (16) ensures that the delivery quantity of the DC is equal to the total quantity of vehicles picked up. Constraint (17) requires that perishable products delivered to retailers remain within their shelf lives. Constraints (18) and (19) enforce the First-In-First-Out (FIFO) policy for perishable products at the DCs. Constraint (20) states that the quantity of perishable products picked up by each vehicle at the DC must be equal to the total quantity delivered by the served retailer. Constraint (21) indicates that the pickup quantity of each vehicle cannot exceed the capacity limitation. Constraints (22)-(24) calculate the load quantity of vehicles in each arc segment. Constraints (25) and (26) specify that vehicles deliver perishable products according to retailer demand.