# Optimizing Perishable Product Supply Chain Network Using Hybrid Metaheuristic Algorithms
## 2 Problem Description
### Mathematical Model  
According to the problem descriptions, assumptions, and notations, this study formulates the LIRP for a multi-period, multi-product PPSCN as a mixed-integer nonlinear programming (MINLP) model. The formulation contains five costs: fixed cost, production cost, transportation cost, inventory cost, and waste cost. \[minf =\left(\sum\limits_{i\in I}f_{i}^{1}+\sum\limits_{j\in I}f_{j}^{2}x_ {j}\right)\text{(The fixed cost of manufacturers and operated DCs)}\] \[+\sum\limits_{i\in I}\sum\limits_{p\in P}\sum\limits_{t\in T}pc_{ ip}q_{ipt}^{1}\text{(The production cost)}\] \[+\left(\sum\limits_{i\in I}\sum\limits_{j\in P}\sum\limits_{p\in P }\sum\limits_{t\in T}tc_{ip}s_{ij}^{1}w_{ipt}^{1}y_{ipt}^{1}\right.\] \[+\sum\limits_{l\in(I\cup K)}\sum\limits_{n\in(I\cup K)}\sum \limits_{v\in V}\sum\limits_{p\in P}\sum\limits_{t\in T}tc_{mvp}^{2}s_{ln}^{2} w_{mprt}^{2}z_{Inrupt}\right)\text{(The transportation cost)}\] \[+\sum\limits_{j\in I}\sum\limits_{p\in P}\sum\limits_{t\in T}hc_{ ip}w_{ipt}^{2}\text{(The inventory cost)}\] \[+\left(\sum\limits_{j\in I}\sum\limits_{v\in V}\sum\limits_{p\in P }\sum\limits_{t\in T}\alpha\cdot rc_{jp}w_{ijpt}^{1}y_{ipt}^{1}+\sum\limits_ {j\in I}\sum\limits_{p\in P}\sum\limits_{t\in T}rc_{jp}w_{jpt}^{3}\right.\] \[+\sum\limits_{j\in I}\sum\limits_{v\in V}\sum\limits_{p\in P}\sum \limits_{t\in T}\alpha\cdot rc_{jp}w_{jpt}^{6}\bigg{)}\text{(The waste cost)},\]  
subject to  
\[\sum\limits_{i\in I}y_{ipt}^{1}\leq x_{j}\ \ \forall j,p,t, \tag{2}\]  
\[\sum\limits_{j\in I}y_{jkpt}^{2}=1,\forall k,p,t, \tag{3}\]  
\[y_{jkpt}^{2}\leq x_{j},\forall j,k,p,t, \tag{4}\]  
\[+\sum\limits_{i\in I}\sum\limits_{p\in P}\sum\limits_{t\in T}pc_{ ip}q_{ipt}^{1}\text{(The production cost)}\]0  
\[+\sum\limits_{i\in I}\sum\limits_{p\in P}\sum\limits_{t\in T}pc_{ ip}q_{ipt}^{1}\text{(The production cost)}\]1  
\[+\sum\limits_{i\in I}\sum\limits_{p\in P}\sum\limits_{t\in T}pc_{ ip}q_{ipt}^{1}\text{(The production cost)}\]2  
\[+\sum\limits_{i\in I}\sum\limits_{p\in P}\sum\limits_{t\in T}pc_{ ip}q_{ipt}^{1}\text{(The production cost)}\]3  
\[+\sum\limits_{i\in I}\sum\limits_{p\in P}\sum\limits_{t\in T}pc_{ ip}q_{ipt}^{1}\text{(The production cost)}\]4  
\[+\sum\limits_{i\in I}\sum\limits_{p\in P}\sum\limits_{t\in T}pc_{ ip}q_{ipt}^{1}\text{(The production cost)}\]5\[\left\{ \begin{array}{c}z_{j_{k}ypt}\leq y_{j_{k}pt}^{2},\forall j,k,v,p,t\\ z_{j_{k}ypt}\leq y_{j_{k}pt}^{2},\forall j,k,v,p,t\\ z_{lj_{k}ypt}=0,\forall l\in J,j,l\
eq j,v,p,t\\ z_{lkypt}\geq 0,\forall l\in K,k\in K,l\
eq k,v,p,t\\ (1-\alpha)q_{ipt}^{1}=\sum\limits_{j\in I}w_{ijpt}^{1}y_{ijpt}^{1},\forall i, p,t,\\ q_{ipt}^{1}\leq m_{ipt}^{cap}\quad\forall i,p,t,\end{array} \right., \tag{11}\]  
\[w_{ipt}^{2}=\left\{ \begin{array}{c}\sum\limits_{i\in I}w_{ijpt}^{1}y_{ijpt}^{ 1}-w_{jpt}^{3}-w_{ipt}^{4}\quad\forall j,p,t,t=1\\ w_{j_{l}[t-1]}^{2}+\sum\limits_{i\in I}w_{ijpt}^{1}y_{ijpt}^{1}-w_{jpt}^{3}-w_{ jpt}^{4}\quad\forall j,p,t,t\geq 2\end{array} \right., \tag{14}\]  
\[w_{kpt}^{3}=\left\{ \begin{array}{c}\sum\limits_{i}w_{ijp(t-\tau_{p})}^{1}y_{ ijp(t-\tau_{p})}^{1}-\sum\limits_{g=t-\tau_{p}}^{t-1}w_{jp(t-\tau_{p})g}^{5}u_{ jp(t-\tau_{p})g}\quad\forall j,p,t>\tau_{p}\\ 0\quad\forall j,p,t,t\leq\tau_{p}\end{array} \right., \tag{15}\]  
\[w_{ipt}^{4}=\sum\limits_{v\in V}w_{j_{k}ypt}^{6}\ \forall j,p,t, \tag{16}\]  
\[w_{ipt}^{4}=\left\{ \begin{array}{c}\sum\limits_{m=t-\tau_{p}}^{t}w_{jpm}^{5} u_{jpm}^{5}u_{jpm}\quad\forall j,p,t,t>\tau_{p}\\ \sum\limits_{m=1}^{t}w_{jpm}^{5}u_{jpm}\quad\forall j,p,t,t\leq\tau_{p}\end{array} \right., \tag{17}\]  
\[u_{jpmt}\geq u_{jp[m+1]t}\quad\forall j,p,m,t, \tag{18}\]  
\[\left\{