Data Science for Business and Decision Making

$\begin{array}{l}{F}_{\mathrm{obj}}=minz=\sum _{i=1}^m\sum _{j=1}^n\sum _{t=1}^T\left(c_{\mathit{ijt}}{x}_{\mathit{ijt}}+i_{\mathit{ijt}}{I}_{\mathit{ijt}}+\sum _{k=1}^p{y}_{\mathit{ijkt}}{z}_{\mathit{ijkt}}\right)\\ \mathrm{s}.\mathrm{t}.\hfill \\ \sum _{k=1}^p{D}_{\mathit{ikt}}{z}_{\mathit{ijkt}}+I_{\mathit{ijt}}=I_{\mathit{ij},t-1}+x_{\mathit{ijt}},i=1,\dots ,m;j=1,\dots ,n;t=1,\dots ,T\left(1\right)\\ \sum _{j=1}^n{z}_{\mathit{ijkt}}=1,k=1,\dots ,p;\left(2\right)\\ x_{\mathit{ijt}}\le x_{\mathit{ijt}}^{\max },i=1,\dots ,m;j=1,\dots ,n;t=1,\dots ,T\left(3\right)\\ I_{\mathit{ijt}}\le I_{\mathit{ijt}}^{\max },i=1,\dots ,m;j=1,\dots ,n;t=1,\dots ,T\left(4\right)\\ z_{\mathit{ijkt}}\in \left\{0,1\right\},i=1,\dots ,m;j=1,\dots ,n;k=1,\dots ,p;t=1,\dots ,T\left(5\right)\\ x_{\mathit{ijt}},I_{\mathit{ijt}}\ge 0i=1,\dots ,m;j=1,\dots ,n;t=1,\dots ,T\end{array}$

Ex. 17

Decision variables:

x_ijt = quantity of product i to be manufactured in facility j in period t

I_ijt = final stock of product i in facility j in period t

Y_ijkt = quantity of product i to be transported from facility j to retailer k in period t

$z_{\mathit{ijt}}=\left\{\begin{array}{l}1\text{if the manufacturing of product}i\text{in period}t\text{occurs in facility}j\\ 0\text{otherwise}\end{array}\right.$

Model parameters:

D_ikt = demand ...