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

Ex. 17

Decision variables:

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

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

Yijkt = 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:

Dikt = demand ...

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