Integrated Production-Inventory Supply Chain Model
Read this article. An integrated production-inventory model is constructed to address supplier, manufacturer, and retailer uncertainties. According to the author, what are the three types of uncertainties in supply chain management?
Mathematical formulation of the model
Formulation of suppliers' individual average profit
Differential equation for the supplier in Figure 2 in [0,T] is given by
(3)
![](https://learn.saylor.org/pluginfile.php/5619158/mod_book/chapter/50085/2.jpeg)
![](https://learn.saylor.org/pluginfile.php/5619158/mod_book/chapter/50085/40467_2013_Article_6_Fig4_HTML.webp)
![\begin{aligned}&=c_{r} I_{p}\left[n \int_{M}^{T_{R}}\left(D_{R}-D_{c} t\right) d t\right] \\&=\frac{n c_{r} I_{p}}{2}\left[T_{R} D_{R}+D_{c} M^{2}-2 M D_{R}\right]\end{aligned} \begin{aligned}&=c_{r} I_{p}\left[n \int_{M}^{T_{R}}\left(D_{R}-D_{c} t\right) d t\right] \\&=\frac{n c_{r} I_{p}}{2}\left[T_{R} D_{R}+D_{c} M^{2}-2 M D_{R}\right]\end{aligned}](https://learn.saylor.org/filter/tex/pix.php/954a7522958a97abb68408d026e6c572.svg)
Case 2
![\begin{aligned}=& \frac{D_{c}}{p_{m} T_{s}+D_{R}}\left[\left(c_{m}-c_{s}\right) p_{m} T_{s}-h_{s}\left(\frac{p_{s} t_{s}^{2}}{p_{m}}-p_{s} t_{s}^{2}\right)-\mathrm{id}_{s}\left(T_{R}+p_{s} t_{s} \frac{1}{D_{c}}-\frac{1}{p_{m}}\right)-A_{s}\right.\\&+\left(c_{r}-c_{m}\right) p_{m} T_{s}-h_{m}\left(n p_{m} T_{s} T_{R}-\frac{n^{2}+n-2 r-2}{2} T_{R} D_{R}-\frac{p_{s}^{2} t_{s}^{2}}{2 p_{m}}\right) \\&-\operatorname{id}_{m}\left(\frac{p_{m} T_{m}-n D_{R}}{D_{c}}\right) \\&-A_{m}+\left(c_{r 1}-c_{r}\right) p_{m} T_{s}-\frac{h_{r}}{2}\left(\frac{p_{m}^{2} T_{s}^{2}}{D_{c}}-2 n p_{m} T_{s} T_{R}-(2 n+1) T_{R} D_{R}\right) \\&\left.+\frac{n c_{r_{1}} I_{e} D_{c} M^{2}}{2}+\frac{c_{r_{1}} I_{e}}{2}\left(p_{m} T_{s}-n D_{R}\right)(2 M-T)-\mathrm{id}_{r} T_{R}-A_{r}\right] \\=& \frac{D_{c}}{p_{m} T_{s}+D_{R}}\left[A p_{s}^{2} t_{s}^{2}-B p_{s} t_{s}+F\right]\end{aligned} \begin{aligned}=& \frac{D_{c}}{p_{m} T_{s}+D_{R}}\left[\left(c_{m}-c_{s}\right) p_{m} T_{s}-h_{s}\left(\frac{p_{s} t_{s}^{2}}{p_{m}}-p_{s} t_{s}^{2}\right)-\mathrm{id}_{s}\left(T_{R}+p_{s} t_{s} \frac{1}{D_{c}}-\frac{1}{p_{m}}\right)-A_{s}\right.\\&+\left(c_{r}-c_{m}\right) p_{m} T_{s}-h_{m}\left(n p_{m} T_{s} T_{R}-\frac{n^{2}+n-2 r-2}{2} T_{R} D_{R}-\frac{p_{s}^{2} t_{s}^{2}}{2 p_{m}}\right) \\&-\operatorname{id}_{m}\left(\frac{p_{m} T_{m}-n D_{R}}{D_{c}}\right) \\&-A_{m}+\left(c_{r 1}-c_{r}\right) p_{m} T_{s}-\frac{h_{r}}{2}\left(\frac{p_{m}^{2} T_{s}^{2}}{D_{c}}-2 n p_{m} T_{s} T_{R}-(2 n+1) T_{R} D_{R}\right) \\&\left.+\frac{n c_{r_{1}} I_{e} D_{c} M^{2}}{2}+\frac{c_{r_{1}} I_{e}}{2}\left(p_{m} T_{s}-n D_{R}\right)(2 M-T)-\mathrm{id}_{r} T_{R}-A_{r}\right] \\=& \frac{D_{c}}{p_{m} T_{s}+D_{R}}\left[A p_{s}^{2} t_{s}^{2}-B p_{s} t_{s}+F\right]\end{aligned}](https://learn.saylor.org/filter/tex/pix.php/b71242caf17635e7295ddb8b9b2f658b.svg)
(19)
where A and B are given in (16) and (17), respectively and
Differential equation for the supplier in Figure 2 in [0,T] is given by
with boundary conditions
and
. Solving the differential equation with the boundary condition, we have
(3)
Figure 2 Inventory level of supplier.
![](https://learn.saylor.org/pluginfile.php/5619158/mod_book/chapter/50085/2.jpeg)
(4)
Formulation of manufacturer individual average profit
Inventory level of manufacturer in Figure 3 in [0,T] is given by![](https://learn.saylor.org/pluginfile.php/5619158/mod_book/chapter/50085/f3.jpeg)
Formulation of retailer individual average profit
Inventory level of retailer in Figure 4 in [0,T] is given byFigure 4 Inventory level of retailer.
![](https://learn.saylor.org/pluginfile.php/5619158/mod_book/chapter/50085/40467_2013_Article_6_Fig4_HTML.webp)
(10)
(11)
Case 2
(19)
where A and B are given in (16) and (17), respectively and
F=[hmn2+n−2r−22+hr2n+12]TRDR+(n+1)cr1IeDcM22
+(nidm−ids−idr)TR+idsTs−(As+Am+Ar) (20)