Integrated Production-Inventory Supply Chain Model

The integrated inventory model (Figure 1) starts when $t = 0$ and stock is zero. At that time, the suppliers start their production with the rate $p_s$ unit per unit time and purchase at the rate $p_m$ unit per unit time to the manufacturer. When $t = t_s$ , suppliers stop their production, and at $t = T_s$ , the inventory level of suppliers become zero. The total time of the integrated model is $T$, so the idle time for suppliers is $T−T_s$. Similarly, the manufacturers start their production at the same time $t = 0$ with the production rate $p_m$ unit per unit time and purchase this production $D_r$ unit to the retailer in the time gap $T_R$, which is the bulk pattern. At time $t=T_s (=[ \dfrac{T_s}{T_R}])$, manufacturers stop their production, and at $t = (n + 1)T _R ( n=[\dfrac{p_mT_s}{D_r}])$, the stock of manufacturer is zero. Thus, idle period for the manufacturer is $T − (n + 1)T_r$. Retailers start selling this product to the customers at time $t = T_r$ and end selling at $T=(n+1)T_r+\dfrac{{p_m}^{T_s−nD_nr} }{D_c}$. The idle period for retailers is $T_r$.

Figure 1 Inventory level for the integrated model.