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.