Nonlinear modeling and performance analysis of a closed-loop supply chain in the presence of stochastic noise

Figures & data

Figure 1. The supply chain system with hybrid recycling channel and sales game.

Table 1. Notations.

$x_1\left(k\right)$	Manufacturer inventory
$x_2\left(k\right)$	Wholesaler inventory
$x_3\left(k\right)$	Retailer inventory
$x_4\left(k\right)$	Recovery provider inventory
$x_5\left(k\right)$	Customer demand
$u_1\left(k\right)$	Production
$u_2\left(k\right)$	Wholesaler order
$u_3\left(k\right)$	Retailer order
$u_4\left(k\right)$	Price as penalty
$a$	Disposal rate
$b$	Remanufacturing rate
$c$	Recycling rate
$d$	Return rate
$p$	Acceptable price ceiling of the product for the retailer
$q$	The order quantity of retailer
$\theta$	The impact factor of return rate on the price
$\alpha$	The retailer adjustment speed

Figure 2. Frequency responses of $\frac{{\mathrm{U}}_3\left(\mathrm{z}\right)}{X_5\left(z\right)}$ in the case of complexity dynamic for various lead time.

Figure 3. Frequency responses of $\frac{{\mathrm{U}}_3\left(\mathrm{z}\right)}{X_5\left(z\right)}$ in the case of complexity dynamic for various gain.

Figure 4. Frequency responses of $\frac{{\mathrm{U}}_3\left(\mathrm{z}\right)}{X_5\left(z\right)}$ in the case of complexity without any lead time and gain.

Figure 5. The magnitude ratio of transfer function for all nodes.

Figure 6. Frequency responses of $\frac{U_3\left(z\right)}{X_5\left(z\right)}$ for various variance noise (process and measurement noise).

Figure 7. Frequency responses of $\frac{U_j\left(z\right)}{U_{j+1}\left(z\right)}$ for all nodes with same variance noise.

Figure 8. The bullwhip effect in supply chain without controller and supply chain with controller.

Figure 9. The bullwhip effect in supply chain without controller and supply chain with controller.

Table 2. Simulation data.

	Manufacturer	Wholesaler	Retailer
Initial inventory	200	170	100
Safety inventory	150	160	90
Expected inventory	400	180	120
Maximum inventory	600	200	130

Figure 10. Output of closed-loop supply chain (system with optimal controller).

Figure 11. Optimal policy.

Figure 12. LQG tracker.

Figure 13. Optimal order, production and pricing.