BUS606: Operations and Supply Chain Management

The above simulations suggest that the retailers' sales adjustment speeds in the quasi-stable state would bring less bullwhip effect than the quasi-period doubling and quasi-chaotic states. From the supply management perspective, to keep the system away from chaos and weaken the bullwhip effect are helpful for the retailers and the manufacturer. Therefore, it is important to find a useful method of chaos control. Some literatures in the field of supply chain complexity have successfully used the delayed feedback control method to control the chaos of supply chain system [34,35].

Definition 3. (The Delayed Feedback Control (DFC) Method). For a general chaotic system with an additional feedback force:

$x(t)=Bx(t−1)+f(x(t−1))+u(t)$ (19)

where $x(t)∈R^n$ is the state vector, $u(t)$ is feedback control input vector, $B$ is constant system matrix representing the linear parts of the system, $f$ is the nonlinear parts of the system. If the control input is presented as $u(t)=K(x(t−ΔT)−x(t))$, the method is called the delayed feedback control method, where ΔT
is the length of the lag time, K is the control parameter.

We make the online retailer control the chaos when making decisions on the sales with the help of the control parameter K. With the controlling, the sales (demands) of the online retailer can be rewritten as:

$d_{Or,t} = βd_{Or,t−1}(a−cOr+θ(φ¯−φ)−b_{21}d_{Tr,t−1}−2b_{22}d_{Or,t−1})(1−φ) +d_{Or,t−1}+K(d_{Or,t−ΔT}−d_{Or,t})+τε_{Or,t}$ (20)

We consider the control parameter in the system with $ΔT=1$ , and then we can obtain a new sales demand game system for the two retailers of the supply chain, under the control of the online retailer:

$\left\{ \begin{array} {ll} d_{Tr,t}=αd_{Tr,t−1}(a+\overline{a}−cT_r−2b_{11}d_{Tr,t−1}−b_{12}d_{Or,t−1})+d_{Tr,t−1}+τεT_r,t \\ d_{Or,t} = βd_{Or,t−1}(a−cO_r+θ(\overline {φ} φ)−b_{21}d_{Tr,t−1}−2b_{22}d_{Or,t−1})(1−φ)+d_{Or,t−1}+K(d_{Or,t−1}−d_{Or,t})+τεO_r,t \end{array} \right.$ (21)

First, we make the control parameter $K=0.5$ and the adjustment parameter $α=1.5$ to investigate the effect of the DFC method. Figure 10a shows the demands quasi-bifurcation diagram of the two retailers as the adjustment parameter $β$ increasing from 4 to 6.5. The small window in Figure 10a shows the whole bifurcation diagram when $β∈(0,6.5)$. We can see that the quasi-stable state is extended due to the feedback parameter ($K=0.5$). Figure 11b shows the bullwhip effect of the online retailer as the sales adjustment speed $β$ varies from 0 to 6.5. We can see that the period with little bullwhip effect is extended obviously.

Figure 10. (a) Quasi-bifurcation diagram; and (b) the online retailer's bullwhip effect with $K=0.5$, $α=1.5$ and $β∈[0, 6.5]$.

Figure 11. (a) Quasi-bifurcation diagram; and (b) the online retailer's bullwhip effect with $α=1.5$ and $β=4.15$,  $K∈(0, 1)$.

Secondly, we observe the control process of the system and the bullwhip effect, when the adjustment parameters $α=1.5$ and $β=4.15$. According to the previous numerical simulation and analysis, we know that in this case the system is in a state of chaos. Next we will observe the bifurcation diagram and bullwhip effect of the online retailer in the controlled system after entering the control parameter K.

Based on the numerical simulation results, we can find that the system ($α=1.5$,  $β=4.15$) is in a quasi-chaotic state and the system is gradually changed from quasi-chaos to quasi-stable state under the action of the DFC parameter K.

From Figure 11a, we can see that when $K > 0.13$, the system is in a quasi-two-fold period state; when K > 0.4, the competition system is out of chaos and enters into a quasi-stable state. Figure 11b shows the impact of the control parameter K on OVR. As the control parameter grows from zero, the OVR of the online retailer remains high and volatile until K > 0.2. There is a rapid descent of OVR when the control parameter is in the interval [0.2, 0.4], especially, OVR is equal to one when K = 0.4, and less than one when K > 0.4, i.e., the bullwhip effect is mitigated thanks to K.

The effect of the control parameter can also be found in Figure 12. As the growth of K, the entropy of the online retailer will decrease significantly until near to 0.4, then it will be stable at zero. The entropy characters of Figure 12 are consistent with the quasi-bifurcation diagram and bullwhip effect in Figure 11.

Figure 12. Entropy diagram of the online retailer with $α=1.5$ and $β=4.15$,  $K∈(0, 1)$.

This shows that the DFC method has achieved a good effect, and effectively alleviated the supply chain bullwhip effect and entropy under a sales game demand model.