Bullwhip Entropy Analysis and Chaos Control

From the above analysis, we can find with the continuous growth of the sales adjustment speed of retailers, demand model enters into the quasi-periodic state from the quasi-stable state, and finally evolves into the quasi-chaotic state. In this section, we will design experiments to investigate the bullwhip effect, when the demand model is quasi-stable, quasi-periodic, and quasi-chaotic.

Measure of Bullwhip Effect

This experiment mainly inspects the order variance ratio (bullwhip effect). We take the common measure proposed by Chen et al.

Order variance ratio (OVR):

$OVR = \dfrac{σ^2_{q} /μ_q}{σ^{2}_{d}/μ_d}$ (18)

We also set the parameters in the inventory policy and the demand prediction for the following experiments. Considering the online retailer has got the products sooner the traditional retailer, We let lead time $L_{Or}=2$ and $L_{Tr}=3$, and the whole time length $T=52$
for every numerical simulation. Experiments 1 and 2 investigate how bullwhip effect is affected by the risk neutral retailers' sales game strategies.

Experiment 1: To Investigate the Dynamical Evolution of Bullwhip Effect over Time

We design to calculate and compare the order variance ratio of each cycle ($T=52$) when the system is in the different state. As can be seen from Figure 5, when $α=1.5$, for the online retailer, with the increasing of the sales adjustment speed β, system (8) from the quasi-stable, quasi-period enters into quasi-chaos. We set three different β values, respectively, making the system (8) be in a quasi-stable, quasi-periodic and quasi-chaotic state, as follows: (1) Quasi-stable state: $β=1.5$; (2) Quasi-periodic state: $β=3.5$; (3) Quasi-chaos state: $β=4.15$.

In the three different states, we can find some properties by calculating the order variance ratio. Figure 8a gives the order variance ratio diagram of the traditional retailer when the system is in the quasi-stable, quasi-periodic or quasi-chaotic states, respectively. The small window in the north-east corner of Figure 8a is the whole diagram, while the big figure is a magnified image when $OVR ∈ [0,5]$. In the quasi-stable state, with the passage of time, the order variance ratio will decrease rapidly close to 1 during the first ten periods, and be stable at 1 in the time later. While in the quasi-periodic and quasi-chaotic state, the order variance ratio will gradually increase after a rapid reduction, since time $t=10$. Figure 8b gives the order variance ratio diagram of the online retailer when the system is in the quasi-stable, quasi-periodic or quasi-chaotic states. The small window in the north-east corner of Figure 8b is the whole diagram, while the big figure is a magnified image when $OVR ∈ [0,12]$. There is the similar trend for the OVR of the online retailer in the quasi-stable state. Unlike the traditional retailer, the OVR of the online retailer is greater in the quasi-periodic and quasi-chaotic states. After time $t=17$, the $OVR$ of the online retailer in the quasi-periodic state is larger than that in quasi-chaotic state.

Figure 8. (a) Timing diagram of the traditional retailer's bullwhip effect; (b) Timing diagram of the online retailer's bullwhip effect.

Experiment 2: To Investigate the Impact of the Sales Adjustment Speed on the Bullwhip Effect

We design to calculate the order variance ratio of the total 52 cycles and investigate the changes of the OVR as the sales adjustment speed of the online retailer changes. In order to further study how the adjustment speed of sales volume for retailers affects the bullwhip effect, we set $α=1.5$, and make β increase. We turn to observe the change trends of bullwhip effect in the process of system evolution for two different retailers.

Figure 9a gives the influence of adjustment speed on the bullwhip effect for the traditional retailer. From the figure, we can find that when the online retailer begins to adopt an adjustment mechanism in a quasi-stable state, there is a little bullwhip effect from the traditional retailer's orders when $β < 1.6$. There is no bullwhip effect when $1.6 < β < 3$. When system (8) is beginning to enter into a quasi-periodic state, the bullwhip effect appears and increases rapidly, with some waving phenomena. Bullwhip effect will be in the high turbulence when the system is in a quasi-chaotic state.

Figure 9. (a) Effect of β on the traditional retailer's OVR; (b) Effect of β on the online retailer's OVR.

As shown in Figure 9b, the order variance ratio will show a similar trend to that of the traditional retailer when the online retailer adopts an adjustment mechanism. There is a little bullwhip effect from the online retailer's orders when $β < 0.94$ and no bullwhip effect when $0.94 < β < 2.9$. There is a rapid increase once $β > 2.9$,
with a high turbulence in a quasi-chaotic state.

Through the comparison of the two maps, we find that both retailers will suffer little bullwhip effect when the system is in the quasi-stable state. Bullwhip effect can be mitigated completely for the two retailers when $1.6 < β < 2.9$. The largest OVR of the traditional retailer is less than 2, while the OVR of the online retailer is larger than 2 in the vast majority of cases when the system is in a quasi-periodic state or a quasi-chaotic state. This system instability caused by the online retailer has a greater impact on itself, leading to greater bullwhip effect in its own channel.