BUS606: Operations and Supply Chain Management

The nonlinear demand process with random disturbance shown in Equation (8) consists of a deterministic system and random items. When an attractor is disturbed by noises, the trajectories will deviate from this attractor temporarily, but most of them can be attracted back and form a cloud or bundle around the deterministic attractor, which is called stochastic attractor (Bashkirtseva and Ryashko [29]). For systems perturbed by weak noises, the response can be seen as stochastic attractors around the deterministic attractors and transitions between them. If $τ$ is small enough compared with the value of di, the disturbance from the white noise doesn't change the properties completely. The time-consuming direct numerical simulation is a basic tool for analyzing the effects of small random perturbations in the stochastic dynamics.

In this paper, we investigate a nonlinear demand process with weak random disturbance. In the following subsection, we study the equilibrium and the local stability of the deterministic system firstly, and then analyze the complexity of the demand model.

Equilibrium and the Local Stability

Definition 1. (Equilibrium Point). An equilibrium point of the difference system $d_{i,t+1}=f(d_{i,t})$,
is $(d^{∗}_{1},d^{∗}_{2})$ such that $d^{∗}_{i} =f(d^{∗}_{i} ) ( i=1,2)$.

Definition 2. (Nash Equilibrium Point). An equilibrium point $(d^{∗}_{1},d^{∗}_{2})$
is a Nash equilibrium point of the difference system, if no retailer has anything to gain by changing only his or her own strategy.

There are four equilibrium points in the deterministic system of Equation (8):

$E_0(0,0), E_1(\dfrac{a+\overline{a}−c_{Tr}} {2b_{11},0)}, E_2(0,\dfrac{a−c_{Or}+θ(\overline{φ}−φ)}{2b_{22}})$

$E^∗(\dfrac{b_{12}(a−c_{Or}+θ(\overline{φ}−φ))−2b_{22}(a+\overline{a}−c_{Tr})}{b_{12}b_{21}−4b_{11}b_{22}} , \dfrac{b_{21}(a+\overline{a}−c_{Tr})−2b_{11}(a−c_{Or}+θ(\overline{φ}−φ))}{b_{12}b_{21}−4b_{11}b_{22})}$

where $E_0,  E_1, E_2$ are boundary equilibrium points, only $E^∗$ is the Nash equilibrium point.

Proposition 1. In the deterministic system of Equation (8), the boundary equilibrium point $E_0, E_1$ and $E_2$ are not stable.

At the equilibrium point $E^∗$, the Jacobian matrix has the form:

$J(E^∗)= j_0 \left[\begin{array}{ll} j_{11} & j_{12}\\ j_{21} & j_{22}\end{array} \right]$ (14)

here, $j_0=1/(b_{12}b_{21}−4b_{11}b_{22})$,

$j_{11}=4b_{11}b_{22}(−1+α(a+\overline{a}−c_{Tr}))+b_{12}(b_{21}−2b_{11}α(a−c_{Or}+θ(\overline{φ}−φ)))$,

$j_{12}=b_{12}α(2b_{22}(a+\overline{a}−c_{Tr})−b_{12}(a−c_{Or}+θ(\overline{φ}−φ)))$,

$j_{21}=b_{21}β(−1+φ)(b_{21}(a+\overline{a}−c_{Tr})−b_{21}c_{Tr}−2b_{11}(a−c_{Or}+θ(\overline{φ}−φ)))$,

$j_{22}=j_0+b_{21}β(−1+φ)(b_{21}(a+\overline{a}−c_{Tr})−2b_{11}(a−c_{Or}+θ(\overline{φ}−φ)))$. In order to guarantee the sales at the equilibrium point are positive, we can set: $b_{12}b_{21}−4b_{11}b_{22} < 0$.

From the reality of the social economic activity perspective, the Nash equilibrium point $E^∗$ is more meaningful. At this point the two retailers' sales volumes are not equal to zero. We can derive the sufficient and necessary condition for the stability by means of the Jury criterion:

$\left\{ \begin{array}{lll} 1+T_r(J(E^∗))+Det(J(E^∗)) > 0 \\ 1−T_r(J(E^∗))+Det(J(E^∗)) > 0 \\ 1−Det(J(E^∗)) > 0 \end{array} \right.$ (15)

In order to obtain a better understanding of the stability characteristics of the system, we assign fixed values to the parameters. Unless otherwise specified, the parameter values will be still used in the following article. Without loss of generality, we set the parameters for $c_{Or}=0.2$, $c_{Tr}=0.2$,  $a=1$, $\overline{a}=0.1$,  $b_{11}=1.1$, $b_{22}=1$, $b_{12}=0.25$, $b_{21}=0.3$, $\overline{φ}=0.05$, $φ=0.07$. At this moment, the stability region of the deterministic system of Equation (8) is shown as the green part in Figure 2a. The stability region of system is determined by both retailers' adjustment speed parameter. For both retailers, when the speed of the sales adjustment is in the stable region, the system tends to a Nash equilibrium point after a finite game; otherwise, the system is unstable.

Figure 2. (a) Stability region; and (b) basin of attraction of system.

Figure 2b shows the basin of attraction in a stable system. The x-axis represents the traditional retailer's sales game decision and the y-axis represents the sales decision of the online retailer. The initial decision variables in the domain of attraction will converge to the same equilibrium point called an attractor. Here, the equilibrium attractor is $E^∗=(0.37, 0.34)$.

This means that the stable system is not sensitive to the initial decision values taken from the domain of attraction. However, not every initial sales volume can converge to that equilibrium point. Once the decision variables aren't in the basin of attraction, the systems will not converge. Therefore, both retailers should be mindful of the initial sales volume and avoid making sales out of the green area to maintain the market stability.

Complexity of the Demand Model

In this subsection, we investigate the complexity of the demand system with weak noises. Disturbed by weak noises, the stable, periodic and chaotic states of the deterministic system of Equation (8) will evolve into the quasi-stable, quasi-periodic and quasi-chaotic states. We may investigate the properties of the demand model with the help of the deterministic system of Equation (8). Firstly, we want to find the impact of both retailers' sales adjustment speeds on the system.

Figure 3 shows parameter basin of the deterministic system of Equation (8), in which different colors represent different states. Besides the green stable state, the yellow area is cycles of period 2, purple for period 4, gray for chaos, and white for divergence. With the growth of α and β, green, yellow, red and purple constitute the road from period doubling bifurcation to chaos. In the discrete system, it is also known as the flip bifurcation. It is worth noting that there are also some intermittent odd periodic points in the chaotic region of the deterministic system of Equation (8), and we call them topological chaos. In Figure 3, we can find that there are two roads leading to the chaos:

1. When the adjustment speed of sales for retailers pass from the green area, sequentially through the yellow, red and purple areas to the gray area, the system enters into a chaotic state through the flip bifurcation. In the state of chaos, if retailers continue to increase the speed of adjustment, the system will overflow, which means one retailer will withdraw from the market.
2. When the adjustment speed of sales for retailers from the green area sequentially pass the yellow and red area to the gray area, the system from 4 cycle period enters into a chaotic state through the Neimark-Sacker bifurcation.
   

Figure 3. Parameter basin.

Secondly, we investigate the change of two retailers' sales volumes with respect to the adjustment speed parameter of online retailer via the bifurcation of the deterministic system. Figure 4 shows the bifurcation diagram of the deterministic system with $α=1.5$ and β varying from 2.7 to 4.3. The small window in Figure 4 shows the whole bifurcation diagram when $β ∈ (0,4.3)$. The red line is the sales volume of the online retailer and the blue line represents that of the traditional retailer. As shown in the diagram, when the value of β is less than 3.03, the sales volumes of both retailers are stable and the traditional retailer sells much more products than the online retailer. Once the adjustment speed of the online retailer grows larger than 3.03, the system experiences the bifurcation and enters into the two-period state. After four-period and eight-period, the system will finally fall into chaos. Apparently, the online retailer will be affected much more by the growth of β
and the vibration amplitude of its sales volume is more than that of the traditional retailer.

Figure 4. Bifurcation diagram of the deterministic system.

In the following section, we investigate the complexity of the system (8) considering the weak noise with the help of the largest Lyapunov exponent (LLE). The LLE is an effective means for determining and classifying nonlinear system behavior. If the LLE is less than zero, the deterministic system is in a quasi-stable state. With the growth of the adjustment parameter, once the LLE increases to zero, the system will lose its stability and enters into the stage of quasi-periodic bifurcation if the LLE becomes negative again. But if the LLE becomes positive after a zero value, the system will enter into chaos. Therefore, we can depend on the LLE diagram to determine the state of the demand model (quasi-stable, quasi-periodic, and quasi-chaotic).

When the noise intensity $τ$ is set to 0.001, we can plot the quasi-bifurcation diagram of the demand model and LLE of system (7) in a figure. Figure 5a,b give the quasi-bifurcation diagram and the corresponding LLE, with $α=1.5$ and β  from 0 to 4.5. As shown in the figures, the LLE is less than zero when $β < 3.03$. Therefore, the deterministic system of Equation (8) is in the stable state. However, system (8) is in a quasi-stable state rather than a pure stable state, i.e., the sales volumes of retailers are not determined at fixed values and fluctuate around their equilibriums slightly. The same phenomenon occurs in the period-doubling bifurcation stage, and the system is more chaotic in the quasi-chaos state.

Figure 5. (a) Quasi-bifurcation; (b) Largest Lyapunov exponent; and (c) Entropy diagram.

According to the theory of entropy (Han et al. [33]), we plot the entropy diagram to show the complexity of the system. Figure 5c gives the entropy diagram of the sales volume for the online retailer as β
varying from 0 to 4.5. Combining this with Figure 5b, we can find that the entropy is equal to zero when system is in the quasi-stable state ($β < 3.03$), simultaneously, the LLE is less than zero. Once the system enters into a period-doubling bifurcation stage or the quasi-chaos state ($β>3.03$) , the entropy will be positive. The larger the entropy is, the more chaotic the system is.

Next, we study the complexity with the help of the system's attractor. Figure 6 shows attractors of system (8) and its deterministic system when $β=4.15$. In Figure 6a, the blue line is the chaos attractor of the deterministic system, while the red points around the blue line is a quasi-chaos attractor with weak noise in Figure 6b.

Figure 6. (a) The attractor of the deterministic system; and (b) the quasi-chaos attractor of the system with weak noise.

Impact of the Return Rate

In this subsection, the profits of two retailers will be analyzed in the stable state. The optimal sales volumes and profits of retailers can be derived via the first-order optimal condition.
The optimal sales volumes can be expressed as:
$\left\{ \begin{array}{ll} d^{∗}_{Tr}=\dfrac{b^{12}(a−c_{Or}+θ(\overline{φ}−φ))−2b_{22}(a+\overline{a}−c_{Tr})}{b_{12}b_{21}−4b_{11}b_{22}} \\ d^{∗}_{Or}=\dfrac{b_{21}(a+\overline{a}−c_{Tr})−2b_{11}(a−c_{Or}+θ(\overline{φ}−φ))} {b_{12}b_{21}−4b_{11}b_{22}} \end{array} \right.$ (16)

Substituting Equation (16) into Equations (3) and (4), the optimal profits of two retailers can be derived:

$\left\{ \begin{array}{ll} π^{∗}_{Tr}=\dfrac{b_{11}(b_{12}(a−c_{Or}+θ(\overline{φ}−φ))−2b_{22}(a+\overline{a}−c_{Tr}))^2}{(b_{12}b_{21}−4b_{11}b_{22})^2} \\ π^{∗}_{Or}=\dfrac{b_{22}(1−φ)(b_{21}(a+\overline{a}−c_{Tr})−2b_{11}(a−c_{Or}+θ(\overline{φ}−φ)))^2}{(b_{12}b_{21}−4b_{11}b_{22})^2}\end{array} \right.$ (17)

Proposition 2. When the online retailer employs a return policy, the sales of the traditional retailer increase, but those of the online retailer will be reduced by the growth of the return rate in the online channel. The optimal sales volume in the online channel will be affected by its return rate much more than that of a traditional retailer.

Proposition 3. When the online retailer employs a return policy, the profits of the traditional retailer and the online retailer are both decreasing functions with respect to the return rate.

Based on the above parameter hypothesis, a numerical simulation is proposed to investigate the impact of the return rate on profits. Figure 7 shows how the two retailers' optimal profits are affected by the return rate. The red line is the profit of the online retailer and that blue line is that of the traditional retailer. It is clear that the profit of the online retailer will be reduced as the return rate grows. However, it is interesting that the profit of the traditional retailer will be reduced by the growth of the online retailer's return rate.

Figure 7. The optimal profits with respect to the return rate.