Bullwhip Entropy Analysis and Chaos Control

In this paper, we establish a supply chain system that consists of one manufacturer and two retailers, one of which is a traditional retailer and the other is an online retailer. The two retailers sell similar products to a common market, and there is a competitive relationship between them as shown in Figure 1.

Figure 1. Supply chain model.

Customers who purchase a product through online retailers can return it within 7 days without any reason because the 7 days no reason for return policy is statutory for online retailers and not applicable to traditional retailers. The returned products are assumed to satisfy the conditions to be sold again without any other costs. The products meeting the conditions will be returned to the online retailer, after a return cycle through its reverse logistics. In order to pursue greater profit and market share, retailers tend to adopt some strategy to increase sales. For example, they usually use advertising, promotions, and providing gifts and other ways to stimulate the desire of consumers to buy more, so as to achieve the purpose of improving the market demand for their products. We are working on the assumption that they both take the bounded rational decision to determine their sales volume. The retailers are both familiar with the market and can obtain the demand information of the current period for their products in a timely way. They use a sales game to forecast the sales volume (demand forecast) for the next period according to the bounded rationality expectation. The order-up-to inventory policy is assumed to be employed by the two retailers. Based on the current and the forecasted demand for the future, retailers estimate the order-up-to point and place their own orders. According to the orders submitted by retailers, the manufacturer delivers the products to the retailers and after a lead time, both retailers receive the products. Forecasting method, demand process, inventor-policy and some symbol notations are introduced in the following subsections.

Notation

Table 1 presents the operations management-related variables and parameters of the proposed supply chain model.

Table 1. Notations.

$a$    the price ceiling of the product for the online retailer

$\overline{a}$    the price gap between two retailers

$θ$    the impact factor of return rate on the price

$φ$    the return rate

$\overline{φ}$    the benchmark of the return rate

$b_{mn}$    the impact of the sales volume on the price

$p_i$    the price of retailer $i (i=T_r,O_r)$

$d_{i,t}$    demand (or sales) of retailer $i (i=T_r,O_r)$ at period $t$

$c_i$    the unit wholesale price for retailer $i (i=T_r,O_r)$

$ε_{i,t}$    Gaussian white noise for retailer $i$

$τ$    the intensity of the noise

$q_{i,t}$    the order quantity of retailer $i$ at period $t$

$S_{i,t}$    the order-up-to point of retailer $i$ at period $t$

$\widehat{D}^{L}_{i,t}$    the estimated demand of retailer $i$ in the lead-time at period $t$

$\widehat{d} _{i,t}$    the prediction of retailer i's demand at period $t$

$π_{i,t}$    the profit of retailer $i$ at period $t$

$α$    the adjustment parameter of the traditional retailer in the game system

$β$    the adjustment parameter of the online retailer in the game system

$L_i$    the lead-time of retailer $i$

$σ ^{L_i}_{t}$    an estimate of the standard deviation of the forecasting error of the lead-time

$z$    retailers' service level

$σ^{2}_{q}$    the variance of the order

$σ^{2}_{d}$    the variance of the demand

$μ_d$    the mean of the demand

$μ_q$    the mean of the order

$T$    the time length of the numerical experiment

Demand Forecasting with Bounded Rationality

Suppose that retailers determine the demand forecasting in the next period with bounded rationality, on the basis of the partial estimation of the marginal profit of their current period, i.e., the retailer will reduce its sales volume forecast for the next period if the current marginal profit is negative, the retailer will maintain its sales volume, if the current marginal profit is zero. Otherwise, the retailer will increase the sales volume in the next period.

In the same market, sales volume of both retailers have negative effect on their prices because of the similarity of their products, so the inverse demand function can be written as:

$p_{Tr}=a+\overline{a}−b_{11}d_{Tr}−b_{12}d_{Or}$ (1)

$p_{Or}=a−b_{21}d_{T_r}−b_{22}d_{Or}+θ(\overline{φ}−φ)$ (2)

here, the parameter a represents the acceptable price ceiling of the product for the online retailer. $\overline{a}$ is the price gap between the traditional retailer and the online retailer and the physical traditional retail price is usually higher than that of online retailer, i.e.,$\overline{a} > 0$. The parameters $b_{mn}$ determine the relationship between prices and sales, $b_{mm}(m=1, 2)$ determine the impact of sales on their own prices, and $b_{mn}(m≠n)$ determines the impact of their rival's sales on their own prices. The quotient $b_{mn}/b_{mm}∈(0, 1)$, denotes the index of supply chain differentiation or supply chain substitution. The degree of supply chain differentiation will increase as bmn/bmm. The traditional retailer and online retailer are homogeneous when $b_{mn}/b_{mm}=1. φ$ is the return rate and $\overline{φ}$ is the benchmark of the return rate which is considered normal by most consumers. $θ$ is the impact factor of return rate on the price.

We assume that both retailers purchase goods from the manufacturer in accordance with the price of $c_i$, so we can respectively obtain the profit functions of the retailers:

$π_{Tr}=d_{Tr}(a+\overline{a}−b_{11}d_{Tr}−b_{12}d_{O_rt}−c_{T_r})$ (3)

$π_{Or}=(1−φ)d_{Or}(a−b_{21}d_{Tr}−b_{22}d_{Or}+θ(\overline{φ}−φ)−c_{Or})$ (4)

The marginal profits of two retailers can be calculated using the following expressions:

$\dfrac{∂πT_r}{∂dT_r}=a+\overline{a}−cT_r−2b_{11}d_{Tr}−b_{12}d_{Or}$ (5)

$\dfrac{∂πO_r}{∂dO_r}=(a−c_{Or}−b_{21}d_{Tr}+θ(\overline{φ}−φ)−2b_{22}d_{Or})(1−φ)$ (6)

When the retailers adopt a sales game, they consider some existing sales information and their own marginal profit. If the marginal profit is greater than zero, the retailer can increase sales on the basis of the current sales volume. Retailers can give the forecast demand for the next period according to the bounded rational expectation:

$\left\{ \begin{array}{ll} \widehat {d}_{Tr,t}=d_{Tr,t−1}+αd_{Tr,t−1}(a+\overline{a}−c_{Tr}−2b_{11}d_{Tr,t−1}−b_{12}d_{Or,t−1}) \\\widehat {d}_{Or,t}=d_{Or,t−1}+βd_{Or,t−1}(a−c_{Or}+θ(\overline{φ}−φ)−b_{21}d_{Tr,t−1}−2b_{22}d_{Or,t−1})(1−φ) .\end{array} \right.$ (7)

The parameters $α,β( > 0)$ in Equation (7) represent the two retailers' adjustment speed in each supply chain, respectively. The values of the adjustment speed ($α,β$) depend on the enthusiasm of the retailer's pursuit of profit, and their ability to regulate and control the sales. The retailer who has a strong drive to increase profits can take a bigger adjustment parameter and needs to have a stronger ability to regulate and control his sales.

Demand Model

In this paper, we consider the retailer sales volume is equal to the consumer demand for goods. $d_t$ is the whole demand of consumer ($d_t=d_{1,t}+d_{2,t})$, and $q_t$ is the total orders of the two retailers ($q_t=q_{1,t}+q_{2,t}$.

Besides the retailers' sales game, the demand for one product is also affected by some incidental and random factors. Therefore, a demand model with random disturbance term is more representative of the actual supply chain:

$\left\{ \begin{array}{ll} d-{Tr,t}=d_{Tr,t−1}+αd_{Tr,t−1}(a+\overline{a}−c_{Tr}−2b_{11}d_{Tr,t−1}−b_{12}d_{Or,t−1})+τε_{1,t} \\ d_{Or,t}=d_{Or,t−1}+βd_{Or,t−1}(a−c_{Or}+θ(\overline{φ}−φ)−b_{21}d_{Tr,t−1}−2b_{22}d_{Or,t−1})(1−φ)+τε_{2,t} .\end{array} \right.$
(8)

here, $ε_{i,t}$ is Gaussian white noise with zero mean and $τ$ is intensity of the noise. System (8) is a nonlinear demand process with random disturbance which has much more complex nonlinear characteristics. Its complexity will be analyzed in Section 3.

Order-Up-To Inventory Policy

We assume that both retailers employ an order-up-to policy as their inventory policy. At the start of period $t$, retailer i knows its own demand of period $t−1$ denoted by $d_{i,t−1}$, and must estimate the order-up-to point $S_{i,t}$, and send the order $q_{1,t}$ to the manufacturer. After the lead time $L_i$, Retailer i receives the products from the manufacturer in the beginning of the period $t+L_i$. A retailer's expected inventory of the period t, is determined by the demand prediction of the lead time and the returned products:

$S_{i,t}=\widehat {D}^{L_i}_{i,t} +\widehat {zσ}^{L_i}_{i,t}$ (9)

where $\widehat{σ}^{L_i}_{i,t}=\sqrt{Var(D^{L_i}_{i,t}−\widehat{D}^{L_i}_{i,t}})$ represents the standard deviation between the actual demand and demand prediction of the lead time. $z (≥0)$ is called safety factor, on behalf of the expected level of service. If $z=0$, the retailer is risk neutral.

$R_{i,t}$ is the volume of returned products that the online retailer receives from consumers. The returned products are assumed to satisfy the conditions to be sold again without any other costs. The products meets the conditions will be returned to the online retailer, after a return cycle $L_i$ through its reverse logistics.

At the end of period $t−1$, the online retailer has gotten the information about the volume of the returned products in time $t$:

$R_{i,t}=φd_{i,t−Li}$

Retailer $i$ can make its order $q_{i,t}$ based on the demand of period $t$ and the desired inventory goal, and then send the order to the manufacturer at the beginning of the period $t$. Retailer i will receive the products at the start of period $t+L_i$. The order $q_{i,t}$ can be written as:

$q_{i,t}=S_{i,t}−S_{i,t−1}−R_{i,t}+d_{i,t−1}= \widehat {D} ^{L_i}_{i,t}−\widehat{D}^{L_i}_{i,t−1})−R_{i,t}+z(\widehat{σ}^{L}_{i,t}−\widehat{σ}^{L}_{i,t−1})+d_{i,t−1}$ (10)

In the beginning of the period $t$, Retailer $i$ knew the actual demands of consumers for its products $d_{i,t−1}$, but the actual demand of the lead time $d_{i,t}, d_{i,t+1},…, d_{i,t+Li−1}$ are unknown, where $\widehat{d}_{i,t}$, $\widehat{d}_{i,t+1,…}$, $\widehat{d}_{i,t+Li−1}$ are the corresponding predicted values for $d_{i,t}$, $d_{i,t+1,…}$, $d_{i,t+L−1}$. The whole demand predictions of the lead time can be written as:

$\widehat{D}^{L}_{i,t}=\widehat{d}_{i,t}+\widehat{d}_{i,t+1}+⋯+\widehat{d}_{i,t+L−1}$ (11)

On the basis of the assumption that the retailers use the bounded rationality to estimate $\widehat{d}_{i,t}$ according to the demand observations from the previous period. The demand prediction in the lead time can be written as:

$\widehat{D}^{L}_{i,t}=L_i \widehat{d}_{i,t}$ (12)

Substituting Equation (12) into (10), then we can get the orders of the two retailers:

$\left\{ \begin{array}{lll} q_{Tr,t}=L_{Tr}(αd_{Tr,t−1}(a+\overline{a}−c_{Tr}−2b_{11}d_{Tr,t−1}−b_{12}d_{Or,t−1})−\widehat{d}_{Tr,t−1}) \\ +(L_{Tr+1})d_{Tr,t−1}+z(\widehat{σ}^{L}_{Tr,t}−\widehat{σ}^{L}_{Tr,t−1}) \\ q_{Or,t}=L_{Or}(βd_{Or,t−1}(a−c_{Or}+θ(\overline{φ}−φ)−b_{21}d_{Tr,t−1}−2b_{22}d_{Or,t−1})(1−φ)−\widehat{d}_{Or,t−1}) \\ −R_{Or,t} +(L_{Or}+1)d_{Or,t−1}+z(\widehat{σ}^{L}_{Or,t}−\widehat{σ}^{L}_{Or,t−1}) \end{array} \right.$