An agricultural supply chain, consists of farmers, processing enterprises, distributors, retailers and consumers, has the characteristics of multiple network structures. Because there is no information platform in ASC, so we suppose that there is no information sharing between members of ASC. Retailers do not provide the real consumer information they have to upstream members in the ASC, and dealers do not provide the correct ordering information of retailers to processing enterprises, and so forth.

Members can only predict the demand with the order quantity of former members. Meanwhile, members use the moving average forecasting technology to predict the future order of its downstream members and determine their order quantity to the upstream members with the order-up-to strategy.

Now we assume that there is one produce in the ASC, and suppose that commercial behaviors between the upstream and downstream members occur in an infinite discrete time range (*t* = − ∞, …, −1, 0, 1, …, ∞). On the basis of past sales information and stock replenishment period (L), members predict the future order (t+L) of its downstream members with moving average forecasting in the end of t, and use order-up-to strategy to determine their order quantity to the upstream members. And then they make an order *q*_{t} to the upstream members at the beginning of the next period (t+L).

To facilitate the analysis, we take a two-level ASC including retailers and dealers for example. Suppose that the market demands (*d*_{t}) retailers face is a simple stationary autocorrelation time series AR(1):
$${d}_{t}=\mu +\rho {d}_{(}t-1)+{\epsilon}_{t},t=0,\pm 1,2,\cdots \phantom{\rule{0ex}{0ex}}$$(1)

In the equation, *μ* is a non-negative constant, *ρ* is an autocorrelation coefficient, which indicates the correlation of consumers demand in two adjacent periods, and we suppose |*ρ*| < 1. And *ε*_{t} are independent random variables with the same distribution, which shows the fluctuation error of market consumption demand. The mean value is 0 and variance is (*σ*^{2}). We can get the mean value and variance of (*d*_{t}):
$$\begin{array}{r}\left\{\begin{array}{l}(E({d}_{t})=\frac{\mu}{(1-\rho )}\\ var({d}_{t})=\frac{{\sigma}^{2}}{(1-{\rho}^{2})}\end{array}\right.\end{array}$$(2)

According to the foregoing hypothesis, order quantity (*q*_{t}) issued by the retailers to the dealers will meet the following conditions:
$${y}_{t}={\hat{d}}_{t}^{L}+z{\hat{\sigma}}_{t}^{L}$$(3)
$${q}_{t}={y}_{t}-{y}_{(}t-1)+{d}_{t}$$(4)

In the equation, ${\hat{d}}_{t}^{L}$ represents the estimated value of market demand in the stock replenishment period. And if the estimated value (${\hat{d}}_{t}^{L}$) is based on consumer demands, retailers observe in the former few periods, we can get the following equation.

$${\hat{d}}_{t}^{L}=L\times {\hat{d}}_{t}=L\left(\frac{\sum _{i=0}^{p-1}{d}_{t-i}}{p}\right)$$(5)

We define *y*_{t} as the maximum inventory level, and $z{\hat{\sigma}}_{t}^{L}$ as the safety stock (z is the service level coefficient and ${\hat{\sigma}}_{t}^{L}$ is the estimated value of the standard deviation of prediction error). To simplify analysis, we suppose the value of z is 0. Then we can get the mathematical expression of the bullwhip effect $\frac{var({q}_{t})}{var({d}_{t})}$. According to formula (3), formula (4) and formula (5), we can get order quantity (*q*_{t}) of retailers.

$$\begin{array}{rl}{q}_{t}& ={\hat{d}}_{t}^{L}-{\hat{d}}_{t-1}^{L}+{d}_{t}\\ & =L\left(\frac{\sum _{i=0}^{p-1}{d}_{t-i}}{p}\right)-L\left(\frac{\sum _{i=0}^{p-1}{d}_{t-i-1}}{p}\right)+{d}_{t}\\ & =\left(1+\frac{L}{p}\right){d}_{t}-\frac{L}{p}{d}_{t-p}\end{array}$$(6)

Bring formula (1) into formula (6),
$$\begin{array}{rl}{q}_{t}& =\left(1+\frac{L}{p}\right)\left(\mu +\rho {d}_{t-1}+{\epsilon}_{t}\right)-\frac{L}{p}\left(\mu +\rho {d}_{t-p-1}+{\epsilon}_{t-p}\right)\\ & =\mu +\rho \left[\left(1+\frac{L}{p}\right){d}_{t-1}-\frac{L}{p}{d}_{t-p-1}\right]+\left(1+\frac{L}{p}\right){\epsilon}_{t}\\ & -\frac{L}{p}{\epsilon}_{t-p}=\mu +\rho {q}_{t-1}+\left(1+\frac{L}{p}\right){\epsilon}_{t}-\frac{L}{p}{\epsilon}_{t-p}\end{array}$$(7)

We define ${\epsilon}_{t}^{\prime}$ and *Θ′* as follows.

$${\epsilon}_{t}^{\prime}=\left(1+\frac{L}{p}\right){\epsilon}_{t}$$(8)
$${\mathrm{\Theta}}^{\prime}=\frac{L}{(P+L)}$$(9)

Then formula (7) can be simplified as:
$${q}_{t}=\mu +\rho {q}_{t-1}+{\epsilon}_{t}^{\prime}-{\mathrm{\Theta}}^{\prime}{\epsilon}_{t-p}^{\prime}$$(10)

Among the formula, the mean and variance of ${\epsilon}_{t}^{\prime}$ can be calculated.

$$\begin{array}{r}\left\{\begin{array}{l}E({\epsilon}_{t}^{\prime})=0\\ var({\epsilon}_{t}^{\prime})={\left(1+\frac{L}{p}\right)}^{2}var({\epsilon}_{t})={\left(1+\frac{L}{p}\right)}^{2}{\sigma}^{2}\end{array}\right.\end{array}$$(11)

And then, we can deduce the mean and variance of (*q*_{t}).

$$E({q}_{t})=\frac{\mu}{(1-\rho )}=E({d}_{t})$$(12)
$$\begin{array}{rl}var{q}_{t}& =var(\mu +\rho {q}_{t-1}+{\epsilon}_{t}^{\prime}-{\mathrm{\Theta}}^{\prime}{\epsilon}_{t-p}^{\prime})\\ & ={\rho}^{2}var{q}_{t-1}+var({\epsilon}_{t}^{\prime}-{\mathrm{\Theta}}^{\prime}{\epsilon}_{t-p}^{\prime})\\ & +2\rho cov({q}_{t-1},{\epsilon}_{t}^{\prime}-{\mathrm{\Theta}}^{\prime}{\epsilon}_{t-p}^{\prime})\end{array}$$(13)

Then we can get:
$$\begin{array}{rl}(1-{\rho}^{2})var{q}_{t}& =(1+\frac{2L}{p}+\frac{2{L}^{2}}{{p}^{2}}){\sigma}^{2}\\ & -2\rho {\mathrm{\Theta}}^{\prime}cov({q}_{t-1},{\epsilon}_{t-p}^{\prime})\end{array}$$(14)

If *p* ≥ 1, we can prove the following relationship.

$$cov({q}_{t-1},{\epsilon}_{t-p}^{\prime})={\rho}^{p-1}\left(1+\frac{L}{p}\right){\sigma}^{2}$$(15)

Bring formula (15) into (14),
$$var{q}_{t}=\left[1+\left(\frac{2L}{p}+\frac{2{L}^{2}}{{p}^{2}}\right)(1-{\rho}^{p})\right]var{d}_{t}$$(16)

In summary, our analysis shows that: when the market demands (*d*_{t}) retailers face is a simple stationary autocorrelation time series AR(1), and retailers forecast their order quantity with moving average forecasting technology and order-up-to strategy, the order quantity (*q*_{t}) will be a random process ARMA(1,P).

Similarly, dealers forecast their demand (${\hat{d}}_{t}^{(2)}$) with moving average forecasting technology on the basis of retailers’ previous orders.

$${\hat{d}}_{t}^{(2)}=\frac{\left(\sum _{i=0}^{p-1}{q}_{t-i}^{(1)}\right)}{p}$$(17)

Meanwhile, dealers use order-up-to strategy to determine their order amount (${q}_{t}^{(2)}$) to processing enterprises.

$$\begin{array}{rl}{q}_{t}^{(2)}& ={L}_{2}({\hat{d}}_{t}^{(2)}-{\hat{d}}_{t-1}^{(2)})+{q}_{t}^{(1)}\\ & =\left(1+\frac{{L}_{2}}{p}\right){q}_{t}^{(1)}-\frac{{L}_{2}}{p}{q}_{t-p}^{(1)}\end{array}$$(18)

The above formula implies the following relationship.

$${q}_{t-1}^{(2)}=\left(1+\frac{{L}_{2}}{p}\right){q}_{t-1}^{(1)}-\frac{{L}_{2}}{p}{q}_{t-p-1}^{(1)}$$(19)

According to formula (10) and (19),
$$\begin{array}{rl}{q}_{t}^{(2)}& =\mu +\rho {q}_{t-1}^{(2)}+\left(1+\frac{{L}_{2}}{p}\right)\left({\epsilon}_{t}^{(1)}-{\mathrm{\Theta}}^{(1)}{\epsilon}_{t-p}^{(1)}\right)\\ & -\frac{{L}_{2}}{p}\left({\epsilon}_{t-p}^{(1)}-{\mathrm{\Theta}}^{(1)}{\epsilon}_{t-2p}^{(1)}\right)\end{array}$$(20)

We define ${\epsilon}_{t}^{(2)}$ and *Θ*^{(2)} as follows.

$$\begin{array}{r}\left\{\begin{array}{l}{\epsilon}_{t}^{(2)}=\left(1+\frac{{L}_{2}}{p}\right)\left({\epsilon}_{t}^{(1)}-{\mathrm{\Theta}}^{(1)}{\epsilon}_{t-p}^{(1)}\right)\\ {\mathrm{\Theta}}^{(2)}=\frac{{L}_{2}}{{L}_{2}+p}\end{array}\right.\end{array}$$(21)

Then the order quantity ${q}_{t}^{(2)}$ has the same expression with ${q}_{t}^{(1)}$, which is a random process ARMA(1,P).

$${q}_{t}^{(2)}=\mu +\rho {q}_{t-1}^{(2)}+{\epsilon}_{t}^{(2)}-{\mathrm{\Theta}}^{(2)}{\epsilon}_{t-p}^{(2)}$$(22)

And then, $E({\epsilon}_{t}^{(2)})=0$,
$$\begin{array}{rl}var({\epsilon}_{t}^{(2)})& ={\left(1+\frac{{L}_{2}}{p}\right)}^{2}\left[var\left({\epsilon}_{t}^{(1)}\right)\right.\\ & \left.-2{\mathrm{\Theta}}^{(1)}cov\left({\epsilon}_{t}^{(1)},{\epsilon}_{t-p}^{(1)}\right)+{\left({\mathrm{\Theta}}^{(1)}\right)}^{2}var\left({\epsilon}_{t-p}^{(1)}\right)\right]\\ & =\left[1+{\left({\mathrm{\Theta}}^{(1)}\right)}^{2}\right]{\left(1+\frac{{L}_{1}}{p}\right)}^{2}{\left(1+\frac{{L}_{2}}{p}\right)}^{2}{\sigma}^{2}\end{array}$$(23)

According to the basic theory of probability theory, no matter how much the value of P, we can get the following relationship.

$$\begin{array}{rl}var\left({q}_{t}^{(2)}\right)& ={\rho}^{2}var\left({q}_{(}t-1{)}^{(2)}\right)+var\left({\epsilon}_{t}^{(2)}\right)\\ & +{\left({\mathrm{\Theta}}^{(2)}\right)}^{2}var\left({\epsilon}_{t-p}^{(2)}\right)\\ & -2{\mathrm{\Theta}}^{(1)}{\mathrm{\Theta}}^{(2)}{\left(1+\frac{{L}_{2}}{p}\right)}^{2}var\left({\epsilon}_{t-p}^{(1)}\right)\\ & -2{\rho}^{p}{\mathrm{\Theta}}^{(1)}{\left(1+\frac{{L}_{2}}{p}\right)}^{2}var\left({\epsilon}_{t-p}^{(1)}\right)\\ & -2{\rho}^{p}{\mathrm{\Theta}}^{(1)}{\left(\frac{{L}_{2}}{p}\right)}^{2}var\left({\epsilon}_{t-2p}^{(1)}\right)\\ & +2{\epsilon}^{2}p{\mathrm{\Theta}}^{(1)}\left(\frac{{L}_{2}}{p}\right)\left(1+\frac{{L}_{2}}{p}\right)var\left({\epsilon}_{t-2p}^{(1)}\right)\\ & -2{\rho}^{p}\left(\frac{{L}_{2}}{p}\right)\left(1+\frac{{L}_{2}}{p}\right)\\ & \cdot \left[var\left({\epsilon}_{t-p}^{(1)}\right)+{\left({\mathrm{\Theta}}^{(1)}\right)}^{2}var\left({\epsilon}_{t-2p}^{(1)}\right)\right]\end{array}$$(24)

Then,
$$\begin{array}{rl}\frac{var\left({q}_{t}^{(2)}\right)}{var{d}_{t}}& =\left[1+\left(\frac{2{L}_{1}}{p}+\frac{2{L}_{1}^{2}}{{p}^{2}}\right)\left(1-{\rho}^{p}\right)\right]\\ & \cdot \left[1+\left(\frac{2{L}_{2}}{p}+\frac{2{L}_{2}^{2}}{{p}^{2}}\right)\left(1-{\rho}^{p}\right)\right]\\ & +2\left(\frac{2{L}_{1}}{p}+\frac{{L}_{1}^{2}}{{p}^{2}}\right)\left(\frac{{L}_{2}}{p}+\frac{{L}_{2}^{2}}{{p}^{2}}\right)\left(1-{\rho}^{2}p\right)\end{array}$$(25)

Similarly, we can deduce the bullwhip effect processing enterprises $\left(\frac{var{q}_{t}^{(3)}}{var{d}_{t}}\right)$ and farmers $\left(\frac{var{q}_{t}^{(4)}}{var{d}_{t}}\right)$ encounter. In summary, when there is no information sharing, the bullwhip effect' general mathematical model in the ASC can be expressed as:
$$\begin{array}{rl}\frac{var{q}_{t}^{(k)}}{var{d}_{t}}& =\left\{\begin{array}{l}1+\left(\frac{2{L}_{1}}{p}+\frac{2{L}_{1}^{2}}{{p}^{2}}\right)(1-{\rho}^{p}),k=1\\ \prod _{i=1}^{k}\left[1+\left(\frac{2{L}_{i}}{p}+\frac{2{L}_{i}^{2}}{{p}^{2}}\right)(1-{\rho}^{p})\right]\\ +2(1-{\rho}^{2}p)\prod _{i=1}^{k}\left(\frac{{L}_{i}}{p}+\frac{{L}_{i}^{2}}{{p}^{2}}\right),k=2,3,4\end{array}\right.\end{array}$$(26)

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