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Quantifying the impact of demand substitution on the bullwhip effect in a supply chain

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Logistics Research

Abstract

In a supply chain, the distorted demand information when it goes upstream is commonly known as the bullwhip effect. In this paper, the impact of demand substitution on the bullwhip effect in a two-stage supply chain is investigated. In our model, a single retailer observes inventory levels for two products, among which product 1 can be used to substitute product 2. The retailer places orders to a single manufacturer following an order-up-to inventory policy and uses a simple moving average forecasting method to estimate the lead-time demand. The customers’ demands are modeled by an autoregressive process. By analyzing the bullwhip effect in such settings, quantitative relations between the bullwhip effect and the forecasting method, lead time, demand process, and the product substitution are obtained. Numerical results show that demand substitution can reduce the bullwhip effect in most cases.

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Notes

  1. The explicit technique details are shown in the Appendix.

  2. The explicit technique details are shown in the Appendix.

  3. Chen et al (2000a) oversimplified the situation to claim that for ρ < 0 the lower bound on the increase in variability will be large for odd values of p than for even values of p.

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Authors and Affiliations

  1. Department of Industrial and Information Engineering, University of Tennessee, Knoxville 408 East Stadium Hall, Knoxville, TN, 37996-0700, USA

    Xueping Li, Laigang Song & Zhaoxiao Zhao

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  1. Xueping Li
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  2. Laigang Song
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  3. Zhaoxiao Zhao
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Correspondence to Xueping Li.

Appendix

Appendix

1.1 The derivation process of E(Dt,i) and Var(Dt,i)

When the autoregressive demand process is stationary, we have E(Dt,i) = E(Dt−1,i) = E(Dt−2,i) = ⋯ = E(D i ) and Var(Dt,i) = Var(Dt−1,i) = Var(Dt−2,i) = ⋯ = Var(D i )

$$ \begin{aligned} D_{t,1} &=\mu_1+\rho_1D_{t-1,1}+\epsilon_{t,1}+\lambda D_{t,1}\\ (1-\lambda) D_{t,1} &=\mu_1+\rho_1D_{t-1,1}+\epsilon_{t,1} \end{aligned} $$
$$ \begin{aligned} (1-\lambda) E(D_{t,1})&=E(\mu_1)+\rho_1 E(D_{t-1,1})+E(\epsilon_{t,1})\\ (1-\lambda)E(D_{1})&=\mu_1+\rho_1 E(D_{1})+0 \\ E(D_{1})&= \frac{\mu_1}{1-\rho_1-\lambda} \end{aligned} $$
$$ \begin{aligned} (1-\lambda)^2 {\rm Var}(D_{t,1})&={\rm Var}(\mu_1)+{\rho_1}^2 {\rm Var}(D_{t-1,1})+{\rm Var}(\epsilon_{t,1}) \\ (1-\lambda)^2 {\rm Var}(D_{1})&=0+{\rho_1}^2 {\rm Var}(D_{1})+\sigma_1^2 \\ {\rm Var}(D_{1}) &= \frac{\sigma_1^2}{(1-\lambda)^2-\rho_1^2} \end{aligned} $$
$$ D_{t,2} = \mu_2+\rho_2D_{t-1,2}+\epsilon_{t,2}-\lambda D_{t,1} $$
$$ \begin{aligned} E(D_{t,2}) &=E(\mu_2)+\rho_2 E(D_{t-1,2})+E(\epsilon_{t,2})-\lambda E(D_{t,1})\\ E(D_{2}) &=\mu_2+\rho_2 E(D_{2})+0-\lambda E(D_{1}) \\ (1-\rho_2)E(D_{2}) &=\mu_2-\lambda \frac{\mu_1}{1-\rho_1-\lambda} \\ E(D_{t,2}) &= \frac{\mu_2(1-\rho_1-\lambda)-\lambda\mu_1}{(1-\rho_1-\lambda)(1-\rho_2)} \end{aligned} $$
$$ \begin{aligned} {\rm Var}(D_{t,2}) &={\rm Var}(\mu_2)+{\rho_2}^2 {\rm Var}(D_{t-1,2})+{\rm Var}(\epsilon_{t,2})-\lambda^2 {\rm Var}(D_{t,1})\\ {\rm Var}(D_{2}) &=0+{\rho_2}^2 var(D_{2})+{\sigma_2}^2-\lambda^2 {\rm Var}(D_{1}) \\ (1-\rho_2^2){\rm Var}(D_{2}) &={\sigma_2}^2-\lambda^2 \frac{\sigma_1^2}{(1-\lambda)^2-\rho_1^2} \\ {\rm Var}(D_{t,2})&=\frac{[(1-\lambda)^2-\rho_1^2]\sigma_2^2-\lambda^2\sigma_1^2}{[(1-\lambda)^2-\rho_1^2](1-\rho_2^2)} \end{aligned} $$

1.2 The derivation process of the further expression of qt,1

$$ \begin{aligned} q_{t,1}&=y_{t,1}-y_{t-1,1}+D_{t-1,1}\\ &=(\hat{D}_{t,1}^{L}+z_1\hat{\sigma}_{t,1}^L)-(\hat{D}_{t-1,1}^{L}+z_1\hat{\sigma}_{t-1,1}^L)+D_{t-1,1}\\ &=(\hat{D}_{t,1}^{L}-\hat{D}_{t-1,1}^{L})+D_{t-1,1}+z_1(\hat{\sigma}_{t,1}^L-\hat{\sigma}_{t-1,1}^L)\\ &=\frac{L}{p}\left(\sum_{i=1}^{p}D_{t-i,1}-\sum_{i=1}^{p}D_{t-1-i,1}\right)+D_{t-1,1}+z_1(\hat{\sigma}_{t,1}^L-\hat{\sigma}_{t-1,1}^L)\\ &=\frac{L}{p}(D_{t-1,1}-D_{t-p-1,1})+D_{t-1,1}+z_1(\hat{\sigma}_{t,1}^L-\hat{\sigma}_{t-1,1}^L)\\ &=(1+L/p)D_{t-1,1}-(L/p)D_{t-p-1,1}+z_1(\hat{\sigma}_{t,1}^L-\hat{\sigma}_{t-1,1}^L) \end{aligned} $$

1.3 The derivation process of the expression of Var(qt,1)

$$ \begin{aligned} {\rm Var}(q_{t,1})&={\rm Var}[(1+L/p)D_{t-1,1}-(L/p)D_{t-p-1,1}+z_1(\hat{\sigma}_{t,1}^L-\hat{\sigma}_{t-1,1}^L)]\\ &=(1+L/p)^2Var(D_{t-1,1})-2(L/p)(1+L/p)\times {\rm Cov}(D_{t-1,1},D_{t-p-1,1})\\ &\;\;\;\;\;+(L/p)^2Var(D_{t-p-1,1})+z_1^2Var(\hat{\sigma}_{t,1}^L-\hat{\sigma}_{t-1,1}^L)\\ &\;\;\;\;\;+2z_1(1+2L/p)\times {\rm Cov}(D_{t-1,1},\hat{\sigma}_{t,1}^{L})\\ &=\left(1+2\frac{L}{p}+2\frac{L^2}{p^2}\right){\rm Var}(D_{1})-\left(\frac{2L}{p}+\frac{2L^2}{p^2}\right) {\rm Cov}(D_{t-1,1},D_{t-p-1,1})\\ &\;\;\;\;\;+z_1^2Var(\hat{\sigma}_{t,1}^L-\hat{\sigma}_{t-1,1}^L)+2z_1(1+2L/p) {\rm Cov}(D_{t-1,1},\hat{\sigma}_{t,1}^{L}) \end{aligned} $$
(13)

To further simplify Equation 9, we need to calculate Cov(Dt-1,1, Dt-p-1,1) and \(Cov(D_{t-1,1},\hat{\sigma}_{t,1}^{L})\)

$$ \begin{aligned} {\rm Cov}(D_{t-1,1},D_{t-p-1,1})&={\rm Cov}\left(\frac{1}{1-\lambda}(\mu_1+\rho_1D_{t-2,1}+\epsilon_{t,1}),D_{t-p-1,1}\right)\\ &=\underbrace{{\rm Cov}\left(\frac{\mu_1}{1-\lambda},D_{t-p-1,1}\right)}_{=0}+\frac{\rho_1}{1-\lambda}{\rm Cov}(D_{t-2,1},D_{t-p-1,1})\\ &\;\;\;\;\;+\frac{1}{1-\lambda}\underbrace{{\rm Cov}(\epsilon_{t,1},D_{t-p-1,1})}_{=0}\\ &=\frac{\rho_1}{1-\lambda}{\rm Cov}(D_{t-2,1},D_{t-p-1,1})\\ &\vdots\\ &=\left(\frac{\rho_1^p}{(1-\lambda)^p}\right){\rm Cov}(D_{t-p-1,1},D_{t-p-1,1})\\ &=\frac{\rho_1^p}{(1-\lambda)^p}{\rm Var}(D_{1}) \end{aligned} $$
(14)

Note that \(Cov(\frac{\mu_1}{1-\lambda},D_{t-p-1,1})=0\), because \(\frac{\mu_1}{1-\lambda}\) is a constant (Cov(X,a) = 0). \(Cov(\epsilon_{t,1},D_{t-p-1,1})=0\), because \(\epsilon_{t,1},D_{t-p-1,1}\) are independent from each other.

Ryan [20] proved the following result that can further simplify Equation 9. Assume the customer demands seen by a retailer are random variables of the form as \(D_{t}=\mu+\rho D_{t-1}+\epsilon_{t}\) and the error terms \(\epsilon_t\) are i.i.d. from a symmetric distribution with mean 0 and variance σ2. Let the estimate of the standard deviation of forecast error of lead-time demand be \(\hat{\sigma}_t^L=C_{L,\rho}\sqrt{\frac{\sum_{i=j}^p(D_{t-j}-\hat{D}_{t-j})^2}{p}},\) then

$$ {\rm Cov}(D_{t-j},\hat{\sigma}_t^L)=0, \forall i=1,\ldots,p. $$
(15)

By applying the results of Equation 28 and Equation 29, the expression about Var(qt,1) can be further simplified.

$$ \begin{aligned} {\rm Var}(q_{t,1})&=\left(1+2\frac{L}{p}+2\frac{L^2}{p^2}\right){\rm Var}(D_{1})-\left(\frac{2L}{p}+\frac{2L^2}{p^2}\right){\rm Cov}(D_{t-1,1},D_{t-p-1,1})\\ &\;\;\;\;\;+z_1^2Var(\hat{\sigma}_{t,1}^L-\hat{\sigma}_{t-1,1}^L)+2z_1(1+2L/p){\rm Cov}(D_{t-1,1},\hat{\sigma}_{t,1}^{L})\\ &=\left(1+2\frac{L}{p}+2\frac{L^2}{p^2}\right){\rm Var}(D_{1})-\left(\frac{2L}{p}+\frac{2L^2}{p^2}\right)\frac{\rho_1^p}{(1-\lambda)^p}{\rm Var}(D_{1})\\ &\;\;\;\;\;+z_1^2Var(\hat{\sigma}_{t,1}^L-\hat{\sigma}_{t-1,1}^L)+0\\ &={\rm Var}(D_1)\left[1+\left(\frac{2L}{p}+\frac{2L^2}{p^2}\right)\left(1-\frac{\rho_1^p}{(1-\lambda)^p}\right)\right]+z_1^2Var(\hat{\sigma}_{t,1}^L-\hat{\sigma}_{t-1,1}^L) \end{aligned} $$
(16)

1.4 The derivation process of the further expression of qt,2

$$ \begin{aligned} q_{t,2}&=y_{t,2}-y_{t-1,2}+D_{t-1,2} \\ &=(\hat{D}_{t,2}^{L}+z_1\hat{\sigma}_{t,2}^L)-(\hat{D}_{t-1,2}^{L}+z_1\hat{\sigma}_{t-1,2}^L)+D_{t-1,2}\\ &=(\hat{D}_{t,2}^{L}-\hat{D}_{t-1,2}^{L})+D_{t-1,2}+z_2(\hat{\sigma}_{t,2}^L-\hat{\sigma}_{t-1,2}^L)\\ &=\frac{L}{p}\left(\sum_{i=1}^{p}D_{t-i,2}-\sum_{i=1}^{p}D_{t-1-i,2}\right)+D_{t-1,2}+z_2(\hat{\sigma}_{t,2}^L-\hat{\sigma}_{t-1,2}^L) \\ &=\frac{L}{p}(D_{t-1,2}-D_{t-p-1,2})+D_{t-1,2}+z_2(\hat{\sigma}_{t,2}^L-\hat{\sigma}_{t-1,2}^L) \\ &=(1+L/p)D_{t-1,2}-(L/p)D_{t-p-1,2}+z_2(\hat{\sigma}_{t,2}^L-\hat{\sigma}_{t-1,2}^L) \end{aligned} $$

1.5 The derivation process of the expression of Var(qt,2)

$$ \begin{aligned} {\rm Var}(q_{t,2})&={\rm Var}[(1+L/p)D_{t-1,2}-(L/p)D_{t-p-1,2}+z_2(\hat{\sigma}_{t,2}^L-\hat{\sigma}_{t-1,2}^L)]\\ &=(1+L/p)^2Var(D_{t-1,2})-2(L/p)(1+L/p)\times {\rm Cov}(D_{t-1,2},D_{t-p-1,2})\\ &\;\;\;\;\;+(L/p)^2Var(D_{t-p-1,2})+z_2^2Var(\hat{\sigma}_{t,2}^L-\hat{\sigma}_{t-1,2}^L)\\ &\;\;\;\;\;+2z_2(1+2L/p)\times {\rm Cov}(D_{t-1,2},\hat{\sigma}_{t,2}^{L})\\ &=\left(1+2\frac{L}{p}+2\frac{L^2}{p^2}\right){\rm Var}(D_{2})-\left(\frac{2L}{p}+\frac{2L^2}{p^2}\right){\rm Cov}(D_{t-1,2},D_{t-p-1,2})\\ &\;\;\;\;\;+z_2^2Var(\hat{\sigma}_{t,2}^L-\hat{\sigma}_{t-1,2}^L)+\underbrace{2z_2(1+2L/p){\rm Cov}(D_{t-1,2},\hat{\sigma}_{t,2}^{L})}_{=0} \end{aligned} $$
(17)

Now we calculate Cov(Dt-1,2, Dt-p-1,2)

$$ \begin{aligned} {\rm Cov}(D_{t-1,2},D_{t-p-1,2})&={\rm Cov}(\mu_2+\rho_2D_{t-2,2}+\epsilon_{t-1,2}-\lambda D_{t-1,1},D_{t-p-1,2})\\ &=\underbrace{{\rm Cov}(\mu_2,D_{t-p-1,2})}_{=0}+\rho_2Cov(D_{t-2,2},D_{t-p-1,2})\\ &\;\;\;\;\;+\underbrace{{\rm Cov}(\epsilon_{t-1,2},D_{t-p-1,2})}_{=0}-\lambda {\rm Cov}(D_{t-1,1},D_{t-p-1,2})\\ &=\rho_2Cov(D_{t-2,2},D_{t-p-1,2})-\lambda {\rm Cov}(D_{t-1,1},D_{t-p-1,2})\\ &\vdots \\ &=\rho_2^pVar(D_2)-\lambda\sum_{i=0}^{p-1}\rho_2^iCov(D_{t-1-i,1},D_{t-p-1,2}) \end{aligned} $$

We assume that the covariance is only affected by the number of periods which are taken into consideration. That is

$$ \begin{aligned} {\rm Cov}(D_{t,1},D_{t-p,2})&={\rm Cov}(D_{t-1,1},D_{t-p-1,2})\\ D_{t,1}&=\frac{1}{1-\lambda}(\mu_1+\rho_1D_{t-1,1}+\epsilon_{t,1}) \\ D_{t-p,2}&=\mu_2+\rho_2D_{t-p-1,2}+\epsilon_{t-p,2}-\lambda D_{t-p,1} \\ {\rm Cov}(D_{t,1},D_{t-p,2})&={\rm Cov}(\frac{1}{1-\lambda}(\mu_1+\rho_1D_{t-1,1}+\epsilon_{t,1}),\mu_2+\rho_2D_{t-p-1,2}+\epsilon_{t-p,2}-\lambda D_{t-p,1}) \\ &=\frac{\rho_1\rho_2}{1-\lambda}{\rm Cov}(D_{t-1,1},D_{t-p-1,2})-\frac{\lambda\rho_1}{1-\lambda}{\rm Cov}(D_{t-1,1},D_{t-p,1}) \\ {\rm Cov}(D_{t-1,1},D_{t-p-1,2})&=\frac{\rho_1\rho_2}{1-\lambda}{\rm Cov}(D_{t-1,1},D_{t-p-1,2})-\frac{\lambda\rho_1}{1-\lambda}{\rm Cov}(D_{t-1,1},D_{t-p,1}) \\ {\rm Cov}(D_{t-1,1},D_{t-p-1,2})&=\frac{-\lambda(1-\lambda)\rho_1^p}{(1-\lambda-\rho_1\rho_2)(1-\lambda)^{p}}{\rm Var}(D_{1}) \\ {\rm Cov}(D_{t-2,1},D_{t-p-1,2})&=\frac{-\lambda(1-\lambda)\rho_1^{p-1}}{(1-\lambda-\rho_1\rho_2)(1-\lambda)^{p-1}}{\rm Var}(D_{1}) \\ &\vdots \\ {\rm Cov}(D_{t-i,1},D_{t-p-1,2})&=\frac{-\lambda(1-\lambda)\rho_1^{p-i+1}}{(1-\lambda-\rho_1\rho_2)(1-\lambda)^{p-i+1}}{\rm Var}(D_{1}) \\ &\vdots \\ {\rm Cov}(D_{t-i,1},D_{t-p-1,2})&=\frac{-\lambda(1-\lambda)\rho_1^{2}}{(1-\lambda-\rho_1\rho_2)(1-\lambda)^{2}}{\rm Var}(D_{1}) \end{aligned} $$

Thus

$$ \begin{aligned} {\rm Cov}(D_{t-1,2},D_{t-p-1,2})&=\rho_2^pVar(D_2)-\lambda {\rm Cov}(D_{t-1,1},D_{t-p-1,2})\sum_{i=0}^{p}\rho_2^i \left(\frac{1-\lambda}{\rho_1}\right)^{i}\\ &=\rho_2^pVar(D_2)-\lambda\sum_{i=0}^{p}\rho_2^iCov(D_{t-1-i,1},D_{t-p-1,2}) \\ &=\rho_2^pVar(D_2)+\frac{\lambda^2(1-\lambda){\rm Var}(D_1)}{(1-\lambda-\rho_1\rho_2)}\sum_{i=0}^{p-1}\left(\frac{\rho_2^{i}\rho_1^{p-i+1}}{(1-\lambda)^{p-i+1}}\right) \end{aligned} $$

Substitute the above equation to Eq. 17, we have

$$ \begin{aligned} {\rm Var}(q_{t,2})&=\left(1+2\frac{L}{p}+2\frac{L^2}{p^2}\right){\rm Var}(D_{2})-\left(\frac{2L}{p}+\frac{2L^2}{p^2}\right){\rm Cov}(D_{t-1,2},D_{t-p-1,2})\\ &\;\;\;\;\;+z_2^2Var(\hat{\sigma}_{t,2}^L-\hat{\sigma}_{t-1,2}^L)\\ &=\left(1+2\frac{L}{p}+2\frac{L^2}{p^2}\right){\rm Var}(D_{2})-\left(\frac{2L}{p}+\frac{2L^2}{p^2}\right)\rho_2^pVar(D_2)\\ &\;\;\;\;\;-\left(\frac{2L}{p}+\frac{2L^2}{p^2}\right)\frac{\lambda^2(1-\lambda){\rm Var}(D_1)}{(1-\lambda-\rho_1\rho_2)}\sum_{i=0}^{p-1}\left(\frac{\rho_2^{i}\rho_1^{p-i+1}}{(1-\lambda)^{p-i+1}}\right)\\ &\;\;\;\;\;+z_2^2Var(\hat{\sigma}_{t,2}^L-\hat{\sigma}_{t-1,2}^L) \end{aligned} $$
(18)

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Li, X., Song, L. & Zhao, Z. Quantifying the impact of demand substitution on the bullwhip effect in a supply chain. Logist. Res. 3, 221–232 (2011). https://doi.org/10.1007/s12159-011-0060-y

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