A Measure of SCM Bullwhip Effect Under Mixed Autoregressive-Moving Average with Errors Heteroscedasticity (ARMA(1,1)–GARCH(1,1)) Model

Abstract

Measuring the bullwhip effect, a phenomenon in which demand variability increases as one moves up the supply chain, is a major issue in supply chain management. In this paper, we quantify the impact of the bullwhip effect on a simple two-stage supply chain consisting of one supplier and one retailer, where the retailer employed a base-stock policy to replenish their inventory. The demand forecast was performed via a mixed autoregressive moving average model, ARMA(1,1), in which ARMA model errors have the GARCH process and the model’s variance changes with time i.e. the model has conditional heteroscedasticity in order to simulate the bullwhip effect which has a non-linear behavior. The definition of bullwhip effect has been expanded to “over time bullwhip effect” (conditional bullwhip effect). We use the minimum mean-square error forecasting technique and also investigate the effects of the autoregressive coefficient, the moving average parameter and the lead time on the bullwhip effect. Moreover, bullwhip effect has been compared in linear demand ARMA and none linear demand ARMA–GARCH process. The results show that the bullwhip effect can be decreased by choosing correct coefficients in demand process through none linear demand process.

Appendices

Appendix 1

$$\begin{aligned} \hbox {cov}(d_{t-1} ,\varepsilon _{t-1} )= & {} E(d_{t-1} \cdot \varepsilon _{t-1} )-E(d_{t-1} )E(\varepsilon _{t-1} ) \\= & {} E(d_{t-1} \cdot \varepsilon _{t-1} ) \\= & {} E((\delta +\phi d_{t-2} +\varepsilon _{t-1} -\theta \varepsilon _{t-2} )\cdot \varepsilon _{t-1} ) \\= & {} \delta E(\varepsilon _{t-1} ) +\phi E(d_{t-2} \cdot \varepsilon _{t-1} )+E(\varepsilon _{t-1}^{2} )-\theta E(\varepsilon _{t-2} \cdot \varepsilon _{t-1} ) \\= & {} E(\varepsilon _{t-1}^{2} )=\hbox {var}(\varepsilon _{t} )=\frac{c_{0} }{1-\alpha -\beta } \\ E(d_{t-2} \cdot \varepsilon _{t-1} )= & {} E(E(d_{t-2} \cdot \varepsilon _{t-1} \vert \psi _{t-2} )) \\= & {} E(d_{t-2} E(\varepsilon _{t-1} \vert \psi _{t-2} )) \\= & {} 0 \end{aligned}$$

Appendix 2

$$\begin{aligned} d_{t+i}= & {} \delta +\phi d_{t+i-1} +\varepsilon _{t+i} -\theta \varepsilon _{t+i-1} \\= & {} \delta +\phi (\delta +\phi d_{t+i-2} +\varepsilon _{t+i-1} -\theta \varepsilon _{t+i-2} )+\varepsilon _{t+i} -\theta \varepsilon _{t+i-1} \\= & {} \delta +\phi (\delta +\phi (\delta +\phi d_{t+i-3} +\varepsilon _{t+i-2} -\theta \varepsilon _{t+i-3} )+\varepsilon _{t+i-1} -\theta \varepsilon _{t+i-2} )\\&\quad +\,\varepsilon _{t+i} -\theta \varepsilon _{t+i-1} \\&\quad \quad \quad \vdots \\= & {} (1+\phi +\phi ^{2}\cdots +\phi ^{i-1})\delta +\phi ^{i}d_{t} +\varepsilon _{t+i} +(\phi -\theta )\varepsilon _{t+i-1} \\&\quad +\,\phi (\phi -\theta )\varepsilon _{t+i-2} +\cdots -\phi ^{i-1}\theta \varepsilon _{t} \\ \end{aligned}$$

According to Pindyck and Rubinfeld [34], for the ARMA(1,1) process, $\hat{{d}}_{t}$ can be determined as

$$\begin{aligned} \hat{{d}}_{t+i}= & {} E(d_{t+i} \vert \psi _{t} ) \\= & {} E((1+\phi +\phi ^{2}\cdots +\phi ^{i-1})\delta +\phi ^{i}d_{t} +\varepsilon _{t+i} +(\phi -\theta )\varepsilon _{t+i-1} \\&+\,\phi (\phi -\theta )\varepsilon _{t+i-2} +\cdots +\phi ^{i-2}(\phi -\theta )\varepsilon _{t+1} -\phi ^{i-1}\theta \varepsilon _{t} )\vert \psi _{t} ) \\= & {} (1+\phi +\phi ^{2}\cdots +\phi ^{i-1})\delta +\phi ^{i}d_{t} -\phi ^{i-1}\theta \varepsilon _{t} \\= & {} \delta \frac{(1-\phi ^{i})}{\left( {1-\phi } \right) }+\phi ^{i}d_{t} -\phi ^{i-1}\theta \varepsilon _{t} \\= & {} \frac{\delta }{\left( {1-\phi } \right) }(1-\phi ^{i})+\phi ^{i}d_{t} -\phi ^{i-1}\theta \varepsilon _{t} \\= & {} \mu _{d} +\phi ^{i}\left( {d_{t-1} -\mu _{d} } \right) -\phi ^{i-1}\theta \varepsilon _{t} \end{aligned}$$

Because $E(d_{t} \vert \psi _{t} )=d_{t}$, $E(\varepsilon _{t} \vert \psi _{t} )=\varepsilon _{t}$ and $E(\varepsilon _{t+i} \vert \psi _{t} )=0 , {\forall i>0}$

Appendix 3

$$\begin{aligned} \left\{ \begin{array}{l} q_{t} = S_{t} - S_{{t - 1}} + d_{{t - 1}} \\ S_{t} = \hat{d}_{t}^{L} + z\hat{\sigma }_{t}^{L}\\ \end{array}\right. \Rightarrow q_{t}= & {} (\hat{d}_{t}^{L} + z\hat{\sigma }_{t}^{L} ) - (\hat{d}_{{t - 1}}^{L} + z\hat{\sigma }_{{t - 1}}^{L} ) +d_{{t - 1}}\\= & {} \hat{d}_{t}^{L} - \hat{d}_{{t - 1}}^{L} + d_{{t - 1}} \end{aligned}$$

$$\begin{aligned}= & {} \left( {L\mu _{d} + \frac{{\left( {1 - \phi ^{L} } \right) }}{{(1 - \phi )}}(d_{t} - \mu _{d} ) - \frac{{\theta \left( {1 - \phi ^{L} } \right) }}{{\phi (1 - \phi )}}\varepsilon _{t} } \right) \\&- \left( {L\mu _{d} + \frac{{\left( {1 - \phi ^{L} } \right) }}{{(1 - \phi )}}(d_{{t - 1}} - \mu _{d} ) - \frac{{\theta \left( {1 - \phi ^{L} } \right) }}{{\phi (1 - \phi )}}\varepsilon _{{t - 1}} } \right) + d_{{t - 1}} \\= & {} \frac{{\left( {1 - \phi ^{L} } \right) }}{{(1 - \phi )}}d_{t} - \frac{{\left( {1 - \phi ^{L} } \right) }}{{(1 - \phi )}}d_{{t - 1}} + d_{{t - 1}} - \frac{{\theta \left( {1 - \phi ^{L} } \right) }}{{\phi (1 - \phi )}}\varepsilon _{t} + \frac{{\theta \left( {1 - \phi ^{L} } \right) }}{{\phi (1 - \phi )}}\varepsilon _{{t - 1}} \\= & {} \frac{{1 - \phi ^{L} }}{{1 - \phi }}d_{t} - \frac{{\phi \left( {1 - \phi ^{{L - 1}} } \right) }}{{(1 - \phi )}}d_{{t - 1}} - \frac{{\theta \left( {1 - \phi ^{L} } \right) }}{{\phi (1 - \phi )}}\varepsilon _{t} + \frac{{\theta \left( {1 - \phi ^{L} } \right) }}{{\phi (1 - \phi )}}\varepsilon _{{t - 1}} \\ \text {var} (q_{{t + i}} |\psi _{t} )= & {} \text {var} \left( {\frac{{1 - \phi ^{L} }}{{1 - \phi }}d_{{t + i}} - \frac{{\phi \left( {1 - \phi ^{{L - 1}} } \right) }}{{(1 - \phi )}}d_{{t + i - 1}} - \frac{{\theta \left( {1 - \phi ^{L} } \right) }}{{\phi (1 - \phi )}}\varepsilon _{{t + i}} + \frac{{\theta \left( {1 - \phi ^{L} } \right) }}{{\phi (1 - \phi )}}\varepsilon _{{t + i - 1}} |\psi _{t} } \right) \\= & {} \left( {\frac{{1 - \phi ^{L} }}{{1 - \phi }}} \right) ^{2} \text {var} (d_{{t + i}} |\psi _{t} ) + \left( {\frac{{\phi \left( {1 - \phi ^{{L - 1}} } \right) }}{{(1 - \phi )}}} \right) ^{2} \text {var} (d_{{t + i - 1}} |\psi _{t} ) + \left( {\frac{{\theta \left( {1 - \phi ^{L} } \right) }}{{\phi (1 - \phi )}}} \right) ^{2} \\&\qquad \text {var} (\varepsilon _{{t + i}} |\psi _{t} ) + \left( {\frac{{\theta \left( {1 - \phi ^{L} } \right) }}{{\phi (1 - \phi )}}} \right) ^{2} \text {var} (\varepsilon _{{t + i - 1}} |\psi _{t} ) \\&-\, 2\frac{{\phi \left( {1 - \phi ^{{L - 1}} } \right) \left( {1 - \phi ^{L} } \right) }}{{\left( {1 - \phi } \right) ^{2} }}\text {cov} (d_{{t + i}} ,d_{{t + i - 1}} |\psi _{t} ) \\&-\, 2\frac{{\theta \left( {1 - \phi ^{L} } \right) ^{2} }}{{\phi (1 - \phi )^{2} }}\text {cov} (d_{{t + i}} ,\varepsilon _{{t + i}} |\psi _{t} ) \\&+\,2\frac{{\theta \left( {1 - \phi ^{L} } \right) \left( {1 - \phi ^{{L - 1}} } \right) }}{{(1 - \phi )^{2} }}\text {cov} (d_{{t + i - 1}} ,\varepsilon _{{t + i - 1}} |\psi _{t} ) \\&+\, 2\frac{{\theta \left( {1 - \phi ^{L} } \right) ^{2} }}{{\phi (1 - \phi )^{2} }}\text {cov} (d_{{t + i}} ,\varepsilon _{{t + i - 1}} |\psi _{t} ) \\&+\, 2\frac{{\theta \left( {1 - \phi ^{L} } \right) \left( {1 - \phi ^{{L - 1}} } \right) }}{{(1 - \phi )^{2} }}\text {cov} (d_{{t + i - 1}} ,\varepsilon _{{t + i}} |\psi _{t} ) \\= & {} \left( {\frac{{1 - \phi ^{L} }}{{1 - \phi }}} \right) ^{2} \text {var} (d_{{t + i}} |\psi _{t} ) + \frac{{\phi ^{2} \left( {1 - \phi ^{{L - 1}} } \right) }}{{(1 - \phi )^{2} }}(3 - \phi ^{{L - 1}} - 2\phi ^{L} )\\&\qquad \text {var} (d_{{t + i - 1}} |\psi _{t} ) + \frac{{\theta \left( {1 - \phi ^{L} } \right) ^{2} }}{{\phi ^{2} (1 - \phi )^{2} }}\left[ {\theta - 2\phi } \right] \text {var} (\varepsilon _{{t + i}} |\psi _{t} ) \\&+ \frac{{\theta \left( {1 - \phi ^{L} } \right) }}{{\phi ^{2} (1 - \phi )^{2} }}\left[ {\theta \left( {1 - \phi ^{L} } \right) \left( {1 - 2\phi } \right) + 2\phi ^{2} \left( {2 - \phi ^{L} - \phi ^{{L - 1}} } \right) } \right] \text {var} (\varepsilon _{{t + i - 1}} |\psi _{t} ) \end{aligned}$$

Because

$$\begin{aligned} \hbox {cov}(d_{t+i} ,d_{t+i-1} \vert \psi _{t} )= & {} \hbox {cov}(\delta +\phi d_{t+i-1} +\varepsilon _{t+i} -\theta \varepsilon _{t+i-1} ,d_{t+i-1} \vert \psi _{t} ) \\= & {} \hbox {cov}(\delta ,d_{t+i-1} \vert \psi _{t} )+\phi \hbox {var}(d_{t+i-1} \vert \psi _{t} )+\hbox {cov}(\varepsilon _{t+i} ,d_{t+i-1} \vert \psi _{t} )\\&-\,\theta \hbox {cov}(\varepsilon _{t+i-1} ,d_{t+i-1} \vert \psi _{t} ) \\= & {} \phi \hbox {var}(d_{t+i-1} \vert \psi _{t} )-\theta \hbox {var}(\varepsilon _{t+i-1} \psi _{t} ) \\ \hbox {cov}(d_{t+i} ,\varepsilon _{t+i} \vert \psi _{t} )= & {} \hbox {cov}(\delta +\phi d_{t+i-1} +\varepsilon _{t+i} -\theta \varepsilon _{t+i-1} ,\varepsilon _{t+i} \vert \psi _{t} ) \\= & {} \hbox {cov}(\delta ,\varepsilon _{t+i} \vert \psi _{t} )+\phi \hbox {cov}(d_{t+i-1} ,\varepsilon _{t+i} \vert \psi _{t} )+\hbox {var}(\varepsilon _{t+i} \vert \psi _{t} )\\&-\,\theta \hbox {cov}(\varepsilon _{t+i} ,\varepsilon _{t+i-1} \vert \psi _{t} ) \\= & {} \hbox {var}(\varepsilon _{t+i} \vert \psi _{t} ) \\ \end{aligned}$$