Abstract
The paper derives the impact of inventory sharing policy on the bullwhip effect in two-stage supply chains with two independent suppliers and two integrated retailers. There exists an inventory sharing policy between two retailers. Under inventory sharing policy, when demand in one retailer exceeds its inventory, this retailer can ask for a product sharing volume from the other in order to satisfy customer demand. With certain assumptions, the bullwhip effect is quantified in both cases, with inventory sharing policy and without inventory sharing policy. We found that inventory sharing has significant impact on the bullwhip effect in the supply system. However, inventory sharing policy does not synchronously reduce or increase the bullwhip effect in both suppliers in the same period. A numerical example is given to illustrate the study model.
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References
Bassok Y, Anupindi R, Akella R (1999) Single-period multi-product inventory models with substitution. Oper Res 47(4):632–642
Bitran G, Dasu S (1992) Ordering policies in an environment of stochastic yields and substitutable demands. Oper Res 40(5):177–185
Chen F, Ryan JK, Simchi-Levi D (2000) The impact of exponential smoothing forecasts on the bullwhip effect. Naval Res Logist 47(4):269–286
Chatfield DC, Kim JG, Harrison TP, Hayya JC (2004) The bullwhip effect—impact of stochastic lead time, information quality, and information sharing: a simulation study. Prod Oper Manag 13(4):340–353
Drezner Z, Gurnani H, Pasternack BA (1995) An eoq model with substitutions between products. J Oper Res Soc 46(7):887–891
Evers PT (2001) Heuristics for assessing emergency transshipments. Eur J Oper Res 129:311–316
Kutanoglu E (2009) An inventory sharing and allocation method for a multi-location service parts logistics network with time-based service levels. Eur Oper Res 194(3):728–742
Forrester J (1958) Industrial dynamics, a major breakthrough for decision makers. Harvard Bus Rev 36:37–66
Chen F, Drezner Z, Ryan JK, Levi DS (2000) Quantifying the bullwhip effect in a simple supply chain. Manag Sci 46(3):436–443
Gilbert K (2005) An ARIMA supply chain model. Manag Sci 51(2):305–310
Luong HT, Phien NH (2010) Measure of bullwhip effect in supply chains: the case of high order autoregressive demand process. Eur J Oper Res 183(1):197–209
Luong HT (2007) Measure of the bullwhip effect in supply chains with autoregressive demand process. Eur J Oper Res 180(3):1086–1097
Sun HX, Ren YT (2005) The impact of forecasting methods on bullwhip effect in supply chain management. Proc Eng Manage Conf 1:215–219
Zhao H, Deshpande V, Ryan K (2005) Inventory sharing and rationing in decentralized dealer networks. Manage Sci 51(4):531–547
Kim JG, Chatfield D, Harrison TP, Hayya JC (2006) Quantifying the bullwhip effect in a supply chain with stochastic lead time. Eur J Oper Res 173:617–636
Heydari J, Karemzadeh RB, Chaharsooghi SK (2009) A study of lead time variation impact on supply chain performance. Int J Adv Manuf Technol 40:1206–1215
Karaesmen I, van Ryzin G (2004) Overbooking with substitutable inventory classes. Oper Res 52(1):83–104
So KC, Zheng X (2003) Impact of supplier’s lead time and forecast demand updating on retailer’s order quantity variability in a two-level supply chain. Int J Prod Econ 86:169–179
Xu K, Dong Y, Evers PT (2000) “Towards better coordination of the supply chain”, Transport. Res Part E Logist Trans Rev 37(1):35–54
Lee HL, Padmanabhan V, Whang S (1997) Information distortion in a supply chain: the bullwhip effect. Manag Sci 43(4):546–558
McGillivray AR, Silver EA (1978) Some concepts for inventory control under substitutable demand. Inf Syst Oper Res 16(1):47–63
Netessine S, Rudi N (2003) Centralized and competitive inventory models with demand substitution. Oper Res 51(2):329–335
Rudi N, Kapur S, Pyke DF (2001) A two-location inventory model with transshipment and local decision making. Manage Sci 47(12):1668–1680
Parlar M, Goyal S (1984) Optimal ordering decisions for twosubstitutable products with stochastic demands. Opsearch 21:1–15
Graves SC (1999) A single-item inventory model for a non-stationary demand process. Manuf Serv Oper Manage 1(1):50–61
Ryan JK (1997) Analysis of inventory models with limited demand information. PhD thesis, Northwestern University, Evanston
Tagaras G (1989) Effects of pooling on the optimization and service levels of two-location inventory systems. IIE Trans 21:250–257
Duc TTH, Luong HT, Kim YD (2008) A measure of bullwhip effect in supply chains with a mixed autore-gressive-moving average demand process. Eur J Oper Res 187(1):243–256
Duc TTH, Luong HT, Kim YD (2010) Investigate the impact of third-Party warehouse on bullwhip effect in supply chains with autoregressive demand process. SIMTech technical reports (STR_V11_N3_08_POM). 11(3)
Zhang X (2004) The impact of forecasting methods on the bullwhip effect. Int J Prod Econ 88:15–27
Li X, Song L, Zhao Z (2011) Quantifying the impact of demand substitution on the bullwhip effect in a supply chain. Logist Res 3:221–232
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Appendix
Appendix
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1.
The derivation process of E(Dt,i) and \( \text{var} (D_{t,i} ) \)When the autoregressive demand process is stationary, we have
$$ E(D_{t,i} ) = E(D_{t - 1,i} ) = E(D_{t - 2,i} ) = \ldots = E(D_{i} ), $$and
$$ \text{var} (D_{t,i} ) = \text{var} (D_{t - 1,i} ) = \text{var} (D_{t - 2,i} ) = \ldots = \text{var} (D_{i} ), $$where
$$ \begin{aligned} & D_{t,1} = \mu_{1} + \rho_{1} D_{t - 1,1} + \varepsilon_{t,1} , \\ & E(D_{t,1} ) = E(\mu_{1} ) + \rho_{1} E(D_{t - 1,1} ) + E(\varepsilon_{t,1} ) \\ & E(D_{1} ) = \,\mu_{1} + \rho_{1} E(D_{1} ) + 0 \\ & \quad \Rightarrow E(D_{1} ) = \frac{{\mu_{1} }}{{1 - \rho_{1} }}. \\ & \,\text{var} (D_{t,1} ) = \text{var} (\mu_{1} ) + \rho_{1}^{2} \text{var} (D_{t - 1,1} ) + \text{var} (\varepsilon_{t,1} ) \\ \, & \text{var} (D_{1} ) = 0 + \rho_{1}^{2} \text{var} (D_{1} ) + \sigma_{1}^{2} \\ & \quad \Rightarrow \text{var} (D_{1} ) = \frac{{\sigma_{1}^{2} }}{{1 - \rho_{1}^{2} }}. \\ \end{aligned} $$Similarly, we have
$$ \begin{aligned} E(D_{2} ) & = \frac{{\mu_{2} }}{{1 - \rho_{2} }}, \\ \text{var} (D_{1} ) & = \frac{{\sigma_{2}^{2} }}{{1 - \rho_{2}^{2} }}. \\ \end{aligned} $$ -
2.
The derivation process of the further equation of qt,1.
$$ \begin{aligned} q_{t,1} &= y_{t,1} - y_{t - 1,1} + D_{t -1,1} + \lambda q_{t,1} , \\ \Rightarrow q_{t,1} & = \frac{1}{1 -\lambda }\left[ {y_{t,1} - y_{t - 1,1} + D_{t - 1,1} } \right] \\& = \frac{1}{1 - \lambda }\left[ {\left({\widehat{D}_{t,1}^{{L_{1} }} + z_{1} \widehat{\sigma}_{t,1}^{{L_{1} }} } \right) - \left( {\widehat{D}_{t - 1,1}^{{L_{1}}} + z_{1} \widehat{\sigma }_{t - 1,1}^{{L_{1} }} } \right) + D_{t -1,1} } \right] \\ & = \frac{1}{1 - \lambda }\left[ {\left({\widehat{D}_{t,1}^{{L_{1} }} - \widehat{D}_{t - 1,1}^{{L_{1} }} }\right) + D_{t - 1,1} - z_{1} \left( {\widehat{\sigma}_{t,1}^{{L_{1} }} - \widehat{\sigma }_{t - 1,1}^{{L_{1} }} }\right)} \right] \\ & = \frac{L}{p(1 - \lambda )}\left({\sum\limits_{i = 1}^{p} {D_{t - i,1} - \sum\limits_{i = 1}^{p}{D_{t - 1 - i,1} } } } \right) \\ & \quad + \frac{{D_{t - 1,1}}}{1 - \lambda } + \frac{{z_{1} }}{1 - \lambda }\left({\widehat{\sigma }_{t,1}^{{L_{1} }} - \widehat{\sigma }_{t -1,1}^{{L_{1} }} } \right) \\ \, & = \frac{L}{p(1 - \lambda)}\left( {D_{t - 1,1} - D_{t - p - 1,1} } \right) + \frac{{D_{t -1,1} }}{1 - \lambda } + \frac{{z_{1} }}{1 - \lambda }\left({\widehat{\sigma }_{t,1}^{{L_{1} }} - \widehat{\sigma }_{t -1,1}^{{L_{1} }} } \right) \\ & = \frac{1}{1 - \lambda }\left(1 +\frac{{L_{1} }}{p}\right)D_{t - 1,1} - \frac{{L_{1} }}{p\left(1 - \lambda\right)}D_{t - p - 1,1} \\ & \quad + \frac{{z_{1} }}{1 - \lambda}\left( {\widehat{\sigma }_{t,1}^{{L_{1} }} - \widehat{\sigma }_{t -1,1}^{{L_{1} }} } \right), \\ \end{aligned} $$$$\text{var} (q_{t,1} ) = \text{var} \left( \begin{gathered}\frac{1}{1 - \lambda }\left(1 + \frac{{L_{1}}}{p}\right)D_{t - 1,1} -\frac{{L_{1} }}{p(1 - \lambda )}D_{t - p - 1,1} \hfill \\ \, +\frac{{z_{1} }}{1 - \lambda }\left( {\widehat{\sigma }_{t,1}^{{L_{1}}} - \widehat{\sigma }_{t - 1,1}^{{L_{1} }} } \right) \hfill \\\end{gathered} \right) $$$$\begin{aligned} &= \frac{1}{{(1 - \lambda )^{2} }}\left(\begin{gathered} \left(1 + \frac{{L_{1} }}{p}\right)^{2} \text{var} (D_{t -1,1} ) \hfill \\ - 2\left(\frac{{L_{1} }}{p}\right)\left(1 + \frac{{L_{1}}}{p}\right)\text{cov} (D_{t - 1,1} ,D_{t - p - 1,1} ) \hfill \\ +\left(\frac{{L_{1} }}{p}\right)^{2} \text{var} (D_{t - p - 1,1} ) + z_{1}^{2}\text{var} (\widehat{\sigma }_{t,1}^{{L_{1} }} - \widehat{\sigma}_{t - 1,1}^{{L_{1} }} ) \hfill \\ + 2z_{1} \left(1 + \frac{{2L_{1}}}{p}\right)\text{cov} (D_{t - 1,1} ,\widehat{\sigma }_{t,1}^{{L_{1} }} )\hfill \\ \end{gathered} \right) \\ & = \frac{1}{{\left(1 - \lambda\right)^{2} }}\left( \begin{gathered} \left(1 + \frac{{2L_{1} }}{p} +\left(\frac{{2L_{1} }}{p}\right)^{2} \right)\text{var} (D_{1} ) \hfill \\ -\left(\frac{{2L_{1} }}{p} + \frac{{2L_{1}^{2} }}{p}\right)\text{cov} (D_{t -1,1} ,D_{t - p - 1,1} ) \hfill \\ + z_{1}^{2} \text{var}(\widehat{\sigma }_{t,1}^{{L_{1} }} - \widehat{\sigma }_{t -1,1}^{{L_{1} }} ) \hfill \\ + 2z_{1} \left(1 + \frac{{2L_{1}}}{p}\right)\text{cov} (D_{t - 1,1} ,\widehat{\sigma }_{t,1}^{{L_{1} }} )\hfill \\ \end{gathered} \right). \\ \end{aligned} $$Now we will determine \( \text{cov} (D_{t - 1,1} ,D_{t - p - 1,1} ) \) and \( \text{cov} (D_{t - 1,1} ,\widehat{\sigma }_{t,1}^{{L_{1} }} ) \) We have
$$ \begin{gathered} \text{cov} (D_{t - 1,1} ,D_{t - p - 1,1} ) \hfill \\ = \text{cov} \left( {(\mu_{1} + \rho_{1} D_{t - 2,1} + \varepsilon_{t,1} ),D_{t - p - 1,1} } \right) \hfill \\ = \text{cov} \left( {\mu_{1} ,D_{t - p - 1,1} } \right) + \rho_{1} \text{cov} \left( {D_{t - 2,1} ,D_{t - p - 1,1} } \right) \hfill \\ + \text{cov} \left( {\varepsilon_{t,1} ,D_{t - p - 1,1} } \right). \hfill \\ \end{gathered} $$$$ \begin{gathered} ({\text{Since}}\,\text{cov} \left( {\mu_{1} ,D_{t - p - 1,1} } \right) = 0\,{\text{and}}\,\text{cov} \left( {\varepsilon_{t,1} ,D_{t - p - 1,1} } \right) = 0), \hfill \\ \text{cov} (D_{t - 1,1} ,D_{t - p - 1,1} ) = \rho_{1} \text{cov} \left( {D_{t - 2,1} ,D_{t - p - 1,1} } \right) \hfill \\ \ldots \hfill \\ = \rho_{1}^{p} \text{cov} \left( {D_{t - p,1} ,D_{t - p - 1,1} } \right) \hfill \\ = \rho_{1}^{p} \text{var} (D_{1} ). \hfill \\ \end{gathered} $$We assume that forecasting customer demands by retailers are random variables of the form as \( D_{t} = \mu + \rho D_{t - 1} + \varepsilon_{t} \), and the error terms ɛ t are identically independent distribution with mean 0 and variance σ2. Let the estimate of the standard deviation of forecast error of the lead time demand be
$$ \widehat{\sigma }_{t}^{L} = C_{L,p} \sqrt {\frac{{\sum\nolimits_{i = j}^{p} {(D_{t - j} - \widehat{D}_{t - j} )^{2} } }}{p}} . $$Applying the result proved in Ryan [21], we have
$$ \text{cov} (D_{t - j} ,\widehat{\sigma }_{t}^{L} ) = 0,\quad \forall i = 1,2, \ldots ,p. $$Hence,
$$ \begin{gathered} \text{var} (q_{t,1} ) \hfill \\ = \frac{1}{{(1 -\lambda )^{2} }}\left[ \begin{gathered} \left(1 + \frac{{2L_{1} }}{p} +2\left(\frac{{L_{1} }}{p}\right)^{2} \right)\text{var} (D_{1} ) \hfill \\ -\left(\frac{{2L_{1} }}{p} + \frac{{2L_{1}^{2} }}{p}\right)\text{cov} (D_{t -1,1} ,D_{t - p - 1,1} ) \hfill \\ + z_{1}^{2} \text{var}(\widehat{\sigma }_{t,1}^{{L_{1} }} - \widehat{\sigma }_{t -1,1}^{{L_{1} }} ) + \,2z_{1} \left(1 + \frac{{2L_{1} }}{p}\right)\text{cov}(D_{t - 1,1} ,\widehat{\sigma }_{t,1}^{{L_{1} }} ) \hfill \\\end{gathered} \right] \hfill \\ = \frac{1}{{(1 - \lambda )^{2}}}\left[ \begin{gathered} \left(1 + \frac{{2L_{1} }}{p} + 2\left(\frac{{L_{1}}}{p}\right)^{2} \right)\text{var} (D_{1} ) - \,\left(\frac{{2L_{1} }}{p} +\frac{{2L_{1}^{2} }}{p}\right)\rho_{1}^{p} \text{var} (D_{1} ) \hfill \\ +z_{1}^{2} \text{var} (\widehat{\sigma }_{t,1}^{{L_{1} }} -\widehat{\sigma }_{t - 1,1}^{{L_{1} }} ) \hfill \\ \end{gathered}\right] \hfill \\ = \frac{{\text{var} (D_{1} )}}{{(1 - \lambda )^{2}}}\left[ {1 + \left(\frac{{2L_{1} }}{p} + \frac{{2L_{1}^{2} }}{p}\right)(1 -\rho_{1}^{p} )} \right] + \frac{{z_{1}^{2} \text{var}(\widehat{\sigma }_{t,1}^{{L_{1} }} - \widehat{\sigma }_{t -1,1}^{{L_{1} }} )}}{{(1 - \lambda )^{2} }}. \hfill \\ \end{gathered}$$ -
3.
The derivation process of the further expression of qt,2:
$$ \begin{aligned} q_{t,2} & = \left(1 + \frac{{L_{2} }}{p}\right)D_{t -1,2} - \left(\frac{{L_{2} }}{p}\right)D_{t - p - 1,2} + z_{2} (\widehat{\sigma}_{t,2}^{{L_{2} }} - \widehat{\sigma }_{t - 1,2}^{{L_{2} }} ) -\lambda q_{t,1} , \\ \Rightarrow \text{var} (q_{t,2} ) \\ & =\text{var} \left[ {\left(1 + \frac{{L_{2} }}{p}\right)D_{t - 1,2} -\left(\frac{{L_{2} }}{p}\right)D_{t - p - 1,2} + z_{2} (\widehat{\sigma}_{t,2}^{{L_{2} }} - \widehat{\sigma }_{t - 1,2}^{{L_{2} }} )}\right] \\ & \quad - \lambda^{2} \text{var} (q_{t,1} ) \\ &= \left(1 + \frac{{L_{2} }}{p}\right)^{2} \text{var} (D_{t - 1,2} ) -2\left(\frac{{L_{2} }}{p}\right)\left(1 + \frac{{L_{2} }}{p}\right)\text{cov} (D_{t - 1,2},D_{t - p - 1,2} ) \\ & \quad + \left(\frac{{L_{2} }}{p}\right)^{2}\text{var} (D_{t - p - 1,2} ) + z_{2}^{2} \text{var}(\widehat{\sigma }_{t,2}^{{L_{2} }} - \widehat{\sigma }_{t -1,2}^{{L_{2} }} ) \\ & \quad + 2z_{2} \left(1 + 2\frac{{L_{2}}}{p}\right)\text{cov} (D_{t - 1,2} ,\widehat{\sigma }_{t,2}^{{L_{2} }} )- \lambda^{2} \text{var} (q_{t,1} ) \\ & = \left( {1 +2\frac{{L_{2} }}{p} + 2\left(\frac{{L_{2} }}{p}\right)^{2} } \right)\text{var}(D_{2} ) \\ & \quad - \left( {\frac{{2L_{2} }}{p} +2(\frac{{L_{2} }}{p})^{2} } \right)\text{cov} (D_{t - 1,2} ,D_{t - p- 1,2} ) \\ \,\quad + z_{2}^{2} \text{var} (\widehat{\sigma}_{t,2}^{{L_{2} }} - \widehat{\sigma }_{t - 1,2}^{{L_{2} }} ) +2z_{2} \left(1 + \frac{{2L_{2} }}{p}\right)\text{cov} (D_{t - 1,2},\widehat{\sigma }_{t,2}^{{L_{2} }} ) \\ & \quad - \lambda^{2}\text{var} (q_{t,1} ). \\ \end{aligned} $$Furthermore, we have \( \text{cov} (D_{t - 1,2} ,\widehat{\sigma }_{t,2}^{{L_{2} }} ) = 0 \) and
$$ \begin{gathered} \text{cov} (D_{t - 1,2} ,D_{t - p - 1,2} ) \hfill \\ = \text{cov} (\mu_{2} + \rho_{2} D_{t - 2,2} + \varepsilon_{t - 1,2} ,D_{t - p - 1,2} ) \hfill \\ = \text{cov} (\mu_{2} ,D_{t - p - 1,2} ) + \rho_{2} \text{cov} (D_{t - 2,2} ,D_{t - p - 1,2} ) \hfill \\ \quad + \text{cov} (\varepsilon_{t - 1,2} ,D_{t - p - 1,2} ) \hfill \\ = \rho_{2} \text{cov} (D_{t - 1,2} ,D_{t - p - 1,2} ) \hfill \\ \ldots \hfill \\ = \rho_{2}^{p} \text{cov} (D_{t - p - 1,2} ,D_{t - p - 1,2} ) \hfill \\ = \rho_{2}^{p} \text{var} (D_{2} ). \hfill \\ \end{gathered}$$$$({\text{Note}}\,{\text{that}}\,\text{cov} (\mu_{2} ,D_{t - p - 1,2} ) = 0\quad {\text{and}}\quad \text{cov} (\varepsilon_{t - 1,2} ,D_{t - p - 1,2} ) = 0) $$Hence,
$$ \begin{aligned} \text{var} (q_{t,2} ) & = \left( {1 + 2\frac{{L_{2} }}{p} + 2\left(\frac{{L_{2} }}{p}\right)^{2} } \right)\text{var} (D_{2} ) \\ & \quad - \left( {\frac{{2L_{2} }}{p} + 2\left(\frac{{L_{2} }}{p}\right)^{2} } \right)\rho_{2}^{2} \text{var} (D_{2} ) \\ & \quad + z_{2}^{2} \text{var} (\widehat{\sigma }_{t,2}^{{L_{2} }} - \widehat{\sigma }_{t - 1,2}^{{L_{2} }} )\, - \lambda^{2} \text{var} (q_{t,1} ) \\ & = \text{var} (D_{2} )\left[ {1 + \left( {2\frac{{L_{2} }}{p} + 2\left(\frac{{L_{2} }}{p}\right)^{2} } \right)(1 - \rho_{2}^{2} )} \right] \\ & \quad + z_{2}^{2} \text{var} (\widehat{\sigma }_{t,2}^{{L_{2} }} - \widehat{\sigma }_{t - 1,2}^{{L_{2} }} )\, - \lambda^{2} \text{var} (q_{t,1} ). \\ \end{aligned} $$
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Le, D.V., Huynh, L.T., Claudiu, K.V. et al. The impact of inventory sharing on the bullwhip effect in decentralized inventory systems. Logist. Res. 6, 89–98 (2013). https://doi.org/10.1007/s12159-012-0096-7
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DOI: https://doi.org/10.1007/s12159-012-0096-7