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January  2017, 13(1): 449-475. doi: 10.3934/jimo.2016026

## Impact of reorder option in supply chain coordination

 1 School of Management and Economics, University of Electronic Science and Technology, Chengdu, China 2 Scheller College of Business, Georgia Institute of Technology, Atlanta, USA 3 Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong 4 Department of Applied Finance and Actuarial Studies, Faculty of Business and Economics, Macquarie University, Sydney, Australia

Received  June 2015 Published  March 2016

This paper studies the impacts of some reorder options on the performance as well as the coordination issues in a supply chain. A large category of products requires a long procurement lead time yet only has a relatively short selling season. Hence the purchase decisions usually have to be made well in advance of the opening of the sales. However, when uncertainty exists, the actual market demand may turn out to severely deviate from the initial order amount. To make up for the deficiency arising from this situation, a reorder option is introduced which renders a second manufacturing chance available shortly before he selling season. This reorder option facilitates an adjustment of the inventory level according to the realization of market demand. Since the market under investigation is facing a downward sloping demand curve, the effect of implementing this option is multi-fold. Moreover, the launch of the reorder option may also affect the decision makings at other levels of operations, such as altering the size of the initial order. Therefore, the overall impact of such option is not immediately clear. In this paper, it is shown that a properly designed reorder option is able to bring in profit growth and stabilize the fluctuations in the market retail price. Besides, quantity discount contracts are constructed to coordinate decisions on the initial inventory amount within the supply chain, so as to achieve higher economic efficiency. Finally, numerical examples are given to demonstrate the conclusions obtained in this paper.

Citation: Na Song, Ximin Huang, Yue Xie, Wai-Ki Ching, Tak-Kuen Siu. Impact of reorder option in supply chain coordination. Journal of Industrial and Management Optimization, 2017, 13 (1) : 449-475. doi: 10.3934/jimo.2016026
##### References:
 [1] P. L. Abad and V. Aggarwal, Incorporating transport cost in the lot size and pricing decisions with downward sloping demand, International Journal of Production Economics, 95 (2005), 297-305. [2] K. J. Arrow and G. Debreu, Existence of an equilibrium for a competitive economy, Econometrica, 22 (1954), 265-290.  doi: 10.2307/1907353. [3] A. Burnetas and P. Ritchken, Option pricing with downward-sloping demand curves: The case of supply chain options, Management Science, 51 (2005), 566-580. [4] M. Baxter and A. Rennie, Financial Calculus An Introduction to Derivative Pricing, Cambridge University Press, 1996. [5] K. Donohue, Efficient supply contracts for fashion goods with forecast updating and two production modes, Management Science, 46 (2000), 1397-1411. [6] G. P. Cachon, Supply chain coordination with contracts, in Handbooks in Operations Research and Management Science: Supply Chain Management (eds. S. Graves and T. de Kok), NorthHolland, Amsterdam, The Netherlands, 2003, 227–339. [7] G. Cachon and M. Lariviere, Supply chain coordination with revenue-sharing contracts: Strengths and limitations, Management Science, 51 (2005), 30-44. [8] S. Dasu and L. Li, Optimal operating policies in the presence of exchange rate variability, Management Science, 43 (1997), 705-722. [9] R. J. Dolan, Quantity discounts: Managerial issues and research opportunities, Marketing Science, 6 (1987), 1-27. [10] S. M. Gilbert and R. H. Ballou, Supply chain benefits from advanced customer commitments, Journal of Operations Management, 18 (1999), 61-73. [11] X. Huang, S. Choi, W. Ching, T. Siu and M. Huang, On supply chain coordination for false failure returns: A quantity discount contract approach, International Journal of Production Economics, 133 (2011), 634-644. [12] X. Huang, N. Song, W. Ching, T. Siu and K. Yiu, A real option approach to optimal inventory management of retail products, Journal of Industrial and Management Optimization, 8 (2012), 379-389.  doi: 10.3934/jimo.2012.8.379. [13] X. Huang, J. Gu, W. Ching and T. Siu, Impact of secondary market on consumer return policies and supply chain coordination, Omega, 45 (2014), 57-70. [14] X. Huang, S. Choi and W. Ching, On improving incentive in a supply chain: Wholesale price contract vs quantity dependent contract, in Computers and Industrial Engineering (CIE), 2010 40th International Conference on, (2010), 1-6. [15] H. Hishamuddin, R. A. Sarker and D. Essam, A disruption recovery model for a single stage production-inventory system, European Journal of Operational Research, 222 (2012), 464-473.  doi: 10.1016/j.ejor.2012.05.033. [16] A. P. Jeuland and S. M. Shugan, Managing channel profits, Marketing Science, 2 (1983), 239-272. [17] A. Kaul, V. Mehrotra and R. Morck, Demand curves for stocks do slope down: New evidence from an index weights adjustment, The Journal of Finance, 55 (2000), 893-912. [18] H. Krishman, R. Kapuscinski and D. Butz, Coordinating contracts for decentralized supply chains with retailer promotional effort, Management Science, 50 (2004), 48-63. [19] B. Kogut and N. Kulatilaka, Operating flexibility, global manufacturing, and the option value of a multinational network, Management Science, 40 (1994), 123-139. [20] S. Kolay, G. Shaffer and J. A. Ordover, All-unit discounts in retail contracts, Journal of Economics and Management Strategy, 13 (2004), 429-459. [21] C. Li and P. Kouvelis, Flexible and risk-sharing supply contracts under price uncertainty, Management Science, 45 (1999), 1378-1398. [22] L. Liang, X. Wang and J. Gao, An option contract pricing model of relief material supply chain, Omega, 40 (2012), 594-600. [23] B. Pasternack, Optimal pricing and returns policies for perishable commodities, Marketing Science, 4 (1985), 166-176. [24] O. D. Palsule-Desai, Supply chain coordination using revenue-dependent revenue sharing contracts, Omega, 41 (2013), 780-796. [25] S. K. Paul, A. Azeem, R. Sarker and D. Essam, Development of a production inventory model with uncertainty and reliability considerations, Optimization and Engineering, 15 (2014), 697-720.  doi: 10.1007/s11081-013-9218-6. [26] S. K. Paul, R. Sarker and D. Essam, Managing risk and disruption in production-inventory and supply chain systems: A review, Journal of Industrial and Management Optimization, 12 (2016), 1009-1029.  doi: 10.3934/jimo.2016.12.1009. [27] S. K. Paul, R. Sarker and D. Essam, Managing disruption in an imperfect production-inventory system, Computers & Industrial Engineering, 84 (2015), 101-112. [28] S. K. Paul, R. Sarker and D. Essam, A disruption recovery plan in a three-stage production-inventory system, Computers and Operations Research, 57 (2015), 60-72.  doi: 10.1016/j.cor.2014.12.003. [29] S. K. Paul, R. Sarker and D. Essam, Real time disruption management for a two-stage batch production inventory system with reliability considerations, European Journal of Operational Research, 237 (2014), 113-128.  doi: 10.1016/j.ejor.2014.02.005. [30] S. K. Paul, R. Sarker and D. Essam, A disruption recovery model in a production-inventory system with demand uncertainty and process reliability, Computer Information Systems and Industrial Management, (2013), 511-522. [31] J. Spengler, Vertical integration and anti-trust policy, Journal of Political Economy, 58 (1950), 347-352. [32] T. A. Taylor, Supply chain coordination under channel rebates with dales effort effects, Management Science, 48 (2002), 992-1007. [33] A. Tsay, S. Nahmias and N. Agrawal, Modeling supply chain contracts: A review in Quantitative Models for Supply Chain Management (eds. S. Tayur, R. Ganeshan and M. Magazine), Kluwer Academic Publishers, Dordrecht (Chapter 10), (1999), 1339-1358. [34] R. Wilson, Nonlinear Pricing, Oxford University Press, Oxford, 1993. [35] S. M. Wagner, S. S. Padhi and I. Zanger, A real option-based supply chain project evaluation and scheduling method, International Journal of Production Research, 52 (2014), 3725-3743.

show all references

##### References:
 [1] P. L. Abad and V. Aggarwal, Incorporating transport cost in the lot size and pricing decisions with downward sloping demand, International Journal of Production Economics, 95 (2005), 297-305. [2] K. J. Arrow and G. Debreu, Existence of an equilibrium for a competitive economy, Econometrica, 22 (1954), 265-290.  doi: 10.2307/1907353. [3] A. Burnetas and P. Ritchken, Option pricing with downward-sloping demand curves: The case of supply chain options, Management Science, 51 (2005), 566-580. [4] M. Baxter and A. Rennie, Financial Calculus An Introduction to Derivative Pricing, Cambridge University Press, 1996. [5] K. Donohue, Efficient supply contracts for fashion goods with forecast updating and two production modes, Management Science, 46 (2000), 1397-1411. [6] G. P. Cachon, Supply chain coordination with contracts, in Handbooks in Operations Research and Management Science: Supply Chain Management (eds. S. Graves and T. de Kok), NorthHolland, Amsterdam, The Netherlands, 2003, 227–339. [7] G. Cachon and M. Lariviere, Supply chain coordination with revenue-sharing contracts: Strengths and limitations, Management Science, 51 (2005), 30-44. [8] S. Dasu and L. Li, Optimal operating policies in the presence of exchange rate variability, Management Science, 43 (1997), 705-722. [9] R. J. Dolan, Quantity discounts: Managerial issues and research opportunities, Marketing Science, 6 (1987), 1-27. [10] S. M. Gilbert and R. H. Ballou, Supply chain benefits from advanced customer commitments, Journal of Operations Management, 18 (1999), 61-73. [11] X. Huang, S. Choi, W. Ching, T. Siu and M. Huang, On supply chain coordination for false failure returns: A quantity discount contract approach, International Journal of Production Economics, 133 (2011), 634-644. [12] X. Huang, N. Song, W. Ching, T. Siu and K. Yiu, A real option approach to optimal inventory management of retail products, Journal of Industrial and Management Optimization, 8 (2012), 379-389.  doi: 10.3934/jimo.2012.8.379. [13] X. Huang, J. Gu, W. Ching and T. Siu, Impact of secondary market on consumer return policies and supply chain coordination, Omega, 45 (2014), 57-70. [14] X. Huang, S. Choi and W. Ching, On improving incentive in a supply chain: Wholesale price contract vs quantity dependent contract, in Computers and Industrial Engineering (CIE), 2010 40th International Conference on, (2010), 1-6. [15] H. Hishamuddin, R. A. Sarker and D. Essam, A disruption recovery model for a single stage production-inventory system, European Journal of Operational Research, 222 (2012), 464-473.  doi: 10.1016/j.ejor.2012.05.033. [16] A. P. Jeuland and S. M. Shugan, Managing channel profits, Marketing Science, 2 (1983), 239-272. [17] A. Kaul, V. Mehrotra and R. Morck, Demand curves for stocks do slope down: New evidence from an index weights adjustment, The Journal of Finance, 55 (2000), 893-912. [18] H. Krishman, R. Kapuscinski and D. Butz, Coordinating contracts for decentralized supply chains with retailer promotional effort, Management Science, 50 (2004), 48-63. [19] B. Kogut and N. Kulatilaka, Operating flexibility, global manufacturing, and the option value of a multinational network, Management Science, 40 (1994), 123-139. [20] S. Kolay, G. Shaffer and J. A. Ordover, All-unit discounts in retail contracts, Journal of Economics and Management Strategy, 13 (2004), 429-459. [21] C. Li and P. Kouvelis, Flexible and risk-sharing supply contracts under price uncertainty, Management Science, 45 (1999), 1378-1398. [22] L. Liang, X. Wang and J. Gao, An option contract pricing model of relief material supply chain, Omega, 40 (2012), 594-600. [23] B. Pasternack, Optimal pricing and returns policies for perishable commodities, Marketing Science, 4 (1985), 166-176. [24] O. D. Palsule-Desai, Supply chain coordination using revenue-dependent revenue sharing contracts, Omega, 41 (2013), 780-796. [25] S. K. Paul, A. Azeem, R. Sarker and D. Essam, Development of a production inventory model with uncertainty and reliability considerations, Optimization and Engineering, 15 (2014), 697-720.  doi: 10.1007/s11081-013-9218-6. [26] S. K. Paul, R. Sarker and D. Essam, Managing risk and disruption in production-inventory and supply chain systems: A review, Journal of Industrial and Management Optimization, 12 (2016), 1009-1029.  doi: 10.3934/jimo.2016.12.1009. [27] S. K. Paul, R. Sarker and D. Essam, Managing disruption in an imperfect production-inventory system, Computers & Industrial Engineering, 84 (2015), 101-112. [28] S. K. Paul, R. Sarker and D. Essam, A disruption recovery plan in a three-stage production-inventory system, Computers and Operations Research, 57 (2015), 60-72.  doi: 10.1016/j.cor.2014.12.003. [29] S. K. Paul, R. Sarker and D. Essam, Real time disruption management for a two-stage batch production inventory system with reliability considerations, European Journal of Operational Research, 237 (2014), 113-128.  doi: 10.1016/j.ejor.2014.02.005. [30] S. K. Paul, R. Sarker and D. Essam, A disruption recovery model in a production-inventory system with demand uncertainty and process reliability, Computer Information Systems and Industrial Management, (2013), 511-522. [31] J. Spengler, Vertical integration and anti-trust policy, Journal of Political Economy, 58 (1950), 347-352. [32] T. A. Taylor, Supply chain coordination under channel rebates with dales effort effects, Management Science, 48 (2002), 992-1007. [33] A. Tsay, S. Nahmias and N. Agrawal, Modeling supply chain contracts: A review in Quantitative Models for Supply Chain Management (eds. S. Tayur, R. Ganeshan and M. Magazine), Kluwer Academic Publishers, Dordrecht (Chapter 10), (1999), 1339-1358. [34] R. Wilson, Nonlinear Pricing, Oxford University Press, Oxford, 1993. [35] S. M. Wagner, S. S. Padhi and I. Zanger, A real option-based supply chain project evaluation and scheduling method, International Journal of Production Research, 52 (2014), 3725-3743.
Time line of the base case without the reorder option
Time line of the base case with the reorder option
Total profits with and without reorder option under $\sigma_1=10$.
Optimal initial order $Q^*$ with a reorder option under $\sigma_1=10$.
Incomes with exercising or dropping the reorder option at the states of $a_H$ and $a_L$ under $\sigma_1=10$.
Total profits with and without a reorder option under $\sigma_2=25$.
Optimal initial order $Q^*$ with a reorder option under $\sigma_2=25$.
Incomes with exercising or dropping the reorder option at the states of $a_H$ and $a_L$ under $\sigma_2=25$.
Supply chain's total profit and the retailer's profit with respect to $Q$ and $Q_R$, respectively.
Distribution of the maximised supply chain total profit between the retailer and the supplier with respect to $\eta$.
Incomes of the retailer with respect to $X$.
A List of Notations
 $P$ market clearing price of the product $Q$ amount of products at time $0$ with the reorder option $Q^*$ optimal amount of products at time $0$ with the reorder option $Q_0$ amount of products at time $0$ without the reorder option $Q_0^*$ optimal amount of products at time $0$ without the reorder option $\delta$ the slope of the demand curve $a_H$ ($a_L$) the indicator of the market condition in the high (low) state $e_H$ ($e_L$) Arrow-Debreu state price for the high (low) state $\rho$ $\rho=e_H/e_L$ $B$ present value of the risk-free coupon that pays 1 dollar regardless of the state $A$ present value of the security that pays $a_H$ ($a_L$) in the high (low) state $p_H$ ($p_L$) risk-neutral probability of the occurrence of the high (low) state $\mu$ the mean of the uncertain factor $a$ ${\sigma}^2$ the variance of the uncertain factor $a$ $c$ the unit production cost of a product at time $0$ $c_1$ the unit production cost of a product at time $T$ $R$ a pre-determined amount of products in the reorder option
 $P$ market clearing price of the product $Q$ amount of products at time $0$ with the reorder option $Q^*$ optimal amount of products at time $0$ with the reorder option $Q_0$ amount of products at time $0$ without the reorder option $Q_0^*$ optimal amount of products at time $0$ without the reorder option $\delta$ the slope of the demand curve $a_H$ ($a_L$) the indicator of the market condition in the high (low) state $e_H$ ($e_L$) Arrow-Debreu state price for the high (low) state $\rho$ $\rho=e_H/e_L$ $B$ present value of the risk-free coupon that pays 1 dollar regardless of the state $A$ present value of the security that pays $a_H$ ($a_L$) in the high (low) state $p_H$ ($p_L$) risk-neutral probability of the occurrence of the high (low) state $\mu$ the mean of the uncertain factor $a$ ${\sigma}^2$ the variance of the uncertain factor $a$ $c$ the unit production cost of a product at time $0$ $c_1$ the unit production cost of a product at time $T$ $R$ a pre-determined amount of products in the reorder option
A List of Notations
 $w$ a fixed wholesale price which is higher than $c$ $X$ strike price which is pre-determined by the supplier $Q_R$ amount of products ordered by the retailer at time $0$ with the reorder option $Q_R^*$ optimal amount of products ordered by the retailer at time $0$ with the reorder option $Q_{max}$ maximum size of products ordered by the retailer at time $0$ with the reorder option $a, b$ parameters in the function of wholesale price $w$ $\eta$ ($\overline{\eta}$) the portion of the maximized supply chain total profit earned by the retailer (supplier)
 $w$ a fixed wholesale price which is higher than $c$ $X$ strike price which is pre-determined by the supplier $Q_R$ amount of products ordered by the retailer at time $0$ with the reorder option $Q_R^*$ optimal amount of products ordered by the retailer at time $0$ with the reorder option $Q_{max}$ maximum size of products ordered by the retailer at time $0$ with the reorder option $a, b$ parameters in the function of wholesale price $w$ $\eta$ ($\overline{\eta}$) the portion of the maximized supply chain total profit earned by the retailer (supplier)
A summary of parameters
 A B c $c_1$ $\rho$ $\delta$ 16 dollars 0.8 dollar 4 dollars 7 dollars 4 1 $e_L$ $e_H$ $\mu$ 0.64 dollar 0.16 dollar 20 dollars
 A B c $c_1$ $\rho$ $\delta$ 16 dollars 0.8 dollar 4 dollars 7 dollars 4 1 $e_L$ $e_H$ $\mu$ 0.64 dollar 0.16 dollar 20 dollars
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