# American Institute of Mathematical Sciences

July  2017, 13(3): 1553-1586. doi: 10.3934/jimo.2017007

## Optimum pricing strategy for complementary products with reservation price in a supply chain model

 Department of Industrial & Management Engineering, Hanyang University, Ansan Gyeonggi-do, 15588, South Korea

* Corresponding author: mitalisarkar.ms@gmail.com (Mitali Sarkar), Phone: +82-1074901981, Fax: +82-31-436-8146

Received  November 2015 Published  December 2016

This paper describes a two-echelon supply chain model with two manufacturers and one common retailer. Two types of complementary products are produced by two manufacturers, and the common retailer buys products separately using a reservation price and bundles them for sale. The demands of manufacturers and retailer are assumed to be stochastic in nature. When the retailer orders for products, any one of manufacturers agrees to allow those products, and the rest of the manufacturers have to provide the same amount. The profits of two manufacturers and the retailer are maximized by using Stackelberg game policy. By applying a game theoretical approach, several analytical solutions are obtained. For some cases, this model obtains quasi-closed-form solutions, for others, it finds closed-form solutions. Some numerical examples, sensitivity analysis, managerial insights, and graphical illustrations are given to illustrate the model.

Citation: Mitali Sarkar, Young Hae Lee. Optimum pricing strategy for complementary products with reservation price in a supply chain model. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1553-1586. doi: 10.3934/jimo.2017007
##### References:
 [1] A. Banerjee, A joint economic-lot-size model for purchaser and vendor, Decision Sciences, 17 (1986), 292-311. doi: 10.1111/j.1540-5915.1986.tb00228.x. [2] L. E. Cárdenas-Barrón and S. S. Sana, A production-inventory model for a two-echelon supply chain when demand is dependent on sales teams' initiatives, International Journal of Production Economics, 155 (2014), 249-258. [3] C. S. Choi, Price competition in a channel structure with a common retailer, Marketing Science, 10 (1991), 271-296. doi: 10.1287/mksc.10.4.271. [4] A. L. EI-Ansary and L. W. Stern, Power measurement in the distribution channel, Journal of Marketing Research, 9 (1972), 47-52. [5] G. Ertek and P. M. Griffin, Supplier-and buyer-driven channels in a two-stage supply chain, IIE Transactions, 34 (2002), 691-700. doi: 10.1080/07408170208928905. [6] J. Gabszewicz, N. Sonnac and X. Wauthy, On price competition with complementary goods, Economics Letters, 70 (2001), 431-437. doi: 10.1016/S0165-1765(00)00383-9. [7] S. K. Goyal, An integrated inventory model for a single supplier-single customer problem, International Journal of Production Research, 15 (1977), 107-111. doi: 10.1080/00207547708943107. [8] S. K. Goyal, A joint economic-lot-size model for purchaser and vendor: A comment, Decision Sciences, 19 (1988), 236-241. doi: 10.1111/j.1540-5915.1988.tb00264.x. [9] C. C. Hsieh and C. H. Wu, Coordinated decisions for substitutable products in a common retailer supply chain, European Journal of Operational Research, 196 (2009), 273-288. doi: 10.1016/j.ejor.2008.02.019. [10] K. F. McCardle, K. Rajaram and C. S. Tang, Bundling retail products: Models and analysis, European Journal of Operational Research, 177 (2007), 1197-1217. doi: 10.1016/j.ejor.2005.11.009. [11] N. M. Modak, S. Panda and S. S. Sana, Three-echelon supply chain coordination considering duopolistic retailers with perfect quality products, International Journal of Production Economics, 182 (2016), 564-578. doi: 10.1016/j.ijpe.2015.05.021. [12] S. Mukhopadhyay, X. Yue and X. Zhu, A Stackelberg model of pricing of complementary goods under information asymmetry, International Journal of Production Economics, 134 (2011), 424-433. doi: 10.1016/j.ijpe.2009.11.015. [13] K. Pan, K. K. Lai, S. C. H. Leung and D. Xiao, Revenue-sharing versus wholesale price mechanisms under different channel power structures, European Journal of Operational Research, 203 (2010), 532-538. doi: 10.1016/j.ejor.2009.08.010. [14] B. Sarkar, Supply chain coordination with variable backorder, inspections, and discount policy for fixed lifetime products Mathematical Problems in Engineering 2016 (2016), Article ID 6318737, 14 pages. doi: 10.1155/2016/6318737. [15] B. Sarkar, A production-inventory model with probabilistic deterioration in two-echelon supply chain management, Applied Mathematical Modelling, 37 (2013), 3138-3151. doi: 10.1016/j.apm.2012.07.026. [16] B. Sarkar and A. Majumder, Integrated vendor-buyer supply chain model with vendors setup cost reduction, Applied Mathematics and Computation, 224 (2013), 362-371. doi: 10.1016/j.amc.2013.08.072. [17] B. Sarkar, S. Saren, D. Sinha and S. Hur, Effect of unequal lot sizes, variable setup cost, and carbon emission cost in a supply chain model Mathematical Problems in Engineering 2015 (2015), Article ID 469486, 13 pages. doi: 10.1155/2015/469486. [18] J. Wei, J. Zhao and Y. Li, Pricing decisions for complementary products with firms' different market powers, European Journal of Operational Research, 224 (2013), 507-519. doi: 10.1016/j.ejor.2012.09.011. [19] J. Wei, J. Zhao and Y. Li, Price and warranty period decisions for complementary products with horizontal firms' cooperation/noncooperation strategies, Journal of Cleaner Production, 105 (2015), 86-102. doi: 10.1016/j.jclepro.2014.09.059. [20] C. H. Wu, C. W. Chen and C. C. Hsieh, Competitive pricing decisions in a two echelon supply chain with horizontal and vertical competition, International Journal of Production Economics, 135 (2012), 265-274. doi: 10.1016/j.ijpe.2011.07.020. [21] Z. Yao, S. C. H. Leung and K. K. Lai, Manufacturer's revenue-sharing contract and retail competition, European Journal of Operational Research, 186 (2008), 637-651. doi: 10.1016/j.ejor.2007.01.049. [22] X. Yue, S. Mukhopadhyay and X. Zhu, A Bertrand model of pricing of complementary goods under information asymmetry, Journal of Business Research, 59 (2006), 1182-1192. doi: 10.1016/j.jbusres.2005.06.005. [23] J. Zhao, W. Tang, R. Zhao and J. Wei, Pricing decisions for substitutable products with a common retailer in fuzzy environments, European Journal of Operational Research, 216 (2012), 409-419. doi: 10.1016/j.ejor.2011.07.026.

show all references

##### References:
 [1] A. Banerjee, A joint economic-lot-size model for purchaser and vendor, Decision Sciences, 17 (1986), 292-311. doi: 10.1111/j.1540-5915.1986.tb00228.x. [2] L. E. Cárdenas-Barrón and S. S. Sana, A production-inventory model for a two-echelon supply chain when demand is dependent on sales teams' initiatives, International Journal of Production Economics, 155 (2014), 249-258. [3] C. S. Choi, Price competition in a channel structure with a common retailer, Marketing Science, 10 (1991), 271-296. doi: 10.1287/mksc.10.4.271. [4] A. L. EI-Ansary and L. W. Stern, Power measurement in the distribution channel, Journal of Marketing Research, 9 (1972), 47-52. [5] G. Ertek and P. M. Griffin, Supplier-and buyer-driven channels in a two-stage supply chain, IIE Transactions, 34 (2002), 691-700. doi: 10.1080/07408170208928905. [6] J. Gabszewicz, N. Sonnac and X. Wauthy, On price competition with complementary goods, Economics Letters, 70 (2001), 431-437. doi: 10.1016/S0165-1765(00)00383-9. [7] S. K. Goyal, An integrated inventory model for a single supplier-single customer problem, International Journal of Production Research, 15 (1977), 107-111. doi: 10.1080/00207547708943107. [8] S. K. Goyal, A joint economic-lot-size model for purchaser and vendor: A comment, Decision Sciences, 19 (1988), 236-241. doi: 10.1111/j.1540-5915.1988.tb00264.x. [9] C. C. Hsieh and C. H. Wu, Coordinated decisions for substitutable products in a common retailer supply chain, European Journal of Operational Research, 196 (2009), 273-288. doi: 10.1016/j.ejor.2008.02.019. [10] K. F. McCardle, K. Rajaram and C. S. Tang, Bundling retail products: Models and analysis, European Journal of Operational Research, 177 (2007), 1197-1217. doi: 10.1016/j.ejor.2005.11.009. [11] N. M. Modak, S. Panda and S. S. Sana, Three-echelon supply chain coordination considering duopolistic retailers with perfect quality products, International Journal of Production Economics, 182 (2016), 564-578. doi: 10.1016/j.ijpe.2015.05.021. [12] S. Mukhopadhyay, X. Yue and X. Zhu, A Stackelberg model of pricing of complementary goods under information asymmetry, International Journal of Production Economics, 134 (2011), 424-433. doi: 10.1016/j.ijpe.2009.11.015. [13] K. Pan, K. K. Lai, S. C. H. Leung and D. Xiao, Revenue-sharing versus wholesale price mechanisms under different channel power structures, European Journal of Operational Research, 203 (2010), 532-538. doi: 10.1016/j.ejor.2009.08.010. [14] B. Sarkar, Supply chain coordination with variable backorder, inspections, and discount policy for fixed lifetime products Mathematical Problems in Engineering 2016 (2016), Article ID 6318737, 14 pages. doi: 10.1155/2016/6318737. [15] B. Sarkar, A production-inventory model with probabilistic deterioration in two-echelon supply chain management, Applied Mathematical Modelling, 37 (2013), 3138-3151. doi: 10.1016/j.apm.2012.07.026. [16] B. Sarkar and A. Majumder, Integrated vendor-buyer supply chain model with vendors setup cost reduction, Applied Mathematics and Computation, 224 (2013), 362-371. doi: 10.1016/j.amc.2013.08.072. [17] B. Sarkar, S. Saren, D. Sinha and S. Hur, Effect of unequal lot sizes, variable setup cost, and carbon emission cost in a supply chain model Mathematical Problems in Engineering 2015 (2015), Article ID 469486, 13 pages. doi: 10.1155/2015/469486. [18] J. Wei, J. Zhao and Y. Li, Pricing decisions for complementary products with firms' different market powers, European Journal of Operational Research, 224 (2013), 507-519. doi: 10.1016/j.ejor.2012.09.011. [19] J. Wei, J. Zhao and Y. Li, Price and warranty period decisions for complementary products with horizontal firms' cooperation/noncooperation strategies, Journal of Cleaner Production, 105 (2015), 86-102. doi: 10.1016/j.jclepro.2014.09.059. [20] C. H. Wu, C. W. Chen and C. C. Hsieh, Competitive pricing decisions in a two echelon supply chain with horizontal and vertical competition, International Journal of Production Economics, 135 (2012), 265-274. doi: 10.1016/j.ijpe.2011.07.020. [21] Z. Yao, S. C. H. Leung and K. K. Lai, Manufacturer's revenue-sharing contract and retail competition, European Journal of Operational Research, 186 (2008), 637-651. doi: 10.1016/j.ejor.2007.01.049. [22] X. Yue, S. Mukhopadhyay and X. Zhu, A Bertrand model of pricing of complementary goods under information asymmetry, Journal of Business Research, 59 (2006), 1182-1192. doi: 10.1016/j.jbusres.2005.06.005. [23] J. Zhao, W. Tang, R. Zhao and J. Wei, Pricing decisions for substitutable products with a common retailer in fuzzy environments, European Journal of Operational Research, 216 (2012), 409-419. doi: 10.1016/j.ejor.2011.07.026.
Graphical representation for Case 1.1, total profit of manufacturer 1 versus selling-price and lot size
Graphical representation for Case 1.1, total profit of manufacturer 2 versus selling-price
Graphical representation for Case 1.1, total profit of retailer versus selling-price of bundle product
Graphical representation for Case 1.2, total profit of manufacturer 1 versus selling-price and lot size
Graphical representation for Case 1.2, total profit of manufacturer 2 and retailer versus selling-price and selling-price of bundle product
Graphical representation for Case 2.1, total profit of manufacturer 2 versus selling-price and lot size
Graphical representation for Case 2.1, total profit of manufacturer 1 versus sellingprice
Graphical representation for Case 2.1, total profit of retailer versus selling-price of bundle product
Graphical representation for Case 2.2, total profit of manufacturer 2 versus selling price and lot size
Graphical representation for Case 2.2, total profit of manufacturer 1 and retailer versus selling-price and selling-price of bundle product
Comparative studies of cooperation and non-cooperation for the selling-price of product 2 of manufacturer 2 in Case 1. Blue ink of the graphical representation indicates under cooperative strategy and the red ink of the graphical representation indicates under noncooperative strategy
Comparative studies of cooperation and non-cooperation for the selling-price of bundle product of retailer in Case 1. Blue ink of the graphical representation indicates under cooperative strategy and the red ink of the graphical representation indicates under noncooperative strategy
Comparative studies of cooperation and non-cooperation for the selling-price of product 1 of manufacturer 1 in Case 2. Blue ink of the graphical representation indicates under cooperative strategy and the red ink of the graphical representation indicates under noncooperative strategy
Comparative studies of cooperation and non-cooperation for the selling-price of bundle product of retailer in Case 2. Blue ink of the graphical representation indicates under cooperative strategy and the red ink of the graphical representation indicates under noncooperative strategy
Comparison between the contributions of different authors
 Author (s) SCM Competitive price study Reservation price Game approach Stochastic demand Choi [3] √ √ Yue et al. [22] √ √ Mukhopadhyay et al. [12] √ √ Wei et al. [18] √ √ √ Cárdenas-Barrón and Sana [2] √ √ Sarkar [14] √ √ McCardle et al. [10] √ √ √ This Model √ √ √ √ √
 Author (s) SCM Competitive price study Reservation price Game approach Stochastic demand Choi [3] √ √ Yue et al. [22] √ √ Mukhopadhyay et al. [12] √ √ Wei et al. [18] √ √ √ Cárdenas-Barrón and Sana [2] √ √ Sarkar [14] √ √ McCardle et al. [10] √ √ √ This Model √ √ √ √ √
 Decision variables $Q$ order quantity (units) $P_{i}$ selling-price of product j, j=1, 2 (＄/unit) $P_{r}$ selling-price of the bundle product (＄/unit) Random variables $D_{m_{i}}$ demand for product j, j=1, 2 (units) $D_{r}$ demand for the bundle product (units) Parameters $C_{i}$ manufacturing cost of product j, j=1, 2 (＄/unit) $h_{m_{i}}$ holding cost of product j per unit per unit time, j=1, 2 (＄/unit/unit time) $h_{r}$ holding cost of the bundle product per unit per unit time (＄/unit/unit time) $S_{m_{i}}$ setup cost per setup of product j, j=1, 2 (＄/unit) $K_{m_{i}}$ production rate of product j, j=1, 2 (units) $M$ known market size (units) $A$ ordering cost per order of the retailer (＄/order) $I_{m_{i1}}$ inventory of manufacturer i at $t \in[0, t_{m_{i}}]$, i=1, 2 $I_{m_{i2}}$ inventory of manufacturer i at $t \in [t_{m_{i}}, T_{m_{i}}]$, i=1, 2 $AP_{m_{i}}$ expected average profit of manufacturer i, i=1, 2 $AP_{r}$ expected average profit of the retailer $t_{m_{i}}$ time required for maximum inventory of manufacturer i, i=1, 2 $T_{m_{i}}$ cycle time of manufacturer i, i=1, 2 $R_{i}^{a}$ lower limit of reservation price of manufacturer i, i=1, 2 $R_{i}^{b}$ upper limit of reservation price of manufacturer i, i=1, 2
 Decision variables $Q$ order quantity (units) $P_{i}$ selling-price of product j, j=1, 2 (＄/unit) $P_{r}$ selling-price of the bundle product (＄/unit) Random variables $D_{m_{i}}$ demand for product j, j=1, 2 (units) $D_{r}$ demand for the bundle product (units) Parameters $C_{i}$ manufacturing cost of product j, j=1, 2 (＄/unit) $h_{m_{i}}$ holding cost of product j per unit per unit time, j=1, 2 (＄/unit/unit time) $h_{r}$ holding cost of the bundle product per unit per unit time (＄/unit/unit time) $S_{m_{i}}$ setup cost per setup of product j, j=1, 2 (＄/unit) $K_{m_{i}}$ production rate of product j, j=1, 2 (units) $M$ known market size (units) $A$ ordering cost per order of the retailer (＄/order) $I_{m_{i1}}$ inventory of manufacturer i at $t \in[0, t_{m_{i}}]$, i=1, 2 $I_{m_{i2}}$ inventory of manufacturer i at $t \in [t_{m_{i}}, T_{m_{i}}]$, i=1, 2 $AP_{m_{i}}$ expected average profit of manufacturer i, i=1, 2 $AP_{r}$ expected average profit of the retailer $t_{m_{i}}$ time required for maximum inventory of manufacturer i, i=1, 2 $T_{m_{i}}$ cycle time of manufacturer i, i=1, 2 $R_{i}^{a}$ lower limit of reservation price of manufacturer i, i=1, 2 $R_{i}^{b}$ upper limit of reservation price of manufacturer i, i=1, 2
Input data
 Player Market size (units) Manufacturer 1 $M=1500$ Manufacturer 2 $M=1500$ Retailer $M=1500$ Setup cost (＄/setup) $S_{m_{1}}=20$ $S_{m_{2}}=20$ $A=1$ Holding cost (＄/unit/year) $h_{m_{1}}=0.015$ $h_{m_{2}}=0.015$ $h_r=0.01$ Production rate (units/year) $K_{m_{1}}=2000$ $K_{m_{2}}=2000$ - Purchasing cost (＄/unit) $C_{1}=0.25$ $C_{2}=0.15$ - Reservation interval [0, 1] [0.1, 0.9] [0.1, 0.9] -indicates that the parameter is not available for this case.
 Player Market size (units) Manufacturer 1 $M=1500$ Manufacturer 2 $M=1500$ Retailer $M=1500$ Setup cost (＄/setup) $S_{m_{1}}=20$ $S_{m_{2}}=20$ $A=1$ Holding cost (＄/unit/year) $h_{m_{1}}=0.015$ $h_{m_{2}}=0.015$ $h_r=0.01$ Production rate (units/year) $K_{m_{1}}=2000$ $K_{m_{2}}=2000$ - Purchasing cost (＄/unit) $C_{1}=0.25$ $C_{2}=0.15$ - Reservation interval [0, 1] [0.1, 0.9] [0.1, 0.9] -indicates that the parameter is not available for this case.
Optimum results of Example 1
 Case $Q^*$ units $P_{1}^{*}$ ＄/unit $P_{2}^{*}$ ＄/unit $P_{r}^{*}$ ＄/unit $AP_{m_{1}}^{*}$ ＄/year $AP_{m_{2}}^{*}$ ＄/year $AP_{{r}}^{*}$ ＄/year $AP_{m_{1}r}^{*}$ ＄/year $AP_{m_{2}r}^{*}$ ＄/year 1.1 1433.14 0.63 0.53 1.33 195.39 246.92 36.37 - - 1.2 1433.14 0.63 0.44 1.29 195.39 - - - 294.18 2.1 1690.88 0.63 0.53 1.33 195.18 247.15 35.90 - - 2.2 1690.88 0.51 0.53 1.27 - 247.15 - 245.87 - -indicates that the average profit is not available for this case.
 Case $Q^*$ units $P_{1}^{*}$ ＄/unit $P_{2}^{*}$ ＄/unit $P_{r}^{*}$ ＄/unit $AP_{m_{1}}^{*}$ ＄/year $AP_{m_{2}}^{*}$ ＄/year $AP_{{r}}^{*}$ ＄/year $AP_{m_{1}r}^{*}$ ＄/year $AP_{m_{2}r}^{*}$ ＄/year 1.1 1433.14 0.63 0.53 1.33 195.39 246.92 36.37 - - 1.2 1433.14 0.63 0.44 1.29 195.39 - - - 294.18 2.1 1690.88 0.63 0.53 1.33 195.18 247.15 35.90 - - 2.2 1690.88 0.51 0.53 1.27 - 247.15 - 245.87 - -indicates that the average profit is not available for this case.
Input data from McCardle et al. [10]
 Player Market size (units) Manufacturer 1 $M=100$ Manufacturer 2 $M=100$ Retailer $M=100$ Setup cost (＄/setup) $S_{m_{1}}=0$ $S_{m_{2}}=0$ $A=0$ Holding cost (＄/unit/year) $h_{m_{1}}=0$ $h_{m_{2}}=0$ $h_r=0$ Production rate (units/year) $K_{m_{1}}=0$ $K_{m_{2}}=0$ - Purchasing cost (＄/unit) $C_{1}=0.25$ $C_{2}=0.25$ - Reservation interval [0, 1] [0.1, 0.9] [0.1, 0.9] -indicates that the parameter is not available for this case.
 Player Market size (units) Manufacturer 1 $M=100$ Manufacturer 2 $M=100$ Retailer $M=100$ Setup cost (＄/setup) $S_{m_{1}}=0$ $S_{m_{2}}=0$ $A=0$ Holding cost (＄/unit/year) $h_{m_{1}}=0$ $h_{m_{2}}=0$ $h_r=0$ Production rate (units/year) $K_{m_{1}}=0$ $K_{m_{2}}=0$ - Purchasing cost (＄/unit) $C_{1}=0.25$ $C_{2}=0.25$ - Reservation interval [0, 1] [0.1, 0.9] [0.1, 0.9] -indicates that the parameter is not available for this case.
Optimum results of Example 2
 Case $Q^*$ units $P_{1}^{*}$ ＄/unit $P_{2}^{*}$ ＄/unit $P_{r}^{*}$ ＄/unit $AP_{m_{1}}^{*}$ ＄/year $AP_{m_{2}}^{*}$ ＄/year $AP_{{r}}^{*}$ ＄/year $AP_{m_{1}r}^{*}$ ＄/year $AP_{m_{2}r}^{*}$ ＄/year 1.1 $300$ $0.625$ $0.625$ $1.375$ $14.0625$ $14.0625$ $1.5625$ - - 1.2 $300$ $0.625$ $0.541667$ $1.33$ $14.0625$ - - - $16.1458$ 2.1 $300$ $0.625$ $0.625$ $1.375$ $14.0625$ $14.0625$ $1.5625$ - - 2.2 $300$ $0.625$ $0.541667$ $1.33333$ - $14.0625$ - 16.1458 - -indicates that the average profit is not available for this case.
 Case $Q^*$ units $P_{1}^{*}$ ＄/unit $P_{2}^{*}$ ＄/unit $P_{r}^{*}$ ＄/unit $AP_{m_{1}}^{*}$ ＄/year $AP_{m_{2}}^{*}$ ＄/year $AP_{{r}}^{*}$ ＄/year $AP_{m_{1}r}^{*}$ ＄/year $AP_{m_{2}r}^{*}$ ＄/year 1.1 $300$ $0.625$ $0.625$ $1.375$ $14.0625$ $14.0625$ $1.5625$ - - 1.2 $300$ $0.625$ $0.541667$ $1.33$ $14.0625$ - - - $16.1458$ 2.1 $300$ $0.625$ $0.625$ $1.375$ $14.0625$ $14.0625$ $1.5625$ - - 2.2 $300$ $0.625$ $0.541667$ $1.33333$ - $14.0625$ - 16.1458 - -indicates that the average profit is not available for this case.
Sensitivity analysis for Case 1.1
 Parameter change(in %) $AP_{m_{1}}$ (in %) $AP_{m_{2}}$ (in %) $AP_{r}$ (in %) $M$ -50% -52.76 -52.18 -59.85 -25% -26.38 -26.09 -29.93 +25% +26.38 +26.09 +29.93 +50% +52.75 +52.18 +59.85 Parameter change(in %) $AP_{m_{1}}$ (in %) Parameter change(in %) $AP_{m_{2}}$ (in %) $S_{m_{1}}$ -50% +1.99 $S_{m_{2}}$ -50% +1.97 -25% +0.99 -25% +0.98 +25% -0.99 +25% -0.98 +50% -1.98 +50% -1.95 $h_{m_{1}}$ -50% +2.33 $h_{m_{2}}$ -50% +1.42 -25% +1.06 -25% +1.42 +25% -0.94 +25% -0.71 +50% -1.78 +50% -1.42 $C_{1}$ -50% +38.63 $C_{2}$ -50% +22.18 -25% +18.54 -25% +10.82 +25% -17.04 +25% -10.29 +50% -32.58 +50% -20.04 Parameter change(in %) $AP_{r}$ (in %) Parameter change(in %) $AP_{r}$ (in %) $A$ -50% +0.25 $h_{r}$ -50% +9.85 -25% +0.12 -25% +4.93 +25% -0.12 +25% -4.93 +50% -0.25 +50% -9.85
 Parameter change(in %) $AP_{m_{1}}$ (in %) $AP_{m_{2}}$ (in %) $AP_{r}$ (in %) $M$ -50% -52.76 -52.18 -59.85 -25% -26.38 -26.09 -29.93 +25% +26.38 +26.09 +29.93 +50% +52.75 +52.18 +59.85 Parameter change(in %) $AP_{m_{1}}$ (in %) Parameter change(in %) $AP_{m_{2}}$ (in %) $S_{m_{1}}$ -50% +1.99 $S_{m_{2}}$ -50% +1.97 -25% +0.99 -25% +0.98 +25% -0.99 +25% -0.98 +50% -1.98 +50% -1.95 $h_{m_{1}}$ -50% +2.33 $h_{m_{2}}$ -50% +1.42 -25% +1.06 -25% +1.42 +25% -0.94 +25% -0.71 +50% -1.78 +50% -1.42 $C_{1}$ -50% +38.63 $C_{2}$ -50% +22.18 -25% +18.54 -25% +10.82 +25% -17.04 +25% -10.29 +50% -32.58 +50% -20.04 Parameter change(in %) $AP_{r}$ (in %) Parameter change(in %) $AP_{r}$ (in %) $A$ -50% +0.25 $h_{r}$ -50% +9.85 -25% +0.12 -25% +4.93 +25% -0.12 +25% -4.93 +50% -0.25 +50% -9.85
Sensitivity analysis for Case 1.2
 Parameter change(in %) $AP_{m_{1}}$ (in %) $AP_{m_{2}r}$ (in %) $M$ -50% -52.15 -53.04 -25% -26.24 -26.52 +25% +26.49 +26.52 +50% +53.18 +53.04 Parameter change(in %) $AP_{m_{1}}$ (in %) Parameter change(in %) $AP_{m_{2}r}$ (in %) $S_{m_{1}}$ -50% +1.99 $S_{m_{2}}$ -50% +2.04 -25% +0.99 -25% +1.02 +25% -0.99 +25% -1.01 +50% -1.98 +50% -2.02 $h_{m_{1}}$ -50% +2.33 $h_{m_{2}}$ -50% +1.05 -25% +1.06 -25% +0.52 +25% -0.94 +25% -0.52 +50% -1.78 +50% -1.04 $C_{1}$ -50% +38.63 $C_{2}$ -50% +22.91 -25% +18.55 -25% +11.18 +25% -17.02 +25% -10.62 +50% -32.52 +50% -20.67
 Parameter change(in %) $AP_{m_{1}}$ (in %) $AP_{m_{2}r}$ (in %) $M$ -50% -52.15 -53.04 -25% -26.24 -26.52 +25% +26.49 +26.52 +50% +53.18 +53.04 Parameter change(in %) $AP_{m_{1}}$ (in %) Parameter change(in %) $AP_{m_{2}r}$ (in %) $S_{m_{1}}$ -50% +1.99 $S_{m_{2}}$ -50% +2.04 -25% +0.99 -25% +1.02 +25% -0.99 +25% -1.01 +50% -1.98 +50% -2.02 $h_{m_{1}}$ -50% +2.33 $h_{m_{2}}$ -50% +1.05 -25% +1.06 -25% +0.52 +25% -0.94 +25% -0.52 +50% -1.78 +50% -1.04 $C_{1}$ -50% +38.63 $C_{2}$ -50% +22.91 -25% +18.55 -25% +11.18 +25% -17.02 +25% -10.62 +50% -32.52 +50% -20.67
Sensitivity analysis for Case 2.1
 Parameters change(in %) $AP_{m_{1}}$ (in %) $AP_{m_{2}}$ (in %) $AP_{r}$ (in %) $M$ -50% -53.25 -52.57 -61.77 -25% -26.62 -26.28 -30.89 +25% +26.62 +26.28 +30.89 +50% +53.25 +52.57 +61.77 Parameter change(in %) $AP_{r}$ (in %) Parameter change(in %) $AP_{r}$ (in %) $A$ -50% +0.21 $h_{r}$ -50% +11.77 -25% +0.11 -25% +5.89 +25% -0.11 +25% -5.89 +50% -0.21 +50% -11.77 Parameter change(in %) $AP_{m_{1}}$ (in %) Parameter change(in %) $AP_{m_{2}}$ (in %) $S_{m_{1}}$ -50% +1.70 $S_{m_{2}}$ -50% +1.96 -25% +0.85 -25% +0.90 +25% -0.84 +25% -0.79 +50% -1.69 +50% -1.50 $h_{m_{1}}$ -50% +2.34 $h_{m_{2}}$ -50% +1.96 -25% +1.17 -25% +0.90 +25% -1.17 +25% -0.79 +50% -2.34 +50% -1.50 $C_{1}$ -50% +38.76 $C_{2}$ -50% +22.27 -25% +18.63 -25% +10.86 +25% -17.13 +25% -10.32 +50% -32.76 +50% -20.09
 Parameters change(in %) $AP_{m_{1}}$ (in %) $AP_{m_{2}}$ (in %) $AP_{r}$ (in %) $M$ -50% -53.25 -52.57 -61.77 -25% -26.62 -26.28 -30.89 +25% +26.62 +26.28 +30.89 +50% +53.25 +52.57 +61.77 Parameter change(in %) $AP_{r}$ (in %) Parameter change(in %) $AP_{r}$ (in %) $A$ -50% +0.21 $h_{r}$ -50% +11.77 -25% +0.11 -25% +5.89 +25% -0.11 +25% -5.89 +50% -0.21 +50% -11.77 Parameter change(in %) $AP_{m_{1}}$ (in %) Parameter change(in %) $AP_{m_{2}}$ (in %) $S_{m_{1}}$ -50% +1.70 $S_{m_{2}}$ -50% +1.96 -25% +0.85 -25% +0.90 +25% -0.84 +25% -0.79 +50% -1.69 +50% -1.50 $h_{m_{1}}$ -50% +2.34 $h_{m_{2}}$ -50% +1.96 -25% +1.17 -25% +0.90 +25% -1.17 +25% -0.79 +50% -2.34 +50% -1.50 $C_{1}$ -50% +38.76 $C_{2}$ -50% +22.27 -25% +18.63 -25% +10.86 +25% -17.13 +25% -10.32 +50% -32.76 +50% -20.09
Sensitivity analysis for Case 2.2
 Parameter change(in %) $AP_{m_{2}}$ (in %) $AP_{m_{1}r}$ (in %) $M$ -50% -52.57 -54.30 -25% -26.28 -27.15 +25% +26.28 +27.15 +50% +52.57 +54.30 Parameter change(in %) $AP_{m_{1r}}$ (in %) Parameter change(in %) $AP_{m_{2}}$ (in %) $S_{m_{1}}$ -50% +1.76 $S_{m_{2}}$ -50% +1.96 -25% +0.88 -25% +0.90 +25% -0.88 +25% -0.79 +50% -1.75 +50% -1.50 $h_{m_{1}}$ -50% +1.64 $h_{m_{2}}$ -50% +1.96 -25% +0.82 -25% +0.90 +25% -0.82 +25% -0.79 +50% -1.64 +50% -1.50 $C_{1}$ -50% +40.31 $C_{2}$ -50% +22.27 -25% +19.36 -25% +10.86 +25% -17.77 +25% -10.32 +50% -33.95 +50% -20.09 Parameter change(in %) $AP_{m_{1r}}$ (in %) Parameter change(in %) $AP_{r}$ (in %) $A$ -50% +0.04 $h_{r}$ -50% +1.72 -25% +0.02 -25% +0.86 +25% -0.02 +25% -0.86 +50% -0.04 +50% -1.72
 Parameter change(in %) $AP_{m_{2}}$ (in %) $AP_{m_{1}r}$ (in %) $M$ -50% -52.57 -54.30 -25% -26.28 -27.15 +25% +26.28 +27.15 +50% +52.57 +54.30 Parameter change(in %) $AP_{m_{1r}}$ (in %) Parameter change(in %) $AP_{m_{2}}$ (in %) $S_{m_{1}}$ -50% +1.76 $S_{m_{2}}$ -50% +1.96 -25% +0.88 -25% +0.90 +25% -0.88 +25% -0.79 +50% -1.75 +50% -1.50 $h_{m_{1}}$ -50% +1.64 $h_{m_{2}}$ -50% +1.96 -25% +0.82 -25% +0.90 +25% -0.82 +25% -0.79 +50% -1.64 +50% -1.50 $C_{1}$ -50% +40.31 $C_{2}$ -50% +22.27 -25% +19.36 -25% +10.86 +25% -17.77 +25% -10.32 +50% -33.95 +50% -20.09 Parameter change(in %) $AP_{m_{1r}}$ (in %) Parameter change(in %) $AP_{r}$ (in %) $A$ -50% +0.04 $h_{r}$ -50% +1.72 -25% +0.02 -25% +0.86 +25% -0.02 +25% -0.86 +50% -0.04 +50% -1.72
 [1] Yeong-Cheng Liou, Siegfried Schaible, Jen-Chih Yao. Supply chain inventory management via a Stackelberg equilibrium. Journal of Industrial & Management Optimization, 2006, 2 (1) : 81-94. doi: 10.3934/jimo.2006.2.81 [2] Jun Li, Hairong Feng, Kun-Jen Chung. Using the algebraic approach to determine the replenishment optimal policy with defective products, backlog and delay of payments in the supply chain management. Journal of Industrial & Management Optimization, 2012, 8 (1) : 263-269. doi: 10.3934/jimo.2012.8.263 [3] Lisha Wang, Huaming Song, Ding Zhang, Hui Yang. Pricing decisions for complementary products in a fuzzy dual-channel supply chain. Journal of Industrial & Management Optimization, 2019, 15 (1) : 343-364. doi: 10.3934/jimo.2018046 [4] Jing Feng, Yanfei Lan, Ruiqing Zhao. Impact of price cap regulation on supply chain contracting between two monopolists. Journal of Industrial & Management Optimization, 2017, 13 (1) : 349-373. doi: 10.3934/jimo.2016021 [5] Gang Xie, Wuyi Yue, Shouyang Wang. Optimal selection of cleaner products in a green supply chain with risk aversion. Journal of Industrial & Management Optimization, 2015, 11 (2) : 515-528. doi: 10.3934/jimo.2015.11.515 [6] Tinggui Chen, Yanhui Jiang. Research on operating mechanism for creative products supply chain based on game theory. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1103-1112. doi: 10.3934/dcdss.2015.8.1103 [7] Ximin Huang, Na Song, Wai-Ki Ching, Tak-Kuen Siu, Ka-Fai Cedric Yiu. A real option approach to optimal inventory management of retail products. Journal of Industrial & Management Optimization, 2012, 8 (2) : 379-389. doi: 10.3934/jimo.2012.8.379 [8] Fei Cheng, Shanlin Yang, Ram Akella, Xiaoting Tang. An integrated approach for selection of service vendors in service supply chain. Journal of Industrial & Management Optimization, 2011, 7 (4) : 907-925. doi: 10.3934/jimo.2011.7.907 [9] Hamid Norouzi Nav, Mohammad Reza Jahed Motlagh, Ahmad Makui. Modeling and analyzing the chaotic behavior in supply chain networks: a control theoretic approach. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1123-1141. doi: 10.3934/jimo.2018002 [10] Lianju Sun, Ziyou Gao, Yiju Wang. A Stackelberg game management model of the urban public transport. Journal of Industrial & Management Optimization, 2012, 8 (2) : 507-520. doi: 10.3934/jimo.2012.8.507 [11] Bibhas C. Giri, Bhaba R. Sarker. Coordinating a multi-echelon supply chain under production disruption and price-sensitive stochastic demand. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-21. doi: 10.3934/jimo.2018115 [12] Amin Aalaei, Hamid Davoudpour. Two bounds for integrating the virtual dynamic cellular manufacturing problem into supply chain management. Journal of Industrial & Management Optimization, 2016, 12 (3) : 907-930. doi: 10.3934/jimo.2016.12.907 [13] Katherinne Salas Navarro, Jaime Acevedo Chedid, Whady F. Florez, Holman Ospina Mateus, Leopoldo Eduardo Cárdenas-Barrón, Shib Sankar Sana. A collaborative EPQ inventory model for a three-echelon supply chain with multiple products considering the effect of marketing effort on demand. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-14. doi: 10.3934/jimo.2019020 [14] Jiuping Xu, Pei Wei. Production-distribution planning of construction supply chain management under fuzzy random environment for large-scale construction projects. Journal of Industrial & Management Optimization, 2013, 9 (1) : 31-56. doi: 10.3934/jimo.2013.9.31 [15] Ali Naimi Sadigh, S. Kamal Chaharsooghi, Majid Sheikhmohammady. A game theoretic approach to coordination of pricing, advertising, and inventory decisions in a competitive supply chain. Journal of Industrial & Management Optimization, 2016, 12 (1) : 337-355. doi: 10.3934/jimo.2016.12.337 [16] Azam Moradi, Jafar Razmi, Reza Babazadeh, Ali Sabbaghnia. An integrated Principal Component Analysis and multi-objective mathematical programming approach to agile supply chain network design under uncertainty. Journal of Industrial & Management Optimization, 2019, 15 (2) : 855-879. doi: 10.3934/jimo.2018074 [17] Juliang Zhang, Jian Chen. Information sharing in a make-to-stock supply chain. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1169-1189. doi: 10.3934/jimo.2014.10.1169 [18] Juliang Zhang. Coordination of supply chain with buyer's promotion. Journal of Industrial & Management Optimization, 2007, 3 (4) : 715-726. doi: 10.3934/jimo.2007.3.715 [19] Na Song, Ximin Huang, Yue Xie, Wai-Ki Ching, Tak-Kuen Siu. Impact of reorder option in supply chain coordination. Journal of Industrial & Management Optimization, 2017, 13 (1) : 449-475. doi: 10.3934/jimo.2016026 [20] Liping Zhang. A nonlinear complementarity model for supply chain network equilibrium. Journal of Industrial & Management Optimization, 2007, 3 (4) : 727-737. doi: 10.3934/jimo.2007.3.727

2017 Impact Factor: 0.994