# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2020109

## Optimal ordering and pricing models of a two-echelon supply chain under multipletimes ordering

 1 School of Management and Economics, Beijing Institute of Technology, Beijing, 100081, China, Springfield, MO 65801-2604, USA 2 School of Economics and Management, Xi'an University of Posts and Telecommunications, Xi'an Shaanxi, 710121, China, Springfield, MO 65810, USA

* Corresponding author: Fujun Hou

Received  October 2019 Revised  February 2020 Published  June 2020

Fund Project: This work is supported by the Natural Science Foundation of China (71571019)

This paper studies ordering and pricing issues under multiple times ordering. A manufacturer and a retailer are involved in our discussion. The definition of a reasonable price is given based on the practical requirement. First, we construct a Stackelberg model in which the manufacturer and the retailer make their decisions respectively. During the process of derivation, both ordering time-points and optimal prices are expressed as functions of number of times of ordering. By solving a quadratic programming model with an undetermined parameter, we demonstrate that the optimal ordering time-points of the retailer are equidistant time points on the given selling period. Second, a cooperative model is developed in which the manufacturer and the retailer jointly make decisions. It is shown that the optimal retail price is lower and the number of times of ordering is more in the cooperative situation than the noncooperative one. Further, an allocation method based on revenue proportions is proposed.

Citation: Zhenkai Lou, Fujun Hou, Xuming Lou. Optimal ordering and pricing models of a two-echelon supply chain under multipletimes ordering. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020109
##### References:
 [1] S. A. H. S. Amiri, A. Herbon, A. Zahedi, M. Kazemi, J. Soroor and M. Hajiaghaei-Keshteli, Determination of the optimal sales level of perishable goods in a two-echelon supply chain network, Computers & Industrial Engineering, 139 (2020), 147-156.   Google Scholar [2] T. Avinadav, A. Herbon and U. Spiegel, Optimal ordering and pricing policy for demand functions that are separable into price and inventory age, International Journal of Production Economics, 155 (2014), 406-417.   Google Scholar [3] H. J. Chang, J. T. Teng, L. Y. Ouyang and C. Y. Dye, Retailer's optimal pricing and lot-sizing policies for deteriorating items with partial backlogging, European Journal of Operational Research, 168 (2005), 51-64.  doi: 10.1016/j.ejor.2004.05.003.  Google Scholar [4] B. X. Chen, X. L. Chao and H. S. Ahn, Coordinating pricing and inventory replenishment with nonparametric demand learning, Operations Research, 67 (2019), 1035-1052.  doi: 10.1287/opre.2018.1808.  Google Scholar [5] X. Chen and P. Hu, Joint pricing and inventory management with deterministic demand and costly price adjust, Operations Research Letters, 40 (2012), 385-389.  doi: 10.1016/j.orl.2012.05.011.  Google Scholar [6] Y. H. Chen, S. B. Ray and Y. Y. Song, Optimal pricing and inventory control policy in periodic-review systems with fixed ordering cost and lost sales, Naval Research Logistics, 53 (2006), 117-136.  doi: 10.1002/nav.20127.  Google Scholar [7] W. M. Chung, S. Talluri and R. Narasimhan, Optimal pricing and inventory strategies with multiple price markdowns over time, European Journal of Operational Research, 243 (2015), 130-141.  doi: 10.1016/j.ejor.2014.11.020.  Google Scholar [8] O. C. Demirag, Y. H. Chen and Y. Yang, Ordering policies for periodic-review inventory systems with quantity-dependent fixed costs, Operations Research, 60 (2012), 785-796.  doi: 10.1287/opre.1110.1033.  Google Scholar [9] C. Y. Dye, Joint pricing and ordering policy for a deteriorating inventory with partial backlogging, Omega, 35 (2007), 184-189.   Google Scholar [10] O. Kaya and A. L. Polat, Coordinated pricing and inventory decisions for perishable products, OR Spectrum, 39 (2017), 589-606.  doi: 10.1007/s00291-016-0467-6.  Google Scholar [11] J. Li, S. Y. Wang and T. C. E. Cheng, Competition and cooperation in a single-retailer two-supplier supply chain with supply disruption, International Journal of Production Economics, 124 (2010), 137-150.   Google Scholar [12] R. H. Li, Y. L. Chan and C. T. Chang, Pricing and lot-sizing policies for perishable products with advance-cash-credit payments by a discounted cash-flow analysis, International Journal of Production Economics, 193 (2017), 578-589.   Google Scholar [13] B. Mondal, A. K. Bhunia and M. Maiti, An inventory system of ameliorating items for price dependent demand rate, Computers & Industrial Engineering, 45 (2003), 443-456.   Google Scholar [14] M. S. Sajadieh and M. R. A. Jokar, Optimizing shipment, ordering and pricing policies in a two-stage supply chain with price-sensitive demand, Transportation Research Part E, 45 (2009), 564-571.   Google Scholar [15] D. A. Serel, Optimal ordering and pricing in a quick response system, International Journal of Production Economics, 121 (2009), 700-714.   Google Scholar [16] J. M. Shi, G. Q. Zhang and K. K. Lai, Optimal ordering and pricing policy with supplier quantity discounts and price-dependent stochastic demand, Optimization, 61 (2012), 151-162.  doi: 10.1080/02331934.2011.590485.  Google Scholar [17] A. A. Taleizadeh, H. R. Zarei and B. R. Sarker, An optimal control of inventory under probablistic replenishment intervals and known price increase, European Journal of Operational Research, 257 (2017), 777-791.  doi: 10.1016/j.ejor.2016.07.041.  Google Scholar [18] S. Tiwari, L. E. Cardenas-Barron and M. Goh, Joint pricing and inventory model for deteriorating items with expiration dates and partial backlogging under two-level partial trade credits in supply chain, International Journal of Production Economics, 200 (2018), 16-36.   Google Scholar [19] W. Xing and S. Y. Wang, Optimal ordering and pricing strategies in the presence of a B2B spot market, European Journal of Operational Research, 221 (2012), 87-98.  doi: 10.1016/j.ejor.2012.03.017.  Google Scholar [20] S. L. Yang, C. M. Shi and X. Zhao, Optimal ordering and pricing decisions for a target oriented newsvendor, Omega, 39 (2011), 110-115.   Google Scholar [21] P. S. You and M. T. Wu, Optimal ordering and pricing policy for an inventory system with order cancellations, OR Spectrum, 29 (2007), 661-679.  doi: 10.1007/s00291-006-0067-y.  Google Scholar [22] P. S. You, S. Ikuta and Y. C. Hsieh, Optimal ordering and pricing policy for an inventory system with trial period, Applied Mathematical Modelling, 34 (2010), 3179-3188.  doi: 10.1016/j.apm.2010.02.008.  Google Scholar [23] X. B. Zhang, S. Huang and Z. Wan, Optimal pricing and ordering in global supply chain management with constraints under random demand, Applied Mathematical Modelling, 40 (2016), 10105-10130.  doi: 10.1016/j.apm.2016.06.054.  Google Scholar

show all references

##### References:
 [1] S. A. H. S. Amiri, A. Herbon, A. Zahedi, M. Kazemi, J. Soroor and M. Hajiaghaei-Keshteli, Determination of the optimal sales level of perishable goods in a two-echelon supply chain network, Computers & Industrial Engineering, 139 (2020), 147-156.   Google Scholar [2] T. Avinadav, A. Herbon and U. Spiegel, Optimal ordering and pricing policy for demand functions that are separable into price and inventory age, International Journal of Production Economics, 155 (2014), 406-417.   Google Scholar [3] H. J. Chang, J. T. Teng, L. Y. Ouyang and C. Y. Dye, Retailer's optimal pricing and lot-sizing policies for deteriorating items with partial backlogging, European Journal of Operational Research, 168 (2005), 51-64.  doi: 10.1016/j.ejor.2004.05.003.  Google Scholar [4] B. X. Chen, X. L. Chao and H. S. Ahn, Coordinating pricing and inventory replenishment with nonparametric demand learning, Operations Research, 67 (2019), 1035-1052.  doi: 10.1287/opre.2018.1808.  Google Scholar [5] X. Chen and P. Hu, Joint pricing and inventory management with deterministic demand and costly price adjust, Operations Research Letters, 40 (2012), 385-389.  doi: 10.1016/j.orl.2012.05.011.  Google Scholar [6] Y. H. Chen, S. B. Ray and Y. Y. Song, Optimal pricing and inventory control policy in periodic-review systems with fixed ordering cost and lost sales, Naval Research Logistics, 53 (2006), 117-136.  doi: 10.1002/nav.20127.  Google Scholar [7] W. M. Chung, S. Talluri and R. Narasimhan, Optimal pricing and inventory strategies with multiple price markdowns over time, European Journal of Operational Research, 243 (2015), 130-141.  doi: 10.1016/j.ejor.2014.11.020.  Google Scholar [8] O. C. Demirag, Y. H. Chen and Y. Yang, Ordering policies for periodic-review inventory systems with quantity-dependent fixed costs, Operations Research, 60 (2012), 785-796.  doi: 10.1287/opre.1110.1033.  Google Scholar [9] C. Y. Dye, Joint pricing and ordering policy for a deteriorating inventory with partial backlogging, Omega, 35 (2007), 184-189.   Google Scholar [10] O. Kaya and A. L. Polat, Coordinated pricing and inventory decisions for perishable products, OR Spectrum, 39 (2017), 589-606.  doi: 10.1007/s00291-016-0467-6.  Google Scholar [11] J. Li, S. Y. Wang and T. C. E. Cheng, Competition and cooperation in a single-retailer two-supplier supply chain with supply disruption, International Journal of Production Economics, 124 (2010), 137-150.   Google Scholar [12] R. H. Li, Y. L. Chan and C. T. Chang, Pricing and lot-sizing policies for perishable products with advance-cash-credit payments by a discounted cash-flow analysis, International Journal of Production Economics, 193 (2017), 578-589.   Google Scholar [13] B. Mondal, A. K. Bhunia and M. Maiti, An inventory system of ameliorating items for price dependent demand rate, Computers & Industrial Engineering, 45 (2003), 443-456.   Google Scholar [14] M. S. Sajadieh and M. R. A. Jokar, Optimizing shipment, ordering and pricing policies in a two-stage supply chain with price-sensitive demand, Transportation Research Part E, 45 (2009), 564-571.   Google Scholar [15] D. A. Serel, Optimal ordering and pricing in a quick response system, International Journal of Production Economics, 121 (2009), 700-714.   Google Scholar [16] J. M. Shi, G. Q. Zhang and K. K. Lai, Optimal ordering and pricing policy with supplier quantity discounts and price-dependent stochastic demand, Optimization, 61 (2012), 151-162.  doi: 10.1080/02331934.2011.590485.  Google Scholar [17] A. A. Taleizadeh, H. R. Zarei and B. R. Sarker, An optimal control of inventory under probablistic replenishment intervals and known price increase, European Journal of Operational Research, 257 (2017), 777-791.  doi: 10.1016/j.ejor.2016.07.041.  Google Scholar [18] S. Tiwari, L. E. Cardenas-Barron and M. Goh, Joint pricing and inventory model for deteriorating items with expiration dates and partial backlogging under two-level partial trade credits in supply chain, International Journal of Production Economics, 200 (2018), 16-36.   Google Scholar [19] W. Xing and S. Y. Wang, Optimal ordering and pricing strategies in the presence of a B2B spot market, European Journal of Operational Research, 221 (2012), 87-98.  doi: 10.1016/j.ejor.2012.03.017.  Google Scholar [20] S. L. Yang, C. M. Shi and X. Zhao, Optimal ordering and pricing decisions for a target oriented newsvendor, Omega, 39 (2011), 110-115.   Google Scholar [21] P. S. You and M. T. Wu, Optimal ordering and pricing policy for an inventory system with order cancellations, OR Spectrum, 29 (2007), 661-679.  doi: 10.1007/s00291-006-0067-y.  Google Scholar [22] P. S. You, S. Ikuta and Y. C. Hsieh, Optimal ordering and pricing policy for an inventory system with trial period, Applied Mathematical Modelling, 34 (2010), 3179-3188.  doi: 10.1016/j.apm.2010.02.008.  Google Scholar [23] X. B. Zhang, S. Huang and Z. Wan, Optimal pricing and ordering in global supply chain management with constraints under random demand, Applied Mathematical Modelling, 40 (2016), 10105-10130.  doi: 10.1016/j.apm.2016.06.054.  Google Scholar
Model parameters
 Parameters Definition [0, $T$] The given selling period $t_{i }$ The ordering time-point, where $t_{i} \in$[0, $T$] and $i \in M$, $M$ = {$1, \ldots, m$} $p_{b}$ The wholesale price determined by the manufacturer $p_{c}$ The retail price determined by the retailer $q$ The total procurement volume of the retailer $e$ The production cost per item of the manufacturer $m$ The number of times of ordering of the retailer $k$ The fixed ordering cost of each order $\lambda$ The linear price-sensitive coefficient of the demand rate $r(p_{c})$ The demand rate under price $p_{c}$: $r(p_{c}) = a$ – $\lambda p_{c}$, $a$ ¿ 0 $h$ The stock-holding cost per item per unit time $W$ The revenue of the manufacturer $Z$ The revenue of the retailer $S$ The total revenue incurred by centralized decision-making
 Parameters Definition [0, $T$] The given selling period $t_{i }$ The ordering time-point, where $t_{i} \in$[0, $T$] and $i \in M$, $M$ = {$1, \ldots, m$} $p_{b}$ The wholesale price determined by the manufacturer $p_{c}$ The retail price determined by the retailer $q$ The total procurement volume of the retailer $e$ The production cost per item of the manufacturer $m$ The number of times of ordering of the retailer $k$ The fixed ordering cost of each order $\lambda$ The linear price-sensitive coefficient of the demand rate $r(p_{c})$ The demand rate under price $p_{c}$: $r(p_{c}) = a$ – $\lambda p_{c}$, $a$ ¿ 0 $h$ The stock-holding cost per item per unit time $W$ The revenue of the manufacturer $Z$ The revenue of the retailer $S$ The total revenue incurred by centralized decision-making
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