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November  2021, 17(6): 3113-3129. doi: 10.3934/jimo.2020110

## Pricing and lot-sizing decisions for perishable products when demand changes by freshness

 1 Eskisehir Technical University, Department of Industrial Engineering, Eskisehir, Turkey 2 Koc University, Department of Industrial Engineering, Istanbul, Turkey

* Corresponding author: Onur Kaya, onur_kaya@eskisehir.edu.tr

Received  November 2019 Revised  March 2020 Published  November 2021 Early access  June 2020

Perishable products like dairy products, vegetables, fruits, pharmaceuticals, etc. lose their freshness over time and become completely obsolete after a certain period. Customers generally prefer the fresh products over aged ones, leading the perishable products to have a decreasing demand function with respect to their age. We analyze the inventory management and pricing decisions for these products, considering an age-and-price-dependent stochastic demand function. A stochastic dynamic programming model is developed in order to decide when and how much inventory to order and how to price these products considering their freshness over time. We prove the characteristics of the optimal solution of the developed model and extract managerial insights regarding the optimal inventory and pricing strategies. The numerical studies show that dynamic pricing can lead to significant savings over static pricing under certain parameter settings. In addition, longer replenishment cycles are seen under dynamic pricing compared to static pricing, even though similar quantities are ordered in each replenishment.

Citation: Onur Kaya, Halit Bayer. Pricing and lot-sizing decisions for perishable products when demand changes by freshness. Journal of Industrial & Management Optimization, 2021, 17 (6) : 3113-3129. doi: 10.3934/jimo.2020110
##### References:
 [1] P. L. Abad, Optimal pricing and lot sizing under conditions of perishability and partial backordering, Management Science, 42 (1996), 1093-1227.  doi: 10.1287/mnsc.42.8.1093.  Google Scholar [2] P. L. Abad, Optimal policy for a reseller when the supplier offers a temporary reduction in price, Decision Sciences, 28 (1997), 637-653.  doi: 10.1111/j.1540-5915.1997.tb01325.x.  Google Scholar [3] P. L. Abad, Optimal price and order size for a reseller under partial backordering, Computers and Operations Research, 28 (2001), 53-65.  doi: 10.1016/S0305-0548(99)00086-6.  Google Scholar [4] P. L. Abad, Optimal pricing and lot sizing under conditions of perishability, finite production and partial backordering and lost sale, European Journal of Operational Research, 144 (2003), 677-685.  doi: 10.1016/S0377-2217(02)00159-5.  Google Scholar [5] B. Adenso-Diaz, S. Lozano and A. Palacio, Effects of dynamic pricing of perishable products on revenue and waste, Applied Mathematical Modelling, 45 (2017), 148-164.  doi: 10.1016/j.apm.2016.12.024.  Google Scholar [6] M. Bakker, J. Riezebos and R. H. Teunter, Review of inventory systems with deterioration since 2001., European Journal Operational Research, 221 (2012), 275-284.  doi: 10.1016/j.ejor.2012.03.004.  Google Scholar [7] L. Benkherouf, On an inventory model with deteriorating items and decreasing time-varying demand and shortages, European Journal of Operational Research, 86 (1995), 293-299.  doi: 10.1016/0377-2217(94)00101-H.  Google Scholar [8] E. Berk and Ü. Gürler, Analysis of the (Q, r) inventory model for perishables with positive lead times and lost sales, Operations Research, 56 (2008), 1238-1246.  doi: 10.1287/opre.1080.0582.  Google Scholar [9] D. Bertsekas, Dynamic Programming and Optimal Control, , 2$^nd$ edition, Athena Scientific, 2001.  Google Scholar [10] R. A. C. M. Broekmeulen and K. H. van Donselaar, A heuristic to manage perishable inventory with batch ordering, positive lead-times, and time-varying demand, Computers and Operations Research, 36 (2009), 3013-3018.  doi: 10.1016/j.cor.2009.01.017.  Google Scholar [11] V. Chaudhary, R. Kulshrestha and S. Routroy, State-of-the-art literature review on inventory models for perishable products, Journal of Advances in Management Research, 15 (2018), 306-346.  doi: 10.1108/JAMR-09-2017-0091.  Google Scholar [12] L. M. Chen and A. Sapra, Joint inventory and pricing decisions for perishable products with two-period lifetime, Naval Research Logistics, 60 (2013), 343-366.  doi: 10.1002/nav.21538.  Google Scholar [13] X. Chen, Z. Pang and L. Pan, Coordinating inventory control and pricing strategies for perishable products, Operations Research, 62 (2014), 284-300.  doi: 10.1287/opre.2014.1261.  Google Scholar [14] E. Chew Peng and L. Chulung, Joint inventory allocation and pricing decisions for perishable products, International Journal of Production Economics, 120 (2009), 139-150.   Google Scholar [15] P. Chintapalli, Simultaneous pricing and inventory management of deteriorating perishable products, Annals of Operations Research, 229 (2015), 287-301.  doi: 10.1007/s10479-014-1753-9.  Google Scholar [16] G. A. Chua, R. Mokhlesi and A. Sainathan, Optimal discounting and replenishment policies for perishable products, International Journal of Production Economics, 186 (2017), 8-20.  doi: 10.1016/j.ijpe.2017.01.016.  Google Scholar [17] Y. Duan, G. Li, J. M. Tien and J. Hu, Inventory models for perishable items with inventory level dependent demand rate, Applied Mathematical Modelling, 36) (2012), 5015-5028.  doi: 10.1016/j.apm.2011.12.039.  Google Scholar [18] W. Elmaghraby and P. Keskinocak, Dynamic pricing in the presence of inventory considerations: Research overview, current practices, and future directions, Management Science, 49 (2003), 1287-1309.   Google Scholar [19] L. Feng, Y.-L. Chan and L. E. Cárdenas-Barrón, Pricing and lot-sizing polices for perishable goods when the demand depends on selling price, displayed stocks, and expiration date, International Journal of Production Economics, 185 (2017), 11-20.   Google Scholar [20] N. Ferguson and M. E. Ketzenberg, Informations sharing to improve retail product freshness of perishables, Production and Operations Management, 15 (2006), 57-73.   Google Scholar [21] S. K. Ghosh, S. Khanra and K. S. Chaudhuri, Optimal price and lot size determination for a Perishable product under conditions of finite production, partial backordering and lost sale, Applied Mathematics and Computation, 217 (2011), 6047-6053.  doi: 10.1016/j.amc.2010.12.050.  Google Scholar [22] R. Haijema, Optimal ordering, issuance and disposal policies for inventory management of perishable products, International Journal of Production Economics, 157 (2014), 158-169.  doi: 10.1016/j.ijpe.2014.06.014.  Google Scholar [23] R. Haijema and S. Minner, Stock-level dependent ordering of perishables: A comparison of hybrid base-stock and constant order policies, International Journal of Production Economics, 181 (2016), 215-225.  doi: 10.1016/j.ijpe.2015.10.013.  Google Scholar [24] R. Haijema and S. Minner, Improved ordering of perishables: The value of stock-age information., International Journal of Production Economics, 209 (2019), 316-324.  doi: 10.1016/j.ijpe.2018.03.008.  Google Scholar [25] L. Hengyu, J. Zhang, C. Zhou and Y. Ru, Optimal purchase and inventory retrieval policies for perishable seasonal agricultural products., Omega, 79 (2018), 133-145.   Google Scholar [26] A. Herbon, Should retailers hold a perishable product having different ages? the case of a homogeneous market and multiplicative demand model, International Journal of Production Economics, 193 (2017), 479-490.  doi: 10.1016/j.ijpe.2017.08.008.  Google Scholar [27] A. Herbon and E. Khmelnitsky, Optimal dynamic pricing and ordering of a perishable product under additive effects of price and time on demand, European Journal of Operational Research, 260 (2017), 546-556.  doi: 10.1016/j.ejor.2016.12.033.  Google Scholar [28] A. Herbon, E. Levner and T. C. E. Cheng, Perishable inventory management with dynamic pricing using time-temperature indicators linked to automatic detecting devices, International Journal of Production Economics, 147 (2014), 605-613.  doi: 10.1016/j.ijpe.2013.07.021.  Google Scholar [29] L. Janssen, T. Claus and J. Sauser, Literature review of deteriorating inventory models by key topics from 2012–2015., International Journal of Production Economics, 182 (2016), 86-112.  doi: 10.1016/j.ijpe.2016.08.019.  Google Scholar [30] F. Jing, Z. Lan and Y. Pan, Forecast horizon of dynamic lot size model for perishable inventory with minimum order quantities, Journal of Industrial & Management Optimization, 16 (2020), 1435-1456.  doi: 10.3934/jimo.2019010.  Google Scholar [31] I. Karaesmen, A. Scheller-Wolf and B. Deniz, Managing perishable and aging inventories: Review and future research directions, in Planning Production and Inventories in the Extended Enterprise (eds. K. Kempf, P. Keskinocak and R. Uzsoy), Springer, Boston, MA, (2010), 393–436. doi: 10.1007/978-1-4419-6485-4_15.  Google Scholar [32] O. Kaya and S. R. Ghahroodi, Inventory control and pricing for perishable products under age and price dependent stochastic demand, Mathematical Methods of Operations Research, 88 (2018), 1-35.  doi: 10.1007/s00186-017-0626-9.  Google Scholar [33] Y. Li, A. Lim and B. Rodrigues, Pricing and inventory dontrol for a perishable product, Manufacturing & Service Operations Management, 11 (2009), 538-542.   Google Scholar [34] Y. Li, B. Cheang and A. Lim, Grocery perishables management, Production and Operations Management, 21 (2012), 504-517.  doi: 10.1111/j.1937-5956.2011.01288.x.  Google Scholar [35] R. Li and J.-T. Teng, Pricing and lot-sizing decisions for perishable goods when demand depends on selling price, reference price, product freshness, and displayed stocks, European Journal of Operational Research, 270 (2018), 1099-1108.  doi: 10.1016/j.ejor.2018.04.029.  Google Scholar [36] G. Liu, J. Zhang and W. Tang, Joint dynamic pricing and investment strategy for perishable foods with price-quality dependent demand, Annals of Operations Research, 226 (2015), 397-416.  doi: 10.1007/s10479-014-1671-x.  Google Scholar [37] S. Minner and S. Transchel, Periodic review inventory-control for perishable products under service-level constraints, OR Spectrum, 32 (2010), 979-996.  doi: 10.1007/s00291-010-0196-1.  Google Scholar [38] V. K. Mishra and S. L. Singh, Deteriorating inventory model with time dependent demand and partial backlogging, Applied Mathematical Sciences, 4 (2010), 3611-3619.   Google Scholar [39] S. Mukhopadhyay, R. N. Mukherjeea and K. S. Chaudhuri, Joint pricing and ordering policy for a deteriorating inventory, Computers and Industrial Engineering, 47 (2004), 339-349.  doi: 10.1016/j.cie.2004.06.007.  Google Scholar [40] S. Nahmias, Optimal ordering policies for perishable inventory, Operations Research, 23 (1975), 603-840.  doi: 10.1287/opre.23.4.735.  Google Scholar [41] S. Nahmias, Perishable inventory theory: A review, Operations Research, 30 (1982), 601-796.  doi: 10.1287/opre.30.4.680.  Google Scholar [42] R. A. Rajan and R. Steinberg, Dynamic pricing and ordering decisions by a monopolist, Management Science, 38 (1992), 157-305.  doi: 10.1287/mnsc.38.2.240.  Google Scholar [43] S. M. Ross, Introduction to Probability Models, 9$^{th}$ edition, Academic Press Inc., Orlando, FL, USA, 1980.  Google Scholar [44] S. S. Sana, Optimal selling price and lot-size with time varying deterioration and partial backlogging, Applied Mathematics and Computation, 217 (2010), 185-194.  doi: 10.1016/j.amc.2010.05.040.  Google Scholar [45] A. Sen, Competitive markdown timing for perishable and substitutable products, Omega, 64 (2016), 24-41.   Google Scholar [46] S. Transchel and S. Minner, The impact of dynamic pricing on the economic order decision, European Journal of Operational Research, 198 (2009), 773-789.  doi: 10.1016/j.ejor.2008.10.011.  Google Scholar [47] G. Van Zyl, Inventory Control for Perishable Commodities, Ph.D thesis, University of North-Carolina, Chapel Hill, NC, 1964.  Google Scholar [48] X. Wang and D. Li, A dynamic product quality evaluation based pricing model for perishable food supply chains, Omega, 40 (2012), 906-917.  doi: 10.1016/j.omega.2012.02.001.  Google Scholar [49] K-J. Wang and Y-S. Lin, Optimal inventory replenishment strategy for deteriorating items in a demand-declining market with the retailer's price manipulation, Annals of Operations Research, 201 (2012), 475-494.  doi: 10.1007/s10479-012-1213-3.  Google Scholar [50] H. M. Wee, Deteriorating inventory model with quantity discount, pricing and partial backordering, International Journal of Production Economics, 59 (1999), 511-518.  doi: 10.1016/S0925-5273(98)00113-3.  Google Scholar [51] X. Xu and X. Cai, Price and delivery-time competition of perishable products: Existence and uniqueness of Nash equilibrium, Journal of Industrial & Management Optimization, 4 (2008), 843-859.  doi: 10.3934/jimo.2008.4.843.  Google Scholar [52] P.-S. You, Inventory policy for products with price and time-dependent demands, Journal of the Operational Research Society, 56 (2005), 870-873.   Google Scholar

show all references

##### References:
 [1] P. L. Abad, Optimal pricing and lot sizing under conditions of perishability and partial backordering, Management Science, 42 (1996), 1093-1227.  doi: 10.1287/mnsc.42.8.1093.  Google Scholar [2] P. L. Abad, Optimal policy for a reseller when the supplier offers a temporary reduction in price, Decision Sciences, 28 (1997), 637-653.  doi: 10.1111/j.1540-5915.1997.tb01325.x.  Google Scholar [3] P. L. Abad, Optimal price and order size for a reseller under partial backordering, Computers and Operations Research, 28 (2001), 53-65.  doi: 10.1016/S0305-0548(99)00086-6.  Google Scholar [4] P. L. Abad, Optimal pricing and lot sizing under conditions of perishability, finite production and partial backordering and lost sale, European Journal of Operational Research, 144 (2003), 677-685.  doi: 10.1016/S0377-2217(02)00159-5.  Google Scholar [5] B. Adenso-Diaz, S. Lozano and A. Palacio, Effects of dynamic pricing of perishable products on revenue and waste, Applied Mathematical Modelling, 45 (2017), 148-164.  doi: 10.1016/j.apm.2016.12.024.  Google Scholar [6] M. Bakker, J. Riezebos and R. H. Teunter, Review of inventory systems with deterioration since 2001., European Journal Operational Research, 221 (2012), 275-284.  doi: 10.1016/j.ejor.2012.03.004.  Google Scholar [7] L. Benkherouf, On an inventory model with deteriorating items and decreasing time-varying demand and shortages, European Journal of Operational Research, 86 (1995), 293-299.  doi: 10.1016/0377-2217(94)00101-H.  Google Scholar [8] E. Berk and Ü. Gürler, Analysis of the (Q, r) inventory model for perishables with positive lead times and lost sales, Operations Research, 56 (2008), 1238-1246.  doi: 10.1287/opre.1080.0582.  Google Scholar [9] D. Bertsekas, Dynamic Programming and Optimal Control, , 2$^nd$ edition, Athena Scientific, 2001.  Google Scholar [10] R. A. C. M. Broekmeulen and K. H. van Donselaar, A heuristic to manage perishable inventory with batch ordering, positive lead-times, and time-varying demand, Computers and Operations Research, 36 (2009), 3013-3018.  doi: 10.1016/j.cor.2009.01.017.  Google Scholar [11] V. Chaudhary, R. Kulshrestha and S. Routroy, State-of-the-art literature review on inventory models for perishable products, Journal of Advances in Management Research, 15 (2018), 306-346.  doi: 10.1108/JAMR-09-2017-0091.  Google Scholar [12] L. M. Chen and A. Sapra, Joint inventory and pricing decisions for perishable products with two-period lifetime, Naval Research Logistics, 60 (2013), 343-366.  doi: 10.1002/nav.21538.  Google Scholar [13] X. Chen, Z. Pang and L. Pan, Coordinating inventory control and pricing strategies for perishable products, Operations Research, 62 (2014), 284-300.  doi: 10.1287/opre.2014.1261.  Google Scholar [14] E. Chew Peng and L. Chulung, Joint inventory allocation and pricing decisions for perishable products, International Journal of Production Economics, 120 (2009), 139-150.   Google Scholar [15] P. Chintapalli, Simultaneous pricing and inventory management of deteriorating perishable products, Annals of Operations Research, 229 (2015), 287-301.  doi: 10.1007/s10479-014-1753-9.  Google Scholar [16] G. A. Chua, R. Mokhlesi and A. Sainathan, Optimal discounting and replenishment policies for perishable products, International Journal of Production Economics, 186 (2017), 8-20.  doi: 10.1016/j.ijpe.2017.01.016.  Google Scholar [17] Y. Duan, G. Li, J. M. Tien and J. Hu, Inventory models for perishable items with inventory level dependent demand rate, Applied Mathematical Modelling, 36) (2012), 5015-5028.  doi: 10.1016/j.apm.2011.12.039.  Google Scholar [18] W. Elmaghraby and P. Keskinocak, Dynamic pricing in the presence of inventory considerations: Research overview, current practices, and future directions, Management Science, 49 (2003), 1287-1309.   Google Scholar [19] L. Feng, Y.-L. Chan and L. E. Cárdenas-Barrón, Pricing and lot-sizing polices for perishable goods when the demand depends on selling price, displayed stocks, and expiration date, International Journal of Production Economics, 185 (2017), 11-20.   Google Scholar [20] N. Ferguson and M. E. Ketzenberg, Informations sharing to improve retail product freshness of perishables, Production and Operations Management, 15 (2006), 57-73.   Google Scholar [21] S. K. Ghosh, S. Khanra and K. S. Chaudhuri, Optimal price and lot size determination for a Perishable product under conditions of finite production, partial backordering and lost sale, Applied Mathematics and Computation, 217 (2011), 6047-6053.  doi: 10.1016/j.amc.2010.12.050.  Google Scholar [22] R. Haijema, Optimal ordering, issuance and disposal policies for inventory management of perishable products, International Journal of Production Economics, 157 (2014), 158-169.  doi: 10.1016/j.ijpe.2014.06.014.  Google Scholar [23] R. Haijema and S. Minner, Stock-level dependent ordering of perishables: A comparison of hybrid base-stock and constant order policies, International Journal of Production Economics, 181 (2016), 215-225.  doi: 10.1016/j.ijpe.2015.10.013.  Google Scholar [24] R. Haijema and S. Minner, Improved ordering of perishables: The value of stock-age information., International Journal of Production Economics, 209 (2019), 316-324.  doi: 10.1016/j.ijpe.2018.03.008.  Google Scholar [25] L. Hengyu, J. Zhang, C. Zhou and Y. Ru, Optimal purchase and inventory retrieval policies for perishable seasonal agricultural products., Omega, 79 (2018), 133-145.   Google Scholar [26] A. Herbon, Should retailers hold a perishable product having different ages? the case of a homogeneous market and multiplicative demand model, International Journal of Production Economics, 193 (2017), 479-490.  doi: 10.1016/j.ijpe.2017.08.008.  Google Scholar [27] A. Herbon and E. Khmelnitsky, Optimal dynamic pricing and ordering of a perishable product under additive effects of price and time on demand, European Journal of Operational Research, 260 (2017), 546-556.  doi: 10.1016/j.ejor.2016.12.033.  Google Scholar [28] A. Herbon, E. Levner and T. C. E. Cheng, Perishable inventory management with dynamic pricing using time-temperature indicators linked to automatic detecting devices, International Journal of Production Economics, 147 (2014), 605-613.  doi: 10.1016/j.ijpe.2013.07.021.  Google Scholar [29] L. Janssen, T. Claus and J. Sauser, Literature review of deteriorating inventory models by key topics from 2012–2015., International Journal of Production Economics, 182 (2016), 86-112.  doi: 10.1016/j.ijpe.2016.08.019.  Google Scholar [30] F. Jing, Z. Lan and Y. Pan, Forecast horizon of dynamic lot size model for perishable inventory with minimum order quantities, Journal of Industrial & Management Optimization, 16 (2020), 1435-1456.  doi: 10.3934/jimo.2019010.  Google Scholar [31] I. Karaesmen, A. Scheller-Wolf and B. Deniz, Managing perishable and aging inventories: Review and future research directions, in Planning Production and Inventories in the Extended Enterprise (eds. K. Kempf, P. Keskinocak and R. Uzsoy), Springer, Boston, MA, (2010), 393–436. doi: 10.1007/978-1-4419-6485-4_15.  Google Scholar [32] O. Kaya and S. R. Ghahroodi, Inventory control and pricing for perishable products under age and price dependent stochastic demand, Mathematical Methods of Operations Research, 88 (2018), 1-35.  doi: 10.1007/s00186-017-0626-9.  Google Scholar [33] Y. Li, A. Lim and B. Rodrigues, Pricing and inventory dontrol for a perishable product, Manufacturing & Service Operations Management, 11 (2009), 538-542.   Google Scholar [34] Y. Li, B. Cheang and A. Lim, Grocery perishables management, Production and Operations Management, 21 (2012), 504-517.  doi: 10.1111/j.1937-5956.2011.01288.x.  Google Scholar [35] R. Li and J.-T. Teng, Pricing and lot-sizing decisions for perishable goods when demand depends on selling price, reference price, product freshness, and displayed stocks, European Journal of Operational Research, 270 (2018), 1099-1108.  doi: 10.1016/j.ejor.2018.04.029.  Google Scholar [36] G. Liu, J. Zhang and W. Tang, Joint dynamic pricing and investment strategy for perishable foods with price-quality dependent demand, Annals of Operations Research, 226 (2015), 397-416.  doi: 10.1007/s10479-014-1671-x.  Google Scholar [37] S. Minner and S. Transchel, Periodic review inventory-control for perishable products under service-level constraints, OR Spectrum, 32 (2010), 979-996.  doi: 10.1007/s00291-010-0196-1.  Google Scholar [38] V. K. Mishra and S. L. Singh, Deteriorating inventory model with time dependent demand and partial backlogging, Applied Mathematical Sciences, 4 (2010), 3611-3619.   Google Scholar [39] S. Mukhopadhyay, R. N. Mukherjeea and K. S. Chaudhuri, Joint pricing and ordering policy for a deteriorating inventory, Computers and Industrial Engineering, 47 (2004), 339-349.  doi: 10.1016/j.cie.2004.06.007.  Google Scholar [40] S. Nahmias, Optimal ordering policies for perishable inventory, Operations Research, 23 (1975), 603-840.  doi: 10.1287/opre.23.4.735.  Google Scholar [41] S. Nahmias, Perishable inventory theory: A review, Operations Research, 30 (1982), 601-796.  doi: 10.1287/opre.30.4.680.  Google Scholar [42] R. A. Rajan and R. Steinberg, Dynamic pricing and ordering decisions by a monopolist, Management Science, 38 (1992), 157-305.  doi: 10.1287/mnsc.38.2.240.  Google Scholar [43] S. M. Ross, Introduction to Probability Models, 9$^{th}$ edition, Academic Press Inc., Orlando, FL, USA, 1980.  Google Scholar [44] S. S. Sana, Optimal selling price and lot-size with time varying deterioration and partial backlogging, Applied Mathematics and Computation, 217 (2010), 185-194.  doi: 10.1016/j.amc.2010.05.040.  Google Scholar [45] A. Sen, Competitive markdown timing for perishable and substitutable products, Omega, 64 (2016), 24-41.   Google Scholar [46] S. Transchel and S. Minner, The impact of dynamic pricing on the economic order decision, European Journal of Operational Research, 198 (2009), 773-789.  doi: 10.1016/j.ejor.2008.10.011.  Google Scholar [47] G. Van Zyl, Inventory Control for Perishable Commodities, Ph.D thesis, University of North-Carolina, Chapel Hill, NC, 1964.  Google Scholar [48] X. Wang and D. Li, A dynamic product quality evaluation based pricing model for perishable food supply chains, Omega, 40 (2012), 906-917.  doi: 10.1016/j.omega.2012.02.001.  Google Scholar [49] K-J. Wang and Y-S. Lin, Optimal inventory replenishment strategy for deteriorating items in a demand-declining market with the retailer's price manipulation, Annals of Operations Research, 201 (2012), 475-494.  doi: 10.1007/s10479-012-1213-3.  Google Scholar [50] H. M. Wee, Deteriorating inventory model with quantity discount, pricing and partial backordering, International Journal of Production Economics, 59 (1999), 511-518.  doi: 10.1016/S0925-5273(98)00113-3.  Google Scholar [51] X. Xu and X. Cai, Price and delivery-time competition of perishable products: Existence and uniqueness of Nash equilibrium, Journal of Industrial & Management Optimization, 4 (2008), 843-859.  doi: 10.3934/jimo.2008.4.843.  Google Scholar [52] P.-S. You, Inventory policy for products with price and time-dependent demands, Journal of the Operational Research Society, 56 (2005), 870-873.   Google Scholar
Optimal Ordering and Pricing Decisions
 q/t 1 2 ... 148 149 150 ... 0 30 (5.49) 30 (5.49) ... 30 (5.49) 30 (5.49) 30 (5.49) ... 1 0 (5.82) 0 (5.81) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 2 0 (5.81) 0 (5.80) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 3 0 (5.80) 0 (5.79) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 4 0 (5.78) 0 (5.77) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 5 0 (5.77) 0 (5.76) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 6 0 (5.76) 0 (5.75) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 7 0 (5.75) 0 (5.74) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 8 0 (5.74) 0 (5.73) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 9 0 (5.73) 0 (5.72) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 10 0 (5.71) 0 (5.70) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 11 0 (5.70) 0 (5.69) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 12 0 (5.69) 0 (5.68) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 13 0 (5.68) 0 (5.67) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 14 0 (5.67) 0 (5.66) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 15 0 (5.66) 0 (5.65) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 16 0 (5.65) 0 (5.64) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 17 0 (5.63) 0 (5.62) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 18 0 (5.62) 0 (5.61) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 19 0 (5.61) 0 (5.60) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 20 0 (5.60) 0 (5.59) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 21 0 (5.59) 0 (5.58) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 22 0 (5.58) 0 (5.57) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 23 0 (5.57) 0 (5.56) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 24 0 (5.55) 0 (5.54) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 25 0 (5.54) 0 (5.53) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 26 0 (5.53) 0 (5.52) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 27 0 (5.52) 0 (5.51) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 28 0 (5.51) 0 (5.50) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 29 0 (5.50) 0 (5.49) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 30 0 (5.49) 0 (5.48) ... 0 (4.36) 30 (5.49) 30 (5.49) ... ... ... ... ... ... ... ... ...
 q/t 1 2 ... 148 149 150 ... 0 30 (5.49) 30 (5.49) ... 30 (5.49) 30 (5.49) 30 (5.49) ... 1 0 (5.82) 0 (5.81) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 2 0 (5.81) 0 (5.80) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 3 0 (5.80) 0 (5.79) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 4 0 (5.78) 0 (5.77) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 5 0 (5.77) 0 (5.76) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 6 0 (5.76) 0 (5.75) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 7 0 (5.75) 0 (5.74) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 8 0 (5.74) 0 (5.73) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 9 0 (5.73) 0 (5.72) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 10 0 (5.71) 0 (5.70) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 11 0 (5.70) 0 (5.69) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 12 0 (5.69) 0 (5.68) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 13 0 (5.68) 0 (5.67) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 14 0 (5.67) 0 (5.66) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 15 0 (5.66) 0 (5.65) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 16 0 (5.65) 0 (5.64) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 17 0 (5.63) 0 (5.62) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 18 0 (5.62) 0 (5.61) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 19 0 (5.61) 0 (5.60) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 20 0 (5.60) 0 (5.59) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 21 0 (5.59) 0 (5.58) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 22 0 (5.58) 0 (5.57) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 23 0 (5.57) 0 (5.56) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 24 0 (5.55) 0 (5.54) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 25 0 (5.54) 0 (5.53) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 26 0 (5.53) 0 (5.52) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 27 0 (5.52) 0 (5.51) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 28 0 (5.51) 0 (5.50) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 29 0 (5.50) 0 (5.49) ... 0 (4.36) 30 (5.49) 30 (5.49) ... 30 0 (5.49) 0 (5.48) ... 0 (4.36) 30 (5.49) 30 (5.49) ... ... ... ... ... ... ... ... ...
Dynamic Price Model Results for Varying Parameters
 Unit Fixed Freshness Price Order Order Initial Expected cost(c) cost(A) Sens.(k) Sens.(b) Time Quantity Price Profit 1 10 0.001 0.1 149 30 5.49 1.7283 0.2 266 39 3.00 0.6037 0.002 0.1 90 21 5.49 1.6067 0.2 157 27 3.00 0.5246 20 0.001 0.1 179 42 5.49 1.6080 0.2 312 55 3.00 0.5251 0.002 0.1 111 29 5.50 1.4384 0.2 192 38 3.00 0.4171 2 10 0.001 0.1 254 28 5.99 1.3204 0.2 486 33 3.50 0.2817 0.002 0.1 143 20 5.98 1.2064 0.2 273 23 3.50 0.2148 20 0.001 0.1 286 39 6.00 1.2071 0.2 545 47 3.50 0.2160 0.002 0.1 167 27 6.00 1.0495 0.2 321 32 3.50 0.1273
 Unit Fixed Freshness Price Order Order Initial Expected cost(c) cost(A) Sens.(k) Sens.(b) Time Quantity Price Profit 1 10 0.001 0.1 149 30 5.49 1.7283 0.2 266 39 3.00 0.6037 0.002 0.1 90 21 5.49 1.6067 0.2 157 27 3.00 0.5246 20 0.001 0.1 179 42 5.49 1.6080 0.2 312 55 3.00 0.5251 0.002 0.1 111 29 5.50 1.4384 0.2 192 38 3.00 0.4171 2 10 0.001 0.1 254 28 5.99 1.3204 0.2 486 33 3.50 0.2817 0.002 0.1 143 20 5.98 1.2064 0.2 273 23 3.50 0.2148 20 0.001 0.1 286 39 6.00 1.2071 0.2 545 47 3.50 0.2160 0.002 0.1 167 27 6.00 1.0495 0.2 321 32 3.50 0.1273
Static Price Model Results for Varying Parameters
 Unit Fixed Freshness Price Order Order Best Expected Percent cost cost Sens. Sens. Time Quantity Price Profit Diff. 1 10 0.001 0.1 133 30 5.28 1.7269 0.081 0.2 202 39 2.86 0.6024 0.215 0.002 0.1 80 21 5.19 1.6037 0.187 0.2 119 26 2.83 0.5205 0.782 20 0.001 0.1 157 41 5.27 1.6053 0.168 0.2 237 53 2.82 0.5224 0.514 0.002 0.1 99 29 5.11 1.4329 0.382 0.2 147 37 2.72 0.4085 2.062 2 10 0.001 0.1 186 28 5.78 1.3187 0.129 0.2 252 34 3.29 0.2786 1.101 0.002 0.1 105 19 5.73 1.2018 0.381 0.2 144 22 3.23 0.2050 4.562 20 0.001 0.1 207 38 5.77 1.2041 0.249 0.2 282 45 3.25 0.2101 2.731 0.002 0.1 124 27 5.61 1.0407 0.838 0.2 170 30 3.12 0.1056 17.046
 Unit Fixed Freshness Price Order Order Best Expected Percent cost cost Sens. Sens. Time Quantity Price Profit Diff. 1 10 0.001 0.1 133 30 5.28 1.7269 0.081 0.2 202 39 2.86 0.6024 0.215 0.002 0.1 80 21 5.19 1.6037 0.187 0.2 119 26 2.83 0.5205 0.782 20 0.001 0.1 157 41 5.27 1.6053 0.168 0.2 237 53 2.82 0.5224 0.514 0.002 0.1 99 29 5.11 1.4329 0.382 0.2 147 37 2.72 0.4085 2.062 2 10 0.001 0.1 186 28 5.78 1.3187 0.129 0.2 252 34 3.29 0.2786 1.101 0.002 0.1 105 19 5.73 1.2018 0.381 0.2 144 22 3.23 0.2050 4.562 20 0.001 0.1 207 38 5.77 1.2041 0.249 0.2 282 45 3.25 0.2101 2.731 0.002 0.1 124 27 5.61 1.0407 0.838 0.2 170 30 3.12 0.1056 17.046
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