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Optimal ordering and pricing models of a two-echelon supply chain under multipletimes ordering
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 |
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.
Keywords:
Dynamic Pricing,
Inventory Control,
Perishable Products,
Age-Dependent Demand,
Dynamic Programming.
Mathematics Subject Classification:
Primary: 90B05, 90C39; Secondary: 60J28.
Citation:
Onur Kaya, Halit Bayer. Pricing and lot-sizing decisions for perishable products when demand changes by freshness. Journal of Industrial and 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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [9] |
D. Bertsekas, Dynamic Programming and Optimal Control, , 2$^nd$ edition, Athena Scientific, 2001. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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.
|
| [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.
|
| [20] |
N. Ferguson and M. E. Ketzenberg,
Informations sharing to improve retail product freshness of perishables, Production and Operations Management, 15 (2006), 57-73.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [33] |
Y. Li, A. Lim and B. Rodrigues,
Pricing and inventory dontrol for a perishable product, Manufacturing & Service Operations Management, 11 (2009), 538-542.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [40] |
S. Nahmias,
Optimal ordering policies for perishable inventory, Operations Research, 23 (1975), 603-840.
doi: 10.1287/opre.23.4.735. |
| [41] |
S. Nahmias,
Perishable inventory theory: A review, Operations Research, 30 (1982), 601-796.
doi: 10.1287/opre.30.4.680. |
| [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. |
| [43] |
S. M. Ross, Introduction to Probability Models, 9$^{th}$ edition, Academic Press Inc., Orlando, FL, USA, 1980. |
| [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. |
| [45] |
A. Sen,
Competitive markdown timing for perishable and substitutable products, Omega, 64 (2016), 24-41.
|
| [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. |
| [47] |
G. Van Zyl, Inventory Control for Perishable Commodities, Ph.D thesis, University of North-Carolina, Chapel Hill, NC, 1964. |
| [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. |
| [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. |
| [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. |
| [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. |
| [52] |
P.-S. You,
Inventory policy for products with price and time-dependent demands, Journal of the Operational Research Society, 56 (2005), 870-873.
|
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [9] |
D. Bertsekas, Dynamic Programming and Optimal Control, , 2$^nd$ edition, Athena Scientific, 2001. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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.
|
| [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.
|
| [20] |
N. Ferguson and M. E. Ketzenberg,
Informations sharing to improve retail product freshness of perishables, Production and Operations Management, 15 (2006), 57-73.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [33] |
Y. Li, A. Lim and B. Rodrigues,
Pricing and inventory dontrol for a perishable product, Manufacturing & Service Operations Management, 11 (2009), 538-542.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [40] |
S. Nahmias,
Optimal ordering policies for perishable inventory, Operations Research, 23 (1975), 603-840.
doi: 10.1287/opre.23.4.735. |
| [41] |
S. Nahmias,
Perishable inventory theory: A review, Operations Research, 30 (1982), 601-796.
doi: 10.1287/opre.30.4.680. |
| [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. |
| [43] |
S. M. Ross, Introduction to Probability Models, 9$^{th}$ edition, Academic Press Inc., Orlando, FL, USA, 1980. |
| [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. |
| [45] |
A. Sen,
Competitive markdown timing for perishable and substitutable products, Omega, 64 (2016), 24-41.
|
| [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. |
| [47] |
G. Van Zyl, Inventory Control for Perishable Commodities, Ph.D thesis, University of North-Carolina, Chapel Hill, NC, 1964. |
| [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. |
| [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. |
| [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. |
| [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. |
| [52] |
P.-S. You,
Inventory policy for products with price and time-dependent demands, Journal of the Operational Research Society, 56 (2005), 870-873.
|
| 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) | ... |
| ... | ... | ... | ... | ... | ... | ... | ... |
Table 2.
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 |
Table 3.
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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