doi: 10.3934/jimo.2021233
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The effect of rebate value and selling price-dependent demand for a four-level production manufacturing system

1. 

Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, 632014, India

2. 

Department of Mathematics, Hajee Mohammad Danesh Science and Technology University, Dinajpur-5200, Bangladesh

3. 

School of Engineering and IT, University of New South Wales (UNSW), Canberra, Australia

4. 

Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore-721102, WB, India

5. 

Department of Industrial Engineering, Prince Sattam bin Abdulaziz University, Alkharj, KSA, 16273

6. 

Department of Mathematics, Jahangirnagar University, Savar, Dhaka-1342, Bangladesh

*Corresponding author: Abu Hashan Md Mashud

Received  March 2021 Revised  September 2021 Early access January 2022

Price rebate is only permitted when purchases made by the customer exceed a predefined limit and they later buy other items from the purchaser. There are various forms of rebate used by production companies. This study provides a deteriorating inventory model of four-level production rates and derives the rebate-value-based demand with the product selling price under shortages. This model gives preference to optimal replenishment time, ordering quantity, rebate value, and selling price while maximizing total profit. This model first explores and discusses the demand function, which discretely hinges on the selling price of rebate value, followed by discussions on demand based on the selling price. This study proposes a solution through unique propositions and the construction of two algorithms that are suitable for four-level production; this has not yet been explored in-depth in the literature. Illustrative examples and a sensitivity analysis demonstrate the applicability of the proposed algorithms; the customer decides to buy a product that is larger than the minimum suitable for a price rebate and the buyer can then deal with a higher price rebate. The benefit of rebate marketing helps production companies increases conversion rates and encourages customers to purchase goods. This model demonstrates that proposing rebates can consume substantial pricing and inventory inferences and can result in a substantial increase in profit.

Citation: Umakanta Mishra, Abu Hashan Md Mashud, Sankar Kumar Roy, Md Sharif Uddin. The effect of rebate value and selling price-dependent demand for a four-level production manufacturing system. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021233
References:
[1]

P. L. Abad, Optimal pricing and lot sizing under conditions of perishability and partial backordering, Management Science, 42 (1996), 1093-1104.  doi: 10.1287/mnsc.42.8.1093.

[2]

P. L. Abad and C. K. Jaggi, A joint approach for setting unit price and the length of the credit period for a seller when end demand is price sensitive, International Journal of Production Economics, 83 (2003), 115-122.  doi: 10.1016/S0925-5273(02)00142-1.

[3]

A. AbdulA. Jolson Marin and Y. Darmon Reney, A model for optimizing the refund value in rebate promotions, Journal of Business Research, 29 (1994), 239-245. 

[4]

F. A. ArcelusS. Kumar and G. Srinivasan, Pricing and rebate policies in the two-echelon supply chain with asymmetric information under-price-dependent stochastic demand, International Journal of Production Economics, 113 (2008), 598-618. 

[5]

O. Caliskan-DemiragY. (Frank) Chen and J. Li, Customer and retailer rebates under risk aversion, International Journal of Production Economics, 133 (2011), 736-750.  doi: 10.1016/j.ijpe.2011.06.002.

[6]

K. CaoG. HanB. Xu and J. Wang, Gift card payment or cash payment: Which payment is suitable for trade-in rebate?, Transportation Research Part E: Logistics and Transportation Review, 134 (2020), 101857.  doi: 10.1016/j.tre.2020.101857.

[7]

H. J. ChangJ. T. TengL. Y. Ouyang and C. Y. Dye, Retailer's optimal pricing and lot-sizing policies for deteriorating items with partial backlogging, European J. Oper. Res., 168 (2005), 51-64.  doi: 10.1016/j.ejor.2004.05.003.

[8]

T. H. Chen, Optimizing pricing replenishment and rework decision for imperfect and deteriorating items in a manufacturer-retailer channel, International Journal of Production Economics, 183 (2017), 539-550.  doi: 10.1016/j.ijpe.2016.08.015.

[9]

T. Chernonog and T. Avinadav, Pricing and advertising in a supply chain of perishable products under asymmetric information, International Journal of Production Economics, 209 (2019), 249-264.  doi: 10.1016/j.ijpe.2017.10.002.

[10]

R. Hasan, A. H. M. Mashud, Y. Daryanto and H. M. Wee, A non-instantaneous inventory model of agricultural products considering deteriorating impacts and pricing policies, Kybernetes, 50 (2020). doi: 10.1108/K-05-2020-0288.

[11]

C. H. Ho, The optimal integrated inventory policy with price-and-credit-linked demand under two-level trade credit, Computers and Industrial Engineering, 60 (2011), 117-126.  doi: 10.1016/j.cie.2010.10.009.

[12]

S. HuZ.-J. Ma and J. B. Sheu, Optimal prices and trade-in rebates for successive-generation products with strategic consumers and limited trade-in duration, Transportation Research Part E: Logistics and Transportation Review, 124 (2019), 92-107.  doi: 10.1016/j.tre.2019.02.004.

[13]

O. M. JadidiY. Jaber and S. Zolfaghari, Joint pricing and inventory problem with price dependent stochastic demand and price discounts, Computers and Industrial Engineering, 114 (2017), 45-53. 

[14]

A. Khakzad and M. R. Gholamian, The effect of inspection on deterioration rate: An inventory model for deteriorating items with advanced payment, Journal of Cleaner Production, 2541 (2020), 120117. 

[15]

M. Khouja, A joint optimal pricing, rebate value, and lot-sizing model, European Journal of Operational Research, 174 (2006), 706-723. 

[16]

M. KhoujaM. Hadzikadic and M. A. Zaffar, An agent based modeling approach for determining optimal price-rebate schemes, Simulation Modelling Practice and Theory, 16 (2008), 111-126. 

[17]

C. LiuxinC. XianM. F. Keblis and L. Gen, Optimal pricing and replenishment policy for deteriorating inventory under stock-level-dependent, time-varying and price-dependent demand, Computers and Industrial Engineering, 135 (2019), 1294-1299. 

[18]

L. LuJ. Zhang and W. Tang, Optimal dynamic pricing and replenishment policy for perishable items with inventory-level-dependent demand, Internat. J. Systems Sci., 47 (2016), 1480-1494.  doi: 10.1080/00207721.2014.938784.

[19]

A. K. MannaJ. K. Dey and S. K. Mondal, Imperfect production inventory model with production rate dependent defective rate and advertisement dependent demand, Computers and Industrial Engineering, 104 (2017), 9-22.  doi: 10.1016/j.cie.2016.11.027.

[20]

U. Mishra, An inventory model for controllable probabilistic deterioration rate under shortages, Evolving Systems, 7 (2016), 287-307.  doi: 10.1007/s12530-016-9150-z.

[21]

A. H. M. MashudD. RoyY. DaryantoR. K. Chakrabortty and M.-L. Tseng, A sustainable inventory model with controllable carbon emissions, deterioration and advance payments, Journal of Cleaner Production, 296 (2021), 126608.  doi: 10.1016/j.jclepro.2021.126608.

[22]

U. Mishra, Optimizing three-rates-of-production inventory model under market selling price and advertisement cost with deteriorating items, International Journal of Management Science and Engineering Management, 13 (2018), 295-305. 

[23]

U. MishraL. E. Cardenas-BarronS. TiwariA. A. Shaikh and G. Treviño-Garza, An inventory model under price and stock dependent demand for controllable deterioration rate with shortages and preservation technology investment, Ann. Oper. Res., 254 (2017), 165-190.  doi: 10.1007/s10479-017-2419-1.

[24]

A. H. M. MashudD. RoyY. Daryanto and M. H. Ali, A Sustainable inventory model with imperfect products, deterioration, and controllable emissions, Mathematics, 8 (2020), 2049.  doi: 10.3390/math8112049.

[25]

U. MishraJ.-Z. Wu and M.-L. Tseng, Effects of a hybrid-price-stock dependent demand on the optimal solutions of a deteriorating inventory system and trade credit policy on re-manufactured product, Journal of Cleaner Production, 241 (2019), 118282.  doi: 10.1016/j.jclepro.2019.118282.

[26]

A. H. M. MashudM. ParvinU. MishraM.-L. Tseng and M. K. Lim, A sustainable inventory model with controllable carbon emissions for green-warehouse farms, Journal of Cleaner Production, 298 (2021), 126777.  doi: 10.1016/j.jclepro.2021.126777.

[27]

U. MishraA. H. M. MashudM. L. Tseng and J.-Z. Wu, Optimizing a sustainable supply chain inventory model for controllable deterioration and emission rates in a greenhouse farm, Mathematics, 9 (2021), 495.  doi: 10.3390/math9050495.

[28]

A. H. M. Mashud, A deteriorating inventory model with different types of demand and fully backlogged shortages, International Journal of Logistics systems and Management, 36 (2020), 16-45.  doi: 10.1504/IJLSM.2020.107220.

[29]

A. H. M. MashudH. M. WeeC. V. Huang and J.-Z. Wu, Optimal replenishment policy for deteriorating products in a newsboy problem with multiple just-in-time deliveries, Mathematics, 8 (2020), 1981.  doi: 10.3390/math8111981.

[30]

A. MuzaffarM. N. Malik and S. Deng, Efficacy of retailer rebates and delayed incentives under customer heterogeneity, RAIRO Oper. Res., 55 (2021), 1695-1713.  doi: 10.1051/ro/2021070.

[31]

M. ÖnalA. Yenipazarli and O. E. Kundakcioglu, A mathematical model for perishable products with price-and displayed-stock-dependent demand, Computers and Industrial Engineering, 102 (2016), 246-258. 

[32]

B. O'Neill and S. Sanni, Profit optimisation for deterministic inventory systems with linear cost, Computers and Industrial Engineering, 122 (2018), 303-317.  doi: 10.1016/j.cie.2018.05.032.

[33]

L. Y. OuyangC. H. Ho and C. H. Su, An optimization approach for joint pricing and ordering problem in an integrated inventory system with order-size dependent trade credit, Computers and Industrial Engineering, 57 (2009), 920-930.  doi: 10.1016/j.cie.2009.03.011.

[34]

M. Shafieezadeh and A. Sadegheih, Developing an integrated inventory management model for multi-item multi-echelon supply chain, International Journal of Advanced Manufacturing Technology, 72 (2014), 1099-1119.  doi: 10.1007/s00170-014-5684-z.

[35]

C. K. Sivashankari and S. Panayappan, Production inventory model for two-level production with deteriorative items and shortages, International Journal of Advanced Manufacturing Technology, 76 (2015), 2003-2014. 

[36]

A. A. ShaikhA. H. M. MashudM. S. Uddin and M. A. A. Khan, Non-instantaneous deterioration inventory model with price and stock dependent demand for fully backlogged shortages under inflation, International. Journal of Business Forecasting and Marketing Intelligence, 3 (2017), 152-164.  doi: 10.1504/IJBFMI.2017.084055.

[37]

H. N. Soni and K. A. Patel, Joint pricing and replenishment policies for non-instantaneous deteriorating items with imprecise deterioration free time and credibility constraint, Computers and Industrial Engineering, 66 (2013), 944-951.  doi: 10.1016/j.cie.2013.08.022.

[38]

S. SrinivasanK. PawelsD. M. Hanssens and M. Dekimpe, Do promotions benefit manufacturer, retailers or both?, Management Science, 50 (2004), 617-629. 

[39]

N. TashakkorS. H. Mirmohammadi and M. Iranpoor, Joint optimization of dynamic pricing and replenishment cycle considering variable non-instantaneous deterioration and stock-dependent demand, Computers and Industrial Engineering, 123 (2018), 232-241.  doi: 10.1016/j.cie.2018.06.029.

[40]

H. Teunter, Analysis of Sales Promotion Effects on Household Purchasing Behavior, ERIM Ph. D. research series in management, Erasmus University, Rotterdam, 2002.

[41]

C. Wang and R. Huang, Pricing for seasonal deteriorating products with price- and ramp-type time-dependent demand, Computers and Industrial Engineering, 77 (2014), 29-34. 

[42]

H. M. Wee, Joint pricing and replenishment policy for deteriorating inventory with declining market, International Journal of Production Economics, 40 (1995), 163–171. doi: 10.1016/0925-5273(95)00053-3.

[43]

H. M. Wee, A replenishment policy for items with a price-dependent demand and a varying rate of deterioration, Production Planning and Control, 8 (1997), 494-499.  doi: 10.1080/095372897235073.

[44]

W. K. WongJ. Qi and S. Y. S. Leung, Coordinating supply chains with sales rebate contracts and vendor-managed inventory, International Journal of Production Economics, 120 (2009), 151-161.  doi: 10.1016/j.ijpe.2008.07.025.

[45]

L. Yang and S. Dong, Rebate strategy to stimulate online customer reviews, International Journal of Production Economics, 204 (2018), 99-107.  doi: 10.1016/j.ijpe.2018.07.032.

[46]

J. ZhanX. Chen and Q. Hu, The value of trade credit with rebate contract in a capital-constrained supply chain, International Journal of Production Research, 57 (2019), 379-396. 

show all references

References:
[1]

P. L. Abad, Optimal pricing and lot sizing under conditions of perishability and partial backordering, Management Science, 42 (1996), 1093-1104.  doi: 10.1287/mnsc.42.8.1093.

[2]

P. L. Abad and C. K. Jaggi, A joint approach for setting unit price and the length of the credit period for a seller when end demand is price sensitive, International Journal of Production Economics, 83 (2003), 115-122.  doi: 10.1016/S0925-5273(02)00142-1.

[3]

A. AbdulA. Jolson Marin and Y. Darmon Reney, A model for optimizing the refund value in rebate promotions, Journal of Business Research, 29 (1994), 239-245. 

[4]

F. A. ArcelusS. Kumar and G. Srinivasan, Pricing and rebate policies in the two-echelon supply chain with asymmetric information under-price-dependent stochastic demand, International Journal of Production Economics, 113 (2008), 598-618. 

[5]

O. Caliskan-DemiragY. (Frank) Chen and J. Li, Customer and retailer rebates under risk aversion, International Journal of Production Economics, 133 (2011), 736-750.  doi: 10.1016/j.ijpe.2011.06.002.

[6]

K. CaoG. HanB. Xu and J. Wang, Gift card payment or cash payment: Which payment is suitable for trade-in rebate?, Transportation Research Part E: Logistics and Transportation Review, 134 (2020), 101857.  doi: 10.1016/j.tre.2020.101857.

[7]

H. J. ChangJ. T. TengL. Y. Ouyang and C. Y. Dye, Retailer's optimal pricing and lot-sizing policies for deteriorating items with partial backlogging, European J. Oper. Res., 168 (2005), 51-64.  doi: 10.1016/j.ejor.2004.05.003.

[8]

T. H. Chen, Optimizing pricing replenishment and rework decision for imperfect and deteriorating items in a manufacturer-retailer channel, International Journal of Production Economics, 183 (2017), 539-550.  doi: 10.1016/j.ijpe.2016.08.015.

[9]

T. Chernonog and T. Avinadav, Pricing and advertising in a supply chain of perishable products under asymmetric information, International Journal of Production Economics, 209 (2019), 249-264.  doi: 10.1016/j.ijpe.2017.10.002.

[10]

R. Hasan, A. H. M. Mashud, Y. Daryanto and H. M. Wee, A non-instantaneous inventory model of agricultural products considering deteriorating impacts and pricing policies, Kybernetes, 50 (2020). doi: 10.1108/K-05-2020-0288.

[11]

C. H. Ho, The optimal integrated inventory policy with price-and-credit-linked demand under two-level trade credit, Computers and Industrial Engineering, 60 (2011), 117-126.  doi: 10.1016/j.cie.2010.10.009.

[12]

S. HuZ.-J. Ma and J. B. Sheu, Optimal prices and trade-in rebates for successive-generation products with strategic consumers and limited trade-in duration, Transportation Research Part E: Logistics and Transportation Review, 124 (2019), 92-107.  doi: 10.1016/j.tre.2019.02.004.

[13]

O. M. JadidiY. Jaber and S. Zolfaghari, Joint pricing and inventory problem with price dependent stochastic demand and price discounts, Computers and Industrial Engineering, 114 (2017), 45-53. 

[14]

A. Khakzad and M. R. Gholamian, The effect of inspection on deterioration rate: An inventory model for deteriorating items with advanced payment, Journal of Cleaner Production, 2541 (2020), 120117. 

[15]

M. Khouja, A joint optimal pricing, rebate value, and lot-sizing model, European Journal of Operational Research, 174 (2006), 706-723. 

[16]

M. KhoujaM. Hadzikadic and M. A. Zaffar, An agent based modeling approach for determining optimal price-rebate schemes, Simulation Modelling Practice and Theory, 16 (2008), 111-126. 

[17]

C. LiuxinC. XianM. F. Keblis and L. Gen, Optimal pricing and replenishment policy for deteriorating inventory under stock-level-dependent, time-varying and price-dependent demand, Computers and Industrial Engineering, 135 (2019), 1294-1299. 

[18]

L. LuJ. Zhang and W. Tang, Optimal dynamic pricing and replenishment policy for perishable items with inventory-level-dependent demand, Internat. J. Systems Sci., 47 (2016), 1480-1494.  doi: 10.1080/00207721.2014.938784.

[19]

A. K. MannaJ. K. Dey and S. K. Mondal, Imperfect production inventory model with production rate dependent defective rate and advertisement dependent demand, Computers and Industrial Engineering, 104 (2017), 9-22.  doi: 10.1016/j.cie.2016.11.027.

[20]

U. Mishra, An inventory model for controllable probabilistic deterioration rate under shortages, Evolving Systems, 7 (2016), 287-307.  doi: 10.1007/s12530-016-9150-z.

[21]

A. H. M. MashudD. RoyY. DaryantoR. K. Chakrabortty and M.-L. Tseng, A sustainable inventory model with controllable carbon emissions, deterioration and advance payments, Journal of Cleaner Production, 296 (2021), 126608.  doi: 10.1016/j.jclepro.2021.126608.

[22]

U. Mishra, Optimizing three-rates-of-production inventory model under market selling price and advertisement cost with deteriorating items, International Journal of Management Science and Engineering Management, 13 (2018), 295-305. 

[23]

U. MishraL. E. Cardenas-BarronS. TiwariA. A. Shaikh and G. Treviño-Garza, An inventory model under price and stock dependent demand for controllable deterioration rate with shortages and preservation technology investment, Ann. Oper. Res., 254 (2017), 165-190.  doi: 10.1007/s10479-017-2419-1.

[24]

A. H. M. MashudD. RoyY. Daryanto and M. H. Ali, A Sustainable inventory model with imperfect products, deterioration, and controllable emissions, Mathematics, 8 (2020), 2049.  doi: 10.3390/math8112049.

[25]

U. MishraJ.-Z. Wu and M.-L. Tseng, Effects of a hybrid-price-stock dependent demand on the optimal solutions of a deteriorating inventory system and trade credit policy on re-manufactured product, Journal of Cleaner Production, 241 (2019), 118282.  doi: 10.1016/j.jclepro.2019.118282.

[26]

A. H. M. MashudM. ParvinU. MishraM.-L. Tseng and M. K. Lim, A sustainable inventory model with controllable carbon emissions for green-warehouse farms, Journal of Cleaner Production, 298 (2021), 126777.  doi: 10.1016/j.jclepro.2021.126777.

[27]

U. MishraA. H. M. MashudM. L. Tseng and J.-Z. Wu, Optimizing a sustainable supply chain inventory model for controllable deterioration and emission rates in a greenhouse farm, Mathematics, 9 (2021), 495.  doi: 10.3390/math9050495.

[28]

A. H. M. Mashud, A deteriorating inventory model with different types of demand and fully backlogged shortages, International Journal of Logistics systems and Management, 36 (2020), 16-45.  doi: 10.1504/IJLSM.2020.107220.

[29]

A. H. M. MashudH. M. WeeC. V. Huang and J.-Z. Wu, Optimal replenishment policy for deteriorating products in a newsboy problem with multiple just-in-time deliveries, Mathematics, 8 (2020), 1981.  doi: 10.3390/math8111981.

[30]

A. MuzaffarM. N. Malik and S. Deng, Efficacy of retailer rebates and delayed incentives under customer heterogeneity, RAIRO Oper. Res., 55 (2021), 1695-1713.  doi: 10.1051/ro/2021070.

[31]

M. ÖnalA. Yenipazarli and O. E. Kundakcioglu, A mathematical model for perishable products with price-and displayed-stock-dependent demand, Computers and Industrial Engineering, 102 (2016), 246-258. 

[32]

B. O'Neill and S. Sanni, Profit optimisation for deterministic inventory systems with linear cost, Computers and Industrial Engineering, 122 (2018), 303-317.  doi: 10.1016/j.cie.2018.05.032.

[33]

L. Y. OuyangC. H. Ho and C. H. Su, An optimization approach for joint pricing and ordering problem in an integrated inventory system with order-size dependent trade credit, Computers and Industrial Engineering, 57 (2009), 920-930.  doi: 10.1016/j.cie.2009.03.011.

[34]

M. Shafieezadeh and A. Sadegheih, Developing an integrated inventory management model for multi-item multi-echelon supply chain, International Journal of Advanced Manufacturing Technology, 72 (2014), 1099-1119.  doi: 10.1007/s00170-014-5684-z.

[35]

C. K. Sivashankari and S. Panayappan, Production inventory model for two-level production with deteriorative items and shortages, International Journal of Advanced Manufacturing Technology, 76 (2015), 2003-2014. 

[36]

A. A. ShaikhA. H. M. MashudM. S. Uddin and M. A. A. Khan, Non-instantaneous deterioration inventory model with price and stock dependent demand for fully backlogged shortages under inflation, International. Journal of Business Forecasting and Marketing Intelligence, 3 (2017), 152-164.  doi: 10.1504/IJBFMI.2017.084055.

[37]

H. N. Soni and K. A. Patel, Joint pricing and replenishment policies for non-instantaneous deteriorating items with imprecise deterioration free time and credibility constraint, Computers and Industrial Engineering, 66 (2013), 944-951.  doi: 10.1016/j.cie.2013.08.022.

[38]

S. SrinivasanK. PawelsD. M. Hanssens and M. Dekimpe, Do promotions benefit manufacturer, retailers or both?, Management Science, 50 (2004), 617-629. 

[39]

N. TashakkorS. H. Mirmohammadi and M. Iranpoor, Joint optimization of dynamic pricing and replenishment cycle considering variable non-instantaneous deterioration and stock-dependent demand, Computers and Industrial Engineering, 123 (2018), 232-241.  doi: 10.1016/j.cie.2018.06.029.

[40]

H. Teunter, Analysis of Sales Promotion Effects on Household Purchasing Behavior, ERIM Ph. D. research series in management, Erasmus University, Rotterdam, 2002.

[41]

C. Wang and R. Huang, Pricing for seasonal deteriorating products with price- and ramp-type time-dependent demand, Computers and Industrial Engineering, 77 (2014), 29-34. 

[42]

H. M. Wee, Joint pricing and replenishment policy for deteriorating inventory with declining market, International Journal of Production Economics, 40 (1995), 163–171. doi: 10.1016/0925-5273(95)00053-3.

[43]

H. M. Wee, A replenishment policy for items with a price-dependent demand and a varying rate of deterioration, Production Planning and Control, 8 (1997), 494-499.  doi: 10.1080/095372897235073.

[44]

W. K. WongJ. Qi and S. Y. S. Leung, Coordinating supply chains with sales rebate contracts and vendor-managed inventory, International Journal of Production Economics, 120 (2009), 151-161.  doi: 10.1016/j.ijpe.2008.07.025.

[45]

L. Yang and S. Dong, Rebate strategy to stimulate online customer reviews, International Journal of Production Economics, 204 (2018), 99-107.  doi: 10.1016/j.ijpe.2018.07.032.

[46]

J. ZhanX. Chen and Q. Hu, The value of trade credit with rebate contract in a capital-constrained supply chain, International Journal of Production Research, 57 (2019), 379-396. 

Figure 1.  Schematic view of rebate pays to the buyer
Figure 2.  Representing inventory vs time
Figure 3.  Staple manufacturing product
Figure 4.  The concave nature of $ \Pi $ with respect to $ P $ Example 1 and 2
Figure 5.  The concave nature of $ \Pi $ with respect to $ t_7 $ for Example 1 and 2
Figure 6.  The concave nature of $ \Pi $ with respect to $ t_7 $ for Example 3 and 4
Figure 7.  The concave nature of $ \Pi $ with respect to $ p $ for Example 3 and 4
Figure 8.  The concave nature of $ \Pi $ with respect to $ r $ for Example 1 and 2
Figure 9.  Comparison of profit, ordering quantity and shortages concerning altered demand
Figure 10.  The consequence of $ \Pi $ concerning $ a $ for Example 1, 2, 3 and 4
Figure 11.  The consequence of $ \Pi $ concerning $ b $ for Example 1, 2, 3 and 4
Figure 12.  The consequence of $ \Pi $ concerning $ c_p $ for Example 1, 2, 3 and 4
Figure 13.  The consequence of $ \Pi $ concerning $ c_s $ for Example 1, 2, 3 and 4
Figure 14.  The consequence of $ \Pi $ concerning $ d $ for Example 1, 2, 3 and 4
Figure 15.  The consequence of $ \Pi $ concerning $ w $ for Example 1, 2, 3 and 4
Figure 16.  The consequence of $ \Pi $ concerning $ P_1 $ for Example 1, 2, 3 and 4
Table 1.  Prior studies and Assessments among the research
Reference Inventory type Production Level Demand based on rebate value and selling price Backorder Profit
Khouja [15] Integrated x x
khouja et al. [16] Integrated x x
Arcelus, et al. [4] Integrated x
Wong et al. [44] Integrated x x
Caliskan-Demirag et al. [5] Integrated x x
Ho [11] Integrated x x x
Sivashankari and Panayappan [35] EPQ x x x
Mishra [20] EPQ x x
Lu et al. [18] EOQ x x x
Mishra et al. [23] EOQ x x x
Manna et al. [19]] EPQ x x x
Jadidi et al. [13] Integrated x x
Chen [8] Integrated x x
Mishra [22] EPQ x x
Yang and Dong [45] Integrated x x
Hu et al. [12] Integrated x x
Liuxin et al. [17] Integrated x x x
Chernonog and Avinadav [9] Integrated x x x
Zhan et al. [46] Integrated x
Cao et al. [6] Integrated x x
Mishra et al. [25]] EOQ x x x
Khakzad and Gholamian [14] Integrated x x x x
This paper EPQ
Reference Inventory type Production Level Demand based on rebate value and selling price Backorder Profit
Khouja [15] Integrated x x
khouja et al. [16] Integrated x x
Arcelus, et al. [4] Integrated x
Wong et al. [44] Integrated x x
Caliskan-Demirag et al. [5] Integrated x x
Ho [11] Integrated x x x
Sivashankari and Panayappan [35] EPQ x x x
Mishra [20] EPQ x x
Lu et al. [18] EOQ x x x
Mishra et al. [23] EOQ x x x
Manna et al. [19]] EPQ x x x
Jadidi et al. [13] Integrated x x
Chen [8] Integrated x x
Mishra [22] EPQ x x
Yang and Dong [45] Integrated x x
Hu et al. [12] Integrated x x
Liuxin et al. [17] Integrated x x x
Chernonog and Avinadav [9] Integrated x x x
Zhan et al. [46] Integrated x
Cao et al. [6] Integrated x x
Mishra et al. [25]] EOQ x x x
Khakzad and Gholamian [14] Integrated x x x x
This paper EPQ
Table 2.  Computational results
Number of iterations $R$ $t_7$ $r$ $p$ $\Pi (t_7,r,p)$
1 0.216284 3.97675 1.31944 14.028 334.61
2 0.216279 6.36706 0.665818 10.9352 377.988
3 0.0216281 5.34862 0.604776 10.4854 380.469
4 0.0216282 5.23315 0.600339 10.4499 380.643
5 0.0216282 5.2243 0.60018 10.4473 380.656
6 0.0216282 5.22366 0.600007 10.4472 380.657
7 0.0216282 5.22363 0.600007 10.4472 380.657
8 0.0216282 5.22363 0.600007 10.4472 380.657
Number of iterations $R$ $t_7$ $r$ $p$ $\Pi (t_7,r,p)$
1 0.216284 3.97675 1.31944 14.028 334.61
2 0.216279 6.36706 0.665818 10.9352 377.988
3 0.0216281 5.34862 0.604776 10.4854 380.469
4 0.0216282 5.23315 0.600339 10.4499 380.643
5 0.0216282 5.2243 0.60018 10.4473 380.656
6 0.0216282 5.22366 0.600007 10.4472 380.657
7 0.0216282 5.22363 0.600007 10.4472 380.657
8 0.0216282 5.22363 0.600007 10.4472 380.657
Table 3.  Computational results
Number of iterations $R$ $t_7$ $p$ $\Pi (t_7,p)$
1 0.0216284 4.23357 10.1707 377.01
2 0.0216282 4.23357 10.1707 377.01
3 0.0216282 4.23357 10.1707 377.01
Number of iterations $R$ $t_7$ $p$ $\Pi (t_7,p)$
1 0.0216284 4.23357 10.1707 377.01
2 0.0216282 4.23357 10.1707 377.01
3 0.0216282 4.23357 10.1707 377.01
Table 4.  Computational results
Number of iterations $R$ $t_7$ $r$ $p$ $\Pi (t_7,r,p)$
1 0.0216287 1.5758 3.40356 8.37926 301.769
2 0.0216285 3.64039 3.7101 8.37926 290.055
3 0.0216285 3.33961 3.23438 7.73499 334.006
4 0.0216285 3.36767 2.9662 6.58827 276.844
5 0.0216286 2.91956 2.82123 5.97419 393.994
6 0.0216286 2.71482 2.74503 5.66496 399.362
7 0.0216286 2.62285 2.70566 5.5092 401.031
8 0.0216286 2.57939 2.6855 5.43053 401.587
9 0.0216286 2.55819 2.67523 5.39073 401.791
10 0.0216286 2.54765 2.67001 5.37057 401.875
11 0.0216286 2.54236 2.66736 5.36036 401.912
12 0.0216286 2.53969 2.66602 5.35518 401.929
13 0.0216286 2.53834 2.66534 5.35257 401.938
14 0.0216286 2.53766 2.66499 5.35124 401.942
15 0.0216286 2.53732 2.66482 5.35056 401.944
16 0.0216286 2.53713 2.66473 5.35023 401.945
17 0.0216286 2.53705 2.66468 5.35005 401.946
18 0.0216286 2.53701 2.66466 5.34996 401.946
19 0.0216286 2.53698 2.66465 5.34992 401.946
20 0.0216286 2.63697 2.66465 5.3499 401.946
21 0.0216286 2.53696 2.66465 5.3499 401.946
22 0.0216286 2.53696 2.66465 5.3499 401.946
Number of iterations $R$ $t_7$ $r$ $p$ $\Pi (t_7,r,p)$
1 0.0216287 1.5758 3.40356 8.37926 301.769
2 0.0216285 3.64039 3.7101 8.37926 290.055
3 0.0216285 3.33961 3.23438 7.73499 334.006
4 0.0216285 3.36767 2.9662 6.58827 276.844
5 0.0216286 2.91956 2.82123 5.97419 393.994
6 0.0216286 2.71482 2.74503 5.66496 399.362
7 0.0216286 2.62285 2.70566 5.5092 401.031
8 0.0216286 2.57939 2.6855 5.43053 401.587
9 0.0216286 2.55819 2.67523 5.39073 401.791
10 0.0216286 2.54765 2.67001 5.37057 401.875
11 0.0216286 2.54236 2.66736 5.36036 401.912
12 0.0216286 2.53969 2.66602 5.35518 401.929
13 0.0216286 2.53834 2.66534 5.35257 401.938
14 0.0216286 2.53766 2.66499 5.35124 401.942
15 0.0216286 2.53732 2.66482 5.35056 401.944
16 0.0216286 2.53713 2.66473 5.35023 401.945
17 0.0216286 2.53705 2.66468 5.35005 401.946
18 0.0216286 2.53701 2.66466 5.34996 401.946
19 0.0216286 2.53698 2.66465 5.34992 401.946
20 0.0216286 2.63697 2.66465 5.3499 401.946
21 0.0216286 2.53696 2.66465 5.3499 401.946
22 0.0216286 2.53696 2.66465 5.3499 401.946
Table 5.  Computational results
Number of iterations $R$ $t_7$ $p$ $\Pi (t_7,p)$
1 0.0216279 6.30321 2.99373 144.63
2 0.0216284 4.21988 2.84046 151.042
3 0.0216284 4.09248 2.83108 151.35
4 0.0216284 4.08481 2.83052 151.368
5 0.0216284 4.08435 2.83049 151.369
6 0.0216284 4.08433 2.83048 151.369
7 0.0216284 4.08432 2.83048 151.369
8 0.0216284 4.08432 2.83048 151.369
Number of iterations $R$ $t_7$ $p$ $\Pi (t_7,p)$
1 0.0216279 6.30321 2.99373 144.63
2 0.0216284 4.21988 2.84046 151.042
3 0.0216284 4.09248 2.83108 151.35
4 0.0216284 4.08481 2.83052 151.368
5 0.0216284 4.08435 2.83049 151.369
6 0.0216284 4.08433 2.83048 151.369
7 0.0216284 4.08432 2.83048 151.369
8 0.0216284 4.08432 2.83048 151.369
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