# American Institute of Mathematical Sciences

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:

show all references

##### References:
Schematic view of rebate pays to the buyer
Representing inventory vs time
Staple manufacturing product
The concave nature of $\Pi$ with respect to $P$ Example 1 and 2
The concave nature of $\Pi$ with respect to $t_7$ for Example 1 and 2
The concave nature of $\Pi$ with respect to $t_7$ for Example 3 and 4
The concave nature of $\Pi$ with respect to $p$ for Example 3 and 4
The concave nature of $\Pi$ with respect to $r$ for Example 1 and 2
Comparison of profit, ordering quantity and shortages concerning altered demand
The consequence of $\Pi$ concerning $a$ for Example 1, 2, 3 and 4
The consequence of $\Pi$ concerning $b$ for Example 1, 2, 3 and 4
The consequence of $\Pi$ concerning $c_p$ for Example 1, 2, 3 and 4
The consequence of $\Pi$ concerning $c_s$ for Example 1, 2, 3 and 4
The consequence of $\Pi$ concerning $d$ for Example 1, 2, 3 and 4
The consequence of $\Pi$ concerning $w$ for Example 1, 2, 3 and 4
The consequence of $\Pi$ concerning $P_1$ for Example 1, 2, 3 and 4
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 √ √ √ √
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
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
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
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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