The effect of rebate value and selling price-dependent demand for a four-level production manufacturing system

Umakanta Mishra; Abu Hashan Md Mashud; Sankar Kumar Roy; Md Sharif Uddin

doi:10.3934/jimo.2021233

Article Contents

2023, Volume 19, Issue 2: 1367-1394. Doi: 10.3934/jimo.2021233

This issue Previous Article Optimal recycling price strategy of clothing enterprises based on closed-loop supply chain Next Article A novel separate chance-constrained programming model to design a sustainable medical ventilator supply chain network during the Covid-19 pandemic

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
^*Corresponding author: Abu Hashan Md Mashud

Received: March 2021

Revised: September 2021

Early access: January 2022

Published: February 2023

Abstract / Introduction Full Text(HTML) Figure(16) / Table(5) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 90B05; Secondary: 90C26.

Citation:

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Figure 1. Schematic view of rebate pays to the buyer

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Figure 2. Representing inventory vs time

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Figure 3. Staple manufacturing product

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Figure 4. The concave nature of $ \Pi $ with respect to $ P $ Example 1 and 2

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Figure 5. The concave nature of $ \Pi $ with respect to $ t_7 $ for Example 1 and 2

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Figure 6. The concave nature of $ \Pi $ with respect to $ t_7 $ for Example 3 and 4

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Figure 7. The concave nature of $ \Pi $ with respect to $ p $ for Example 3 and 4

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Figure 8. The concave nature of $ \Pi $ with respect to $ r $ for Example 1 and 2

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Figure 9. Comparison of profit, ordering quantity and shortages concerning altered demand

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Figure 10. The consequence of $ \Pi $ concerning $ a $ for Example 1, 2, 3 and 4

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Figure 11. The consequence of $ \Pi $ concerning $ b $ for Example 1, 2, 3 and 4

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Figure 12. The consequence of $ \Pi $ concerning $ c_p $ for Example 1, 2, 3 and 4

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Figure 13. The consequence of $ \Pi $ concerning $ c_s $ for Example 1, 2, 3 and 4

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Figure 14. The consequence of $ \Pi $ concerning $ d $ for Example 1, 2, 3 and 4

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Figure 15. The consequence of $ \Pi $ concerning $ w $ for Example 1, 2, 3 and 4

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Figure 16. The consequence of $ \Pi $ concerning $ P_1 $ for Example 1, 2, 3 and 4

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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	√	√	√	√

| Show Table

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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

| Show Table

DownLoad: CSV

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

| Show Table

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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

| Show Table

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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

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