A production inventory model for high-tech products involving two production runs and a product variation

Subhendu Ruidas; Mijanur Rahaman Seikh; Prasun Kumar Nayak

doi:10.3934/jimo.2022038

Article Contents

2023, Volume 19, Issue 3: 2178-2205. Doi: 10.3934/jimo.2022038

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A production inventory model for high-tech products involving two production runs and a product variation

1.
Department of Mathematics, Sonamukhi College, Bankura-722 207, India
2.
Department of Mathematics, Kazi Nazrul University, Asansol-713 340, India
3.
Department of Mathematics, Midnapore College (Autonomous), Midnapore-721 101, India

^*Corresponding author: Mijanur Rahaman Seikh
^*Corresponding author: Mijanur Rahaman Seikh

Received: August 2021

Revised: February 2022

Early access: March 2022

Published: March 2023

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Abstract

This paper explores a production inventory model considering two high-tech products of the same kind. One is the primary product and the other is the updated version of that primary product. Due to continuous development in technology, the life-cycle of some high-tech products, like, smartphone, tablet, laptop, etc. have become shorter. We witness the launching of new products more frequently in this field. This prompts the manufacturers to release an updated or pro version of their existing products after a certain time to compete in the market. The reputation of the primary product (in terms of quality and performance) plays an important role in generating the demand for the updated product. Due to the short life-cycle of the products, the proposed model considers only two consecutive production runs. One for the primary product and one for the updated product. Here the demands of both the products depend on the respective selling prices. Moreover, the demand of the updated product is also dependent on the quality of the primary product. Shortages for the primary product are allowed. Those shortages are backlogged partially with the updated product. Also, the possibility of imperfect production during regular production runs is considered. The selling prices, production rates, and the production run times for both the products are considered here as decision variables. Due to the complexity in the resulting optimization problem, the quantum-behaved particle swarm optimization technique is applied to derive the optimal profit. The concavity natures of the profit function are shown graphically. A numerical illustration is presented for the economic validation of the model. Finally, sensitivity analysis of the optimal solutions concerning the key inventory parameters is conducted for identifying several managerial implications.

Note: The affiliations of the three authors have been corrected online.

Keywords:

Production inventory model,
high-tech product,
product variation,
imperfect production process,
quantum-behaved particle swarm optimization technique.

Mathematics Subject Classification: Primary: 90B05, 90B30; Secondary: 80M50.

Citation:

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Figure 1. Behaviour of the inventory level over time

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Figure 2. Concavity of the profit function w.r.t. selling price and time

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Figure 3. Concavity of the profit function w.r.t. production rate and time

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Figure 4. Effect of changing $ t_3 $ on optimal solutions

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Figure 5. Effect of changing $ \beta $ on optimal lot size

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Figure 6. Effect of changing $ r_1 $ on optimal solution

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Figure 7. Effect of changing $ b_1 $ on optimal solution

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Table 1. Summary of literature review

Author(s)	EOQ/ EPQ	Nature of demand	Defective product?	Single/ multi product?	product updation considered?	Demand of updated product depends on quality of primary product?	Shor-tages
Datta [11]	EPQ	price & quality dependent	yes	single	no	NA	no
Khara et al. [20]	EPQ	price, reliability & advertisement dependent	yes	single	no	NA	no
Liu et al. [23]	EOQ	price & quality dependent	no	single	no	NA	no
Banerjee & Agarwal [4]	EOQ	price & freshness dependent	no	single	no	NA	yes
Masache [30]	EOQ	deterministic	no	two	no	NA	no
Stavrulaki [62]	EOQ	stochastic	no	two	no	NA	no
Aggarwal et al. [1]	EOQ	deterministic	no	two	yes	no	no
Chanda & Aggarwal [6]	EOQ	deterministic	no	two	yes	no	no
Chiu et al. [9]	EPQ	deterministic	yes	multi	no	no	no
Proposed model	EPQ	price & quality dependent	yes	two	yes	yes	yes

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Table 2. Sensitivity analysis

Parameter	Changes	$ t_1 $	$ t_5 $	$ p_1 $	$ p_2 $	Profit	$ Q_1 $	$ Q_2 $	$ T $
$ t_3 $	7.0	5.23913	11.47573	249.30	318.49	9201.75	1149.88	969.58	12.45178
	8.0	7.04729	13.32953	231.39	314.59	8897.09	1546.74	1154.54	14.35523
	9.0	8.26835	14.60134	227.05	312.29	8744.92	1814.74	1213.42	15.56001
$ \beta $	0.08	7.20770	10.88072	228.66	325.62	8035.87	1581.95	624.05	11.99597
	0.09	7.14791	11.70929	229.03	321.76	8420.30	1568.82	803.54	12.78338
	1.0	7.04729	13.32953	231.39	314.59	8897.09	1546.74	1154.54	14.35523
$ A_1 $	560	7.04729	13.32953	231.39	314.59	8897.09	1546.74	1154.54	14.35523
	570	7.36848	12.03052	233.79	319.83	9687.66	1617.23	873.13	12.93037
	580	7.48057	11.00804	240.83	325.77	10564.71	1641.83	651.63	11.84619
$ A_2 $	520	7.24766	10.74417	228.39	322.95	7921.57	1590.72	594.47	11.76943
	530	7.09583	11.06169	230.14	321.93	8353.59	1557.39	780.23	12.69032
	540	7.04729	13.32953	231.39	314.59	8897.09	1546.74	1154.54	14.35523
$ r_1 $	0.7	7.52533	8.63797	227.84	353.82	10037.19	1651.66	138.20	8.72560
	0.75	7.40191	10.30894	227.36	329.26	9323.61	1624.57	500.19	11.13030
	0.80	7.04729	13.32953	231.39	314.59	8897.09	1546.74	1154.54	14.35523
$ b_1 $	1.3	7.38216	8.36716	264.23	375.81	14380.91	1620.24	123.37	8.41877
	1.4	7.25239	10.00467	248.81	334.35	10932.50	1591.76	434.27	10.62091
	1.5	7.04729	13.32953	231.39	314.59	8897.09	1546.74	1154.54	14.35523
$ b_2 $	1.5	7.04729	13.32953	231.39	314.59	8897.09	1546.74	1154.54	14.35523
	1.6	7.21987	10.47846	229.30	318.94	7653.91	1584.62	536.91	11.40339
	1.7	7.37527	8.50460	228.27	360.32	7093.80	1618.72	112.83	8.54607
$ a $	0.2	7.04729	13.32953	231.39	314.59	8897.09	1546.74	1154.54	14.35523
	0.22	6.25516	12.52256	243.34	324.01	7730.56	1372.88	979.72	13.68868
	0.24	5.33047	11.74581	258.78	334.01	6649.15	1169.93	811.46	13.04623
$ M_1 $	120	7.44172	10.67127	227.19	330.36	10638.59	1633.31	578.68	11.45049
	125	7.33320	11.94581	227.62	321.13	9715.03	1609.49	854.78	12.86613
	130	7.04729	13.32953	231.39	314.59	8897.09	1546.74	1154.54	14.35523
$ M_2 $	150	7.04729	13.32953	231.39	314.59	8897.09	1546.74	1154.54	14.35523
	155	7.07835	11.73647	231.02	327.66	8401.95	1553.56	809.43	12.78843
	160	7.11430	10.76954	230.57	338.94	8009.73	1561.45	599.94	11.81604
$ \delta $	0.6	6.82219	13.03966	234.36	318.36	8928.81	1497.33	1091.74	14.10543
	0.7	7.04729	13.32953	231.39	314.59	8897.09	1546.74	1154.54	14.35523
	0.8	7.27322	13.54918	228.21	311.69	8867.08	1596.33	1202.12	14.53423

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