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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

Received  August 2021 Revised  February 2022 Early access March 2022

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.

Citation: Subhendu Ruidas, Mijanur Rahaman Seikh, Prasun Kumar Nayak. A production inventory model for high-tech products involving two production runs and a product variation. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022038
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show all references

References:
[1]

K. K. Aggarwal, C. K. Jaggi and A. Kumar, An inventory decision model when demand follows innovation diffusion process under effect of technological substitution, Adv. Decis. Sci., 2013 (2013), Article ID 915657, 10 pp. doi: 10.1155/2013/915657.

[2]

A. AlArjani, M. M. Miah, M. S. Uddin, A. H. M. Mashud, H. M. Wee, S. S. Sana and H. M. Srivastava, A sustainable economic recycle quantity model for imperfect production system with shortages, Journal of Risk and Financial Management, 14 (2021). doi: 10.3390/jrfm14040173.

[3]

H. Arjuna and S. Ilmi, Effect of brand image, price, and quality of product on the smartphone purchase decision, Ekbis: Journal Ekonomy dan Bisnis, 3 (2019), 294-305.  doi: 10.14421/EkBis.2019.3.2.1190.

[4]

S. Banerjee and S. Agrawal, Inventory model for deteriorating items with freshness and price dependent demand: Optimal discounting and ordering policies, Appl. Math. Model., 52 (2017), 53-64.  doi: 10.1016/j.apm.2017.07.020.

[5]

M. Bhattacharyya and S. S. Sana, A Mathematical model on eco-friendly manufacturing system under probabilistic demand, RAIRO Oper. Res., 53 (2019), 1899-1913.  doi: 10.1051/ro/2018120.

[6]

U. Chanda and R. Aggarwal, Optimal inventory policies for successive generations of a high technology product, Journal of High Technology Management Research, 25 (2014), 148-162. 

[7]

Z. Chen and B. R. Sarker, Integrated production-inventory and pricing decisions for a single-manufacturer multi-retailer system of deteriorating items under JIT delivery policy, International Journal of Advanced Manufacturing Technology, 89 (2017), 2099-2117. 

[8]

K. ChenH. Zhou and D. Lei, Two-period pricing and ordering decisions of perishable products with a learning period for demand disruption, J. Ind. Manag. Optim., 17 (2021), 3131-3163.  doi: 10.3934/jimo.2020111.

[9]

S. W. Chiu, C.-T. Tseng, M.-F. Wu and P.-C. Sung, Multi-item EPQ model with scrap, rework and multi-delivery using common cycle policy, Journal of Applied Research and Technology, 12 (2014). doi: 10.1016/S1665-6423(14)71641-4.

[10]

M. Clerc and J. Kennedy, The particle swarm-explosion, stability, and convergence in a multi-dimensional complex space, IEEE Transactions on Evolutionary Computations, 6 (2002), 58-73. 

[11]

T. K. Datta, An inventory model with price and quality dependent demand where some items produced are defective, Adv. Oper. Res., 2013 (2013), Article ID 795078, 8 pp. doi: 10.1155/2013/795078.

[12]

T. K. Datta, Inventory system with defective products and investment opportunity for reducing defective proportion, Operational Research International Journal, 17 (2017), 297-312.  doi: 10.1007/s12351-016-0227-z.

[13]

B. K. DeyB. SarkarM. Sarkar and and S. Pareek, An integrated inventory model involving discrete setup cost reduction, variable safety factor, selling price dependent demand, and investment, RAIRO Oper. Res., 53 (2019), 39-57.  doi: 10.1051/ro/2018009.

[14]

C.-Y. Dye and T.-P. Hsieh, A particle swarm optimization for solving joint pricing and lot-sizing problem with fluctuating demand and unit purchasing cost, Comput. Math. Appl., 60 (2010), 1895-1907.  doi: 10.1016/j.camwa.2010.07.023.

[15]

R. C. Eberhart and J. Kennedy, A new optimizer using particle swarm theory, MHS'95. Proceedings of the Sixth International Symposium on Micro Machine and Human Science, Nagoya, Japan, (1995), 39–43. doi: 10.1109/MHS.1995.494215.

[16]

R. C. Eberhart and Y. Shi, Comparison between genetic algorithms and particle swarm optimization, In Proceedings of the IEEE International Conference on Evolutionary Comptation, (1998), 611–616. doi: 10.1007/BFb0040812.

[17]

J. Kennedy and R. Eberhart, Particle swarm optimization, Proceedings of ICNN'95- International Conference On Neural Network, Perth, WA, Australia, 4 (1995), 1942-1948.  doi: 10.1109/ICNN.1995.488968.

[18]

M. A. Khan, A. A. Shaikh, I. Konstantaras, A. K. Bhunia and L. E. C$\acute{a}$rdenas-Barr$\acute{o}$n, Inventory models for perishable items with advanced payment, linearly time-dependent holding cost and demand dependent on advertisement and selling price, International Journal of Production Economics, 230 (2020), Article ID 107804. doi: 10.1016/j.ijpe.2020.107804.

[19]

B. KharaJ. K. Dey and S. K. Mondal, An inventory model under development cost-dependent imperfect production and reliability-dependent demand, Journal of Management Analytics, 4 (2017), 258-275.  doi: 10.1080/23270012.2017.1344939.

[20]

B. KharaJ. K. Dey and S. K. Mondal, Effects of product reliability dependent demand in an EPQ model considering partially imperfect production, Int. J. Math. Oper. Res., 15 (2019), 242-264.  doi: 10.1504/IJMOR.2019.101621.

[21]

N. Kilicay-ErginC.-Y. Lin and G. E. Okudan, Analysis of dynamic pricing scenarios for multiple-generation product lines, Journal of Systems Science and Systems Engineering, 24 (2015), 107-129.  doi: 10.1007/s11518-015-5264-2.

[22]

K. Y. Lee and J. Park, Application of particle swarm Optimization to economic dispatch problem: Advantages and disadvantages, 2006 IEEE PES Power Systems Conference and Exposition, (2006), 188–192. doi: 10.1109/PSCE.2006.296295.

[23]

G. LiuJ. Zhang and W. Tang, Joint dynamic pricing and investment strategy for perishable foods with price-quality dependent demand, Ann. Oper. Res., 226 (2015), 397-416.  doi: 10.1007/s10479-014-1671-x.

[24]

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 & Industrial Engineering, 135 (2018), 1294-1299. 

[25]

R. LotfiG. W. WeberS. M. Sajadifar and N. Mardani, Interdependent demand in the two-period newsvendor problem, J. Ind. Manag. Optim., 16 (2020), 117-140.  doi: 10.3934/jimo.2018143.

[26]

T. Maiti and B. C. Giri, A closed loop supply chain under retail price and product quality dependent demand, Journal of Manufacturing Systems, 37 (2015), 624-637.  doi: 10.1016/j.jmsy.2014.09.009.

[27]

A. I. Malik and B. Sarkar, Disruption management in a constrained multi-product imperfect production system, Journal of Manufacturing Systems, 56 (2020), 227-240.  doi: 10.1016/j.jmsy.2020.05.015.

[28]

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

[29]

A. K. MannaJ. K. Dey and S. K. Mondal, Two layers supply chain in an imperfect production inventory model with two storage facilities under reliability consideration, Journal of Industrial and Production Engineering, 35 (2018), 57-73.  doi: 10.1080/21681015.2017.1415230.

[30]

A. Masache, A two-product inventory model with a joint ordering policy, Adv. Decis. Sci., 2013 (2013), Article ID 595074, 5 pp. doi: 10.1155/2013/595074.

[31]

A. H. M. MashudD. RoyY. Daryanto and H. M. Wee, Joint pricing deteriorating inventory model considering product life cycle and advance payment with a discount facility, RAIRO Oper. Res., 55 (2021), 1069-1088.  doi: 10.1051/ro/2020106.

[32]

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Figure 1.  Behaviour of the inventory level over time
Figure 2.  Concavity of the profit function w.r.t. selling price and time
Figure 3.  Concavity of the profit function w.r.t. production rate and time
Figure 4.  Effect of changing $ t_3 $ on optimal solutions
Figure 5.  Effect of changing $ \beta $ on optimal lot size
Figure 6.  Effect of changing $ r_1 $ on optimal solution
Figure 7.  Effect of changing $ b_1 $ on optimal solution
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
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
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
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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