# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021036

## Inventory replenishment policies for two successive generations price-sensitive technology products

 1 Department of Management Studies, BITS Pilani, Pilani Campus, Rajasthan, India 2 Department of Management Studies and Department of Mathematics, BITS Pilani, Pilani Campus, Rajasthan, India

* Corresponding author: Gaurav Nagpal

Received  May 2020 Revised  November 2020 Early access  March 2021

The high technology products come in generations, where the demand for newer technology generations is strongly influenced by the installed base of earlier generations (such as computers, cameras, notebooks, etc). However, the effect of technology substitution on inventory replenishment policies has received little attention in the supply chain literature. In the hi-technology market, consumers' purchasing capability, the utility of a product along with the entry of the advanced generation product influence the market expansion/contraction of the products. In this study, the impact of parallel diffusion of two successive generations' products on inventory policies of the monopolist has been analysed. The demand models have been characterised by considering the life-cycle dynamics for a P-type inventory system. The purpose of this paper is to develop a model for joint pricing and replenishment of technology generation products. The model has been solved by using a genetic algorithm technique. The impact of yearly price drop and the price sensitivity of demand on the profit margins vis-à-vis on replenishment policies has also been studied. The paper also brings forward the dynamics of the launch of newer generations and the pricing strategies on optimal inventory replenishment policies. Numerical illustrations have also been covered in the paper.

Citation: Gaurav Nagpal, Udayan Chanda, Nitant Upasani. Inventory replenishment policies for two successive generations price-sensitive technology products. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021036
##### References:
 [1] F. M. Bass, T. V. Krishnan and D. C. Jain, Why the Bass Model Fits without Decision Variables, Marketing Science, 13 (1994), 203-223.  doi: 10.1287/mksc.13.3.203.  Google Scholar [2] N. Chakraborty, S. Mondal and M. Maiti, A deteriorating multi-item inventory model with price discount and variable demands via fuzzy logic under resource constraints, Computers and Industrial Engineering, 66 (2013), 976-987.  doi: 10.1016/j.cie.2013.08.018.  Google Scholar [3] U. Chanda and R. Aggarwal, Optimal inventory policies for successive generations of a high technology product, The Journal of High Technology Management and Research, 25 (2014), 148-162.   Google Scholar [4] O. Duran and P. P. Luis, Solution of the spare parts joint replenishment problem with quantity discounts using a discrete particle swarm optimization technique, Studies in Informatics and Control, 22 (2013), 319-328.   Google Scholar [5] F. Fang, Joint pricing and inventory decisions for substitutable perishable products under demand uncertainty, Thesis, University of Southampton, (2016). Google Scholar [6] L. Feng, J. Zhang and W. Tang, A joint dynamic pricing and advertising model of perishable products, Journal of the Operational Research Society, 66 (2015), 1341-1351.  doi: 10.1057/jors.2014.89.  Google Scholar [7] M. Ghoreishi, A. Mirzazadeh, G. W. Weber and I. Nakhai-Kamalabadi, Joint pricing and replenishment decisions for non-instantaneous deteriorating items with partial backlogging, inflation- and selling price-dependent demand and customer returns, Journal of Industrial and Management Optimization, 11 (2015), 933-949.  doi: 10.3934/jimo.2015.11.933.  Google Scholar [8] R. N. Giri, S. K. Mondal and M. Maiti, Analysis of pricing decision for substitutable and complimentary products with a common retailer, Pacific Science Review A: Natural Science and Engineering, 18 (2016), 190-202.  doi: 10.1016/j.psra.2016.09.012.  Google Scholar [9] R. N. Giri, S. K. Mondal and M. Maiti, Bundle pricing strategies for two complimentary products with different channel powers, Annals of Operations Research, 287 (2020), 701-725.  doi: 10.1007/s10479-017-2632-y.  Google Scholar [10] M. Jain, G. C. Sharma and R. R. Singh, Multi-item Inventory model with Two Price-breaks under multiple recovery and procurement set-ups, Mathematics Today, 28 (2012), 27-42.   Google Scholar [11] D. K. Jana and B. Das, A two-stage multi-item inventory model with hybrid number and nested price discount via hybrid heuristic algorithm, Annals of Operations Research, 248 (2017), 281-304.  doi: 10.1007/s10479-016-2162-z.  Google Scholar [12] S. Kar, T. Roy and M. Maiti, Multi-item inventory model with probabilistic price dependent demand and imprecise goal and constraints, Yugoslav Journal of Operations Research, 11 (2001), 93-103.   Google Scholar [13] V. B. Kreng and B. J. Wang, An innovation diffusion of successive generations by system dynamics - An empirical study of Nike Golf Company, Technological Forecasting and Social Change, 80 (2013), 77-87.  doi: 10.1016/j.techfore.2012.08.002.  Google Scholar [14] C. W. Kuo and K. L. Huang, Dynamic pricing of limited inventories for multi-generation products, European Journal of Operational Research, 217 (2012), 394-403.  doi: 10.1016/j.ejor.2011.09.020.  Google Scholar [15] C. Y. Lee and D. Lee, An efficient method for solving a correlated multi-item inventory system, Operations Research Perspectives, 5 (2018), 13-21.  doi: 10.1016/j.orp.2017.11.002.  Google Scholar [16] G. Liu, J. Zhang and W. Tang, Joint dynamic pricing and investment strategy for perishable foods with price-quality dependent demand, Annals of Operations Research, 226 (2015), 397-416.  doi: 10.1007/s10479-014-1671-x.  Google Scholar [17] A. Mahmoodi, Joint pricing and inventory control of duopoly retailers with deteriorating items and linear demand, Computers and Industrial Engineering, 132 (2016), 36-46.  doi: 10.1016/j.cie.2019.04.017.  Google Scholar [18] S. M. Mousavi, S. T. A. Niaki, A. Bahreininejad and S. N. Musa, Multi-item multi-periodic inventory control problem with variable demand and discounts: A particle swarm optimization algorithm, The Scientific World Journal, 2014 (2014), Article ID: 136047. doi: 10.1155/2014/136047.  Google Scholar [19] M. Nagarajan and S. Rajagopalan, Inventory models for substitutable products: Optimal policies and heuristics, Management Science, 54 (2008), 1453-1466.   Google Scholar [20] G. Nagpal and U. Chanda, Economic Order Quantity model for two generation consecutive technology products under permissible delay in payments, International Journal of Procurement Management, 14 (2021). doi: 10.1504/IJPM.2020.10027606.  Google Scholar [21] G. Nagpal and U. Chanda, Adoption and diffusion of hi-technology product and related inventory policies - an integrative literature review, International Journal of e-Adoption, 12 (2020), 1-14.   Google Scholar [22] F. Nascimento and W. Vanhonacker, Optimal strategic pricing of reproducible consumer products', Management Science, 34 (1988), 921-937.   Google Scholar [23] T. S. Neda, H. Mirmohammadi and I. Mehdi, Joint optimization of dynamic pricing and replenishment cycle considering variable non-instantaneous deterioration and stock-dependent demand, Computers and Industrial Engineering, 123 (2016), 232-241.   Google Scholar [24] M. Nunez-Lopez, J. X. Velasco-Hernandez and P. A. Marquet, The dynamics of technological change under constraints: Adopters and resources, Discrete and Continuous Dynamical Systems - B, 19 (2014), 3299-3317.  doi: 10.3934/dcdsb.2014.19.3299.  Google Scholar [25] J. A. Norton and F. M. Bass, A diffusion theory model of adoption and substitution for successive generations of high-technology products, Management Science, 33 (1987), 1068-1086.  doi: 10.1287/mnsc.33.9.1069.  Google Scholar [26] F. Otrodi, Y. R. Ghasemy and S. AliTorabi, Joint pricing and lot-sizing for a perishable item under two-level trade credit with multiple demand classes, Computers and Industrial Engineering, 127 (2016), 761-778.   Google Scholar [27] D. Panda and M. Maiti, Multi-item inventory models with price dependent demand under flexibility and reliability consideration and imprecise space constraint: A geometric programming approach, Mathematical and Computer Modelling, 49 (2009), 1733-1749.  doi: 10.1016/j.mcm.2008.10.019.  Google Scholar [28] P. M. Parker, Price elasticity dynamics over the adoption and life cycle, Journal of Marketing Research, 29 (1992), 358-367.   Google Scholar [29] S. Paul, M. I. M. Wahab and P. Ongkunaruk, Joint replenishment with imperfect items and price discount, Computers and Industrial Engineering, 74 (2014), 179-185.  doi: 10.1016/j.cie.2014.05.015.  Google Scholar [30] H. Simon, Dynamic of price elasticity and brand life cycles: An empirical study, Journal of Marketing Research, 16 (1979), 439-452.   Google Scholar [31] K. Y. Tam and K. L. Hui, Price elasticity and the growth of computer spending, IEEE Transactions on Engineering Management, 46 (1999), 190-200.   Google Scholar [32] M. Talebian, N. Boland and M. Savelsbergh, Pricing to accelerate demand learning in dynamic assortment planning, European Journal of Operational Research, 237 (2013), 555-565.  doi: 10.1016/j.ejor.2014.01.045.  Google Scholar [33] A. A. Taleizadeh, S. Tavassoli and A. Bhattacharya, Inventory ordering policies for mixed sale of products under inspection policy, multiple prepayment, partial trade credit, payments linked to order quantity and full backordering, Annals of Operations Research, 287 (2020), 403-437.  doi: 10.1007/s10479-019-03369-x.  Google Scholar [34] S. Tayal, S. R. Singh and R. Sharma, A multi item inventory model for deteriorating items with expiration date and allowable shortages, Indian Journal of Science and Technology, 7 (2014), 463-471.   Google Scholar [35] Y. C. Tsao and G. J. Sheen, A multi-item supply chain with credit periods and weight freight cost discounts, International Journal of Production Economics, 135 (2012), 106-115.  doi: 10.1016/j.ijpe.2010.11.013.  Google Scholar [36] J. Wei and J. Zhao, Pricing decisions for substitutable products with horizontal and vertical competition in fuzzy environments, Annals of Operations Research, 242 (2016), 505-528.  doi: 10.1007/s10479-014-1541-6.  Google Scholar [37] C. T. Yang, L. Y. Ouyang, H. F. Yen and K. L. Lee, Joint pricing and ordering policies for deteriorating item with retail price-dependent demand in response to announced supply price increase, Journal of Industrial and Management Optimization, 9 (2013), 437-454.  doi: 10.3934/jimo.2013.9.437.  Google Scholar

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##### References:
 [1] F. M. Bass, T. V. Krishnan and D. C. Jain, Why the Bass Model Fits without Decision Variables, Marketing Science, 13 (1994), 203-223.  doi: 10.1287/mksc.13.3.203.  Google Scholar [2] N. Chakraborty, S. Mondal and M. Maiti, A deteriorating multi-item inventory model with price discount and variable demands via fuzzy logic under resource constraints, Computers and Industrial Engineering, 66 (2013), 976-987.  doi: 10.1016/j.cie.2013.08.018.  Google Scholar [3] U. Chanda and R. Aggarwal, Optimal inventory policies for successive generations of a high technology product, The Journal of High Technology Management and Research, 25 (2014), 148-162.   Google Scholar [4] O. Duran and P. P. Luis, Solution of the spare parts joint replenishment problem with quantity discounts using a discrete particle swarm optimization technique, Studies in Informatics and Control, 22 (2013), 319-328.   Google Scholar [5] F. Fang, Joint pricing and inventory decisions for substitutable perishable products under demand uncertainty, Thesis, University of Southampton, (2016). Google Scholar [6] L. Feng, J. Zhang and W. Tang, A joint dynamic pricing and advertising model of perishable products, Journal of the Operational Research Society, 66 (2015), 1341-1351.  doi: 10.1057/jors.2014.89.  Google Scholar [7] M. Ghoreishi, A. Mirzazadeh, G. W. Weber and I. Nakhai-Kamalabadi, Joint pricing and replenishment decisions for non-instantaneous deteriorating items with partial backlogging, inflation- and selling price-dependent demand and customer returns, Journal of Industrial and Management Optimization, 11 (2015), 933-949.  doi: 10.3934/jimo.2015.11.933.  Google Scholar [8] R. N. Giri, S. K. Mondal and M. Maiti, Analysis of pricing decision for substitutable and complimentary products with a common retailer, Pacific Science Review A: Natural Science and Engineering, 18 (2016), 190-202.  doi: 10.1016/j.psra.2016.09.012.  Google Scholar [9] R. N. Giri, S. K. Mondal and M. Maiti, Bundle pricing strategies for two complimentary products with different channel powers, Annals of Operations Research, 287 (2020), 701-725.  doi: 10.1007/s10479-017-2632-y.  Google Scholar [10] M. Jain, G. C. Sharma and R. R. Singh, Multi-item Inventory model with Two Price-breaks under multiple recovery and procurement set-ups, Mathematics Today, 28 (2012), 27-42.   Google Scholar [11] D. K. Jana and B. Das, A two-stage multi-item inventory model with hybrid number and nested price discount via hybrid heuristic algorithm, Annals of Operations Research, 248 (2017), 281-304.  doi: 10.1007/s10479-016-2162-z.  Google Scholar [12] S. Kar, T. Roy and M. Maiti, Multi-item inventory model with probabilistic price dependent demand and imprecise goal and constraints, Yugoslav Journal of Operations Research, 11 (2001), 93-103.   Google Scholar [13] V. B. Kreng and B. J. Wang, An innovation diffusion of successive generations by system dynamics - An empirical study of Nike Golf Company, Technological Forecasting and Social Change, 80 (2013), 77-87.  doi: 10.1016/j.techfore.2012.08.002.  Google Scholar [14] C. W. Kuo and K. L. Huang, Dynamic pricing of limited inventories for multi-generation products, European Journal of Operational Research, 217 (2012), 394-403.  doi: 10.1016/j.ejor.2011.09.020.  Google Scholar [15] C. Y. Lee and D. Lee, An efficient method for solving a correlated multi-item inventory system, Operations Research Perspectives, 5 (2018), 13-21.  doi: 10.1016/j.orp.2017.11.002.  Google Scholar [16] G. Liu, J. Zhang and W. Tang, Joint dynamic pricing and investment strategy for perishable foods with price-quality dependent demand, Annals of Operations Research, 226 (2015), 397-416.  doi: 10.1007/s10479-014-1671-x.  Google Scholar [17] A. Mahmoodi, Joint pricing and inventory control of duopoly retailers with deteriorating items and linear demand, Computers and Industrial Engineering, 132 (2016), 36-46.  doi: 10.1016/j.cie.2019.04.017.  Google Scholar [18] S. M. Mousavi, S. T. A. Niaki, A. Bahreininejad and S. N. Musa, Multi-item multi-periodic inventory control problem with variable demand and discounts: A particle swarm optimization algorithm, The Scientific World Journal, 2014 (2014), Article ID: 136047. doi: 10.1155/2014/136047.  Google Scholar [19] M. Nagarajan and S. Rajagopalan, Inventory models for substitutable products: Optimal policies and heuristics, Management Science, 54 (2008), 1453-1466.   Google Scholar [20] G. Nagpal and U. Chanda, Economic Order Quantity model for two generation consecutive technology products under permissible delay in payments, International Journal of Procurement Management, 14 (2021). doi: 10.1504/IJPM.2020.10027606.  Google Scholar [21] G. Nagpal and U. Chanda, Adoption and diffusion of hi-technology product and related inventory policies - an integrative literature review, International Journal of e-Adoption, 12 (2020), 1-14.   Google Scholar [22] F. Nascimento and W. Vanhonacker, Optimal strategic pricing of reproducible consumer products', Management Science, 34 (1988), 921-937.   Google Scholar [23] T. S. Neda, H. Mirmohammadi and I. Mehdi, Joint optimization of dynamic pricing and replenishment cycle considering variable non-instantaneous deterioration and stock-dependent demand, Computers and Industrial Engineering, 123 (2016), 232-241.   Google Scholar [24] M. Nunez-Lopez, J. X. Velasco-Hernandez and P. A. Marquet, The dynamics of technological change under constraints: Adopters and resources, Discrete and Continuous Dynamical Systems - B, 19 (2014), 3299-3317.  doi: 10.3934/dcdsb.2014.19.3299.  Google Scholar [25] J. A. Norton and F. M. Bass, A diffusion theory model of adoption and substitution for successive generations of high-technology products, Management Science, 33 (1987), 1068-1086.  doi: 10.1287/mnsc.33.9.1069.  Google Scholar [26] F. Otrodi, Y. R. Ghasemy and S. AliTorabi, Joint pricing and lot-sizing for a perishable item under two-level trade credit with multiple demand classes, Computers and Industrial Engineering, 127 (2016), 761-778.   Google Scholar [27] D. Panda and M. Maiti, Multi-item inventory models with price dependent demand under flexibility and reliability consideration and imprecise space constraint: A geometric programming approach, Mathematical and Computer Modelling, 49 (2009), 1733-1749.  doi: 10.1016/j.mcm.2008.10.019.  Google Scholar [28] P. M. Parker, Price elasticity dynamics over the adoption and life cycle, Journal of Marketing Research, 29 (1992), 358-367.   Google Scholar [29] S. Paul, M. I. M. Wahab and P. Ongkunaruk, Joint replenishment with imperfect items and price discount, Computers and Industrial Engineering, 74 (2014), 179-185.  doi: 10.1016/j.cie.2014.05.015.  Google Scholar [30] H. Simon, Dynamic of price elasticity and brand life cycles: An empirical study, Journal of Marketing Research, 16 (1979), 439-452.   Google Scholar [31] K. Y. Tam and K. L. Hui, Price elasticity and the growth of computer spending, IEEE Transactions on Engineering Management, 46 (1999), 190-200.   Google Scholar [32] M. Talebian, N. Boland and M. Savelsbergh, Pricing to accelerate demand learning in dynamic assortment planning, European Journal of Operational Research, 237 (2013), 555-565.  doi: 10.1016/j.ejor.2014.01.045.  Google Scholar [33] A. A. Taleizadeh, S. Tavassoli and A. Bhattacharya, Inventory ordering policies for mixed sale of products under inspection policy, multiple prepayment, partial trade credit, payments linked to order quantity and full backordering, Annals of Operations Research, 287 (2020), 403-437.  doi: 10.1007/s10479-019-03369-x.  Google Scholar [34] S. Tayal, S. R. Singh and R. Sharma, A multi item inventory model for deteriorating items with expiration date and allowable shortages, Indian Journal of Science and Technology, 7 (2014), 463-471.   Google Scholar [35] Y. C. Tsao and G. J. Sheen, A multi-item supply chain with credit periods and weight freight cost discounts, International Journal of Production Economics, 135 (2012), 106-115.  doi: 10.1016/j.ijpe.2010.11.013.  Google Scholar [36] J. Wei and J. Zhao, Pricing decisions for substitutable products with horizontal and vertical competition in fuzzy environments, Annals of Operations Research, 242 (2016), 505-528.  doi: 10.1007/s10479-014-1541-6.  Google Scholar [37] C. T. Yang, L. Y. Ouyang, H. F. Yen and K. L. Lee, Joint pricing and ordering policies for deteriorating item with retail price-dependent demand in response to announced supply price increase, Journal of Industrial and Management Optimization, 9 (2013), 437-454.  doi: 10.3934/jimo.2013.9.437.  Google Scholar
The behavior of the price of the technology products with the time elapsed after the launch for different values of annual % price drop "$\Upsilon$"
The P Model of inventory management before the launch of the second generation
The P Model of inventory management in the $m^{th}$ planning horizon with $n$ replenishments before the launch of first-generation
Influence of diffusion rate of second generation product on phase-out timing of 1st generation
Influence of price elasticity of demand on the diffusion pattern and revenues
Tabular review of existing literature on multi-item inventory modeling under price-dependent demand
Optimal number of replenishments ($n_1$ and $n_2$, both of them, say equal n, ) in pooled logistics determined by minimizing the sum of holding cost and carrying costs for each planning horizon (All the costs are in Mn INR)
Total Revenue, Profits and Opportunity Loss for two generations scenario (The Revenue and Profit figure are in Mn INR)
Total Profit values $T{P_{{m^{',{n_1}{n_2}*}}}}$ for different values of yearly price drop % with the change in $\beta$ (The Profit figures are in Mn INR)
Total Profit values $T{P_{{m^{',{n_1}{n_{{2_*}}}}}}}$ for different planning horizons with different values of price sensitivity $\beta$ (The Profit figures are in Mn INR
Comparison of the replenishment dynamics: Joint vs Dis-joint for both the generations of products (The Profit figures are in Mn INR)
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