doi: 10.3934/jimo.2021123
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Optimal replenishment, pricing and preservation technology investment policies for non-instantaneous deteriorating items under two-level trade credit policy

Department of Mathematics, Sidho-Kanho-Birsha University, P.O. – Purulia Sainik School, Dist. – Purulia, PIN – 723104, W.B., India

* Corresponding author: Gour Chandra Mahata

Received  July 2020 Revised  March 2021 Early access July 2021

In the business world, both the supplier and the retailer accept the credit to make their business position strong, because the credit not only strengthens their business relationships but also increases the scale of their profits. In this paper, we consider an inventory model for non-instantaneous deteriorating items with price sensitive demand, time varying deterioration rate under two-level trade credit policy. Besides, to reduce deterioration rate, retailers invest some cost to prevent product degradation/decay, known as preservation technology, is also inserted. Consumption of such items within shelf life prevents to deterioration, which can be achieved by bulk sale. In order to stimulate the selling, trade-credit policy is also considered here. In the sequel, not only the supplier would offer fixed credit period to the retailer, but retailer also adopt the trade credit policy to the customers in order to promote the market competition. The retailer can accumulate revenue and interest after the customer pays for the amount of purchasing cost to the retailer until the end of the trade credit period offered by the supplier. The main objective is to determine the optimal replenishment, pricing and preservation technology investment strategies including whether or not invest in preservation technology and how much to invest in order to maximize the average profit of the system. It is proved that the optimal replenishment policy not only exists but is unique for any given selling price and preservation technology cost. An algorithm is presented to derive the optimal solutions of the model. Numerous theorems and lemmas have been inserted to obtain the optimal solution. Finally, numerical examples and managerial implications are incorporated to validate the proposed model.

Citation: Chandan Mahato, Gour Chandra Mahata. Optimal replenishment, pricing and preservation technology investment policies for non-instantaneous deteriorating items under two-level trade credit policy. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021123
References:
[1]

A. K. BhuniaA. A. ShaikhV. DhakaS. Pareek and L. E. Cárdenas-Barrón, An application of genetic algorithm and PSO in an inventory model for single deteriorating item with variable demand dependent on marketing strategy and displayed stock level, Scientia Iranica, 25 (2018), 1641-1655.   Google Scholar

[2]

A. Cambini and L. Martein, Generalized Convexity and Optimization: Theory and Application, Springer-Verlag Berlin Heidelberg, 2009, USA.  Google Scholar

[3]

C.-T. ChangM.-C. Cheng and L.-Y. Ouyang, Optimal pricing and ordering policies for non-instantaneously deteriorating items under order-size-dependent delay in payments, Appl. Math. Model., 39 (2015), 747-763.  doi: 10.1016/j.apm.2014.07.002.  Google Scholar

[4]

C.-T. ChangJ.-T. Teng and S. K. Goyal, Inventory lot-size models under trade credits: A review, Asia-Pac. J. Oper. Res., 25 (2008), 89-112.  doi: 10.1142/S0217595908001651.  Google Scholar

[5]

T. K. Datta and K. Pal, An inventory system with stock-dependent, price-sensitive demand rate, Production Planning and Control, 12 (2001), 13-20.   Google Scholar

[6]

C. Y. Dye and T. P. Hsieh, Deterministic ordering policy with price and stock-dependent demand under fluctuating cost and limited capacity, Expert Syst. Appl., 38 (2011), 14976-14983.   Google Scholar

[7]

C.-Y. Dye and T.-P. Hsieh, An optimal replenishment policy for deteriorating items with effective investment in preservation technology, European J. Oper. Res., 218 (2012), 106-112.  doi: 10.1016/j.ejor.2011.10.016.  Google Scholar

[8]

C. Y. DyeC. T. Yang and C. C. Wu, Joint dynamic pricing and preservation technology investment for an integrated supply chain with reference price effects, Journal of the Operational Research Society, 69 (2018), 1-14.   Google Scholar

[9]

P. M. Ghare and G. H. Schrader, A Model for Exponentially Decaying Inventory System, International Journal of Production Research, 21 (1963), 449-460.   Google Scholar

[10]

S. K. Goyal and B. C. Giri, Recent trends in modeling of deteriorating inventory, European J. Oper. Res., 134 (2001), 1-16.  doi: 10.1016/S0377-2217(00)00248-4.  Google Scholar

[11]

K.-L. Hou and L.-C. Lin, An EOQ model for deteriorating items with price-and stock-dependent selling rates under inflation and time value of money, Internat. J. Systems Sci., 37 (2006), 1131-1139.  doi: 10.1080/00207720601014206.  Google Scholar

[12]

P. H. HsuH. M. Wee and J. T. Teng, Preservation technology investment for deteriorating inventory, International Journal of Production Economics, 124 (2010), 388-394.   Google Scholar

[13]

Y. F. Huang, Optimal retailer's ordering policies in the EOQ model under trade credit financing, Journal of the Operational research society, 54 (2003), 1011-1015.   Google Scholar

[14]

C. K. Jaggi, P. Gautam and A. Khanna, Inventory Decisions for Imperfect Quality Deteriorating Items with Exponential Declining Demand under Trade Credit and Partially Backlogged Shortages, In: Kapur P., Kumar U., Verma A. (eds) Quality, IT and Business Operations. Springer Proceedings in Business and Economics. Springer, Singapore. doi: 10.1007/978-981-10-5577-5_18.  Google Scholar

[15]

C. JaggiA. Sharma and S. Tiwari, Credit financing in economic ordering policies for non-instantaneous deteriorating items with price dependent demand under permissible delay in payments: A new approach, International Journal of Industrial Engineering Computations, 6 (2015), 481-502.   Google Scholar

[16]

C. JaggiS. Tiwari and S. Goel, Replenishment policy for non-instantaneous deteriorating items in two storage facilities under inflationary conditions, International Journal of Industrial Engineering Computations, 7 (2016), 489-506.   Google Scholar

[17]

C. K. JaggiS. Tiwari and S. K. Goel, Credit financing in economic ordering policies for non-instantaneous deteriorating items with price dependent demand and two storage facilities, Ann. Oper. Res., 248 (2017), 253-280.  doi: 10.1007/s10479-016-2179-3.  Google Scholar

[18]

K. M. KamnaP. Gautam and C. K. Jaggi, Sustainable inventory policy for an imperfect production system with energy usage and volume agility, Int. J. Syst. Assur. Eng. Manag., 12 (2021), 44-52.  doi: 10.1007/s13198-020-01006-6.  Google Scholar

[19]

M. A. A. KhanA. A. ShaikhG. C. Panda and I. Konstantaras, Inventory system with expiration date: Pricing and replenishment decisions, Computers and Industrial Engineering, 132 (2019), 232-247.   Google Scholar

[20]

A. KhannaP. Gautam and C. K. Jaggi, Inventory modeling for deteriorating imperfect quality items with selling price dependent demand and shortage backordering under credit financing, International Journal of Mathematical, Engineering and Management Sciences, 2 (2017), 110-124.   Google Scholar

[21]

M. LashgariA. A. Taleizadeh and S. S. Jafar, Ordering policies for non-instantaneous deteriorating items under hybrid partial prepayment, partial trade credit and partial backordering, Journal of the Operational Research Society, 69 (2018), 1167-1196.   Google Scholar

[22]

J. J. Liao, An EOQ model with noninstantaneous receipt and exponentially deteriorating items under two-level trade credit, Int. J. Prod. Econ., 113 (2008), 852-861.   Google Scholar

[23]

F. LinT. JiaF. Wu and Z. Yang, Impacts of two-stage deterioration on an integrated inventory model under trade credit and variable capacity utilization, European J. Oper. Res., 272 (2019), 219-234.  doi: 10.1016/j.ejor.2018.06.022.  Google Scholar

[24]

G. C. Mahata, An EPQ-based inventory model for exponentially deteriorating items under retailer partial trade credit policy in supply chain, Expert Systems with Applications, 39 (2012), 3537-3550.   Google Scholar

[25]

G. C. Mahata and A. Goswami, Production lot-size model with fuzzy production rate and fuzzy demand rate for deteriorating item under permissible delay in payments, OPSEARCH, 43 (2006), 358-375.   Google Scholar

[26]

P. Mahata and G. C. Mahata, Production and payment policies for an imperfect manufacturing system with discount cash flows analysis in fuzzy random environments, Math. Comput. Model. Dyn. Syst., 26 (2020), 374-408.  doi: 10.1080/13873954.2020.1771380.  Google Scholar

[27]

P. MahataG. C. Mahata and S. K. De, An economic order quantity model under two-level partial trade credit for time varying deteriorating items, Int. J. Syst. Sci. Oper. Log., 7 (2020), 1-17.   Google Scholar

[28]

P. MahataG. C. Mahata and A. Mukherjee, An ordering policy for deteriorating items with price-dependent iso-elastic demand under permissible delay in payments and price inflation, Math. Comput. Model. Dyn. Syst., 25 (2019), 575-601.  doi: 10.1080/13873954.2019.1677724.  Google Scholar

[29]

R. Maihami and I. N. K. Abadi, Joint control of inventory and its pricing for non-instantaneously deteriorating items under permissible delay in payments and partial backlogging, Math. Comput. Modelling, 55 (2012), 1722-1733.  doi: 10.1016/j.mcm.2011.11.017.  Google Scholar

[30]

R. Maihami and B. Karimi, Optimizing the pricing and replenishment policy for non-instantaneous deteriorating items with stochastic demand and promotional efforts, Comput. Oper. Res., 51 (2014), 302-312.  doi: 10.1016/j.cor.2014.05.022.  Google Scholar

[31]

U. Mishra, J. Tijerina-Aguilera, S. Tiwari and L. E. Cárdenas-Barrón, Retailer's joint ordering, pricing and preservation technology investment policies for a deteriorating item under permissible delay in payments, Mathematical Problems in Engineering, 2018 (2018), Article ID 6962417, 14 pages. doi: 10.1155/2018/6962417.  Google Scholar

[32]

A. Mukherjee and G. C. Mahata, Optimal replenishment and credit policy in an inventory model for deteriorating items under two-levels of trade credit policy when demand depends on both time and credit period involving default risk, RAIRO Oper. Res., 52 (2018), 1175-1200.   Google Scholar

[33]

L. Y. OuyangK. S. Wu and C. T. Yang, A study on an inventory model for non-instantaneous deteriorating items with permissible delay in payments, Computers Industrial Engineering, 51 (2006), 637-651.   Google Scholar

[34]

Pr iyamvadaP. GautamA. Khanna and C. K. Jaggi, Preservation technology investment for an inventory system with variable deterioration rate under expiration dates and price sensitive demand, Yugosl. J. Oper. Res., 30 (2020), 287-303.  doi: 10.2298/yjor190410011p.  Google Scholar

[35]

F. Raafat, Survey of literature on continuously deteriorating inventory model, Journal of the Operational Research Society, 42 (1991), 27-37.   Google Scholar

[36]

D. SeifertR. W. Seifert and M. Protopappa-Sieke, A review of trade credit literature: opportunity for research in operations, European Journal of Operational Research, 231 (2013), 245-256.   Google Scholar

[37]

N. H. ShahH. N. Soni and K. A. Patel, Optimizing inventory and marketing policy for non-instantaneous deteriorating items with generalized type deterioration and holding cost rates, Omega, 41 (2013), 421-430.   Google Scholar

[38]

A. A. ShaikhL. E. Cárdenas-BarrónA. K. Bhunia and S. Tiwari, An inventory model of a three parameter Weibull distributed deteriorating item with variable demand dependent on price and frequency of advertisement under trade credit, RAIRO Oper. Res., 53 (2019), 903-916.   Google Scholar

[39]

A. A. ShaikhL. E. Cárdenas-Barrón and S. Tiwari, A two-warehouse inventory model for non-instantaneous deteriorating items with interval valued inventory costs and stock dependent demand under inflationary conditions, Neural Computing and Applications, 31 (2019), 1931-1948.   Google Scholar

[40]

A. ShastriS. R. SinghD. Yadav and S. Gupta, Supply chain management for two-level trade credit financing with selling price dependent demand under the effect of preservation technology, International Journal of Procurement Management, 7 (2014), 695-718.   Google Scholar

[41]

H. N. Soni, Optimal replenishment policies for non-instantaneous deteriorating items with price and stock sensitive demand under permissible delay in payment, International Journal of Production Economics, 146 (2013), 259-268.   Google Scholar

[42]

H. Soni and K. Patel, Optimal pricing and inventory policies for non-instantaneous deteriorating items with permissible delay in payment: Fuzzy expected value model, International Journal of Industrial Engineering Computations, 3 (2012), 281-300.   Google Scholar

[43]

J. T. Teng, Optimal ordering policies for a retailer who offers distinct trade credits to its good and bad credit customers, Int. J. Prod. Econ., 119 (2009), 415-423.   Google Scholar

[44]

J.-T. Teng and C.-T. Chang, Economic production quantity models for deteriorating items with price-and stock-dependent demand, Comput. Oper. Res., 32 (2005), 297-308.  doi: 10.1016/S0305-0548(03)00237-5.  Google Scholar

[45]

J.-T. Teng and S. K. Goyal, Optimal ordering policies for a retailer in a supply chain with up-stream and down-stream trade credits, Journal of Operational Research Society, 58 (2007), 1252-1255.   Google Scholar

[46]

S. TiwariL. E. Cardenas-BarronA. A. Shaikh and M. Goh, Retailer's optimal ordering policy for deteriorating items under order-size dependent trade credit and complete backlogging, Computers and Industrial Engineering, 139 (2020), 105-121.   Google Scholar

[47]

J. WuJ.-T. Teng and and K. Skouri, Optimal inventory policies for deteriorating items with trapezoidal-type demand patterns and maximum lifetimes under upstream and downstream trade credits, Ann. Oper. Res., 264 (2018), 459-476.  doi: 10.1007/s10479-017-2673-2.  Google Scholar

[48]

K. S. WuL. Y. Ouyang and C. T. Yang, An optimal replenishment policy for non-instantaneous deteriorating items with stock-dependent demand and partial backlogging, International Journal of Production Economics, 101 (2006), 369-384.   Google Scholar

[49]

C.-T. YangC.-Y. Dye and J.-F. Ding, Optimal dynamic trade credit and preservation technology allocation for a deteriorating inventory model, Computers and Industrial Engineering, 87 (2015), 356-365.  doi: 10.1016/j.cie.2015.05.027.  Google Scholar

show all references

References:
[1]

A. K. BhuniaA. A. ShaikhV. DhakaS. Pareek and L. E. Cárdenas-Barrón, An application of genetic algorithm and PSO in an inventory model for single deteriorating item with variable demand dependent on marketing strategy and displayed stock level, Scientia Iranica, 25 (2018), 1641-1655.   Google Scholar

[2]

A. Cambini and L. Martein, Generalized Convexity and Optimization: Theory and Application, Springer-Verlag Berlin Heidelberg, 2009, USA.  Google Scholar

[3]

C.-T. ChangM.-C. Cheng and L.-Y. Ouyang, Optimal pricing and ordering policies for non-instantaneously deteriorating items under order-size-dependent delay in payments, Appl. Math. Model., 39 (2015), 747-763.  doi: 10.1016/j.apm.2014.07.002.  Google Scholar

[4]

C.-T. ChangJ.-T. Teng and S. K. Goyal, Inventory lot-size models under trade credits: A review, Asia-Pac. J. Oper. Res., 25 (2008), 89-112.  doi: 10.1142/S0217595908001651.  Google Scholar

[5]

T. K. Datta and K. Pal, An inventory system with stock-dependent, price-sensitive demand rate, Production Planning and Control, 12 (2001), 13-20.   Google Scholar

[6]

C. Y. Dye and T. P. Hsieh, Deterministic ordering policy with price and stock-dependent demand under fluctuating cost and limited capacity, Expert Syst. Appl., 38 (2011), 14976-14983.   Google Scholar

[7]

C.-Y. Dye and T.-P. Hsieh, An optimal replenishment policy for deteriorating items with effective investment in preservation technology, European J. Oper. Res., 218 (2012), 106-112.  doi: 10.1016/j.ejor.2011.10.016.  Google Scholar

[8]

C. Y. DyeC. T. Yang and C. C. Wu, Joint dynamic pricing and preservation technology investment for an integrated supply chain with reference price effects, Journal of the Operational Research Society, 69 (2018), 1-14.   Google Scholar

[9]

P. M. Ghare and G. H. Schrader, A Model for Exponentially Decaying Inventory System, International Journal of Production Research, 21 (1963), 449-460.   Google Scholar

[10]

S. K. Goyal and B. C. Giri, Recent trends in modeling of deteriorating inventory, European J. Oper. Res., 134 (2001), 1-16.  doi: 10.1016/S0377-2217(00)00248-4.  Google Scholar

[11]

K.-L. Hou and L.-C. Lin, An EOQ model for deteriorating items with price-and stock-dependent selling rates under inflation and time value of money, Internat. J. Systems Sci., 37 (2006), 1131-1139.  doi: 10.1080/00207720601014206.  Google Scholar

[12]

P. H. HsuH. M. Wee and J. T. Teng, Preservation technology investment for deteriorating inventory, International Journal of Production Economics, 124 (2010), 388-394.   Google Scholar

[13]

Y. F. Huang, Optimal retailer's ordering policies in the EOQ model under trade credit financing, Journal of the Operational research society, 54 (2003), 1011-1015.   Google Scholar

[14]

C. K. Jaggi, P. Gautam and A. Khanna, Inventory Decisions for Imperfect Quality Deteriorating Items with Exponential Declining Demand under Trade Credit and Partially Backlogged Shortages, In: Kapur P., Kumar U., Verma A. (eds) Quality, IT and Business Operations. Springer Proceedings in Business and Economics. Springer, Singapore. doi: 10.1007/978-981-10-5577-5_18.  Google Scholar

[15]

C. JaggiA. Sharma and S. Tiwari, Credit financing in economic ordering policies for non-instantaneous deteriorating items with price dependent demand under permissible delay in payments: A new approach, International Journal of Industrial Engineering Computations, 6 (2015), 481-502.   Google Scholar

[16]

C. JaggiS. Tiwari and S. Goel, Replenishment policy for non-instantaneous deteriorating items in two storage facilities under inflationary conditions, International Journal of Industrial Engineering Computations, 7 (2016), 489-506.   Google Scholar

[17]

C. K. JaggiS. Tiwari and S. K. Goel, Credit financing in economic ordering policies for non-instantaneous deteriorating items with price dependent demand and two storage facilities, Ann. Oper. Res., 248 (2017), 253-280.  doi: 10.1007/s10479-016-2179-3.  Google Scholar

[18]

K. M. KamnaP. Gautam and C. K. Jaggi, Sustainable inventory policy for an imperfect production system with energy usage and volume agility, Int. J. Syst. Assur. Eng. Manag., 12 (2021), 44-52.  doi: 10.1007/s13198-020-01006-6.  Google Scholar

[19]

M. A. A. KhanA. A. ShaikhG. C. Panda and I. Konstantaras, Inventory system with expiration date: Pricing and replenishment decisions, Computers and Industrial Engineering, 132 (2019), 232-247.   Google Scholar

[20]

A. KhannaP. Gautam and C. K. Jaggi, Inventory modeling for deteriorating imperfect quality items with selling price dependent demand and shortage backordering under credit financing, International Journal of Mathematical, Engineering and Management Sciences, 2 (2017), 110-124.   Google Scholar

[21]

M. LashgariA. A. Taleizadeh and S. S. Jafar, Ordering policies for non-instantaneous deteriorating items under hybrid partial prepayment, partial trade credit and partial backordering, Journal of the Operational Research Society, 69 (2018), 1167-1196.   Google Scholar

[22]

J. J. Liao, An EOQ model with noninstantaneous receipt and exponentially deteriorating items under two-level trade credit, Int. J. Prod. Econ., 113 (2008), 852-861.   Google Scholar

[23]

F. LinT. JiaF. Wu and Z. Yang, Impacts of two-stage deterioration on an integrated inventory model under trade credit and variable capacity utilization, European J. Oper. Res., 272 (2019), 219-234.  doi: 10.1016/j.ejor.2018.06.022.  Google Scholar

[24]

G. C. Mahata, An EPQ-based inventory model for exponentially deteriorating items under retailer partial trade credit policy in supply chain, Expert Systems with Applications, 39 (2012), 3537-3550.   Google Scholar

[25]

G. C. Mahata and A. Goswami, Production lot-size model with fuzzy production rate and fuzzy demand rate for deteriorating item under permissible delay in payments, OPSEARCH, 43 (2006), 358-375.   Google Scholar

[26]

P. Mahata and G. C. Mahata, Production and payment policies for an imperfect manufacturing system with discount cash flows analysis in fuzzy random environments, Math. Comput. Model. Dyn. Syst., 26 (2020), 374-408.  doi: 10.1080/13873954.2020.1771380.  Google Scholar

[27]

P. MahataG. C. Mahata and S. K. De, An economic order quantity model under two-level partial trade credit for time varying deteriorating items, Int. J. Syst. Sci. Oper. Log., 7 (2020), 1-17.   Google Scholar

[28]

P. MahataG. C. Mahata and A. Mukherjee, An ordering policy for deteriorating items with price-dependent iso-elastic demand under permissible delay in payments and price inflation, Math. Comput. Model. Dyn. Syst., 25 (2019), 575-601.  doi: 10.1080/13873954.2019.1677724.  Google Scholar

[29]

R. Maihami and I. N. K. Abadi, Joint control of inventory and its pricing for non-instantaneously deteriorating items under permissible delay in payments and partial backlogging, Math. Comput. Modelling, 55 (2012), 1722-1733.  doi: 10.1016/j.mcm.2011.11.017.  Google Scholar

[30]

R. Maihami and B. Karimi, Optimizing the pricing and replenishment policy for non-instantaneous deteriorating items with stochastic demand and promotional efforts, Comput. Oper. Res., 51 (2014), 302-312.  doi: 10.1016/j.cor.2014.05.022.  Google Scholar

[31]

U. Mishra, J. Tijerina-Aguilera, S. Tiwari and L. E. Cárdenas-Barrón, Retailer's joint ordering, pricing and preservation technology investment policies for a deteriorating item under permissible delay in payments, Mathematical Problems in Engineering, 2018 (2018), Article ID 6962417, 14 pages. doi: 10.1155/2018/6962417.  Google Scholar

[32]

A. Mukherjee and G. C. Mahata, Optimal replenishment and credit policy in an inventory model for deteriorating items under two-levels of trade credit policy when demand depends on both time and credit period involving default risk, RAIRO Oper. Res., 52 (2018), 1175-1200.   Google Scholar

[33]

L. Y. OuyangK. S. Wu and C. T. Yang, A study on an inventory model for non-instantaneous deteriorating items with permissible delay in payments, Computers Industrial Engineering, 51 (2006), 637-651.   Google Scholar

[34]

Pr iyamvadaP. GautamA. Khanna and C. K. Jaggi, Preservation technology investment for an inventory system with variable deterioration rate under expiration dates and price sensitive demand, Yugosl. J. Oper. Res., 30 (2020), 287-303.  doi: 10.2298/yjor190410011p.  Google Scholar

[35]

F. Raafat, Survey of literature on continuously deteriorating inventory model, Journal of the Operational Research Society, 42 (1991), 27-37.   Google Scholar

[36]

D. SeifertR. W. Seifert and M. Protopappa-Sieke, A review of trade credit literature: opportunity for research in operations, European Journal of Operational Research, 231 (2013), 245-256.   Google Scholar

[37]

N. H. ShahH. N. Soni and K. A. Patel, Optimizing inventory and marketing policy for non-instantaneous deteriorating items with generalized type deterioration and holding cost rates, Omega, 41 (2013), 421-430.   Google Scholar

[38]

A. A. ShaikhL. E. Cárdenas-BarrónA. K. Bhunia and S. Tiwari, An inventory model of a three parameter Weibull distributed deteriorating item with variable demand dependent on price and frequency of advertisement under trade credit, RAIRO Oper. Res., 53 (2019), 903-916.   Google Scholar

[39]

A. A. ShaikhL. E. Cárdenas-Barrón and S. Tiwari, A two-warehouse inventory model for non-instantaneous deteriorating items with interval valued inventory costs and stock dependent demand under inflationary conditions, Neural Computing and Applications, 31 (2019), 1931-1948.   Google Scholar

[40]

A. ShastriS. R. SinghD. Yadav and S. Gupta, Supply chain management for two-level trade credit financing with selling price dependent demand under the effect of preservation technology, International Journal of Procurement Management, 7 (2014), 695-718.   Google Scholar

[41]

H. N. Soni, Optimal replenishment policies for non-instantaneous deteriorating items with price and stock sensitive demand under permissible delay in payment, International Journal of Production Economics, 146 (2013), 259-268.   Google Scholar

[42]

H. Soni and K. Patel, Optimal pricing and inventory policies for non-instantaneous deteriorating items with permissible delay in payment: Fuzzy expected value model, International Journal of Industrial Engineering Computations, 3 (2012), 281-300.   Google Scholar

[43]

J. T. Teng, Optimal ordering policies for a retailer who offers distinct trade credits to its good and bad credit customers, Int. J. Prod. Econ., 119 (2009), 415-423.   Google Scholar

[44]

J.-T. Teng and C.-T. Chang, Economic production quantity models for deteriorating items with price-and stock-dependent demand, Comput. Oper. Res., 32 (2005), 297-308.  doi: 10.1016/S0305-0548(03)00237-5.  Google Scholar

[45]

J.-T. Teng and S. K. Goyal, Optimal ordering policies for a retailer in a supply chain with up-stream and down-stream trade credits, Journal of Operational Research Society, 58 (2007), 1252-1255.   Google Scholar

[46]

S. TiwariL. E. Cardenas-BarronA. A. Shaikh and M. Goh, Retailer's optimal ordering policy for deteriorating items under order-size dependent trade credit and complete backlogging, Computers and Industrial Engineering, 139 (2020), 105-121.   Google Scholar

[47]

J. WuJ.-T. Teng and and K. Skouri, Optimal inventory policies for deteriorating items with trapezoidal-type demand patterns and maximum lifetimes under upstream and downstream trade credits, Ann. Oper. Res., 264 (2018), 459-476.  doi: 10.1007/s10479-017-2673-2.  Google Scholar

[48]

K. S. WuL. Y. Ouyang and C. T. Yang, An optimal replenishment policy for non-instantaneous deteriorating items with stock-dependent demand and partial backlogging, International Journal of Production Economics, 101 (2006), 369-384.   Google Scholar

[49]

C.-T. YangC.-Y. Dye and J.-F. Ding, Optimal dynamic trade credit and preservation technology allocation for a deteriorating inventory model, Computers and Industrial Engineering, 87 (2015), 356-365.  doi: 10.1016/j.cie.2015.05.027.  Google Scholar

Figure 1.  Inventory Level
Figure 2.  Inventory system for Case 1 when $ T \leq t_d $
Figure 3.  Inventory system for Case 1 when $ T \leq t_d $
Figure 4.  Inventory system for Case 2 when $ T \leq M-N $
Figure 5.  Inventory system for Case 2 when $ M-N \leq T \leq t_d $
Figure 6.  Inventory system for Case 2 when $ T \geq t_d $
Figure 7.  Inventory system for Case 3 when $ T \leq t_d $
Figure 8.  Inventory system for Case 3 when $ t_d \leq T \leq M-N $
Figure 9.  Inventory system for Case 3 when $ T \leq M-N $
Table 1.  Summary of the related research
Authors Demand factors Demand patterns Deterio-ration Deterio-ration Pattern Non-instan-taneous Trade Credit Level of trade credit Preservation technology
Wu et al. [48] Inventory level Linear Yes Constant Yes No - No
Chang et al. [3] Constant - Yes Constant Yes Yes One No
Hsu et al. [12] Constant - Yes Constant No No - Yes
Shastri et al. [40] Selling price Power form Yes Constant No Yes Two Yes
Dye & Hsieh [7] Constant - Yes Constant No No - Yes
Mahata et al. [28] Selling price Iso-elastic Yes Constant No Yes One No
Maihami and Karimi [29] Selling price General Yes Constant Yes No - No
Mukherjee & Mahata et al. [32] Time & Credit period General Yes General type No Yes Two No
Jaggi et al. [17] Selling price Power form Yes Constant Yes Yes One No
Soni [41] Selling price & stock level General Yes Constant No Yes One No
Shah et al. [37] Advert-isement of an item & selling price power form Yes General type Yes No - Yes
Mishra et al. [31] Selling price Linear Yes Constant No Yes One Yes
Yang et al. [49] Time & Credit period General Yes General type No Yes One Yes
Present paper Selling price General form Yes General type Yes Yes Two Yes
Authors Demand factors Demand patterns Deterio-ration Deterio-ration Pattern Non-instan-taneous Trade Credit Level of trade credit Preservation technology
Wu et al. [48] Inventory level Linear Yes Constant Yes No - No
Chang et al. [3] Constant - Yes Constant Yes Yes One No
Hsu et al. [12] Constant - Yes Constant No No - Yes
Shastri et al. [40] Selling price Power form Yes Constant No Yes Two Yes
Dye & Hsieh [7] Constant - Yes Constant No No - Yes
Mahata et al. [28] Selling price Iso-elastic Yes Constant No Yes One No
Maihami and Karimi [29] Selling price General Yes Constant Yes No - No
Mukherjee & Mahata et al. [32] Time & Credit period General Yes General type No Yes Two No
Jaggi et al. [17] Selling price Power form Yes Constant Yes Yes One No
Soni [41] Selling price & stock level General Yes Constant No Yes One No
Shah et al. [37] Advert-isement of an item & selling price power form Yes General type Yes No - Yes
Mishra et al. [31] Selling price Linear Yes Constant No Yes One Yes
Yang et al. [49] Time & Credit period General Yes General type No Yes One Yes
Present paper Selling price General form Yes General type Yes Yes Two Yes
Table 2.  Sensitivity analysis with respect to different parameters
Parameter Values $ P^{*} $ $ \xi^{*} $ $ T^{*} $ $ Q^{*} $ $ TP^{*} $
80 60.0939 4.7922 0.2097 83.1619 $ TP^{*}_{23}=15543.44 $
100 60.1823 4.9543 0.2514 99.7122 $ TP^{*}_{23}=15456.70 $
A 120 60.2498 5.1039 0.2868 113.7114 $ TP^{*}_{23}=15382.39 $
140 60.3056 5.2469 0.3180 126.0302 $ TP^{*}_{23}=15316.27 $
160 60.3540 5.3864 0.3463 137.1341 $ TP^{*}_{23}=15256.08 $
10 55.2849 5.3094 0.3289 146.8334 $ TP^{*}_{33}=19629.55 $
15 57.7692 5.1910 0.3053 128.6614 $ TP^{*}_{23}=17441.33 $
c 20 60.2498 5.1039 0.2868 113.7114 $ TP^{*}_{23}=15382.39 $
25 62.7284 5.0393 0.2725 101.2497 $ TP^{*}_{23}=13451.34 $
30 65.2071 4.9925 0.2618 90.7764 $ TP^{*}_{23}=11647.08 $
1 60.2001 5.4634 0.3623 144.0668 $ TP^{*}_{33}=15509.14 $
2 60.2272 5.2450 0.3185 126.4466 $ TP^{*}_{23}=15441.97 $
h 3 60.2498 5.1039 0.2868 113.7114 $ TP^{*}_{23}=15382.39 $
4 60.2691 5.0048 0.2626 103.9802 $ TP^{*}_{23}=15328.48 $
5 60.2859 4.9310 0.2433 96.2400 $ TP^{*}_{23}=15279.01 $
0.02 60.2890 5.5741 0.3011 119.4436 $ TP^{*}_{33}=15371.08 $
0.05 60.2768 5.3873 0.2964 117.5873 $ TP^{*}_{23}=15374.53 $
$ t_{0} $ 0.10 60.2498 5.1039 0.2868 113.7114 $ TP^{*}_{23}=15382.39 $
0.15 60.2115 4.8468 0.2742 108.6560 $ TP^{*}_{23}=15393.68 $
0.20 60.1570 4.6081 0.2580 102.1061 $ TP^{*}_{23}=15409.61 $
Parameter Values $ P^{*} $ $ \xi^{*} $ $ T^{*} $ $ Q^{*} $ $ TP^{*} $
80 60.0939 4.7922 0.2097 83.1619 $ TP^{*}_{23}=15543.44 $
100 60.1823 4.9543 0.2514 99.7122 $ TP^{*}_{23}=15456.70 $
A 120 60.2498 5.1039 0.2868 113.7114 $ TP^{*}_{23}=15382.39 $
140 60.3056 5.2469 0.3180 126.0302 $ TP^{*}_{23}=15316.27 $
160 60.3540 5.3864 0.3463 137.1341 $ TP^{*}_{23}=15256.08 $
10 55.2849 5.3094 0.3289 146.8334 $ TP^{*}_{33}=19629.55 $
15 57.7692 5.1910 0.3053 128.6614 $ TP^{*}_{23}=17441.33 $
c 20 60.2498 5.1039 0.2868 113.7114 $ TP^{*}_{23}=15382.39 $
25 62.7284 5.0393 0.2725 101.2497 $ TP^{*}_{23}=13451.34 $
30 65.2071 4.9925 0.2618 90.7764 $ TP^{*}_{23}=11647.08 $
1 60.2001 5.4634 0.3623 144.0668 $ TP^{*}_{33}=15509.14 $
2 60.2272 5.2450 0.3185 126.4466 $ TP^{*}_{23}=15441.97 $
h 3 60.2498 5.1039 0.2868 113.7114 $ TP^{*}_{23}=15382.39 $
4 60.2691 5.0048 0.2626 103.9802 $ TP^{*}_{23}=15328.48 $
5 60.2859 4.9310 0.2433 96.2400 $ TP^{*}_{23}=15279.01 $
0.02 60.2890 5.5741 0.3011 119.4436 $ TP^{*}_{33}=15371.08 $
0.05 60.2768 5.3873 0.2964 117.5873 $ TP^{*}_{23}=15374.53 $
$ t_{0} $ 0.10 60.2498 5.1039 0.2868 113.7114 $ TP^{*}_{23}=15382.39 $
0.15 60.2115 4.8468 0.2742 108.6560 $ TP^{*}_{23}=15393.68 $
0.20 60.1570 4.6081 0.2580 102.1061 $ TP^{*}_{23}=15409.61 $
Table 3.  The numerical results for different values of M and N
M N $ P^{*} $ $ \xi^{*} $ $ T^{*} $ $ Q^{*} $ $ TP^{*} $
0.0 60.4012 5.1072 0.2875 112.4756 $ TP^{*}_{23}=15263.37 $
0.2 60.7039 5.1139 0.2890 112.7563 $ TP^{*}_{12}=15026.68 $
0.0 0.4 61.0067 5.1207 0.2906 113.0335 $ TP^{*}_{12}=14791.82 $
0.6 61.3095 5.1276 0.2921 113.3073 $ TP^{*}_{12}=14558.77 $
0.8 61.6124 5.1345 0.2937 113.5776 $ TP^{*}_{12}=14327.53 $
0.0 60.0984 5.1006 0.2860 112.7563 $ TP^{*}_{33}=15501.87 $
0.2 60.4212 5.1085 0.2879 113.0335 $ TP^{*}_{23}=15287.52 $
0.2 0.4 61.7039 5.1139 0.2890 113.3073 $ TP^{*}_{12}=15026.68 $
0.6 61.0067 5.1207 0.2906 113.5776 $ TP^{*}_{12}=14791.82 $
0.8 61.3095 5.1276 0.2921 113.8444 $ TP^{*}_{12}=14558.77 $
0.0 59.7957 5.0941 0.2845 113.0335 $ TP^{*}_{32}=15742.19 $
0.2 60.0984 5.1006 0.2860 113.3073 $ TP^{*}_{33}=15501.87 $
0.4 0.4 60.4302 5.1092 0.2881 113.5776 $ TP^{*}_{32}=15293.23 $
0.6 60.7039 5.1139 0.2892 113.8444 $ TP^{*}_{12}=15026.68 $
0.8 61.0067 5.1207 0.2906 114.1078 $ TP^{*}_{12}=14791.82 $
0.0 59.4930 5.0877 0.2831 113.3073 $ TP^{*}_{32}=15984.32 $
0.2 59.7957 5.0941 0.2845 113.5776 $ TP^{*}_{32}=15742.19 $
0.6 0.4 60.0984 5.1006 0.2860 113.8444 $ TP^{*}_{33}=15501.87 $
0.6 60.4612 5.1102 0.2878 114.1078 $ TP^{*}_{23}=15301.15 $
0.8 60.7039 5.1139 0.2890 114.3679 $ TP^{*}_{12}=15026.68 $
0.0 59.1903 5.0814 0.2816 113.5776 $ TP^{*}_{32}=16228.27 $
0.2 59.4930 5.0877 0.2831 113.8444 $ TP^{*}_{32}=15984.32 $
0.8 0.4 59.7957 5.0941 0.2845 114.1078 $ TP^{*}_{32}=15742.19 $
0.6 60.0984 5.1006 0.2867 114.3679 $ TP^{*}_{33}=15501.87 $
0.8 60.4825 5.1172 0.2885 114.6246 $ TP^{*}_{23}=15374.12 $
M N $ P^{*} $ $ \xi^{*} $ $ T^{*} $ $ Q^{*} $ $ TP^{*} $
0.0 60.4012 5.1072 0.2875 112.4756 $ TP^{*}_{23}=15263.37 $
0.2 60.7039 5.1139 0.2890 112.7563 $ TP^{*}_{12}=15026.68 $
0.0 0.4 61.0067 5.1207 0.2906 113.0335 $ TP^{*}_{12}=14791.82 $
0.6 61.3095 5.1276 0.2921 113.3073 $ TP^{*}_{12}=14558.77 $
0.8 61.6124 5.1345 0.2937 113.5776 $ TP^{*}_{12}=14327.53 $
0.0 60.0984 5.1006 0.2860 112.7563 $ TP^{*}_{33}=15501.87 $
0.2 60.4212 5.1085 0.2879 113.0335 $ TP^{*}_{23}=15287.52 $
0.2 0.4 61.7039 5.1139 0.2890 113.3073 $ TP^{*}_{12}=15026.68 $
0.6 61.0067 5.1207 0.2906 113.5776 $ TP^{*}_{12}=14791.82 $
0.8 61.3095 5.1276 0.2921 113.8444 $ TP^{*}_{12}=14558.77 $
0.0 59.7957 5.0941 0.2845 113.0335 $ TP^{*}_{32}=15742.19 $
0.2 60.0984 5.1006 0.2860 113.3073 $ TP^{*}_{33}=15501.87 $
0.4 0.4 60.4302 5.1092 0.2881 113.5776 $ TP^{*}_{32}=15293.23 $
0.6 60.7039 5.1139 0.2892 113.8444 $ TP^{*}_{12}=15026.68 $
0.8 61.0067 5.1207 0.2906 114.1078 $ TP^{*}_{12}=14791.82 $
0.0 59.4930 5.0877 0.2831 113.3073 $ TP^{*}_{32}=15984.32 $
0.2 59.7957 5.0941 0.2845 113.5776 $ TP^{*}_{32}=15742.19 $
0.6 0.4 60.0984 5.1006 0.2860 113.8444 $ TP^{*}_{33}=15501.87 $
0.6 60.4612 5.1102 0.2878 114.1078 $ TP^{*}_{23}=15301.15 $
0.8 60.7039 5.1139 0.2890 114.3679 $ TP^{*}_{12}=15026.68 $
0.0 59.1903 5.0814 0.2816 113.5776 $ TP^{*}_{32}=16228.27 $
0.2 59.4930 5.0877 0.2831 113.8444 $ TP^{*}_{32}=15984.32 $
0.8 0.4 59.7957 5.0941 0.2845 114.1078 $ TP^{*}_{32}=15742.19 $
0.6 60.0984 5.1006 0.2867 114.3679 $ TP^{*}_{33}=15501.87 $
0.8 60.4825 5.1172 0.2885 114.6246 $ TP^{*}_{23}=15374.12 $
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