American Institute of Mathematical Sciences

June 2018, 8(2): 169-191. doi: 10.3934/naco.2018010

An integrated inventory model with variable holding cost under two levels of trade-credit policy

 1 Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore-721102, West Bengal, India 2 Faculty of Engineering Management, Chair of Marketing and Economic Engineering, Poznan University of Technology, Ul. Strzelecka 11, 60-965 Poznan, Poland

* Corresponding author: sankroy2006@gmail.com
The reviewing process of this paper was handled by Associate Editors A. (Nima) Mirzazadeh, Kharazmi University, Tehran, Iran, and Gerhard-Wilhelm Weber, Middle East Technical University, Ankara, Turkey. "This paper was for the occasion of The 12th International Conference on Industrial Engineering (ICIE 2016), which was held in Tehran, Iran during 25-26 January, 2016".The author, Magfura Pervin is very much thankful to University Grants Commission (UGC) of India for providing financial support to continue this research work under [MANF(UGC)] scheme: Sanctioned letter number [F1-17.1/2012-13/MANF-2012-13-MUS-WES-19170/(SA-Ⅲ/Website)] dated 28/02/2013

Received  February 2017 Revised  October 2017 Published  May 2018

Fund Project: The research of Gerhard-Wilhelm Weber (Institute of Applied Mathematics, Middle East Technical University, 06800, Ankara, Turkey) is partially supported by the Portuguese Foundation for Science and Technology ("FCT-Fundação para a Ciência e a Tecnologia"), through the CIDMA -Center for Research and Development in Mathematics and Applications, within project UID/MAT/ 04106/2013

This paper presents an integrated vendor-buyer model for deteriorating items. We assume that the deterioration follows a constant rate with respect to time. The vendor allows a certain credit period to buyer in order to promote the market competition. Keeping in mind the competition of modern age, the stock-dependent demand rate is included in the formulated model which is a new policy to attract more customers. Shortages are allowed for the model to give the model more realistic sense. Partial backordering is offered for the interested customers, and there is a lost-sale cost during the shortage interval. The traditional parameter of holding cost is considered here as time-dependent. Henceforth, an easy solution procedure to find the optimal order quantity is presented so that the total relevant cost per unit time will be minimized. The mathematical formation is explored by numerical examples to validate the proposed model. A sensitivity analysis of the optimal solution for important parameters is also carried out to modify the result of the model.

Citation: Magfura Pervin, Sankar Kumar Roy, Gerhard Wilhelm Weber. An integrated inventory model with variable holding cost under two levels of trade-credit policy. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 169-191. doi: 10.3934/naco.2018010
References:
 [1] S. P. Aggarwal and C. K. Jaggi, Ordering policies of deteriorating items under permissible delay in payments, The Journal of the Operational Research Society, 46 (1995), 658-662. [2] Z. T. Balkhi and L. Benkherouf, On an inventory model for deteriorating items with stock dependent and time-varying demand rates, Computers and Operations Research, 31 (2004), 223-240. doi: 10.1016/S0305-0548(02)00182-X. [3] H. Chang and C. Dye, An EOQ model for deteriorating items with time varying demand and partial backlogging, The Journal of the Operational Research Society, 50 (1999), 1176-1182. [4] K. J. Chung and J. J. Liao, Lot-sizing decisions under trade credit depending on the ordering quantity, Computers and Operations Research, 31 (2004), 909-928. doi: 10.1016/S0305-0548(03)00043-1. [5] T. K. Datta and K. Paul, An inventory system with stock-dependent, price-sensitive demand rate, Production Planning and Control, 12 (2001), 13-20. doi: 10.1080/09537280150203933. [6] P. M. Ghare and G. P. Schrader, A model for an exponentially decaying inventory, Journal of Industrial Engineering, 14 (1963), 238-243. [7] B. C. Giri, A. Goswami and K. S. Chaudhuri, An EOQ model for deteriorating items with time-varying demand and costs, The Journal of the Operational Research Society, 47(11) (1996), 1398-1405. [8] M. Goh, EOQ models with general demand and holding costs functions, European Journal of Operational Research, 73 (1994), 50-54. doi: 10.1016/0377-2217(94)90141-4. [9] A. Goswami and K. S. Chaudhuri, An economic order quantity model for items with two levels of storage for a linear trend in demand, The Journal of the Operational Research Society, 43 (1997), 157-167. [10] S. K. Goyal, Economic order quantity under conditions of permissible delay in payments, The Journal of Operational Research Society, 36 (1985), 335-338. [11] S. K. Goyal, A joint economic lot size model for purchaser and vendor: A comment, Decision Sciences, 19 (1988), 236-241. doi: 10.1111/j.1540-5915.1988.tb00264.x. [12] F. W. Harris, Operations and Cost, A. W. Shaw Company, Chicago, 1915. [13] C. K. Huang, D. W. Tsai, J. C. Wu and K. J. Chung, An optimal integrated vendor-buyer inventory policy under conditions of order-processing time reduction and permissible delay in payments, International Journal of Production Economics, 128 (2010), 445-451. doi: 10.1016/j.ijpe.2010.08.001. [14] C. K. Jaggi, S. K. Goel and M. Mittal, Credit financing in economic ordering policies for defective items with allowable shortages, Applied Mathematics and Computation, 219 (2013), 5268-5282. doi: 10.1016/j.amc.2012.11.027. [15] F. Jolai, R. Tavakkoli-Moghaddam, M. Rabbani and M. R. Sadoughian, An economic production lot size model with deteriorating items, stock-dependent demand, inflation and partial backlogging, Applied Mathematics and Computation, 181 (2006), 380-389. doi: 10.1016/j.amc.2006.01.039. [16] S. Khalilpourazari and S. H. R. Pasandideh, Multi-item EOQ model with nonlinear unit holding cost and partial backordering: moth-flame optimization algorithm, Journal of Industrial and Production Engineering, 34 (2017), 42-51. doi: 10.1080/21681015.2016.1192068. [17] S. Khalilpourazari, S. H. R. Pasandideh and S. T. A. Niaki, Optimization of multi-product economic production quantity model with partial backordering and physical constraints: SQP, SFS, SA and WCA, Applied Soft Computing, 49 (2016), 770-791. doi: 10.1016/j.asoc.2016.08.054. [18] S. T. Law and H. M. Wee, An integrated production-inventory model for ameliorating and deteriorating items taking account of time discounting, Mathematical and Computer Modelling, 43 (2006), 673-685. doi: 10.1016/j.mcm.2005.12.012. [19] J. J. Liao, An EOQ model with non instantaneous receipt and exponentially deteriorating items under two-level trade credit, International Journal of Production Economics, 113 (2008), 852-861. [20] C. J. Liao and C. H. Shyu, An analytical determination of lead time with normal demand, International Journal of Operations & Production Management, 11 (1991), 72-78. doi: 10.1108/EUM0000000001287. [21] S. T. Lo, H. M. Wee and W. C. Huang, An integrated production-inventory model with imperfect production process and Weibull distribution deterioration under inflation, International Journal of Production Economics, 106 (2007), 248-260. [22] 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. doi: 10.1016/j.eswa.2011.09.044. [23] L. Y. Ouyang, K. S. Wu and C. H. Ho, An integrated vendor-buyer model with quality improvement and lead time reduction, International Journal of Production Economics, 108 (2007), 349-358. doi: 10.1016/j.ijpe.2006.12.019. [24] M. Pervin, G. C. Mahata and S. K. Roy, An inventory model with demand declining market for deteriorating items under trade credit policy, International Journal of Management Science and Engineering Management, 11 (2016), 243-251. doi: 10.1080/17509653.2015.1081082. [25] M. Pervin, S. K. Roy and G. W. Weber, Analysis of inventory control model with shortage under time-dependent demand and time-varying holding cost including stochastic deterioration, Annals of Operations Research, 260 (2018), 437-460. doi: 10.1007/s10479-016-2355-5. [26] M. Pervin, S. K. Roy and G. W. Weber, A Two-echelon inventory model with stock-dependent demand and variable holding cost for deteriorating items, Numerical Algebra, Control and Optimization, 7 (2017), 21-50. doi: 10.3934/naco.2017002. [27] San José, Sicilia and García-Laguna, Analysis of an inventory system with exponential partial backordering, International Journal of Production Economics, 100 (2006), 76-86. [28] J. T. Teng, J. Chen and S. K. Goyal, A comprehensive note on an inventory model under two levels of trade credit and limited storage space derived without derivatives, Applied Mathematical Modelling, 33 (2009), 4388-4396. doi: 10.1016/j.apm.2009.03.010. [29] J. T. Teng and C. T. Chang, Economic production quantity models for deteriorating items with price-and stock-dependent demand, Computers and Operations Research, 32 (2005), 297-308. doi: 10.1016/S0305-0548(03)00237-5. [30] R. P. Tripathi and H. S. Pandey, An EOQ model for deteriorating item with Weibull time dependent demand rate under trade credits, International Journal of Information and Management Sciences, 24 (2013), 329-347. [31] G. A. Widyadana, L. E. Cárdenas-Barrón and H. M. Wee, Economic order quantity model for deteriorating items with planned backorder level, Mathematical and Computer Modelling, 54 (2011), 1569-1575. doi: 10.1016/j.mcm.2011.04.028.

show all references

References:
 [1] S. P. Aggarwal and C. K. Jaggi, Ordering policies of deteriorating items under permissible delay in payments, The Journal of the Operational Research Society, 46 (1995), 658-662. [2] Z. T. Balkhi and L. Benkherouf, On an inventory model for deteriorating items with stock dependent and time-varying demand rates, Computers and Operations Research, 31 (2004), 223-240. doi: 10.1016/S0305-0548(02)00182-X. [3] H. Chang and C. Dye, An EOQ model for deteriorating items with time varying demand and partial backlogging, The Journal of the Operational Research Society, 50 (1999), 1176-1182. [4] K. J. Chung and J. J. Liao, Lot-sizing decisions under trade credit depending on the ordering quantity, Computers and Operations Research, 31 (2004), 909-928. doi: 10.1016/S0305-0548(03)00043-1. [5] T. K. Datta and K. Paul, An inventory system with stock-dependent, price-sensitive demand rate, Production Planning and Control, 12 (2001), 13-20. doi: 10.1080/09537280150203933. [6] P. M. Ghare and G. P. Schrader, A model for an exponentially decaying inventory, Journal of Industrial Engineering, 14 (1963), 238-243. [7] B. C. Giri, A. Goswami and K. S. Chaudhuri, An EOQ model for deteriorating items with time-varying demand and costs, The Journal of the Operational Research Society, 47(11) (1996), 1398-1405. [8] M. Goh, EOQ models with general demand and holding costs functions, European Journal of Operational Research, 73 (1994), 50-54. doi: 10.1016/0377-2217(94)90141-4. [9] A. Goswami and K. S. Chaudhuri, An economic order quantity model for items with two levels of storage for a linear trend in demand, The Journal of the Operational Research Society, 43 (1997), 157-167. [10] S. K. Goyal, Economic order quantity under conditions of permissible delay in payments, The Journal of Operational Research Society, 36 (1985), 335-338. [11] S. K. Goyal, A joint economic lot size model for purchaser and vendor: A comment, Decision Sciences, 19 (1988), 236-241. doi: 10.1111/j.1540-5915.1988.tb00264.x. [12] F. W. Harris, Operations and Cost, A. W. Shaw Company, Chicago, 1915. [13] C. K. Huang, D. W. Tsai, J. C. Wu and K. J. Chung, An optimal integrated vendor-buyer inventory policy under conditions of order-processing time reduction and permissible delay in payments, International Journal of Production Economics, 128 (2010), 445-451. doi: 10.1016/j.ijpe.2010.08.001. [14] C. K. Jaggi, S. K. Goel and M. Mittal, Credit financing in economic ordering policies for defective items with allowable shortages, Applied Mathematics and Computation, 219 (2013), 5268-5282. doi: 10.1016/j.amc.2012.11.027. [15] F. Jolai, R. Tavakkoli-Moghaddam, M. Rabbani and M. R. Sadoughian, An economic production lot size model with deteriorating items, stock-dependent demand, inflation and partial backlogging, Applied Mathematics and Computation, 181 (2006), 380-389. doi: 10.1016/j.amc.2006.01.039. [16] S. Khalilpourazari and S. H. R. Pasandideh, Multi-item EOQ model with nonlinear unit holding cost and partial backordering: moth-flame optimization algorithm, Journal of Industrial and Production Engineering, 34 (2017), 42-51. doi: 10.1080/21681015.2016.1192068. [17] S. Khalilpourazari, S. H. R. Pasandideh and S. T. A. Niaki, Optimization of multi-product economic production quantity model with partial backordering and physical constraints: SQP, SFS, SA and WCA, Applied Soft Computing, 49 (2016), 770-791. doi: 10.1016/j.asoc.2016.08.054. [18] S. T. Law and H. M. Wee, An integrated production-inventory model for ameliorating and deteriorating items taking account of time discounting, Mathematical and Computer Modelling, 43 (2006), 673-685. doi: 10.1016/j.mcm.2005.12.012. [19] J. J. Liao, An EOQ model with non instantaneous receipt and exponentially deteriorating items under two-level trade credit, International Journal of Production Economics, 113 (2008), 852-861. [20] C. J. Liao and C. H. Shyu, An analytical determination of lead time with normal demand, International Journal of Operations & Production Management, 11 (1991), 72-78. doi: 10.1108/EUM0000000001287. [21] S. T. Lo, H. M. Wee and W. C. Huang, An integrated production-inventory model with imperfect production process and Weibull distribution deterioration under inflation, International Journal of Production Economics, 106 (2007), 248-260. [22] 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. doi: 10.1016/j.eswa.2011.09.044. [23] L. Y. Ouyang, K. S. Wu and C. H. Ho, An integrated vendor-buyer model with quality improvement and lead time reduction, International Journal of Production Economics, 108 (2007), 349-358. doi: 10.1016/j.ijpe.2006.12.019. [24] M. Pervin, G. C. Mahata and S. K. Roy, An inventory model with demand declining market for deteriorating items under trade credit policy, International Journal of Management Science and Engineering Management, 11 (2016), 243-251. doi: 10.1080/17509653.2015.1081082. [25] M. Pervin, S. K. Roy and G. W. Weber, Analysis of inventory control model with shortage under time-dependent demand and time-varying holding cost including stochastic deterioration, Annals of Operations Research, 260 (2018), 437-460. doi: 10.1007/s10479-016-2355-5. [26] M. Pervin, S. K. Roy and G. W. Weber, A Two-echelon inventory model with stock-dependent demand and variable holding cost for deteriorating items, Numerical Algebra, Control and Optimization, 7 (2017), 21-50. doi: 10.3934/naco.2017002. [27] San José, Sicilia and García-Laguna, Analysis of an inventory system with exponential partial backordering, International Journal of Production Economics, 100 (2006), 76-86. [28] J. T. Teng, J. Chen and S. K. Goyal, A comprehensive note on an inventory model under two levels of trade credit and limited storage space derived without derivatives, Applied Mathematical Modelling, 33 (2009), 4388-4396. doi: 10.1016/j.apm.2009.03.010. [29] J. T. Teng and C. T. Chang, Economic production quantity models for deteriorating items with price-and stock-dependent demand, Computers and Operations Research, 32 (2005), 297-308. doi: 10.1016/S0305-0548(03)00237-5. [30] R. P. Tripathi and H. S. Pandey, An EOQ model for deteriorating item with Weibull time dependent demand rate under trade credits, International Journal of Information and Management Sciences, 24 (2013), 329-347. [31] G. A. Widyadana, L. E. Cárdenas-Barrón and H. M. Wee, Economic order quantity model for deteriorating items with planned backorder level, Mathematical and Computer Modelling, 54 (2011), 1569-1575. doi: 10.1016/j.mcm.2011.04.028.
Graphical representation of Inventory model for a Vendor
Graphical representation of a Buyer's Inventory model
The convexity of the integrated cost described in Example 1. Included are $T$ , $M$ and the integrated cost $TC$ , along the x-axis, the y-axis and the z-axis, respectively
The convexity of the integrated cost showed in Example 2. Represented are $T$ , $M$ and the integrated cost $TC$ , along the x-axis, the y-axis and the z-axis, respectively
The convexity of the integrated cost presented in Example 3. Followed are $T$ , $M$ and the integrated cost $TC$ , along the x-axis, the y-axis and the z-axis, respectively
$TC$ vs. $T$ at $M = 0.1091$
$TC$ vs. $M$ at $T = 0.0870$
Sensitivity analysis of $\theta$
Sensitivity analysis of $\alpha$
Sensitivity analysis of $h$
Schematic view of all parameters
Previous works of different authors in this field including our work
 Author(s) Ⅰ Ⅱ Ⅲ Ⅳ Ⅴ Ⅵ Aggarwal and Jaggi [1] $\surd$ $\surd$ Ouyang et al. [23] $\surd$ Huang et al. [13] $\surd$ $\surd$ Tripathi and Pandey [30] $\surd$ $\surd$ Mahata [22] $\surd$ $\surd$ Goswami and Chaudhuri [9] $\surd$ $\surd$ Teng and Chang [29] $\surd$ $\surd$ Law and Wee [18] $\surd$ $\surd$ $\surd$ Jolai et al. [15] $\surd$ $\surd$ $\surd$ Lo et al. [21] $\surd$ $\surd$ Pervin et al. [24] $\surd$ $\surd$ Pervin et al. [25] $\surd$ $\surd$ $\surd$ Our paper $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
 Author(s) Ⅰ Ⅱ Ⅲ Ⅳ Ⅴ Ⅵ Aggarwal and Jaggi [1] $\surd$ $\surd$ Ouyang et al. [23] $\surd$ Huang et al. [13] $\surd$ $\surd$ Tripathi and Pandey [30] $\surd$ $\surd$ Mahata [22] $\surd$ $\surd$ Goswami and Chaudhuri [9] $\surd$ $\surd$ Teng and Chang [29] $\surd$ $\surd$ Law and Wee [18] $\surd$ $\surd$ $\surd$ Jolai et al. [15] $\surd$ $\surd$ $\surd$ Lo et al. [21] $\surd$ $\surd$ Pervin et al. [24] $\surd$ $\surd$ Pervin et al. [25] $\surd$ $\surd$ $\surd$ Our paper $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Sensitivity analysis for different parameters involved in Example 1
 Parameter parametric value after % change $T$ $M$ $TC_V$ $TC_B$ $TC$ $A$ 375 0.0910 0.1129 4527.31 4032.05 3461.12 312.5 0.0873 0.1091 4578.44 4113.10 3478.69 275 0.0821 0.1051 4629.73 4150.03 3510.85 $s$ 1.35 0.0674 0.1104 4655.28 4187.46 3382.38 1.125 0.0763 0.1096 4022.38 3810.14 3412.19 0.99 0.0791 0.1052 4065.19 3843.52 3487.71 $c$ 120 0.0683 0.1027 4122.16 3874.00 3560.31 100 0.0724 0.1035 4203.23 3890.64 3619.41 88 0.0812 0.1037 4247.11 3915.26 3670.10 $a$ 0.9 0.0690 0.1366 3877.35 3682.34 3510.94 0.75 0.0728 0.1380 3852.55 3650.34 3422.30 0.66 0.0749 0.1418 3796.21 3614.57 3392.62 $b$ 1.2 0.0467 0.1510 3785.04 3654.87 3473.22 1.0 0.0511 0.1572 3751.39 3627.95 3376.99 0.88 0.0585 0.1618 3728.45 3567.53 3108.04 $c_1$ 60 0.0814 0.1136 4242.33 3735.10 3630.44 50 0.0889 0.1219 4407.61 3846.72 3780.00 44 0.0926 0.1305 4521.17 3918.34 3822.35 $c_2$ 90 0.0672 0.1158 4176.88 3839.15 3610.23 75 0.0779 0.1227 4366.08 3882.46 3679.14 66 0.0837 0.1293 4541.00 3918.27 3746.22 $\alpha$ 450 0.0832 0.1745 3983.52 3728.61 3555.22 375 0.0811 0.1659 3875.00 3984.17 3487.20 330 0.0768 0.1633 3854.33 3729.26 3390.38 $\beta$ 1.05 0.0577 0.1594 4539.74 4218.40 3671.99 0.875 0.0597 0.1677 4487.23 4179.30 3647.19 0.77 0.0649 0.1626 4435.41 4027.64 3557.73 $I_e$ 0.285 0.0855 0.1547 4923.07 4500.68 3750.08 0.2375 0.0732 0.1588 4846.31 4483.47 3921.39 0.209 0.0611 0.1470 4736.58 4375.07 3982.63 $\theta$ 1.35 0.0769 0.1720 4739.13 4460.53 3699.28 1.125 0.0732 0.1693 4688.57 4379.24 3642.57 0.99 0.0684 0.1647 4635.14 4316.20 3571.26 $\delta$ 0.9 0.0592 0.1281 4037.45 3658.74 3429.50 0.75 0.0633 0.1357 4168.70 3791.26 3557.35 0.66 0.0748 0.1414 4231.05 3834.00 3658.10
 Parameter parametric value after % change $T$ $M$ $TC_V$ $TC_B$ $TC$ $A$ 375 0.0910 0.1129 4527.31 4032.05 3461.12 312.5 0.0873 0.1091 4578.44 4113.10 3478.69 275 0.0821 0.1051 4629.73 4150.03 3510.85 $s$ 1.35 0.0674 0.1104 4655.28 4187.46 3382.38 1.125 0.0763 0.1096 4022.38 3810.14 3412.19 0.99 0.0791 0.1052 4065.19 3843.52 3487.71 $c$ 120 0.0683 0.1027 4122.16 3874.00 3560.31 100 0.0724 0.1035 4203.23 3890.64 3619.41 88 0.0812 0.1037 4247.11 3915.26 3670.10 $a$ 0.9 0.0690 0.1366 3877.35 3682.34 3510.94 0.75 0.0728 0.1380 3852.55 3650.34 3422.30 0.66 0.0749 0.1418 3796.21 3614.57 3392.62 $b$ 1.2 0.0467 0.1510 3785.04 3654.87 3473.22 1.0 0.0511 0.1572 3751.39 3627.95 3376.99 0.88 0.0585 0.1618 3728.45 3567.53 3108.04 $c_1$ 60 0.0814 0.1136 4242.33 3735.10 3630.44 50 0.0889 0.1219 4407.61 3846.72 3780.00 44 0.0926 0.1305 4521.17 3918.34 3822.35 $c_2$ 90 0.0672 0.1158 4176.88 3839.15 3610.23 75 0.0779 0.1227 4366.08 3882.46 3679.14 66 0.0837 0.1293 4541.00 3918.27 3746.22 $\alpha$ 450 0.0832 0.1745 3983.52 3728.61 3555.22 375 0.0811 0.1659 3875.00 3984.17 3487.20 330 0.0768 0.1633 3854.33 3729.26 3390.38 $\beta$ 1.05 0.0577 0.1594 4539.74 4218.40 3671.99 0.875 0.0597 0.1677 4487.23 4179.30 3647.19 0.77 0.0649 0.1626 4435.41 4027.64 3557.73 $I_e$ 0.285 0.0855 0.1547 4923.07 4500.68 3750.08 0.2375 0.0732 0.1588 4846.31 4483.47 3921.39 0.209 0.0611 0.1470 4736.58 4375.07 3982.63 $\theta$ 1.35 0.0769 0.1720 4739.13 4460.53 3699.28 1.125 0.0732 0.1693 4688.57 4379.24 3642.57 0.99 0.0684 0.1647 4635.14 4316.20 3571.26 $\delta$ 0.9 0.0592 0.1281 4037.45 3658.74 3429.50 0.75 0.0633 0.1357 4168.70 3791.26 3557.35 0.66 0.0748 0.1414 4231.05 3834.00 3658.10
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