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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 |
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.
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. |









Author(s) | Ⅰ | Ⅱ | Ⅲ | Ⅳ | Ⅴ | Ⅵ |
Aggarwal and Jaggi [1] | ||||||
Ouyang et al. [23] | ||||||
Huang et al. [13] | ||||||
Tripathi and Pandey [30] | ||||||
Mahata [22] | ||||||
Goswami and Chaudhuri [9] | ||||||
Teng and Chang [29] | ||||||
Law and Wee [18] | ||||||
Jolai et al. [15] | ||||||
Lo et al. [21] | ||||||
Pervin et al. [24] | ||||||
Pervin et al. [25] | ||||||
Our paper |
Author(s) | Ⅰ | Ⅱ | Ⅲ | Ⅳ | Ⅴ | Ⅵ |
Aggarwal and Jaggi [1] | ||||||
Ouyang et al. [23] | ||||||
Huang et al. [13] | ||||||
Tripathi and Pandey [30] | ||||||
Mahata [22] | ||||||
Goswami and Chaudhuri [9] | ||||||
Teng and Chang [29] | ||||||
Law and Wee [18] | ||||||
Jolai et al. [15] | ||||||
Lo et al. [21] | ||||||
Pervin et al. [24] | ||||||
Pervin et al. [25] | ||||||
Our paper |
Parameter | parametric value after % change | |||||
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 | |
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 | |
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 | |
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 | |
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 | |
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 | |
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 | |
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 | |
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 | |
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 | |
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 | |
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 | |||||
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 | |
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 | |
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 | |
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 | |
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 | |
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 | |
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 | |
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 | |
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 | |
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 | |
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 | |
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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