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

Magfura Pervin; Sankar Kumar Roy; Gerhard Wilhelm Weber

doi:10.3934/naco.2018010

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2018, Volume 8, Issue 2: 169-191. Doi: 10.3934/naco.2018010

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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

* 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

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 90B05, 91B70; Secondary: 91B24.

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Figure 1. Graphical representation of Inventory model for a Vendor

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Figure 2. Graphical representation of a Buyer's Inventory model

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Figure 5. 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

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Figure 6. 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

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Figure 7. 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

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Figure 3. $TC$ vs. $T$ at $M = 0.1091$

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Figure 4. $TC$ vs. $M$ at $T = 0.0870$

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Figure 8. Sensitivity analysis of $\theta$

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Figure 9. Sensitivity analysis of $\alpha$

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Figure 10. Sensitivity analysis of $h$

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Figure 11. Schematic view of all parameters

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Table 1. 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	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$

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Table 2. 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

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