Multi-item deteriorating two-echelon inventory model with price- and stock-dependent demand: A trade-credit policy

Magfura Pervin; Sankar Kumar Roy; Gerhard Wilhelm Weber

doi:10.3934/jimo.2018098

Article Contents

2019, Volume 15, Issue 3: 1345-1373. Doi: 10.3934/jimo.2018098

This issue Previous Article An inventory model with imperfect item, inspection errors, preventive maintenance and partial backlogging in uncertainty environment Next Article Stability in mean for fuzzy differential equation

Multi-item deteriorating two-echelon inventory model with price- and stock-dependent demand: A 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
* Corresponding author: sankroy2006@gmail.com

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: August 31, 2017

Revised: November 30, 2017

Early access: June 2018

Published: April 2019

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.

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Abstract

This article is concerned with a multi-item inventory model for deteriorating items. The model is formed on the basis of a two-level supply chain policy, i.e., based on manufacturer's and retailer's perspective. The deterioration rate is considered as constant. The demand factor of any items suffer from a large amount of stock level; so, we consider stock-dependent demand function. The demand of any item is also dependent on its selling price; thus, a price-dependent demand function is introduced here. The retailer adopts the trade-credit policy for his customers in order to promote market competitiveness. He can earn revenue and interest after the customer pays the amount of purchasing cost to the retailer until the end of the trade-credit period, offered by the supplier. Shortages are allowed in the retailer's model as it is a very realistic item, too. A price discount on backordered commodities is offered for those customers who are willing to backorder their demand. Thereafter, we present an easy analytical solution procedure to find the total profit for both manufacturer and retailer. We also use the classical game theory and Nash equilibrium approach to find an optimal solution of the joint profit. The results are discussed with several numerical examples to illustrate our model and to provide some managerial insights related to the model. Furthermore, a parametric sensitivity analysis of the optimal solutions is provided and a concavity figure of our profit function is supplied to stabilize our model. The paper ends with a conclusion and an outlook to future research projects.

Keywords:

Mathematics Subject Classification: Primary: 90B05, 91A10; Secondary: 90C26.

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Figure 1. Proposed inventory control model for manufacturer

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Figure 2. Proposed inventory control model for retailer

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Figure 4. Profit function in respect of deterioration

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Figure 3. The concavity of the total profit. Included are $t_1$, $T$ and the total profit $M_1(T, t_1)$, along the x-axis, the y-axis and the z-axis, respectively

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Table 1. Contributions of some authors related to inventory model

Authors	Multi items	Two-echelon model	Stock- and Price- dependent demand	Deterio- rations	Trade-credit policy	Price discount on backorders
Lenard and Roy (1995)	√
Kar et al. (2001)	√		√
Ben-daya and Raouf (1993)	√
Chen and Bell (2011)		√
Datta and Paul (2001)			√
Pal et al. (2014)			√	√
Teng and Chang (2005)			√	√
Thangam and Uthayakumar (2009)		√		√	√
Pervin et al. (2016)				√	√	√
Goswami and Chaudhuri (1991)				√		√
Wu et al. (2014)				√	√
Sarkar et al. (2017)				√		√
Pervin et al. (2017)				√	√
Ouyang et al. (2015)						√
This paper	√	√	√	√	√	√

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Table 2. Computational parametric configuration for 3 items

Parameters	i=1	i=2	i=3	Joint effects of all 3 items
$A_i$	30	35	40	105
$p_i$	10	15	20	45
$k_i$	500	600	700	1800
$\theta_i$	0.05	0.07	0.09	0.21
$c_i$	10	14	18	42
$q_i$	5	10	15	30
$s_i$	0.12	0.18	0.23	0.53
$l_i$	0.20	0.25	0.30	0.75
$a_i$	0.50	0.55	0.60	1.65
$b_i$	0.30	0.35	0.40	1.05
$\delta_i$	0.7	0.8	0.9	2.4
$\alpha_i$	0.2	0.3	0.4	0.9
$\beta_i$	0.5	0.6	0.7	1.8
$\gamma_i$	0.4	0.6	0.8	1.8
$M$	1.2	1.6	2.0	4.8
$I_e$	0.8	1.0	1.2	3.0
$I_c$	0.5	0.7	0.9	2.1
$t_1$	10.62	17.55	29.37	32.76
$t_2$	9.20	15.72	22.18	29.18
$t_3$	15.11	19.83	26.07	36.30
$t_4$	13.57	22.15	34.49	35.02
$M_r(T, t_1)$	26.6214	42.1753	63.8068	91.4128
$R_{1r}(T, M)$	95.1506	117.3394	182.7139	220.6210
$R_{2r}(T, M)$	165.3726	199.5379	274.2680	326.1660

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Table 3. Sensitivity analysis for the parameters $\theta_i (i = 1, 2, 3)$. $JE$ represents the joint effects, respectively

Parameter	$-30\%$	$-20\%$	$-10\%$	$10\%$	$20\%$	$30\%$
$\theta_1$	0.035	0.04	0.045	0.055	0.06	0.065
$M_1(T, t_1)$	28.6725	28.4218	28.2240	27.9031	27.7633	27.5502
$R_{11}(T, M)$	96.1162	95.9631	95.8812	95.7243	95.5708	95.3276
$R_{21}(T, M)$	166.502	166.372	166.127	165.872	165.609	165.421
$\theta_2$	0.049	0.056	0.063	0.077	0.084	0.091
$M_1(T, t_1)$	43.3213	43.1045	42.8871	42.7490	42.5289	42.3407
$R_{11}(T, M)$	118.6731	118.4830	118.2571	117.9013	117.7856	117.5126
$R_{21}(T, M)$	199.410	199.107	198.884	198.619	198.470	198.159
$\theta_3$	0.063	0.072	0.081	0.099	0.108	0.117
$M_1(T, t_1)$	65.1173	64.8243	64.5740	64.2581	63.8056	63.5731
$R_{11}(T, M)$	183.9920	183.7526	183.4911	183.2354	182.8744	182.8530
$R_{21}(T, M)$	273.576	273.291	272.876	272.651	272.304	272.118
$JE((M_i (T, t_1))_{i=1, 2, 3})$	324.13	307.47	299.53	271.84	256.77	231.64
$JE((R_{1i}(T, M))_{i=1, 2, 3})$	573.38	529.13	478.46	431.47	390.52	358.30
$JE((R_{2i}(T, M))_{i=1, 2, 3})$	642.10	618.34	583.57	549.73	499.05	452.17

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Table 4. Sensitivity analysis for the parameters $b_i (i = 1, 2, 3)$. $JE$ represents the joint effects, respectively

Parameter	$-30\%$	$-20\%$	$-10\%$	$10\%$	$20\%$	$30\%$
$b_1$	0.21	0.24	0.27	0.33	0.36	0.39
$M_1(T, t_1)$	27.6820	27.8921	28.1675	28.4691	28.8722	29.3225
$R_{11}(T, M)$	96.3546	96.6704	96.9124	97.3510	97.6021	97.9146
$R_{21}(T, M)$	165.318	165.702	165.993	166.247	166.583	166.875
$b_2$	0.245	0.28	0.315	0.385	0.420	0.455
$M_1(T, t_1)$	42.6813	42.8904	43.3612	43.6675	43.9130	44.4362
$R_{11}(T, M)$	117.1325	117.4076	117.7451	118.2169	118.5306	118.8539
$R_{21}(T, M)$	198.478	198.825	199.350	199.718	199.958	237.574
$b_3$	0.28	0.32	0.36	0.44	0.48	0.52
$M_1(T, t_1)$	66.3753	66.5206	66.8347	67.2541	67.6088	67.9346
$R_{11}(T, M)$	182.7090	182.9364	183.2674	183.5104	183.7984	184.2576
$R_{21}(T, M)$	273.157	273.409	273.749	273.986	274.370	274.875
$JE((M_i (T, t_1))_{i=1, 2, 3})$	142.39	187.52	224.78	279.70	330.87	367.42
$JE((R_{1i}(T, M))_{i=1, 2, 3})$	413.49	437.61	475.23	493.27	534.17	562.19
$JE((R_{2i}(T, M))_{i=1, 2, 3})$	650.14	688.07	723.47	767.33	784.35	820.69

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Table 5. Sensitivity analysis for the parameters $\beta_i(i = 1, 2, 3)$. $JE$ represents the joint effects, respectively

Parameter	$-30\%$	$-20\%$	$-10\%$	$10\%$	$20\%$	$30\%$
$\beta_1$	0.35	0.40	0.45	0.55	0.60	0.65
$M_1(T, t_1)$	30.2670	30.8305	31.4758	32.0165	32.7736	33.3814
$R_{11}(T, M)$	98.4236	98.9470	99.5024	100.0631	100.5861	101.1425
$R_{21}(T, M)$	167.264	167.752	168.369	168.804	169.362	169.972
$\beta_2$	0.42	0.48	0.54	0.66	0.72	0.78
$M_1(T, t_1)$	45.6149	46.1543	46.5783	47.1462	47.6388	48.3592
$R_{11}(T, M)$	118.4268	118.9005	119.2457	119.8319	120.3670	120.7591
$R_{21}(T, M)$	210.375	210.763	211.136	211.545	211.987	212.307
$\beta_3$	0.49	0.56	0.63	0.77	0.84	0.91
$M_1(T, t_1)$	68.3871	68.8414	69.4206	70.9126	70.4756	70.8225
$R_{11}(T, M)$	185.2477	185.6470	186.0853	186.4800	186.8175	187.2052
$R_{21}(T, M)$	275.8145	276.335	276.857	277.3670	277.8950	277.4352
$JE((M_i (T, t_1))_{i=1, 2, 3})$	150.26	163.73	175.44	189.71	213.46	252.04
$JE((R_{1i}(T, M))_{i=1, 2, 3})$	409.24	422.19	435.07	442.86	453.20	467.15
$JE((R_{2i}(T, M))_{i=1, 2, 3})$	660.14	671.42	684.35	697.23	713.75	729.06

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Table 6. Sensitivity analysis for the parameters $\gamma_i(i = 1, 2, 3)$. $JE$ represents the joint effects, respectively

Parameter	$-30\%$	$-20\%$	$-10\%$	$10\%$	$20\%$	$30\%$
$\gamma_1$	0.28	0.32	0.36	0.44	0.48	0.52
$M_1(T, t_1)$	28.2743	28.6183	29.0179	29.3746	29.6524	29.9032
$R_{11}(T, M)$	97.3176	97.6480	97.9208	98.4635	98.7705	99.1432
$R_{21}(T, M)$	165.418	165.727	166.284	166.546	166.920	167.335
$\gamma_2$	0.42	0.48	0.54	0.66	0.72	0.78
$M_1(T, t_1)$	44.1267	44.4725	44.8213	45.2018	45.5537	45.8715
$R_{11}(T, M)$	116.4503	116.6931	116.9917	117.4361	117.8543	118.5174
$R_{21}(T, M)$	206.421	206.885	207.367	207.753	208.290	208.668
$\gamma_3$	0.56	0.64	0.72	0.88	0.96	1.04
$M_1(T, t_1)$	67.3417	67.8864	68.4011	68.9172	79.5140	79.8711
$R_{11}(T, M)$	189.1106	189.6587	190.2275	190.6340	191.3719	191.5603
$R_{21}(T, M)$	279.9902	280.4275	280.8670	281.2552	282.7732	283.5112
$JE((M_i(T, t_1))_{i=1, 2, 3})$	140.31	153.10	161.49	175.25	183.39	199.37
$JE((R_{1i}(T, M))_{i=1, 2, 3})$	408.72	417.28	425.01	437.26	445.49	467.00
$JE((R_{2i}(T, M))_{i=1, 2, 3})$	657.29	669.81	672.31	685.01	693.33	707.46

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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]	M. Ben-daya and A. Raouf, On the constrained multi-item single period inventory problem, International Journal of Production Management, 13 (1993), 104-112. doi: 10.1108/01443579310046472.
[3]	L. Benkherouf and M. Omar, Optimal integrated policies for a single-vendor single-buyer time-varying demand model, Computers and Mathematics with Applications, 60 (2010), 2066-2077. doi: 10.1016/j.camwa.2010.07.047.
[4]	D. K. Bhattacharya, Production, manufacturing and logistics on multi-item inventory, European Journal of Operational Research, 162 (2005), 786-791. doi: 10.1016/j.ejor.2003.02.004.
[5]	J. Chen and P. C. Bell, Coordinating a decentralized supply chain with customer returns and price-dependent stochastic demand using a buyback policy, European Journal of Operational Research, 212 (2011), 293-300. doi: 10.1016/j.ejor.2011.01.036.
[6]	V. Choudri, M. Venkatachalam and S. Panayappan, Production inventory model with deteriorating items, two rates of production cost and taking account of time value of money, Journal of Industrial and Management Optimization, 12 (2016), 1153-1172. doi: 10.3934/jimo.2016.12.1153.
[7]	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.
[8]	P. M. Ghare and G. P. Schrader, A model for an exponentially decaying inventory, Journal of Industrial Engineering, 14 (1963), 238-243.
[9]	A. Goswami and K. S. Chaudhuri, An EOQ model for deteriorating items with shortages and a linear trend in demand, Journal of the Operational Research Society, 42 (1991), 1105-1110.
[10]	S. K. Goyal, Economic order quantity under conditions of permissible delay in payments, Journal of Operational Research Society, 36 (1985), 335-338.
[11]	S. Kar, T. Roy and M. Maiti, Multi-item inventory model with probabilistic price dependent damand and imprecise goal and constraints, Yugoslav Journal of Operations Research, 11 (2001), 93-103.
[12]	N. A. Kurdhi, J. Prasetyo and S. S. Handajani, An inventory model involving back-order price discount when the amount received is uncertain, International Journal of Systems Science, 47 (2016), 662-671. doi: 10.1080/00207721.2014.900136.
[13]	R. I. Levin, C. P. McLaughlin, R. P. Lamone and J. F. Kottas, Productions Operations Management: Contemporary Policy for Managing Operating System, McGraw-Hill Series in Management, New York, 1972.
[14]	T. Y. Lin, M. T. Chen and K. L. Hou, An inventory model for items with imperfect quality and quantity discounts under adjusted screening rate and earned interest, Journal of Industrial and Management Optimization, 12 (2016), 1333-1347. doi: 10.3934/jimo.2016.12.1333.
[15]	M. Mohammadzadeh, A. A. Khamseh and M. Mohammadi, A multi-objective integrated model for closed-loop supply chain configuration and supplier selection considering uncertain demand and different performance levels, Journal of Industrial and Management Optimization, 13 (2017), 1041-1064. doi: 10.3934/jimo.2016061.
[16]	L. Y. Ouyang, B. R. Chuang and Y. J. Lin, Impact of backorder discounts on periodic review inventory model, International Journal of Information and Management Sciences, 14 (2003), 1-13.
[17]	S. Pal, G. S. Mahapatra and G. P. Samanta, An inventory model of price and stock dependent demand rate with deterioration under inflation and delay in payment, International Journal of System Assurance Engineering and Management, 5 (2014), 591-601. doi: 10.1007/s13198-013-0209-y.
[18]	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.
[19]	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.
[20]	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.
[21]	M. Pervin, S. K. Roy and G. W. Weber, An integrated inventory model with variable holding cost under two levels of trade-credit policy, Numerical Algebra, Control and Optimization, 8 (2018), 169-191.
[22]	M. J. Rosenblatt, Multi-item inventory system with budgetary constraint: A comparison between the Lagrangian and fixed cycle approach, International Journal of Production Research, 19 (1981), 331-339. doi: 10.1080/00207548108956661.
[23]	B. Sarkar, B. Mandal and S. Sarkar, Quality improvement and backorder price discount under controllable lead time in an inventory model, Journal of Manufacturing Systems, 35 (2015), 26-36. doi: 10.1016/j.jmsy.2014.11.012.
[24]	B. Sarkar, B. Mandal and S. Sarkar, Preservation of deteriorating seasonal products with stock-dependent consumption rate and shortages, Journal of Industrial and Management Optimization, 13 (2017), 187-206. doi: 10.3934/jimo.2016011.
[25]	B. Sarkar, A. Majumder, M. Sarkar, B. K. Dey and G. Roy, Two-echelon supply chain model with manufacturing quality improvement and setup cost reduction, Journal of Industrial and Management Optimization, 13 (2017), 1085-1104. doi: 10.3934/jimo.2016063.
[26]	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.
[27]	A. Thangam and R. Uthayakumar, Two echelon trade credits financing for perishable items in a supply chain when demands on both selling price and credit period, Computers and Industrial Engineering, 57 (2009), 773-786. doi: 10.1016/j.cie.2009.02.005.
[28]	Y. C. Tsao, Ordering policy for non-instantaneously deteriorating products under price adjustment and trade credits, Journal of Industrial and Management Optimization, 13 (2017), 327-345. doi: 10.3934/jimo.2016020.
[29]	Q. Wang and M. Parlar, A three-person game theory model arising in stochastic inventory control theory, European Journal of Operational Research, 76 (1994), 83-97. doi: 10.1016/0377-2217(94)90008-6.
[30]	J. Wu, L. Y. Ouyang, L. E. Cárdenas-Barrón and S. K. Goyal, Optimal credit period and lot size for deteriorating items with expiration dates under two-level trade credit financing, European Journal of Operational Research, 237 (2014), 898-908. doi: 10.1016/j.ejor.2014.03.009.