Back-ordered inventory model with inflation in a cloudy-fuzzy environment

Haripriya Barman; Magfura Pervin; Sankar Kumar Roy; Gerhard-Wilhelm Weber

doi:10.3934/jimo.2020052

Article Contents

2021, Volume 17, Issue 4: 1913-1941. Doi: 10.3934/jimo.2020052

This issue Previous Article Optimal mean-variance reinsurance in a financial market with stochastic rate of return Next Article The convergence rate analysis of the symmetric ADMM for the nonconvex separable optimization problems

Back-ordered inventory model with inflation in a cloudy-fuzzy environment

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
3.
Institute of Applied Mathematics, Middle East Technical University, 06800, Ankara, Turkey

^* Corresponding Author: Sankar Kumar Roy
^* Corresponding Author: Sankar Kumar Roy

Received: June 2019

Revised: October 2019

Early access: March 2020

Published: July 2021

The author, Haripriya Barman is very much thankful to University Grants Commission (UGC) of India for providing financial support to continue this research work under Junior Research Fellowship scheme with UGC-reference number: [UGC-Ref.No.: 1166/(CSIR-UGC NET DEC.2017)] dated 08/02/2019

Abstract / Introduction Full Text(HTML) Figure(8) / Table(5) Related Papers Cited by

Abstract

In this paper, an Economic Production Quantity model for deteriorating items with time-dependent demand and shortages including partially back-ordered is developed under a cloudy-fuzzy environment. At first, we develop a crisp model by considering linearly time-dependent demand with constant deterioration rate, constant inflation rate and shortages under partially back-ordered, then we fuzzify the model to archive a decision under the cloudy-fuzzy (extension of fuzziness) demand rate, inflation rate, deterioration rate and the partially back-ordered rate which are followed by their practical applications. In this model, we assume ambiances where cloudy normalized triangular fuzzy number is used to handle the uncertainty in information which is coming from the data. The main purpose of our study is to defuzzify the total inventory cost by applying Ranking Index method of fuzzy numbers as well as cloudy-fuzzy numbers and minimize the total inventory cost of crisp, fuzzy, and cloudy-fuzzy model. Finally, a comparative analysis among crisp, fuzzy and cloudy-fuzzy total cost is carried out in this paper. Numerical example, sensitivity analysis, and managerial insights are elaborated to justify the usefulness of the new approach. A comparative inquiry of the numerical result with a new existing paper is also carried out. This paper ends with a conclusion along with advantages and limitations of our solution approach, and an outlook towards possible future studies.

Keywords:

Mathematics Subject Classification: Primary: 90B05; Secondary: 90C70, 90C25.

Citation:

Full Text(HTML)

Figure 1. Graphical representation of our inventory model

Download: Full-size image PowerPoint slide

Figure 2. Graphical representation of the convexity total cost for crisp model. The figure represents the total cost $ Z(T, T_1, T_2) $, $ T_1 $ and $ T_2 $, along the axis of blue colour, the axis of red colour and the axis of green colour, respectively

Download: Full-size image PowerPoint slide

Figure 3. Graphical representation of the convexity of total cost of our model in fuzzy environment. The figure represents the total cost $ I(\tilde{Z}) $, $ T_1 $ and $ T_2 $, along the axis of blue colour, the axis of red colour and the axis of green colour, respectively

Download: Full-size image PowerPoint slide

Figure 4. Graphical representation of the convexity of total cost of our model in cloudy-fuzzy environment. The figure represents the total cost $ CI(\tilde{Z}) $, $ T_1 $ and $ T_2 $, along the axis of blue colour, the axis of red colour and the axis of green colour, respectively

Download: Full-size image PowerPoint slide

Figure 5. Cost variation with respect to cycle-time variation for cloudy-fuzzy model

Download: Full-size image PowerPoint slide

Figure 6. Comparison of cloudy solution with crisp and fuzzy solution

Download: Full-size image PowerPoint slide

Figure 7. Graph of Set up cost vs inventory total cost

Download: Full-size image PowerPoint slide

Figure 8. Graph of Inflation rate vs inventory total cost

Download: Full-size image PowerPoint slide

Table 1. Contribution of various authors related to inventory models

Author(s)	Variable demand Rate	Fuzzy demand Rate	Cloudy Fuzzy Demand Rate	Fuzzy Deterioration	Cloudy Fuzzy Deterioration	Fuzzy Inflation	cloudy Fuzzy Inflation	Variable Holding cost	Fuzzy Back order Rate	cloudy Back order Rate
De et al. (2003)		$ \surd $		$ \surd $
De and Goswami (2006)			$ \surd $		$ \surd $
Jaggi et al. (2012)		$ \surd $		$ \surd $
Dutta and Kumar (2013)		$ \surd $		$ \surd $
Nadjfi (2013)								$ \surd $
Shaboni et al. (2014)				$ \surd $
Pervin et al. (2015)	$ \surd $
Kumar and Rajput (2015)	$ \surd $			$ \surd $
Pervin et al. (2016)	$ \surd $							$ \surd $
De and Mahata (2017)	$ \surd $	$ \surd $	$ \surd $
Parvin et al. (2017)	$ \surd $							$ \surd $
Kumar and Kumar (2017)		$ \surd $		$ \surd $				$ \surd $
Pervin et al. (2018)	$ \surd $			$ \surd $
Our paper	$ \surd $	$ \surd $	$ \surd $	$ \surd $	$ \surd $	$ \surd $	$ \surd $	$ \surd $	$ \surd $	$ \surd $

| Show Table

DownLoad: CSV

Table 2. Optimal solutions of our models

Model	$T^*$(week)	${T_1}^*$(week)	${T_2}^*$(week)	Minimum cost $ Z^*(＄) $	$ d_f = \frac{u_b-l_b}{2M}$	$ CI=\frac{\mbox{ln}(1+T)}{T} $
Crisp	4.21999	0.0101996	3.16691	206.931	—	—
Fuzzy	4.84194	0.00915884	2.80802	213.634	0.16	—
Cloudy fuzzy	5.72418	0.0101397	3.15275	169.752	—	0.14

| Show Table

DownLoad: CSV

Table 3. Cycle-time variation in crisp model and fuzzy model

Cycle time	Crisp model			Fuzzy model
	$T_1$	$T_2$	$Z^*$	$T_1$	$T_2$	$Z^*$
2	0.0102881	3.18061	253.543	0.00921317	2.82304	265.894
3	0.0102483	3.17444	218.974	0.00919408	2.81776	228.391
4	0.0102084	3.16827	207.867	0.00917496	2.81248	215.966
$\textbf{5}$	$\textbf{0.0101685}$	$\textbf{3.16208}$	$\textbf{206.271}$	$\textbf{0.00915582}$	$\textbf{2.80719}$	$\textbf{213.7}$
6	0.0101286	3.15589	209.536	0.00913665	2.80188	216.623
7	0.0100887	3.14968	215.669	0.00911746	2.79657	222.603
8	0.0100488	3.14347	223.674	0.00909825	2.79125	230.574
9	0.0100089	3.13725	232.997	0.00907901	2.78591	239.944
10	0.00997	3.13101	243.307	0.00905975	2.78057	250.359
* Bold represents optimal solution.

| Show Table

DownLoad: CSV

Table 4. Cycle-time variation in cloudy-fuzzy model

Cycle time	$ T_1 $	$ T_2 $	$ Z^* $
2	0.0102881	3.18061	233.896
3	0.0101824	3.15873	195.051
4	0.0101661	3.1565	177.841
5	0.0101501	3.15431	170.946
6	0.0101357	3.15216	169.901
7	0.0101212	3.15005	172.574
8	0.010107	3.14797	177.808
9	0.0100932	3.14592	184.919
10	0.0100796	3.1439	193.475
* Bold represents optimal solution.

| Show Table

DownLoad: CSV

Table 5. Sensitivity Analysis of the cloudy-fuzzy model

Parameter	$ \% $ change	New value of parameter	$ T $	$ T_1 $	$ T_2 $	$ Z_* $	$ \frac{Z_-Z^}{Z^*}\times100\% $
$ B $	+50	750	8.10737	0.010155	3.14775	246.259	45
	+30	650	7.24503	0.010117	3.14954	217.988	28.3
	+15	575	6.52651	0.010128	3.15104	194.942	14.74
	-15	425	4.80246	0.0101536	3.15474	141.517	-0.17
	-30	350	3.68946	0.0101711	3.15718	108.416	-36.19
	-50	250	1.45517	0.0102102	3.16223	45.2895	-73.34
$ c_1 $	+50	28.5	5.75063	0.0101458	3.15369	169.34	-0.33
	+30	24.7	5.73997	0.0101434	3.15332	169.505	-0.23
	+15	21.85	5.73204	0.0101416	3.15304	169.628	-0.16
	-30	13.3	5.70862	0.0101359	3.15216	169.997	0.06
	-50	9.5	5.69837	0.0101333	3.15175	170.161	0.15
$ \gamma $	+50	70.255	5.72707	0.0101755	3.1383	170.195	0.17
	+30	0.221	5.72593	0.0101611	3.14403	170.018	0.07
	+15	0.1955	5.72507	0.0101504	3.14837	169.906	0.03
	-15	0.1445	5.72326	0.0101292	3.15716	169.617	-0.17
	-30	0.119	5.72231	0.0101187	3.16161	169.302	-0.4
	-50	0.085	5.721	0.010105	3.1676	169.302	-0.4
$ \rho $	+50	0.225	5.72139	0.010109	3.16583	169.355	-0.32
	+30	0.195	5.72254	0.0101212	3.16056	169.514	-0.23
	+15	0.1725	5.72337	0.0101304	3.15664	169.633	-0.16
	-15	0.1275	5.72496	0.0101491	3.14888	169.87	-0.02
	-30	0.105	5.72573	0.0101586	3.14505	169.987	0.05
	-50	0.075	5.72672	0.0101713	3.13998	170.143	0.14
$ \alpha $	+50	1.05	4.41441	0.0119104	3.11271	191.321	12.61
	+30	0.91	4.85547	0.0112016	3.12682	183.808	8.19
	+15	0.805	5.25137	0.0106705	3.13897	177.247	4.32
	-15	0.595	6.30428	0.0096094	3.16852	161.137	-5.16
	-30	0.49	7.0423	0.009079	3.18673	151.136	-11.04
	-50	0.35	8.44384	0.00837292	3.21582	134.889	-20.61
$ \beta $	+50	4.5	4.51458	0.0157385	3.19932	136.373	-19.73
	+30	3.9	5.02358	0.0134956	3.18289	150.54	-11.4
	+15	3.45	5.38207	0.0118164	3.16882	160.415	-5.58
	-15	2.55	6.05227	0.00846561	3.13411	178.618	5.13
	-30	2.1	6.36819	0.00679407	3.11212	187.067	10.1
	-50	1.5	6.77288	0.00456966	3.07548	197.757	16.4
$ \theta $	+50	0.09	5.78754	0.0103801	3.07074	171.68	1.05
	+30	0.078	5.76298	0.0102877	3.10254	170.932	0.61
	+15	0.069	5.74389	0.0102152	3.12725	170.351	0.61
	-15	0.051	5.70382	0.0100611	3.17908	169.133	-0.45
	-30	0.042	5.68276	0.009979	3.20629	168.493	-0.83
	-50	0.03	5.65353	0.00986411	3, 24406	167.606	-1.35
$ s $	+50	45	1.60043	0.0162324	4.01898	70.9432	-58.24
	+30	39	3.51682	0.0138168	3.69291	129.89	-23.55
	+15	34.5	4.6588	0.0119896	3.43165	154.538	-9.04
	-15	24.5	7.00539	0.007835	2.78187	179.177	5.46
	-30	21	7.83848	0.00633401	2.52404	180.747	6.38
	-50	15	9.35126	0.00366725	2.02744	176.399	3.82
$ \delta $	+50	0.0075	5.76822	0.0101149	3.1489	169.312	-0.35
	+30	0.0065	5.75046	0.0101249	3.14046	169.489	-0.24
	+15	0.00575	5.73726	0.0101324	3.15161	169.621	-0.16
	-15	0.00425	5.7112	0.010147	3.15388	169.882	-0.01
	-30	0.0035	5.69833	0.0101543	3.155	170.012	0.07
	-50	0.0025	5.68133	0.0101638	3.15647	170.183	0.17
$ N $	+50	1.05	6.0493	0.00775952	2.76778	179.705	5.77
	+30	0.91	5.93784	0.008571	2.90253	176.282	3.75
	+15	0.805	5.83939	0.00929235	3.01911	173.268	0.02
	-15	0.595	5.58678	0.0111551	3.30832	165.573	-2.55
	-30	.49	5.41893	0.0124013	3.49299	160.49	-5.54
	-50	0.35	5.12323	0.0146078	3.8047	151.594	-10.77
$ \epsilon $	+50	0.3	5.27247	0.0101332	3.15255	159.446	-6.15
	+30	0.26	5.43884	0.0101356	3.15261	163.181	-3.95
	+15	0.23	5.57508	0.0101376	3.15267	166.293	-2.12
	-15	0.17	5.89021	0.0101421	3.15284	173.664	2.21
	-30	0.14	6.07952	0.0101448	3.15295	178.199	4.88
	-50	0.1	6.38786	0.010149	3.15313	185.743	9.32
* Bold shows the most sensitive total inventory cost.

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	S. De, S. Maity and M. Pal, Two decision makers's single decision over a back order eoq model with dense fuzzy demand rate, Finance and Market, 3, https://doi.org/10.18686/fm.v3.1061.
[2]	S. K. De and I. Beg, Triangular dense fuzzy sets and new defuzzification methods, Journal of Intelligent & Fuzzy systems, 31 (2016), 469-477. doi: 10.3233/IFS-162160.
[3]	S. K. De and A. Goswami, An EOQ model with fuzzy inflation rate and fuzzy deterioration rate when a delay in payment is permissible, International Journal of Systems Science, 37 (2006), 323-335. doi: 10.1080/00207720600681112.
[4]	S. K. De and G. C. Mahata, Decision of a fuzzy inventory with fuzzy backorder model under cloudy fuzzy demand rate, International Journal of Applied and Computational Mathematics, 3 (2017), 2593-2609. doi: 10.1007/s40819-016-0258-4.
[5]	S. K. De and G. C. Mahata, A cloudy fuzzy economic order quantity model for imperfect-quality items with allowable proportionate discounts, Journal of Industrial Engineering International, 15 (2019), 571-583. doi: 10.1007/s40092-019-0310-1.
[6]	S. K. De, P. K. Kundu and A. Goswami, An economic production quantity inventory model involving fuzzy demand rate and fuzzy deterioration rate, Journal of Applied Mathematics and Computing, 12 (2003), 251-260. doi: 10.1007/BF02936197.
[7]	D. Dutta and P. Kumar, Fuzzy inventory model for deteriorating items with shortages under fully backlogged condition, International Journal of Soft Computing and Engineering (IJSCE), 3 (2013), 393-398.
[8]	T. Garai and H. Garg, Multi-objective linear fractional inventory model with possibility and necessity constraints under generalised intuitionistic fuzzy set environment, CAAI Transactions on Intelligence Technology, 4 (2019), 175-181. doi: 10.1049/trit.2019.0030.
[9]	H. Garg, Fuzzy inventory models for deteriorating items under different types of lead-time distributions, Intelligent Techniques in Engineering Management, Springer, (2015), 247–274. doi: 10.1007/978-3-319-17906-3_11.
[10]	P. Gautam and A. Khanna, An imperfect production inventory model with setup cost reduction and carbon emission for an integrated supply chain, Uncertain Supply Chain Management, 6 (2018), 271-286. doi: 10.5267/j.uscm.2017.11.003.
[11]	P. Gautam, A. Kishore, A. Khanna and C. K. Jaggi, Strategic defect management for a sustainable green supply chain, Journal of Cleaner Production, 233 (2019), 226-241. doi: 10.1016/j.jclepro.2019.06.005.
[12]	N. Ghasemi and B. Afshar Nadjafi, EOQ models with varying holding cost, Journal of Industrial Mathematics, 2013 (2013), 743921, 1–7. doi: 10.1155/2013/743921.
[13]	P. Guchhait, M. K. Maiti and M. Maiti, Production-inventory models for a damageable item with variable demands and inventory costs in an imperfect production process, International Journal of Production Economics, 144 (2013), 180-188. doi: 10.1016/j.ijpe.2013.02.002.
[14]	K.-C. Hung, An inventory model with generalized type demand, deterioration and backorder rates, European Journal of Operational Research, 208 (2011), 239-242. doi: 10.1016/j.ejor.2010.08.026.
[15]	S. Indrajitsingha, P. Samanta and U. Misra, Fuzzy inventory model with shortages under fully backlogged using signed distance method, International Journal for Research in Applied Science & Engineering Technology, 4 (2016), 197-203.
[16]	C. K. Jaggi, S. Pareek, A. Sharma and Ni dhi, Fuzzy inventory model for deteriorating items with time-varying demand and shortages, American Journal of Operational Research, 2 (2012), 81-92.
[17]	C. K. Jaggi, S. Pareek, A. Khanna and N. Nidhi, Optimal replenishment policy for fuzzy inventory model with deteriorating items and allowable shortages under inflationary conditions, Yugoslav Journal of Operations Research, 26 (2016), 507-526. doi: 10.2298/YJOR150202002Y.
[18]	S. Jain, S. Tiwari, L. E. Cárdenas-Barrón, A. A. Shaikh and S. R. Singh, A fuzzy imperfect production and repair inventory model with time dependent demand, production and repair rates under inflationary conditions, RAIRO-Operations Research, 52 (2018), 217-239. doi: 10.1051/ro/2017070.
[19]	A. Karbassi Yazdi, M. A. Kaviani, A. H. Sarfaraz, L. E. Cárdenas-Barrón, H.-M. Wee and S. Tiwari, A comparative study on economic production quantity (epq) model under space constraint with different kinds of data, Grey Systems: Theory and Application, 9 (2019), 86-100.
[20]	N. Kazemi, E. Shekarian, L. E. Cárdenas-Barrón and E. U. Olugu, Incorporating human learning into a fuzzy eoq inventory model with backorders, Computers & Industrial Engineering, 87 (2015), 540-542. doi: 10.1016/j.cie.2015.05.014.
[21]	A. Khanna, M. Mittal, P. Gautam and C. Jaggi, Credit financing for deteriorating imperfect quality items with allowable shortages, Decision Science Letters, 5 (2016), 45-60. doi: 10.5267/j.dsl.2015.9.001.
[22]	A. Khanna, P. 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. doi: 10.33889/IJMEMS.2017.2.2-010.
[23]	N. Kumar and S. Kumar, An inventory model for deteriorating items with partial backlogging using linear demand in fuzzy environment, Cogent Business & Management, 4 (2017), 1-16. doi: 10.1080/23311975.2017.1307687.
[24]	S. Kumar and U. S. Rajput, Fuzzy inventory model for deteriorating items with time dependent demand and partial backlogging, Applied Mathematics, 6 (2015), 496-509. doi: 10.4236/am.2015.63047.
[25]	A. Moradi, J. Razmi, R. Babazadeh and A. Sabbaghnia, An integrated principal component analysis and multi-objective mathematical programming approach to agile supply chain network design under uncertainty, J. Ind. Manag. Optim., 15 (2019), 855-879.
[26]	P. Muniappan, R. Uthayakumar and S. Ganesh, An EOQ model for deteriorating items with inflation and time value of money considering time-dependent deteriorating rate and delay payments, Systems Science & Control Engineering, 3 (2015), 427-434. doi: 10.1080/21642583.2015.1073638.
[27]	M. Pervin, G. C. Mahata and S. Kumar Roy, An inventory model with declining demand market for deteriorating items under a trade credit policy, International Journal of Management Science and Engineering Management, 11 (2016), 243-251.
[28]	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, Numer. Algebra Control Optim., 7 (2017), 21-50. doi: 10.3934/naco.2017002.
[29]	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.
[30]	M. Pervin, S. K. Roy and G. W. Weber, An integrated inventory model with variable holding cost under two levels of trade-credit policy, Numer. Algebra Control Optim., 8 (2018), 169-191. doi: 10.3934/naco.2018010.
[31]	M. Pervin, S. K. Roy and G. W. Weber, Deteriorating inventory with preservation technology under price-and stock-sensitive demand, Journal of Industrial & Management Optimization, (2019), 275–284, https://doi.org/10.3934/jimo.2019019. doi: 10.3934/jimo.2019019.
[32]	M. Pervin, S. K. Roy and G. W. Weber, An integrated vendor-buyer model with quadratic demand under inspection policy and preservation technology, Hacettepe Journal of Mathematics and Statistics, (2019), 1–22, https://doi.org/10.15672/hujms.476056. doi: 10.15672/hujms.476056.
[33]	M. Pervin, S. K. Roy and G. W. Weber, Multi-item deteriorating two-echelon inventory model with price-and stock-dependent demand: A trade-credit policy, J. Ind. Manag. Optim., 15 (2019), 1345-1373.
[34]	S. K. Roy, M. Pervin and G. W. Weber, A two-warehouse probabilistic model with price discount on backorders under two levels of trade-credit policy, Journal of Industrial & Management Optimization, (2018), 658–662, https://doi.org/10.3934/jimo.2018167.
[35]	S. K. Roy, M. Pervin and G. W. Weber, Imperfection with inspection policy and variable demand under trade-credit: A deteriorating inventory model, Numerical Algebra, Control & Optimization, (2019), 658–662, https://doi.org/10.3934/naco.2019032.
[36]	S. Saha and T. Chakrabarti, Fuzzy inventory model for deteriorating items in a supply chain system with price dependent demand and without backorder, American Journal of Engineering Research, 6 (2017), 183-187.
[37]	B. Sarkar and A. S. Mahapatra, Periodic review fuzzy inventory model with variable lead time and fuzzy demand, International Transactions in Operational Research, 24 (2017), 1197-1227. doi: 10.1111/itor.12177.
[38]	S. Sarkar and T. Chakrabarti, An EPQ model having weibull distribution deterioration with exponential demand and production with shortages under permissible delay in payments, Mathematical Theory and Modelling, 3 (2013), 1-7.
[39]	S. Shabani, A. Mirzazadeh and E. Sharifi, An inventory model with fuzzy deterioration and fully backlogged shortage under inflation, SOP Transactions on Applied Mathematics, 1 (2014), 161-171. doi: 10.15764/AM.2014.02015.
[40]	A. A. Shaikh, A. K. Bhunia, L. E. Cárdenas-Barrón, L. Sahoo and S. Tiwari, A fuzzy inventory model for a deteriorating item with variable demand, permissible delay in payments and partial backlogging with shortage follows inventory (SFI) policy, International Journal of Fuzzy Systems, 20 (2018), 1606-1623. doi: 10.1007/s40815-018-0466-7.
[41]	A. A. Shaikh, L. 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.
[42]	E. Shekarian, M. Y. Jaber, N. Kazemi and E. Ehsani, A fuzzified version of the economic production quantity (EPQ) model with backorders and rework for a single-stage system, European Journal of Industrial Engineering, 8 (2014), 291-324. doi: 10.1504/EJIE.2014.060998.
[43]	E. Shekarian, N. Kazemi, S. H. Abdul-Rashid and E. U. Olugu, Fuzzy inventory models: A comprehensive review, Applied Soft Computing, 55 (2017), 588-621. doi: 10.1016/j.asoc.2017.01.013.
[44]	E. Shekarian, E. U. Olugu, S. H. Abdul-Rashid and N. Kazemi, An economic order quantity model considering different holding costs for imperfect quality items subject to fuzziness and learning, Journal of Intelligent & Fuzzy Systems, 30 (2016), 2985-2997. doi: 10.3233/IFS-151907.
[45]	S. Tiwari, L. E. Cárdenas-Barrón, A. Khanna and C. K. Jaggi, Impact of trade credit and inflation on retailer's ordering policies for non-instantaneous deteriorating items in a two-warehouse environment, International Journal of Production Economics, 176 (2016), 154-169. doi: 10.1016/j.ijpe.2016.03.016.
[46]	R. R. Yager, A procedure for ordering fuzzy subsets of the unit interval, Information Sciences, 24 (1981), 143-161. doi: 10.1016/0020-0255(81)90017-7.
[47]	L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353. doi: 10.1016/S0019-9958(65)90241-X.