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

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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: May 31, 2019

Revised: September 30, 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

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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.

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

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Figure 1. Graphical representation of our inventory model

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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

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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

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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

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

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

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

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Figure 8. Graph of Inflation rate vs inventory total cost

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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 $

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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

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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.

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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.

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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.

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