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July  2021, 17(4): 1913-1941. doi: 10.3934/jimo.2020052

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

Haripriya Barman 1, , Magfura Pervin 1, , Sankar Kumar Roy 1,, and  Gerhard-Wilhelm Weber 2,3,

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

Received  June 2019 Revised  October 2019 Published  July 2021 Early access  March 2020

Fund Project: 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

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: Back-ordered inventory, Inflation rate, Deterioration rate, Cloudy-fuzzy number, Optimization, Operational Research.
Mathematics Subject Classification: Primary: 90B05; Secondary: 90C70, 90C25.
Citation: Haripriya Barman, Magfura Pervin, Sankar Kumar Roy, Gerhard-Wilhelm Weber. Back-ordered inventory model with inflation in a cloudy-fuzzy environment. Journal of Industrial and Management Optimization, 2021, 17 (4) : 1913-1941. doi: 10.3934/jimo.2020052
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.

show all references

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.

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 $, along the axis of blue colour, the axis of red colour and the axis of green colour, respectively
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Figure 5.  Cost variation with respect to cycle-time variation for cloudy-fuzzy model
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Figure 6.  Comparison of cloudy solution with crisp and fuzzy solution
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Figure 7.  Graph of Set up cost vs inventory total cost
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Figure 8.  Graph of Inflation rate vs inventory total cost
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Table 1.  Contribution of various authors related to inventory models
Author(s)Variable demand RateFuzzy demand RateCloudy Fuzzy Demand Rate Fuzzy DeteriorationCloudy Fuzzy DeteriorationFuzzy Inflationcloudy Fuzzy InflationVariable Holding costFuzzy Back order Ratecloudy Back order Rate
De et al. (2003)$ \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 $$ \surd $
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Author(s)Variable demand RateFuzzy demand RateCloudy Fuzzy Demand Rate Fuzzy DeteriorationCloudy Fuzzy DeteriorationFuzzy Inflationcloudy Fuzzy InflationVariable Holding costFuzzy Back order Ratecloudy Back order Rate
De et al. (2003)$ \surd $$ \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 $
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} $
Crisp4.219990.01019963.16691206.931——
Fuzzy4.841940.009158842.80802213.6340.16—
Cloudy fuzzy5.724180.01013973.15275169.752—0.14
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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} $
Crisp4.219990.01019963.16691206.931——
Fuzzy4.841940.009158842.80802213.6340.16—
Cloudy fuzzy5.724180.01013973.15275169.752—0.14
Table 3.  Cycle-time variation in crisp model and fuzzy model
Cycle timeCrisp modelFuzzy model
$T_1$$T_2$$Z^*$$T_1$$T_2$$Z^*$
20.01028813.18061253.5430.009213172.82304265.894
30.01024833.17444218.9740.009194082.81776228.391
40.01020843.16827207.8670.009174962.81248215.966
$\textbf{5}$$\textbf{0.0101685}$$\textbf{3.16208}$$\textbf{206.271}$$\textbf{0.00915582}$$\textbf{2.80719}$$\textbf{213.7}$
60.01012863.15589209.5360.009136652.80188216.623
70.01008873.14968215.6690.009117462.79657222.603
80.01004883.14347223.6740.009098252.79125230.574
90.01000893.13725232.9970.009079012.78591239.944
100.009973.13101243.3070.009059752.78057250.359
* Bold represents optimal solution.
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Cycle timeCrisp modelFuzzy model
$T_1$$T_2$$Z^*$$T_1$$T_2$$Z^*$
20.01028813.18061253.5430.009213172.82304265.894
30.01024833.17444218.9740.009194082.81776228.391
40.01020843.16827207.8670.009174962.81248215.966
$\textbf{5}$$\textbf{0.0101685}$$\textbf{3.16208}$$\textbf{206.271}$$\textbf{0.00915582}$$\textbf{2.80719}$$\textbf{213.7}$
60.01012863.15589209.5360.009136652.80188216.623
70.01008873.14968215.6690.009117462.79657222.603
80.01004883.14347223.6740.009098252.79125230.574
90.01000893.13725232.9970.009079012.78591239.944
100.009973.13101243.3070.009059752.78057250.359
* Bold represents optimal solution.
Table 4.  Cycle-time variation in cloudy-fuzzy model
Cycle time$ T_1 $$ T_2 $$ Z^* $
20.01028813.18061233.896
30.01018243.15873195.051
40.01016613.1565177.841
50.01015013.15431170.946
60.01013573.15216169.901
70.01012123.15005172.574
80.0101073.14797177.808
90.01009323.14592184.919
100.01007963.1439193.475
* Bold represents optimal solution.
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Cycle time$ T_1 $$ T_2 $$ Z^* $
20.01028813.18061233.896
30.01018243.15873195.051
40.01016613.1565177.841
50.01015013.15431170.946
60.01013573.15216169.901
70.01012123.15005172.574
80.0101073.14797177.808
90.01009323.14592184.919
100.01007963.1439193.475
* Bold represents optimal solution.
Table 5.  Sensitivity Analysis of the cloudy-fuzzy model
Parameter$ \% $ changeNew value of parameter$ T $$ T_1 $$ T_2 $$ Z_* $$ \frac{Z_*-Z^*}{Z^*}\times100\% $
$ B $+507508.107370.0101553.14775246.25945
+306507.245030.0101173.14954217.98828.3
+155756.526510.0101283.15104194.94214.74
-154254.802460.01015363.15474141.517-0.17
-303503.689460.01017113.15718108.416-36.19
-502501.455170.01021023.1622345.2895-73.34
$ c_1 $+5028.55.750630.01014583.15369169.34-0.33
+3024.75.739970.01014343.15332169.505-0.23
+1521.855.732040.01014163.15304169.628-0.16
-3013.35.708620.01013593.15216169.9970.06
-509.55.698370.01013333.15175170.1610.15
$ \gamma $+5070.2555.727070.01017553.1383170.1950.17
+300.2215.725930.01016113.14403170.0180.07
+150.19555.725070.01015043.14837169.9060.03
-150.14455.723260.01012923.15716169.617-0.17
-300.1195.722310.01011873.16161169.302-0.4
-500.0855.7210.0101053.1676169.302-0.4
$ \rho $+500.2255.721390.0101093.16583169.355-0.32
+300.1955.722540.01012123.16056169.514-0.23
+150.17255.723370.01013043.15664169.633-0.16
-150.12755.724960.01014913.14888169.87-0.02
-300.1055.725730.01015863.14505169.9870.05
-500.0755.726720.01017133.13998170.1430.14
$ \alpha $+501.054.414410.01191043.11271191.32112.61
+300.914.855470.01120163.12682183.8088.19
+150.8055.251370.01067053.13897177.2474.32
-150.5956.304280.00960943.16852161.137-5.16
-300.497.04230.0090793.18673151.136-11.04
-500.358.443840.008372923.21582134.889-20.61
$ \beta $+504.54.514580.01573853.19932136.373-19.73
+303.95.023580.01349563.18289150.54-11.4
+153.455.382070.01181643.16882160.415-5.58
-152.556.052270.008465613.13411178.6185.13
-302.16.368190.006794073.11212187.06710.1
-501.56.772880.004569663.07548197.75716.4
$ \theta $+500.095.787540.01038013.07074171.681.05
+300.0785.762980.01028773.10254170.9320.61
+150.0695.743890.01021523.12725170.3510.61
-150.0515.703820.01006113.17908169.133-0.45
-300.0425.682760.0099793.20629168.493-0.83
-500.035.653530.009864113, 24406167.606-1.35
$ s $ +50451.600430.01623244.0189870.9432-58.24
+30393.516820.01381683.69291129.89-23.55
+1534.54.65880.01198963.43165154.538-9.04
-1524.57.005390.0078352.78187179.1775.46
-30217.838480.006334012.52404180.7476.38
-50159.351260.003667252.02744176.3993.82
$ \delta $+500.00755.768220.01011493.1489169.312-0.35
+300.00655.750460.01012493.14046169.489-0.24
+150.005755.737260.01013243.15161169.621-0.16
-150.004255.71120.0101473.15388169.882-0.01
-300.00355.698330.01015433.155170.0120.07
-500.00255.681330.01016383.15647170.1830.17
$ N $+501.056.04930.007759522.76778179.7055.77
+300.915.937840.0085712.90253176.2823.75
+150.8055.839390.009292353.01911173.2680.02
-150.5955.586780.01115513.30832165.573-2.55
-30.495.418930.01240133.49299160.49-5.54
-500.355.123230.01460783.8047151.594-10.77
$ \epsilon $+500.35.272470.01013323.15255159.446-6.15
+300.265.438840.01013563.15261163.181-3.95
+150.235.575080.01013763.15267166.293-2.12
-150.175.890210.01014213.15284173.6642.21
-300.146.079520.01014483.15295178.1994.88
-500.16.387860.0101493.15313185.7439.32
* Bold shows the most sensitive total inventory cost.
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Parameter$ \% $ changeNew value of parameter$ T $$ T_1 $$ T_2 $$ Z_* $$ \frac{Z_*-Z^*}{Z^*}\times100\% $
$ B $+507508.107370.0101553.14775246.25945
+306507.245030.0101173.14954217.98828.3
+155756.526510.0101283.15104194.94214.74
-154254.802460.01015363.15474141.517-0.17
-303503.689460.01017113.15718108.416-36.19
-502501.455170.01021023.1622345.2895-73.34
$ c_1 $+5028.55.750630.01014583.15369169.34-0.33
+3024.75.739970.01014343.15332169.505-0.23
+1521.855.732040.01014163.15304169.628-0.16
-3013.35.708620.01013593.15216169.9970.06
-509.55.698370.01013333.15175170.1610.15
$ \gamma $+5070.2555.727070.01017553.1383170.1950.17
+300.2215.725930.01016113.14403170.0180.07
+150.19555.725070.01015043.14837169.9060.03
-150.14455.723260.01012923.15716169.617-0.17
-300.1195.722310.01011873.16161169.302-0.4
-500.0855.7210.0101053.1676169.302-0.4
$ \rho $+500.2255.721390.0101093.16583169.355-0.32
+300.1955.722540.01012123.16056169.514-0.23
+150.17255.723370.01013043.15664169.633-0.16
-150.12755.724960.01014913.14888169.87-0.02
-300.1055.725730.01015863.14505169.9870.05
-500.0755.726720.01017133.13998170.1430.14
$ \alpha $+501.054.414410.01191043.11271191.32112.61
+300.914.855470.01120163.12682183.8088.19
+150.8055.251370.01067053.13897177.2474.32
-150.5956.304280.00960943.16852161.137-5.16
-300.497.04230.0090793.18673151.136-11.04
-500.358.443840.008372923.21582134.889-20.61
$ \beta $+504.54.514580.01573853.19932136.373-19.73
+303.95.023580.01349563.18289150.54-11.4
+153.455.382070.01181643.16882160.415-5.58
-152.556.052270.008465613.13411178.6185.13
-302.16.368190.006794073.11212187.06710.1
-501.56.772880.004569663.07548197.75716.4
$ \theta $+500.095.787540.01038013.07074171.681.05
+300.0785.762980.01028773.10254170.9320.61
+150.0695.743890.01021523.12725170.3510.61
-150.0515.703820.01006113.17908169.133-0.45
-300.0425.682760.0099793.20629168.493-0.83
-500.035.653530.009864113, 24406167.606-1.35
$ s $ +50451.600430.01623244.0189870.9432-58.24
+30393.516820.01381683.69291129.89-23.55
+1534.54.65880.01198963.43165154.538-9.04
-1524.57.005390.0078352.78187179.1775.46
-30217.838480.006334012.52404180.7476.38
-50159.351260.003667252.02744176.3993.82
$ \delta $+500.00755.768220.01011493.1489169.312-0.35
+300.00655.750460.01012493.14046169.489-0.24
+150.005755.737260.01013243.15161169.621-0.16
-150.004255.71120.0101473.15388169.882-0.01
-300.00355.698330.01015433.155170.0120.07
-500.00255.681330.01016383.15647170.1830.17
$ N $+501.056.04930.007759522.76778179.7055.77
+300.915.937840.0085712.90253176.2823.75
+150.8055.839390.009292353.01911173.2680.02
-150.5955.586780.01115513.30832165.573-2.55
-30.495.418930.01240133.49299160.49-5.54
-500.355.123230.01460783.8047151.594-10.77
$ \epsilon $+500.35.272470.01013323.15255159.446-6.15
+300.265.438840.01013563.15261163.181-3.95
+150.235.575080.01013763.15267166.293-2.12
-150.175.890210.01014213.15284173.6642.21
-300.146.079520.01014483.15295178.1994.88
-500.16.387860.0101493.15313185.7439.32
* Bold shows the most sensitive total inventory cost.
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