American Institute of Mathematical Sciences

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doi: 10.3934/dcdss.2020401

A fuzzy inventory model for Weibull deteriorating items under completely backlogged shortages

Deepak Kumar Nayak 1, , Sudhansu Sekhar Routray 2, , Susanta Kumar Paikray 1,, and  Hemen Dutta 3,

1. 

Department of Mathematics, Veer Surendra Sai University of Technology, Burla 768018, Odisha, India
2. 

Department of Mathematics, College of Engineering and Technology, Bhubaneswar 751029, Odisha, India
3. 

Department of Mathematics, Gauhati, Guwahati 781014, India

* Corresponding author: Susanta Kumar Paikray

Received  November 2019 Published  June 2020

In this paper, a fuzzy stock replenishment policy implemented for inventory items that follows linear demand and Weibull deterioration under completely backlogged shortages. Moreover, to minimize the aggregate expense per unit time, the fuzzy optimal solution is obtained using general mathematical techniques by considering hexagonal fuzzy numbers and graded mean preference integration strategy. Finally, the complete exposition of the model is provided by numerical examples and sensitivity behavior of the associated parameters.

Keywords: Fuzzy inventory model, deteriorating items, weibull distribution, defuzzification, linear demand.
Mathematics Subject Classification: Primary: 03E72, 90B05; Secondary: 78M50.
Citation: Deepak Kumar Nayak, Sudhansu Sekhar Routray, Susanta Kumar Paikray, Hemen Dutta. A fuzzy inventory model for Weibull deteriorating items under completely backlogged shortages. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020401
References:
[1]

S. Agarwal, S. Banerjee and S. Papachristos, Inventory model with deteriorating items, ramp-type demand and partially backlogged shortages for a two warehouse system, Appl. Math. Model., 37 (2013), 8912-8929.  doi: 10.1016/j.apm.2013.04.026.  Google Scholar

[2]

S. Barik, S. K. Paikray, S. Mishra and U. K. Misra, An inventory model for deteriorating items under time varying demand condition, Int. J. Appl. Eng. Res., 10 (2015), 35770-35773.   Google Scholar

[3]

S. Barik, S. K. Paikray, S. Mishra and U. K. Misra, A Deteriorating inventory model with shortages under pricedependent demand and inflation, Asian J. Math. Comput. Res., 8 (2016), 14-25.   Google Scholar

[4]

D. Chakraborty, D. K. Jana and T. K. Roy, Two-warehouse partial backlogging inventory model with ramp type demand rate, three-parameter Weibull distribution deterioration under inflation and permissible delay in payments, Comput. Indus. Eng., 123 (2018), 157-179.  doi: 10.1016/j.cie.2018.06.022.  Google Scholar

[5]

R. P. Covert and G. C. Philip, An EOQ model for items with weibull distribution deterioration, AIIE Trans., 5 (1973), 323-326.  doi: 10.1080/05695557308974918.  Google Scholar

[6]

D. S. Dinagar and J. R. Kannan, On fuzzy inventory model with allowable shortage, Internat. J. Pure. Appl. Math., 99 (2015), 65-76.  doi: 10.12732/ijpam.v99i1.6.  Google Scholar

[7]

I. Djordjevic, D. Petrovic and G. Stojic, A fuzzy linear programming model for aggregated production planning (APP) in the automotive industry, Comput. Ind., 110 (2019), 48-63.  doi: 10.1016/j.compind.2019.05.004.  Google Scholar

[8]

S. Faddel, A. T. Al-Awami and M. A. Abido, Fuzzy optimization for the operation of electric vehicle parking lots, Elect. Pow. Syst. Res., 145 (2017), 166-174.  doi: 10.1016/j.epsr.2017.01.008.  Google Scholar

[9]

P. M. Ghare and G. F. Schrader, A Model for exponentially decaying inventories, J. Ind. Eng., 14 (1963), 238-243.   Google Scholar

[10]

S. K. Indrajit, S. Routray, S. K. Paikray and U. K. Misra, Fuzzy economic productionquantity model with time dependent demand rate, Sci. J. Logis., 12 (2016), 193-198.   Google Scholar

[11]

C. K. Jaggi, S. Pareek, A. Khanna and R. Sharma, Credit financing in a two-warehouse environment for deteriorating items with price-sensitive demand and fully backlogged shortages, Appl. Math. Model., 38 (2014), 5315-5333.  doi: 10.1016/j.apm.2014.04.025.  Google Scholar

[12]

S. Jain and M. Kumar, An inventory model with power demand pattern Weibull distribution deterioration and shortages, J. Indian Acad. Math., 30 (2008), 55-61.   Google Scholar

[13]

J. Kacprzyk and P. Stanieski, Long-term inventory policy-making through fuzzy decision-making models, Fuzzy Sets Syst., 8 (1982), 117-132.  doi: 10.1016/0165-0114(82)90002-1.  Google Scholar

[14]

C. Kao and W. K. Hsu, A single-period inventory model with fuzzy demand, Comput. Math. Appl., 43 (2002), 841-848.  doi: 10.1016/S0898-1221(01)00325-X.  Google Scholar

[15]

M. T. Lamata, D. Pelta and J. L. Verdegay, Optimisation problems as decision problems: The case of fuzzy optimisation problems, Inform. Sci., 460/461 (2018), 377-388.  doi: 10.1016/j.ins.2017.07.035.  Google Scholar

[16]

H. M. Lee and J. S. Yao, Economic production quantity for fuzzy demand and fuzzy production quantity., European J. Oper. Res., 109 (1998), 203-211.  doi: 10.1016/S0377-2217(97)00200-2.  Google Scholar

[17]

J. Mezei and S. Nikou, Fuzzy optimization to improve mobile health and wellness recommendation systems, Knowledge-Based Syst., 142 (2018), 108-116.  doi: 10.1016/j.knosys.2017.11.030.  Google Scholar

[18]

S. Mishra, G. Misra, U. K. Misra and L. K. Raju, An inventory model with quadratic demand pattern and deterioration with shortages under the influence of inflation, Math. Financ. Lett., 1 (2012), 57-67.   Google Scholar

[19]

S. Mishra, S. Barik, S. K. Paikray and U. K. Misra, An inventory control model of deteriorating items in fuzzy environment, Global J. Pure Appl. Math., 11 (2015), 1301-1312.   Google Scholar

[20]

S. Misra, U. K. Misra, S. Barik and S. K. Paikray, An inventory model for inflation induced demand and Weibull deteriorating items, Internat. J. Adv. Eng. Technol., 4 (2012), 176-182.   Google Scholar

[21]

H. S. Najafi, S. A. Edalatpanah and H. Dutta, A nonlinear model for fully fuzzy linear programming with fully unrestricted variables and parameters, Alexandria Eng. J., 55 (2016), 2589-2595.   Google Scholar

[22]

S. Pal, G. S. Mahapatra and G. P. Samanta, An EPQ model of ramp type demand with Weibull deterioration under inflation and finite horizon in crisp and fuzzy environment, International Journal of Production Economics, 156 (2014), 159-166.  doi: 10.1016/j.ijpe.2014.05.007.  Google Scholar

[23]

K. S. Park, Fuzzy set theoretical interpretation of economic order quantity, IEEE Trans., 17 (1987), 1082-1084.   Google Scholar

[24]

L. K. Raju, U. K. Misra, S. Mishra and G. Misra, An inventory model for Weibull deteriorating items with linear demand pattern, J. Comput. Math. Sci., 3 (2012), 440-445.   Google Scholar

[25]

S. S. Routray, S. K. Paikray, S. Misra and U. K. Misra, Fuzzy inventory model with single item under time dependent demand and holding cost, Internat. J. Adv. Res. Sci. Eng., 6 (2017), 1604-1618.   Google Scholar

[26]

S. S. Sanni and W. I. E. Chukwu, An economic order quantity model for items with three-parameter Weibull distribution deterioration, ramp-type demand and shortages, Appl. Math. Model., 37 (2013), 9698-9706.  doi: 10.1016/j.apm.2013.05.017.  Google Scholar

[27]

S. Sarbjit and S. S. Raj, An optimal inventory policy for items having linear demand and variable deterioration rate with trade credit, J. Math. Stat., 5 (2009), 330-333.  doi: 10.3844/jmssp.2009.330.333.  Google Scholar

[28]

Y. K. Shah, An order-level lot-size inventory model for deteriorating items, AIEE Trans., 9 (1977), 108-112.  doi: 10.1080/05695557708975129.  Google Scholar

[29]

I. Tomba and K. H. Brojendro, An inventory model with linear demand pattern and deterioration with shortages, J. Indian Acad. Math., 33 (2011), 607-612.   Google Scholar

[30]

C. K. Tripathy, L. M. Pradhan and U. Mishra, An EPQ model for linear deteriorating item with variable holding cost, Int. J. Comput. Appl. Math., 5 (2010), 209-215.   Google Scholar

[31]

G. Viji and K. Karthikeyan, An economic production quantity model for three levels of production with Weibull distribution deterioration and shortage, Ain Shams Eng. J., 9 (2018), 1481-1487.  doi: 10.1016/j.asej.2016.10.006.  Google Scholar

[32] T. M. Whitin, The Theory of Inventory Management, 2$^{nd}$ edition, Princeton University Press, Princeton, New Jersey, 1957.   Google Scholar
[33]

H-L. Yang, Two-warehouse partial backlogging inventory models with three-parameter Weibull distribution deterioration under inflation, Int. J. Prod. Econ., 138 (2012), 107-116.  doi: 10.1016/j.ijpe.2012.03.007.  Google Scholar

[34]

H.-M. Lee and J.-S. Yao, Economic order quantity in fuzzy sense for inventory without back-order model, Fuzzy Sets Syst., 105 (1999), 13-31.  doi: 10.1016/S0165-0114(97)00227-3.  Google Scholar

[35]

L. A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X.  Google Scholar

[36]

H.-J. Zimmerman, Fuzzy programming and linear programming with several objective Functions, Fuzzy Sets Syst., 1 (1978), 45-55.  doi: 10.1016/0165-0114(78)90031-3.  Google Scholar

show all references

References:
[1]

S. Agarwal, S. Banerjee and S. Papachristos, Inventory model with deteriorating items, ramp-type demand and partially backlogged shortages for a two warehouse system, Appl. Math. Model., 37 (2013), 8912-8929.  doi: 10.1016/j.apm.2013.04.026.  Google Scholar

[2]

S. Barik, S. K. Paikray, S. Mishra and U. K. Misra, An inventory model for deteriorating items under time varying demand condition, Int. J. Appl. Eng. Res., 10 (2015), 35770-35773.   Google Scholar

[3]

S. Barik, S. K. Paikray, S. Mishra and U. K. Misra, A Deteriorating inventory model with shortages under pricedependent demand and inflation, Asian J. Math. Comput. Res., 8 (2016), 14-25.   Google Scholar

[4]

D. Chakraborty, D. K. Jana and T. K. Roy, Two-warehouse partial backlogging inventory model with ramp type demand rate, three-parameter Weibull distribution deterioration under inflation and permissible delay in payments, Comput. Indus. Eng., 123 (2018), 157-179.  doi: 10.1016/j.cie.2018.06.022.  Google Scholar

[5]

R. P. Covert and G. C. Philip, An EOQ model for items with weibull distribution deterioration, AIIE Trans., 5 (1973), 323-326.  doi: 10.1080/05695557308974918.  Google Scholar

[6]

D. S. Dinagar and J. R. Kannan, On fuzzy inventory model with allowable shortage, Internat. J. Pure. Appl. Math., 99 (2015), 65-76.  doi: 10.12732/ijpam.v99i1.6.  Google Scholar

[7]

I. Djordjevic, D. Petrovic and G. Stojic, A fuzzy linear programming model for aggregated production planning (APP) in the automotive industry, Comput. Ind., 110 (2019), 48-63.  doi: 10.1016/j.compind.2019.05.004.  Google Scholar

[8]

S. Faddel, A. T. Al-Awami and M. A. Abido, Fuzzy optimization for the operation of electric vehicle parking lots, Elect. Pow. Syst. Res., 145 (2017), 166-174.  doi: 10.1016/j.epsr.2017.01.008.  Google Scholar

[9]

P. M. Ghare and G. F. Schrader, A Model for exponentially decaying inventories, J. Ind. Eng., 14 (1963), 238-243.   Google Scholar

[10]

S. K. Indrajit, S. Routray, S. K. Paikray and U. K. Misra, Fuzzy economic productionquantity model with time dependent demand rate, Sci. J. Logis., 12 (2016), 193-198.   Google Scholar

[11]

C. K. Jaggi, S. Pareek, A. Khanna and R. Sharma, Credit financing in a two-warehouse environment for deteriorating items with price-sensitive demand and fully backlogged shortages, Appl. Math. Model., 38 (2014), 5315-5333.  doi: 10.1016/j.apm.2014.04.025.  Google Scholar

[12]

S. Jain and M. Kumar, An inventory model with power demand pattern Weibull distribution deterioration and shortages, J. Indian Acad. Math., 30 (2008), 55-61.   Google Scholar

[13]

J. Kacprzyk and P. Stanieski, Long-term inventory policy-making through fuzzy decision-making models, Fuzzy Sets Syst., 8 (1982), 117-132.  doi: 10.1016/0165-0114(82)90002-1.  Google Scholar

[14]

C. Kao and W. K. Hsu, A single-period inventory model with fuzzy demand, Comput. Math. Appl., 43 (2002), 841-848.  doi: 10.1016/S0898-1221(01)00325-X.  Google Scholar

[15]

M. T. Lamata, D. Pelta and J. L. Verdegay, Optimisation problems as decision problems: The case of fuzzy optimisation problems, Inform. Sci., 460/461 (2018), 377-388.  doi: 10.1016/j.ins.2017.07.035.  Google Scholar

[16]

H. M. Lee and J. S. Yao, Economic production quantity for fuzzy demand and fuzzy production quantity., European J. Oper. Res., 109 (1998), 203-211.  doi: 10.1016/S0377-2217(97)00200-2.  Google Scholar

[17]

J. Mezei and S. Nikou, Fuzzy optimization to improve mobile health and wellness recommendation systems, Knowledge-Based Syst., 142 (2018), 108-116.  doi: 10.1016/j.knosys.2017.11.030.  Google Scholar

[18]

S. Mishra, G. Misra, U. K. Misra and L. K. Raju, An inventory model with quadratic demand pattern and deterioration with shortages under the influence of inflation, Math. Financ. Lett., 1 (2012), 57-67.   Google Scholar

[19]

S. Mishra, S. Barik, S. K. Paikray and U. K. Misra, An inventory control model of deteriorating items in fuzzy environment, Global J. Pure Appl. Math., 11 (2015), 1301-1312.   Google Scholar

[20]

S. Misra, U. K. Misra, S. Barik and S. K. Paikray, An inventory model for inflation induced demand and Weibull deteriorating items, Internat. J. Adv. Eng. Technol., 4 (2012), 176-182.   Google Scholar

[21]

H. S. Najafi, S. A. Edalatpanah and H. Dutta, A nonlinear model for fully fuzzy linear programming with fully unrestricted variables and parameters, Alexandria Eng. J., 55 (2016), 2589-2595.   Google Scholar

[22]

S. Pal, G. S. Mahapatra and G. P. Samanta, An EPQ model of ramp type demand with Weibull deterioration under inflation and finite horizon in crisp and fuzzy environment, International Journal of Production Economics, 156 (2014), 159-166.  doi: 10.1016/j.ijpe.2014.05.007.  Google Scholar

[23]

K. S. Park, Fuzzy set theoretical interpretation of economic order quantity, IEEE Trans., 17 (1987), 1082-1084.   Google Scholar

[24]

L. K. Raju, U. K. Misra, S. Mishra and G. Misra, An inventory model for Weibull deteriorating items with linear demand pattern, J. Comput. Math. Sci., 3 (2012), 440-445.   Google Scholar

[25]

S. S. Routray, S. K. Paikray, S. Misra and U. K. Misra, Fuzzy inventory model with single item under time dependent demand and holding cost, Internat. J. Adv. Res. Sci. Eng., 6 (2017), 1604-1618.   Google Scholar

[26]

S. S. Sanni and W. I. E. Chukwu, An economic order quantity model for items with three-parameter Weibull distribution deterioration, ramp-type demand and shortages, Appl. Math. Model., 37 (2013), 9698-9706.  doi: 10.1016/j.apm.2013.05.017.  Google Scholar

[27]

S. Sarbjit and S. S. Raj, An optimal inventory policy for items having linear demand and variable deterioration rate with trade credit, J. Math. Stat., 5 (2009), 330-333.  doi: 10.3844/jmssp.2009.330.333.  Google Scholar

[28]

Y. K. Shah, An order-level lot-size inventory model for deteriorating items, AIEE Trans., 9 (1977), 108-112.  doi: 10.1080/05695557708975129.  Google Scholar

[29]

I. Tomba and K. H. Brojendro, An inventory model with linear demand pattern and deterioration with shortages, J. Indian Acad. Math., 33 (2011), 607-612.   Google Scholar

[30]

C. K. Tripathy, L. M. Pradhan and U. Mishra, An EPQ model for linear deteriorating item with variable holding cost, Int. J. Comput. Appl. Math., 5 (2010), 209-215.   Google Scholar

[31]

G. Viji and K. Karthikeyan, An economic production quantity model for three levels of production with Weibull distribution deterioration and shortage, Ain Shams Eng. J., 9 (2018), 1481-1487.  doi: 10.1016/j.asej.2016.10.006.  Google Scholar

[32] T. M. Whitin, The Theory of Inventory Management, 2$^{nd}$ edition, Princeton University Press, Princeton, New Jersey, 1957.   Google Scholar
[33]

H-L. Yang, Two-warehouse partial backlogging inventory models with three-parameter Weibull distribution deterioration under inflation, Int. J. Prod. Econ., 138 (2012), 107-116.  doi: 10.1016/j.ijpe.2012.03.007.  Google Scholar

[34]

H.-M. Lee and J.-S. Yao, Economic order quantity in fuzzy sense for inventory without back-order model, Fuzzy Sets Syst., 105 (1999), 13-31.  doi: 10.1016/S0165-0114(97)00227-3.  Google Scholar

[35]

L. A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X.  Google Scholar

[36]

H.-J. Zimmerman, Fuzzy programming and linear programming with several objective Functions, Fuzzy Sets Syst., 1 (1978), 45-55.  doi: 10.1016/0165-0114(78)90031-3.  Google Scholar

Figure 1.  Inventory level at any time $ t $
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Figure 2.  Effect of parameter α on τ
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Figure 3.  Effect of parameter $ \alpha $ on G$ \mathbb{Z}(\tau) $
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Figure 4.  Effect of parameter $ \alpha $ on $ W1 $
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Figure 5.  Effect of parameter $ \beta $ on $ \tau $
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Figure 6.  Effect of parameter $ \beta $ on G$ \mathbb{Z}(\tau) $
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Figure 7.  Effect of parameter $ \beta $ on $ W1 $
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Figure 8.  Effect of parameter $ \eta $ on $ \tau $
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Figure 9.  Effect of parameter $ \eta $ on G$ \mathbb{Z}(\tau) $
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Figure 10.  Effect of parameter $ \eta $ on $ W1 $
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Figure 11.  Effect of parameter $ \gamma $ on $ \tau $
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Figure 12.  Effect of parameter $ \gamma $ on G$ \mathbb{Z}(\tau) $
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Figure 13.  Effect of parameter $ \gamma $ on $ W1 $
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Figure 14.  Effect of parameter $ T $ on $ \tau $
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Figure 15.  Effect of parameter $ T $ on G$ \mathbb{Z}(\tau) $
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Figure 16.  Effect of parameter $ T $ on $ W1 $
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Table 1.  Sensitivity of parameter $ \alpha $
$ \alpha $ $ \tau $ $ G\mathbb{Z}(t) $ $ W1 $
200 1.23281 6225.45 442.512
250 1.22869 7661.34 543.03
300 1.22586 9097.01 643.548
350 1.2238 10532.6 744.067
400 1.22223 11968. 844.586
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$ \alpha $ $ \tau $ $ G\mathbb{Z}(t) $ $ W1 $
200 1.23281 6225.45 442.512
250 1.22869 7661.34 543.03
300 1.22586 9097.01 643.548
350 1.2238 10532.6 744.067
400 1.22223 11968. 844.586
Table 2.  Sensitivity of parameter $ \beta $
$\beta$$\tau$$G\mathbb{Z}(t)$$W1$
201.232816225.45442.512
251.237786345.58452.624
301.242556465.34462.738
351.247126584.76472.852
401.251516703.86482.968
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$\beta$$\tau$$G\mathbb{Z}(t)$$W1$
201.232816225.45442.512
251.237786345.58452.624
301.242556465.34462.738
351.247126584.76472.852
401.251516703.86482.968
Table 3.  Sensitivity of parameter $ \eta $
$\eta$$\tau$$G\mathbb{Z}(t)$$W1$
0.021.232816225.45442.512
0.031.214816287.83443.496
0.041.198786345.67444.357
0.051.184326399.64445.12
0.061.171176450.29445.804
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$\eta$$\tau$$G\mathbb{Z}(t)$$W1$
0.021.232816225.45442.512
0.031.214816287.83443.496
0.041.198786345.67444.357
0.051.184326399.64445.12
0.061.171176450.29445.804
Table 4.  Sensitivity of parameter $ \gamma $
$\gamma$$\tau$$G\mathbb{Z}(t)$$W1$
41.232816225.45442.512
51.223686232.24442.473
61.21416240.8442.458
71.204336250.49442.45
81.194596260.82442.439
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$\gamma$$\tau$$G\mathbb{Z}(t)$$W1$
41.232816225.45442.512
51.223686232.24442.473
61.21416240.8442.458
71.204336250.49442.45
81.194596260.82442.439
Table 5.  Sensitivity of parameter T
$T$$\tau$$G\mathbb{Z}(t)$$W1$
21.232816225.45442.512
2.11.28846586.13467.245
2.21.342816954.19492.284
2.31.3967330.517.635
2.41.447947713.94543.306
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$T$$\tau$$G\mathbb{Z}(t)$$W1$
21.232816225.45442.512
2.11.28846586.13467.245
2.21.342816954.19492.284
2.31.3967330.517.635
2.41.447947713.94543.306
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