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January  2022, 18(1): 173-193. doi: 10.3934/jimo.2020148

Effect of reliability on varying demand and holding cost on inventory system incorporating probabilistic deterioration

Sudip Adak and  G. S. Mahapatra ,

Department of Mathematics, National Institute of Technology Puducherry, Karaikal-609609, India

*Corresponding author: G. S. Mahapatra

Received  June 2020 Revised  August 2020 Published  January 2022 Early access  September 2020

This paper presents a mathematical framework to derive an inventory model for time, reliability, and advertisement dependent demand. This paper considers the demand rate is high initially, and then the demand rate reduces later stage, which reflects the situation related to cash in hand. The uncertain deterioration of the product presents through Uniform, Triangular, and Double Triangular probability distributions. The holding cost of the proposed inventory system is dependent on the reliability of the item to make this study a more realistic one. This proposed inventory system allows the situation of shortage and partially backlogged at a fixed rate. Numerical examples, along with managerial implications and sensitivity analysis of the inventory parameters, discuss to examine the effect of changes on the optimal total inventory cost.

Keywords: Deterioration, Holding cost, Reliability, Shortage, Advertisement.
Mathematics Subject Classification: Primary: 90B05, 90B25 and Secondary: 90C31.
Citation: Sudip Adak, G. S. Mahapatra. Effect of reliability on varying demand and holding cost on inventory system incorporating probabilistic deterioration. Journal of Industrial & Management Optimization, 2022, 18 (1) : 173-193. doi: 10.3934/jimo.2020148
References:
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show all references

References:
[1]

B. Ahmad and L. Benkherouf, Economic-order-type inventory models for non-instantaneous deteriorating items and backlogging, RAIRO - Operations Research, 52 (2018), 895-901.  doi: 10.1051/ro/2018010.  Google Scholar

[2]

H. K. Alfares and A. M. Ghaithan, Inventory and pricing model with price-dependent demand, time-varying holding cost, and quantity discounts, Computers & Industrial Engineering, 94 (2016), 170-177.   Google Scholar

[3]

Z. T. Balkhi and L. Benkherouf, On an inventory model for deteriorating items with stock dependent and time-varying demand rates, Computers and Operations Research, 31 (2004), 223-240.  doi: 10.1016/S0305-0548(02)00182-X.  Google Scholar

[4]

H. Barman, M. Pervin, S. K. Roy and G. W. Weber, Back-ordered inventory model with inflation in a cloudy-fuzzy environment, Journal of Industrial and Management Optimization, 2020. doi: 10.3934/jimo.2020052.  Google Scholar

[5]

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[6]

C. K. Chan, W. H. Wong, A. Langevin and Y. C. E. Lee, An integrated production-inventory model for deteriorating items with consideration of optimal production rate and deterioration during delivery, International Journal of Production Economics, 189 (2017), 1-13.   Google Scholar

[7]

H. J. Chang and C. Y. Dye, An EOQ model for deteriorating items with time varying demand and partial backlogging, Journal of Operational Research Society, 50 (1999), 1176-1182.   Google Scholar

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R. R. Chowdhury, S. K. Ghosh and K.S. Chaudhuri, An order-level inventory model for a deteriorating item with time-quadratic demand and time-dependent partial backlogging with shortages in all cycles, American Journal of Mathematical and Management Sciences, 33 (2014), 75-97.   Google Scholar

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R. R. Chowdhury, S. K. Ghosh and K. S. Chaudhuri, An inventory model for perishable items with stock and advertisement sensitive demand, Int. J. Appl. Comput. Math., 1 (2015), 187-201.  doi: 10.1007/s40819-014-0011-9.  Google Scholar

[12]

P. S. Deng, R. Lin and P. P. Chu, A note on the inventory models for deteriorating items with ramp type demand rate, European Journal of Operational Research, 178 (2007), 112-120.  doi: 10.1016/j.ejor.2006.01.028.  Google Scholar

[13]

B. K. Dey, B. Sarkar, M. Sarkar and S. Pareek, An integrated inventory model involving discrete setup cost reduction, variable safety factor, selling price dependent demand, and investment, RAIRO - Operations Research, 53 (2019), 39-57.  doi: 10.1051/ro/2018009.  Google Scholar

[14]

T. K. Datta and A. K. Pal, Effects of inflation and time-value of money on an inventory model with linear time dependent demand rate and shortages, European Journal of Operational Research, 52 (1991), 326-333.   Google Scholar

[15]

K. V. Geetha and R. Udayakumar, Optimal replenishment policy for deteriorating items with time sensitive demand under trade credit financing., American Journal of Mathematical and Management Sciences, 34 (2015), 197-212.   Google Scholar

[16]

M. Ghoreishi, G. W. Weber and A. Mirzazadeh, An inventory model for non-instantaneous deteriorating items with partial backlogging, permissible delay in payments, inflation and selling price-dependent demand and customer returns, Annals of Operations Research, 226 (2014), 221-238.  doi: 10.1007/s10479-014-1739-7.  Google Scholar

[17]

S. K. Ghosh, T. Sarkar and K. Chaudhuri, A multi-item inventory model for deteriorating items in limited storage space with stock-dependent demand, American Journal of Mathematical and Management Sciences, 34 (2015), 147-161.   Google Scholar

[18]

R. Haji and H. Tayebi, Comparing four ordering policies in a lost sales inventory model with Poisson demand and zero ordering cost, Scientia Iranica, 22 (2015), 1294-1298.   Google Scholar

[19]

M. Hemmati, S. M. T. Fatemi Ghomi and M. S. Sajadieh, Inventory of complementary products with stock-dependent demand under vendor-managed inventory with consignment policy, Scientia Iranica, 25 (2018), 2347-2360.   Google Scholar

[20]

M. R. A. Jokar and M. S. Sajadieh, Optimizing a joint economic lot sizing problem with price-sensitive demand, Scientia Iranica, 16 (2009), 159-164.   Google Scholar

[21]

B. C. Giri and K. S. Chaudhuri, Deterministic models of perishable inventory with stock-dependent demand rate and nonlinear holding cost, European Journal of Operational Research, 105 (1998), 467-474.   Google Scholar

[22]

A. Goli, H. K. Zare, R. Sadeghieh and A. Tavakkoli-Moghaddam, Multiobjective fuzzy mathematical model for a financially constrained closed-loop supply chain with labor employment, Computational Intelligence, 36 (2020), 4-34.   Google Scholar

[23]

A. Goli, H. K. Zare, R. Tavakkoli-Moghaddam and A. Sadeghieh, Hybrid artificial intelligence and robust optimization for a multi-objective product portfolio problem Case study: The dairy products industry, Computers & Industrial Engineering, 37 (2019), 106090. Google Scholar

[24]

A. Goli and S. M. R. Davoodi, Coordination policy for production and delivery scheduling in the closed loop supply chain, Production Engineering, 12 (2018), 621-631.   Google Scholar

[25]

S. Khanra, S. K. Ghosh and K. S. Chaudhuri, An EOQ model for a deteriorating item with time dependent quadratic demand rate under permissible delay in payment, Applied Mathematics and Computation, 218 (2011), 1-9.  doi: 10.1016/j.amc.2011.04.062.  Google Scholar

[26]

I. P. Krommyda, K. Skouri and I. Konstantaras, Optimal ordering quantities for substitutable products with stock-dependent demand, Applied Mathematical Modelling, 39 (2015), 147-164.  doi: 10.1016/j.apm.2014.05.016.  Google Scholar

[27]

S. Kumar and U. S. Rajput, A probabilistic inventory model for deteriorating items with ramp type demand rate under inflation, American Journal of Operational Research, 6 (2016), 16-31.   Google Scholar

[28]

R. Lotfi, G. W. Weber, S. M. Sajadifar and N. Mardani, Interdependent demand in the two-period newsvendor problem, Journal of Industrial & Management Optimization, 16 (2018), 117-140.  doi: 10.3934/jimo.2018143.  Google Scholar

[29]

G. S. Mahapatra, S. Adak and K. Kaladhar, A fuzzy inventory model with three parameter Weibull deterioration with reliant holding cost and demand incorporating reliability, Journal of Intelligent and Fuzzy Systems, 36 (2019), 5731-5744.   Google Scholar

[30]

K. Maity and M. Maiti, Inventory of deteriorating complementary and substitute items with stock dependent demand, American Journal of Mathematical and Management Sciences, 25 (2005), 83-96.  doi: 10.1080/01966324.2005.10737644.  Google Scholar

[31]

U. Mishra, L. E. Cárdenas-Barrón, S. Tiwari, A. A. Shaikh and G. Trevi ño-Garza, An inventory model under price and stock dependent demand for controllable deterioration rate with shortages and preservation technology investment, Annals of Operations Research, 254 (2017), 165-190.  doi: 10.1007/s10479-017-2419-1.  Google Scholar

[32]

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Figure 1.  Proposed inventory model with inventory vs time
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Figure 2.  Total cost vs. $ T_{1} $ vs. $ T_{2} $ for Uniformly distributed deterioration
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Figure 3.  Total cost vs. $ T_{1} $ vs. $ T_{2} $ for Triangular distribution deterioration
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Figure 4.  Total cost vs. $ T_{1} $ vs. $ T_{2} $ for Triangular distribution deterioration
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Figure 5.  Percentage change of total profit vs change of parameter
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Figure 6.  Percentage change of total profit vs change of parameter
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Figure 7.  Percentage change of total profit vs change of parameter
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Figure 8.  Percentage change of total profit vs change of parameter
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Table 1.  Contributions of the proposed model with compare to previous studies
Author's Cash in hand Demand depend on Holding Cost depend on Deterioration Backlog
Giri & Chaudhuri (1998) NA stock Non linear NA No
Chang (2004) NA stock Non linear constant No
Skouri et al. (2009) NA ramp type NA Weibull Yes
Sana (2010) NA stock NA probabilistic No
Sett et al. (2012) NA time demand NA NA No
Sarkar & Sarkar (2013) NA time NA probabilistic No
Chowdhury et al. (2014) NA time-quadratic NA time demand Yes
Ghoreishi et al. (2014) NA price & time NA non-instantaneous Yes
Ghosh et al. (2015) NA stock Constant constant No
Wu & Zhao (2015) NA inventory & time NA constant No
Bhunia et al. (2015) NA time, advertisement NA constant Yes
Alfares & Ghaithan (2016) NA price time NA No
Chanda & Kumar (2016) NA advertising & price NA NA No
Sanni & Chukwu (2016) NA deterministic NA Weibull Yes
Shah & Vaghela (2016) NA time & advertisement NA constant No
Mahapatra et al. (2017) NA time & reliability NA constant Yes
Mokhtari et al. (2017) NA stochastic NA constant Yes
Pervin et al. (2018) NA time time Weibull Yes
Lotfi et al. (2018) NA interdependent NA NA Yes
Dey et al. (2019) NA selling price NA NA Yes
Pervin et al. (2019) NA price and stock purchasing cost constant Yes
Pervin et al. (2020) NA time & price NA constant Yes
Roy et al. (2020) NA probabilistic Constant Weibull Yes
This paper Consider time, reliability, advertisement reliability probabilistic Yes
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Author's Cash in hand Demand depend on Holding Cost depend on Deterioration Backlog
Giri & Chaudhuri (1998) NA stock Non linear NA No
Chang (2004) NA stock Non linear constant No
Skouri et al. (2009) NA ramp type NA Weibull Yes
Sana (2010) NA stock NA probabilistic No
Sett et al. (2012) NA time demand NA NA No
Sarkar & Sarkar (2013) NA time NA probabilistic No
Chowdhury et al. (2014) NA time-quadratic NA time demand Yes
Ghoreishi et al. (2014) NA price & time NA non-instantaneous Yes
Ghosh et al. (2015) NA stock Constant constant No
Wu & Zhao (2015) NA inventory & time NA constant No
Bhunia et al. (2015) NA time, advertisement NA constant Yes
Alfares & Ghaithan (2016) NA price time NA No
Chanda & Kumar (2016) NA advertising & price NA NA No
Sanni & Chukwu (2016) NA deterministic NA Weibull Yes
Shah & Vaghela (2016) NA time & advertisement NA constant No
Mahapatra et al. (2017) NA time & reliability NA constant Yes
Mokhtari et al. (2017) NA stochastic NA constant Yes
Pervin et al. (2018) NA time time Weibull Yes
Lotfi et al. (2018) NA interdependent NA NA Yes
Dey et al. (2019) NA selling price NA NA Yes
Pervin et al. (2019) NA price and stock purchasing cost constant Yes
Pervin et al. (2020) NA time & price NA constant Yes
Roy et al. (2020) NA probabilistic Constant Weibull Yes
This paper Consider time, reliability, advertisement reliability probabilistic Yes
Table 2.  Comparison of the proposed model for deterioration
Range of $ \theta $ Distribution $ \theta $ $ \mathbf{T} _{1}^{\ast } $ $ \mathbf{T}_{2}^{\ast } $ $ \mathbf{TC}\left( T_{1}^{\ast }, \text{ }T_{2}^{\ast }\right) \mathbf{(＄)} $
$ \mathbf{0.7\leq 0.9} $ Uniform $ \mathbf{0.8} $ $ \mathbf{0.166} $ $ \mathbf{0.418} $ $ \mathbf{151.144} $
$ \mathbf{0.7\leq 0.85\leq 0.9} $ Triangular $ \mathbf{0.82} $ $ \mathbf{\ 0.142} $ $ \mathbf{0.418} $ $ \mathbf{151.615} $
$ \mathbf{0.7\leq 0.85\leq 0.9} $ Double Triangular $ \mathbf{0.83} $ $ \mathbf{0.13} $ $ \mathbf{0.418} $ $ \mathbf{151.795} $
$ \mathbf{0.8\leq 0.9} $ Uniform $ \mathbf{0.85} $ $ \mathbf{0.108} $ $ \mathbf{0.418} $ $ \mathbf{152.07} $
$ \mathbf{0.8\leq 0.83\leq 0.9} $ Triangular $ \mathbf{0.84} $ $ \mathbf{\ 0.119} $ $ \mathbf{0.418} $ $ \mathbf{151.946} $
$ \mathbf{0.8\leq 0.83\leq 0.9} $ Double Triangular $ \mathbf{0.83} $ $ \mathbf{0.13} $ $ \mathbf{0.418} $ $ \mathbf{151.795} $
$ \mathbf{0.76\leq 0.96} $ Uniform $ \mathbf{0.86} $ $ \mathbf{0.098} $ $ \mathbf{0.418} $ $ \mathbf{152.171} $
$ \mathbf{0.76\leq 0.83\leq 0.96} $ Triangular $ \mathbf{0.85} $ $ \mathbf{\ 0.108} $ $ \mathbf{0.418} $ $ \mathbf{152.07} $
$ \mathbf{0.76\leq 0.83\leq 0.96} $ Double Triangular $ \mathbf{0.84} $ $ \mathbf{0.119} $ $ \mathbf{0.418} $ $ \mathbf{151.946} $
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Range of $ \theta $ Distribution $ \theta $ $ \mathbf{T} _{1}^{\ast } $ $ \mathbf{T}_{2}^{\ast } $ $ \mathbf{TC}\left( T_{1}^{\ast }, \text{ }T_{2}^{\ast }\right) \mathbf{(＄)} $
$ \mathbf{0.7\leq 0.9} $ Uniform $ \mathbf{0.8} $ $ \mathbf{0.166} $ $ \mathbf{0.418} $ $ \mathbf{151.144} $
$ \mathbf{0.7\leq 0.85\leq 0.9} $ Triangular $ \mathbf{0.82} $ $ \mathbf{\ 0.142} $ $ \mathbf{0.418} $ $ \mathbf{151.615} $
$ \mathbf{0.7\leq 0.85\leq 0.9} $ Double Triangular $ \mathbf{0.83} $ $ \mathbf{0.13} $ $ \mathbf{0.418} $ $ \mathbf{151.795} $
$ \mathbf{0.8\leq 0.9} $ Uniform $ \mathbf{0.85} $ $ \mathbf{0.108} $ $ \mathbf{0.418} $ $ \mathbf{152.07} $
$ \mathbf{0.8\leq 0.83\leq 0.9} $ Triangular $ \mathbf{0.84} $ $ \mathbf{\ 0.119} $ $ \mathbf{0.418} $ $ \mathbf{151.946} $
$ \mathbf{0.8\leq 0.83\leq 0.9} $ Double Triangular $ \mathbf{0.83} $ $ \mathbf{0.13} $ $ \mathbf{0.418} $ $ \mathbf{151.795} $
$ \mathbf{0.76\leq 0.96} $ Uniform $ \mathbf{0.86} $ $ \mathbf{0.098} $ $ \mathbf{0.418} $ $ \mathbf{152.171} $
$ \mathbf{0.76\leq 0.83\leq 0.96} $ Triangular $ \mathbf{0.85} $ $ \mathbf{\ 0.108} $ $ \mathbf{0.418} $ $ \mathbf{152.07} $
$ \mathbf{0.76\leq 0.83\leq 0.96} $ Double Triangular $ \mathbf{0.84} $ $ \mathbf{0.119} $ $ \mathbf{0.418} $ $ \mathbf{151.946} $
Table 3.  Sensitivity analysis of the proposed inventory system for parameters
Parameter Change(%) $ T_{1} $ $ T_{2} $ $ TC $ Change $ TC(\%) $
$ -20 $ $ 0.166 $ $ 0.334 $ $ 143.064 $ $ -5.346 $
$ T $ $ -10 $ $ 0.166 $ $ 0.376 $ $ 146.444 $ $ -3.110 $
$ 10 $ $ 0.166 $ $ 0.459 $ $ 156.796 $ $ 3.739 $
$ 20 $ $ 0.166 $ $ 0.501 $ $ 163.153 $ $ 7.946 $
$ -20 $ $ 0.166 $ $ 0.418 $ $ 96.319 $ $ -36.274 $
$ A $ $ -10 $ $ 0.166 $ $ 0.418 $ $ 119.950 $ $ -20.639 $
$ 10 $ $ 0.166 $ $ 0.418 $ $ 191.193 $ $ 26.498 $
$ 20 $ $ 0.166 $ $ 0.418 $ $ 241.458 $ $ 59.754 $
$ -20 $ $ 0.085 $ $ 0.442 $ $ 140.954 $ $ -6.742 $
$ C_{d} $ $ -10 $ $ 0.127 $ $ 0.430 $ $ 146.384 $ $ -3.149 $
$ 10 $ $ 0.201 $ $ 0.406 $ $ 155.207 $ $ 2.688 $
$ 20 $ $ 0.235 $ $ 0.396 $ $ 158.566 $ $ 4.911 $
$ -20 $ $ 0.250 $ $ 0.431 $ $ 142.313 $ $ -5.843 $
$ C_{h} $ $ -10 $ $ 0.205 $ $ 0.424 $ $ 147.077 $ $ -2.691 $
$ 10 $ $ 0.131 $ $ 0.411 $ $ 154.715 $ $ 2.363 $
$ 20 $ $ 0.099 $ $ 0.405 $ $ 157.929 $ $ 4.489 $
$ -20 $ $ 0.166 $ $ 0.365 $ $ 116.975 $ $ -22.607 $
$ C_{s} $ $ -10 $ $ 0.166 $ $ 0.392 $ $ 134.803 $ $ -10.812 $
$ 10 $ $ 0.166 $ $ 0.441 $ $ 166.178 $ $ 9.947 $
$ 20 $ $ 0.166 $ $ 0.462 $ $ 180.055 $ $ 19.128 $
$ -20 $ $ 0.166 $ $ 0.430 $ $ 178.469 $ $ 18.079 $
$ C_{l} $ $ -10 $ $ 0.166 $ $ 0.424 $ $ 164.852 $ $ 9.069 $
$ 10 $ $ 0.166 $ $ 0.411 $ $ 137.350 $ $ -9.126 $
$ 20 $ $ 0.166 $ $ 0.405 $ $ 123.473 $ $ -18.308 $
$ -10 $ $ 0.074 $ $ 0.400 $ $ 151.598 $ $ 0.300 $
$ r $ $ -5 $ $ 0.121 $ $ 0.409 $ $ 151.493 $ $ 0.231 $
$ 5 $ $ 0.210 $ $ 0.425 $ $ 150.459 $ $ -0.453 $
$ 10 $ $ 0.252 $ $ 0.431 $ $ 149.375 $ $ -1.171 $
$ -10 $ $ 0.283 $ $ 0.418 $ $ 146.868 $ $ -2.829 $
$ \theta $ $ -5 $ $ 0.220 $ $ 0.418 $ $ 149.608 $ $ -1.016 $
$ 5 $ $ 0.119 $ $ 0.418 $ $ 151.946 $ $ 0.530 $
$ 10 $ $ 0.079 $ $ 0.418 $ $ 152.317 $ $ 0.776 $
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Parameter Change(%) $ T_{1} $ $ T_{2} $ $ TC $ Change $ TC(\%) $
$ -20 $ $ 0.166 $ $ 0.334 $ $ 143.064 $ $ -5.346 $
$ T $ $ -10 $ $ 0.166 $ $ 0.376 $ $ 146.444 $ $ -3.110 $
$ 10 $ $ 0.166 $ $ 0.459 $ $ 156.796 $ $ 3.739 $
$ 20 $ $ 0.166 $ $ 0.501 $ $ 163.153 $ $ 7.946 $
$ -20 $ $ 0.166 $ $ 0.418 $ $ 96.319 $ $ -36.274 $
$ A $ $ -10 $ $ 0.166 $ $ 0.418 $ $ 119.950 $ $ -20.639 $
$ 10 $ $ 0.166 $ $ 0.418 $ $ 191.193 $ $ 26.498 $
$ 20 $ $ 0.166 $ $ 0.418 $ $ 241.458 $ $ 59.754 $
$ -20 $ $ 0.085 $ $ 0.442 $ $ 140.954 $ $ -6.742 $
$ C_{d} $ $ -10 $ $ 0.127 $ $ 0.430 $ $ 146.384 $ $ -3.149 $
$ 10 $ $ 0.201 $ $ 0.406 $ $ 155.207 $ $ 2.688 $
$ 20 $ $ 0.235 $ $ 0.396 $ $ 158.566 $ $ 4.911 $
$ -20 $ $ 0.250 $ $ 0.431 $ $ 142.313 $ $ -5.843 $
$ C_{h} $ $ -10 $ $ 0.205 $ $ 0.424 $ $ 147.077 $ $ -2.691 $
$ 10 $ $ 0.131 $ $ 0.411 $ $ 154.715 $ $ 2.363 $
$ 20 $ $ 0.099 $ $ 0.405 $ $ 157.929 $ $ 4.489 $
$ -20 $ $ 0.166 $ $ 0.365 $ $ 116.975 $ $ -22.607 $
$ C_{s} $ $ -10 $ $ 0.166 $ $ 0.392 $ $ 134.803 $ $ -10.812 $
$ 10 $ $ 0.166 $ $ 0.441 $ $ 166.178 $ $ 9.947 $
$ 20 $ $ 0.166 $ $ 0.462 $ $ 180.055 $ $ 19.128 $
$ -20 $ $ 0.166 $ $ 0.430 $ $ 178.469 $ $ 18.079 $
$ C_{l} $ $ -10 $ $ 0.166 $ $ 0.424 $ $ 164.852 $ $ 9.069 $
$ 10 $ $ 0.166 $ $ 0.411 $ $ 137.350 $ $ -9.126 $
$ 20 $ $ 0.166 $ $ 0.405 $ $ 123.473 $ $ -18.308 $
$ -10 $ $ 0.074 $ $ 0.400 $ $ 151.598 $ $ 0.300 $
$ r $ $ -5 $ $ 0.121 $ $ 0.409 $ $ 151.493 $ $ 0.231 $
$ 5 $ $ 0.210 $ $ 0.425 $ $ 150.459 $ $ -0.453 $
$ 10 $ $ 0.252 $ $ 0.431 $ $ 149.375 $ $ -1.171 $
$ -10 $ $ 0.283 $ $ 0.418 $ $ 146.868 $ $ -2.829 $
$ \theta $ $ -5 $ $ 0.220 $ $ 0.418 $ $ 149.608 $ $ -1.016 $
$ 5 $ $ 0.119 $ $ 0.418 $ $ 151.946 $ $ 0.530 $
$ 10 $ $ 0.079 $ $ 0.418 $ $ 152.317 $ $ 0.776 $
Table 4.  Sensitivity analysis of the proposed inventory model for parameters
ParameterChange(%) $ T_{1} $ $ T_{2} $ $ TC $ Change $ TC(\%) $
$ -20 $ $ 0.306 $ $ 0.418 $ $ 145.956 $ $ -3.432 $
$ a $ $ -10 $ $ 0.227 $ $ 0.418 $ $ 149.419 $ $ -1.141 $
$ 10 $ $ 0.116 $ $ 0.418 $ $ 151.970 $ $ 0.547 $
$ 20 $ $ 0.076 $ $ 0.418 $ $ 152.329 $ $ 0.784 $
$ -10 $ $ 0.047 $ $ 0.436 $ $ 160.574 $ $ 6.239 $
$ b $ $ -5 $ $ 0.106 $ $ 0.427 $ $ 156.287 $ $ 3.403 $
$ 5 $ $ 0.227 $ $ 0.409 $ $ 144.764 $ $ -4.221 $
$ 10 $ $ 0.289 $ $ 0.400 $ $ 136.737 $ $ -9.532 $
$ -20 $ $ 0.058 $ $ 0.418 $ $ 131.929 $ $ -12.713 $
$ c $ $ -10 $ $ 0.111 $ $ 0.418 $ $ 141.825 $ $ -6.166 $
$ 10 $ $ 0.221 $ $ 0.418 $ $ 159.594 $ $ 5.591 $
$ 20 $ $ 0.278 $ $ 0.418 $ $ 166.861 $ $ 10.399 $
$ -20 $ $ 0.166 $ $ 0.418 $ $ 82.784 $ $ -45.784 $
$ \nu $ $ -10 $ $ 0.166 $ $ 0.418 $ $ 107.584 $ $ -28.820 $
$ 10 $ $ 0.166 $ $ 0.418 $ $ 227.656 $ $ 50.622 $
$ 20 $ $ 0.166 $ $ 0.418 $ $ 362.046 $ $ 139.537 $
$ -20 $ $ 0.166 $ $ 0.418 $ $ 155.908 $ $ 3.152 $
$ k $ $ -10 $ $ 0.166 $ $ 0.418 $ $ 153.499 $ $ 1.558 $
$ 10 $ $ 0.166 $ $ 0.418 $ $ 148.843 $ $ -1.523 $
$ 20 $ $ 0.166 $ $ 0.418 $ $ 146.594 $ $ -3.010 $
$ -20 $ $ 0.166 $ $ 0.418 $ $ 151.242 $ $ 0.065 $
$ \lambda $ $ -10 $ $ 0.166 $ $ 0.418 $ $ 151.193 $ $ 0.032 $
$ 10 $ $ 0.166 $ $ 0.417 $ $ 151.095 $ $ -0.032 $
$ 20 $ $ 0.166 $ $ 0.417 $ $ 151.046 $ $ -0.065 $
$ -20 $ $ 0.218 $ $ 0.426 $ $ 145.779 $ $ -3.550 $
$ u $ $ -10 $ $ 0.192 $ $ 0.422 $ $ 148.496 $ $ -1.752 $
$ 10 $ $ 0.140 $ $ 0.413 $ $ 153.738 $ $ 1.716 $
$ 20 $ $ 0.115 $ $ 0.408 $ $ 156.298 $ $ 3.410 $
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ParameterChange(%) $ T_{1} $ $ T_{2} $ $ TC $ Change $ TC(\%) $
$ -20 $ $ 0.306 $ $ 0.418 $ $ 145.956 $ $ -3.432 $
$ a $ $ -10 $ $ 0.227 $ $ 0.418 $ $ 149.419 $ $ -1.141 $
$ 10 $ $ 0.116 $ $ 0.418 $ $ 151.970 $ $ 0.547 $
$ 20 $ $ 0.076 $ $ 0.418 $ $ 152.329 $ $ 0.784 $
$ -10 $ $ 0.047 $ $ 0.436 $ $ 160.574 $ $ 6.239 $
$ b $ $ -5 $ $ 0.106 $ $ 0.427 $ $ 156.287 $ $ 3.403 $
$ 5 $ $ 0.227 $ $ 0.409 $ $ 144.764 $ $ -4.221 $
$ 10 $ $ 0.289 $ $ 0.400 $ $ 136.737 $ $ -9.532 $
$ -20 $ $ 0.058 $ $ 0.418 $ $ 131.929 $ $ -12.713 $
$ c $ $ -10 $ $ 0.111 $ $ 0.418 $ $ 141.825 $ $ -6.166 $
$ 10 $ $ 0.221 $ $ 0.418 $ $ 159.594 $ $ 5.591 $
$ 20 $ $ 0.278 $ $ 0.418 $ $ 166.861 $ $ 10.399 $
$ -20 $ $ 0.166 $ $ 0.418 $ $ 82.784 $ $ -45.784 $
$ \nu $ $ -10 $ $ 0.166 $ $ 0.418 $ $ 107.584 $ $ -28.820 $
$ 10 $ $ 0.166 $ $ 0.418 $ $ 227.656 $ $ 50.622 $
$ 20 $ $ 0.166 $ $ 0.418 $ $ 362.046 $ $ 139.537 $
$ -20 $ $ 0.166 $ $ 0.418 $ $ 155.908 $ $ 3.152 $
$ k $ $ -10 $ $ 0.166 $ $ 0.418 $ $ 153.499 $ $ 1.558 $
$ 10 $ $ 0.166 $ $ 0.418 $ $ 148.843 $ $ -1.523 $
$ 20 $ $ 0.166 $ $ 0.418 $ $ 146.594 $ $ -3.010 $
$ -20 $ $ 0.166 $ $ 0.418 $ $ 151.242 $ $ 0.065 $
$ \lambda $ $ -10 $ $ 0.166 $ $ 0.418 $ $ 151.193 $ $ 0.032 $
$ 10 $ $ 0.166 $ $ 0.417 $ $ 151.095 $ $ -0.032 $
$ 20 $ $ 0.166 $ $ 0.417 $ $ 151.046 $ $ -0.065 $
$ -20 $ $ 0.218 $ $ 0.426 $ $ 145.779 $ $ -3.550 $
$ u $ $ -10 $ $ 0.192 $ $ 0.422 $ $ 148.496 $ $ -1.752 $
$ 10 $ $ 0.140 $ $ 0.413 $ $ 153.738 $ $ 1.716 $
$ 20 $ $ 0.115 $ $ 0.408 $ $ 156.298 $ $ 3.410 $
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