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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

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.

Citation: Sudip Adak, G. S. Mahapatra. Effect of reliability on varying demand and holding cost on inventory system incorporating probabilistic deterioration. Journal of Industrial and Management Optimization, 2022, 18 (1) : 173-193. doi: 10.3934/jimo.2020148
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.

[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. 

[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.

[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.

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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. 

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C.T. Chang, Inventory model with stock-dependent demand and nonlinear holding costs for deteriorating items, Asia-Pacific Journal of Operational Research, 21 (2004), 435-446.  doi: 10.1142/S0217595904000321.

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U. Chanda and A. Kumar, Optimization of fuzzy EOQ model for advertising and price sensitive demand model under dynamic ceiling on potential adoption, International Journal of Systems Science, 4 (2016), 145-165. 

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R. R. ChowdhuryS. 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. 

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[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.

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[31]

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[32]

N. M. Modak and P. Kelle, Managing a dual-channel supply chain under price and delivery-time dependent stochastic demand, European Journal of Operational Research, 272 (2019), 147-161.  doi: 10.1016/j.ejor.2018.05.067.

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H. MokhtariA. Naimi-Sadigh and A. Salmasnia, A computational approach to economic production quantity model for perishable products with backordering shortage and stock-dependent demand, Scientia Iranica, 24 (2017), 2138-2151. 

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S. Pal and G. S. Mahapatra, A manufacturing-oriented supply chain model for imperfect quality with inspection errors, stochastic demand under rework and shortages, Computers & Industrial Engineering, 106 (2017), 299-314.  doi: 10.1016/j.cie.2017.02.003.

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S. PalG. 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. 

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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.

[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. 

[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.

[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.

[5]

S. BarzegarM. SeifbarghyS.H. Pasandideh and M. Arjmand, Development of a joint economic lot size model with stochastic demand within non-equal shipments, Scientia Iranica, 23 (2016), 3026-3034. 

[6]

C. K. ChanW. H. WongA. 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. 

[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. 

[8]

C.T. Chang, Inventory model with stock-dependent demand and nonlinear holding costs for deteriorating items, Asia-Pacific Journal of Operational Research, 21 (2004), 435-446.  doi: 10.1142/S0217595904000321.

[9]

U. Chanda and A. Kumar, Optimization of fuzzy EOQ model for advertising and price sensitive demand model under dynamic ceiling on potential adoption, International Journal of Systems Science, 4 (2016), 145-165. 

[10]

R. R. ChowdhuryS. 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. 

[11]

R. R. ChowdhuryS. 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.

[12]

P. S. DengR. 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.

[13]

B. K. DeyB. SarkarM. 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.

[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. 

[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. 

[16]

M. GhoreishiG. 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.

[17]

S. K. GhoshT. 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. 

[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. 

[19]

M. HemmatiS. 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. 

[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. 

[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. 

[22]

A. GoliH. K. ZareR. 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. 

[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.

[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. 

[25]

S. KhanraS. 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.

[26]

I. P. KrommydaK. 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.

[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. 

[28]

R. LotfiG. W. WeberS. 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.

[29]

G. S. MahapatraS. 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. 

[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.

[31]

U. MishraL. E. Cárdenas-BarrónS. TiwariA. 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.

[32]

N. M. Modak and P. Kelle, Managing a dual-channel supply chain under price and delivery-time dependent stochastic demand, European Journal of Operational Research, 272 (2019), 147-161.  doi: 10.1016/j.ejor.2018.05.067.

[33]

H. MokhtariA. Naimi-Sadigh and A. Salmasnia, A computational approach to economic production quantity model for perishable products with backordering shortage and stock-dependent demand, Scientia Iranica, 24 (2017), 2138-2151. 

[34]

U. Mishra, J. Tijerina-Aguilera, S. Tiwari and L. E. C árdenas-Barrón, Retailer's joint ordering, pricing, and preservation technology investment policies for a deteriorating item under permissible delay in payments, Mathematical Problems in Engineering, 2018 (2018), Art. ID 6962417, 14 pp. doi: 10.1155/2018/6962417.

[35]

S. M. H. MolanaH. Davoudpour and S. Minner, An (r, nQ) inventory model for packaged deteriorating products with compound Poisson demand, Journal of the Operational Research Society, 63 (2012), 1499-1507. 

[36]

S. Pal and G. S. Mahapatra, A manufacturing-oriented supply chain model for imperfect quality with inspection errors, stochastic demand under rework and shortages, Computers & Industrial Engineering, 106 (2017), 299-314.  doi: 10.1016/j.cie.2017.02.003.

[37]

S. PalG. S. Mahapatra and G. P. Samanta, A production inventory model for deteriorating item with ramp type demand allowing inflation and shortages under fuzziness, Economic Modelling, 46 (2015), 334-345. 

[38]

S. PalG. 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. 

[39]

M. Palanivel and R. Uthayakumar, An inventory model with imperfect items, stock dependent demand and permissible delay in payments under inflation, RAIRO - Operations Research, 50 (2016), 473-489.  doi: 10.1051/ro/2015028.

[40]

M. Palanivel and R. Uthayakumar, An EOQ model for non-instantaneous deteriorating items with partial backlogging and permissible delay in payments under inflation, International Journal of Industrial and Systems Engineering, 26 (2017), 63–89.

[41]

N. PakhiraM. K. Maiti and M. Maiti, Two-level supply chain of a seasonal deteriorating item with time, price, and promotional cost dependent demand under finite time horizon, American Journal of Mathematical and Management Sciences, 36 (2017), 292-315. 

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Figure 1.  Proposed inventory model with inventory vs time
Figure 2.  Total cost vs. $ T_{1} $ vs. $ T_{2} $ for Uniformly distributed deterioration
Figure 3.  Total cost vs. $ T_{1} $ vs. $ T_{2} $ for Triangular distribution deterioration
Figure 4.  Total cost vs. $ T_{1} $ vs. $ T_{2} $ for Triangular distribution deterioration
Figure 5.  Percentage change of total profit vs change of parameter
Figure 6.  Percentage change of total profit vs change of parameter
Figure 7.  Percentage change of total profit vs change of parameter
Figure 8.  Percentage change of total profit vs change of parameter
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
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} $
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 $
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 $
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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