January  2020, 16(1): 141-167. doi: 10.3934/jimo.2018144

Application of preservation technology for lifetime dependent products in an integrated production system

1. 

Department of Industrial & Management Engineering , Hanyang University , Ansan Gyeonggi-do, 426 791, South Korea

2. 

Department of Industrial Engineering, Yonsei University, 50 Yonsei-ro, Sinchon-dong, Seodaemun-gu, Seoul, 03722, South Korea

* Corresponding author: bsbiswajitsarkar@gmail.com(Biswajit Sarkar), Phone Number-+82-31-400-5260, Fax No +82-31-436-8146

Received  May 2017 Revised  April 2018 Published  September 2018

It is important to adopt precisely the optimum level of preservation technology for deteriorating products, as with every passing day, a larger number of items deteriorate and cause an economic loss. For earning more profit, industries have a tendency to add more preservatives for long lifetime of products. However, realizing the health issues, there is a boundary that no manufacturer can add huge amount of preservatives for infinite lifetime of products. The correlation between the long lifetime along with the price of the product is introduced in this model to show the benefit of the optimum level of investment in preservation technology. To maintain the environmental sustainability, the deteriorated items, which can no longer be preserved by adding preservatives anywhere, are disposed with proper protection. The objective of the study is to obtain profit to show the application through a non-linear mathematical. The model is solved through Kuhn-Tucker and an algorithm. Robustness of the model is verified through numerical experiments and sensitivity analysis. Some comparative analyses are provided, which support the adoption of preservation technology for deteriorating products. Numerical studies proved that the profit increases significantly with the application of proposed preservation technology. Some important managerial insights are provided to help the decision makers while implementing the proposed model in real-world situations.

Citation: Muhammad Waqas Iqbal, Biswajit Sarkar. Application of preservation technology for lifetime dependent products in an integrated production system. Journal of Industrial & Management Optimization, 2020, 16 (1) : 141-167. doi: 10.3934/jimo.2018144
References:
[1]

M. BesiouP. Georgiadis and L. N. Van Wassenhove, Official recycling and scavengers: Symbiotic or conflicting?, European Journal of Operational Research, 218 (2012), 563-576.   Google Scholar

[2]

S. M. Bragg, Production cost reduction, Cost Reduction Analysis: Tools and Strategies, (2010), 91-105.   Google Scholar

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T. ChakrabartyB. C. Giri and K. Chaudhuri, An EOQ model for items with Weibull distribution deterioration, shortages and trended demand: An extension of Philip's model, Computers & Operations Research, 25 (1998), 649-657.   Google Scholar

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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 the Operational Research Society, (1999), 1176-1182.   Google Scholar

[5]

S.-C. Chen and J.-T. Teng, Retailer's optimal ordering policy for deteriorating items with maximum lifetime under supplier's trade credit financing, Applied Mathematical Modelling, 38 (2014), 4049-4061.  doi: 10.1016/j.apm.2013.11.056.  Google Scholar

[6]

S.-C. Chen and J.-T. Teng, Inventory and credit decisions for time-varying deteriorating items with up-stream and down-stream trade credit financing by discounted cash flow analysis, European Journal of Operational Research, 243 (2015), 566-575.  doi: 10.1016/j.ejor.2014.12.007.  Google Scholar

[7]

E. P. ChewC. Lee and R. Liu, Joint inventory allocation and pricing decisions for perishable products, International Journal of Production Economics, 120 (2009), 139-150.   Google Scholar

[8]

C.-Y. Dye, The effect of preservation technology investment on a non-instantaneous deteriorating inventory model, Omega, 41 (2013), 872-880.   Google Scholar

[9]

G. FauzaY. AmerS.-H. Lee and H. Prasetyo, An integrated single-vendor multi-buyer production-inventory policy for food products incorporating quality degradation, International Journal of Production Economics, 182 (2016), 409-417.   Google Scholar

[10]

L. FengY.-L. Chan and L. E. C$\acute{a}$rdenas-Barr$\acute{o}$n, Pricing and lot-sizing polices for perishable goods when the demand depends on selling price, displayed stocks, and expiration date, International Journal of Production Economics, 185 (2017), 11-20.   Google Scholar

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P. Ghare and G. Schrader, A model for exponentially decaying inventory, Journal of Industrial Engineering, 14 (1963), 238-243.   Google Scholar

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Y. He and H. Huang, Optimizing inventory and pricing policy for seasonal deteriorating products with preservation technology investment, Journal of Industrial Engineering, 2013 (2013), Article ID 793568, 1-7. Google Scholar

[13]

P. HsuH. Wee and H. Teng, Preservation technology investment for deteriorating inventory, International Journal of Production Economics, 124 (2010), 388-394.   Google Scholar

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V. JeyakumarS. Srisatkunrajah and N. Huy, Kuhn-Tucker sufficiency for global minimum of multi-extremal mathematical programming problems, Journal of Mathematical Analysis and Applications, 335 (2007), 779-788.  doi: 10.1016/j.jmaa.2007.02.013.  Google Scholar

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S. Karlin and C. R. Carr, Prices and optimal inventory policy, Studies in Applied Probability and Management Science, 4 (1962), 159-172.   Google Scholar

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Y. LiS. Zhang and J. Han, Dynamic pricing and periodic ordering for a stochastic inventory system with deteriorating items, Automatica, 76 (2017), 200-213.  doi: 10.1016/j.automatica.2016.11.003.  Google Scholar

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E. S. Mills, Uncertainty and price theory, The Quarterly Journal of Economics, 73 (1959), 116-130.   Google Scholar

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M. $\ddot{O}$nalA. Yenipazarli and O. E. Kundakcioglu, A mathematical model for perishable products with price-and displayed-stock-dependent demand, Computers & Industrial Engineering, 102 (2016), 246-258.   Google Scholar

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S. Priyan and R. Uthayakumar, An integrated production-distribution inventory system for deteriorating products involving fuzzy deterioration and variable setup cost, Journal of Industrial and Production Engineering, 31 (2014), 491-503.   Google Scholar

[20]

J. Qin and W. Liu, The optimal replenishment policy under trade credit financing with ramp type demand and demand dependent production rate, Discrete Dynamics in Nature and Society, 2014 (2014), Art. ID 839418, 18 pp. doi: 10.1155/2014/839418.  Google Scholar

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Y. QinJ. Wang and C. Wei, Joint pricing and inventory control for fresh produce and foods with quality and physical quantity deteriorating simultaneously, International Journal of Production Economics, 152 (2014), 42-48.   Google Scholar

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R. Sachan, On (T, S i) policy inventory model for deteriorating items with time proportional demand, Journal of the Operational Research Society, (1984), 1013-1019.   Google Scholar

[23]

S. SahaI. Nielsen and I. Moon, Optimal retailer investments in green operations and preservation technology for deteriorating items, Journal of Cleaner Production, 140 (2017), 1514-1527.   Google Scholar

[24]

B. Sarkar, An EOQ model with delay in payments and time varying deterioration rate, Mathematical and Computer Modelling, 55 (2012), 367-377.  doi: 10.1016/j.mcm.2011.08.009.  Google Scholar

[25]

B. Sarkar, A production-inventory model with probabilistic deterioration in two-echelon supply chain management, Applied Mathematical Modelling, 37 (2013), 3138-3151.  doi: 10.1016/j.apm.2012.07.026.  Google Scholar

[26]

B. SarkarP. Mandal and S. Sarkar, An EMQ model with price and time dependent demand under the effect of reliability and inflation, Applied Mathematics and Computation, 231 (2014), 414-421.  doi: 10.1016/j.amc.2014.01.004.  Google Scholar

[27]

B. SarkarS. S. Sana and K. Chaudhuri, Optimal reliability, production lotsize and safety stock: An economic manufacturing quantity model, International Journal of Management Science and Engineering Management, 5 (2010), 192-202.   Google Scholar

[28]

B. Sarkar and S. Sarkar, An improved inventory model with partial backlogging, time varying deterioration and stock-dependent demand, Economic Modelling, 30 (2013), 924-932.   Google Scholar

[29]

N. H. ShahU. Chaudhari and M. Y. Jani, Optimal policies for time-varying deteriorating item with preservation technology under selling price and trade credit dependent quadratic demand in a supply chain, International Journal of Applied and Computational Mathematics, 3 (2017), 363-379.  doi: 10.1007/s40819-016-0141-3.  Google Scholar

[30]

N. H. ShahH. N. Soni and K. A. Patel, Optimizing inventory and marketing policy for non-instantaneous deteriorating items with generalized type deterioration and holding cost rates, Omega, 41 (2013), 421-430.   Google Scholar

[31]

K. SkouriI. KonstantarasS. Papachristos and I. Ganas, Inventory models with ramp type demand rate, partial backlogging and Weibull deterioration rate, European Journal of Operational Research, 192 (2009), 79-92.  doi: 10.1016/j.ejor.2007.09.003.  Google Scholar

[32]

J. TangY. LiuR. Y. Fung and X. Luo, Industrial waste recycling strategies optimization problem: mixed integer programming model and heuristics, Engineering Optimization, 40 (2008), 1085-1100.  doi: 10.1080/03052150802294573.  Google Scholar

[33]

J.-T. TengL. E. C$\acute{a}$rdenas-Barr$\acute{o}$nH.-J. ChangJ. Wu and Y. Hu, Inventory lot-size policies for deteriorating items with expiration dates and advance payments, Applied Mathematical Modelling, 40 (2016), 8605-8616.  doi: 10.1016/j.apm.2016.05.022.  Google Scholar

[34]

Y.-C. Tsao, Designing a supply chain network for deteriorating inventory under preservation effort and trade credits, International Journal of Production Research, 54 (2016), 3837-3851.   Google Scholar

[35]

M. Tsiros and C. M. Heilman, The effect of expiration dates and perceived risk on purchasing behavior in grocery store perishable categories, Journal of Marketing, 69 (2005), 114-129.   Google Scholar

[36]

W.-C. WangJ.-T. Teng and K.-R. Lou, Seller's optimal credit period and cycle time in a supply chain for deteriorating items with maximum lifetime, European Journal of Operational Research, 232 (2014), 315-321.  doi: 10.1016/j.ejor.2013.06.027.  Google Scholar

[37]

H. M. Wee and G. A. Widyadana, Economic production quantity models for deteriorating items with rework and stochastic preventive maintenance time, International Journal of Production Research, 50 (2012), 2940-2952.   Google Scholar

[38]

T. M. Whitin, Inventory control and price theory, Management Science, 2 (1955), 61-68.   Google Scholar

[39]

J. WuF. B. Al-KhateebJ.-T. Teng and L. E. Cárdenas-Barrón, Inventory models for deteriorating items with maximum lifetime under downstream partial trade credits to credit-risk customers by discounted cash-flow analysis, International Journal of Production Economics, 171 (2016), 105-115.   Google Scholar

[40]

C.-T. YangC.-Y. Dye and J.-F. Ding, Optimal dynamic trade credit and preservation technology allocation for a deteriorating inventory model, Computers & Industrial Engineering, 87 (2015), 356-369.   Google Scholar

[41]

M. F. Yang and W.-C. Tseng, Deteriorating inventory model for chilled food, Mathematical Problems in Engineering, 2015 (2015), Article ID 816876, 10pp. doi: 10.1155/2015/816876.  Google Scholar

show all references

References:
[1]

M. BesiouP. Georgiadis and L. N. Van Wassenhove, Official recycling and scavengers: Symbiotic or conflicting?, European Journal of Operational Research, 218 (2012), 563-576.   Google Scholar

[2]

S. M. Bragg, Production cost reduction, Cost Reduction Analysis: Tools and Strategies, (2010), 91-105.   Google Scholar

[3]

T. ChakrabartyB. C. Giri and K. Chaudhuri, An EOQ model for items with Weibull distribution deterioration, shortages and trended demand: An extension of Philip's model, Computers & Operations Research, 25 (1998), 649-657.   Google Scholar

[4]

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

[5]

S.-C. Chen and J.-T. Teng, Retailer's optimal ordering policy for deteriorating items with maximum lifetime under supplier's trade credit financing, Applied Mathematical Modelling, 38 (2014), 4049-4061.  doi: 10.1016/j.apm.2013.11.056.  Google Scholar

[6]

S.-C. Chen and J.-T. Teng, Inventory and credit decisions for time-varying deteriorating items with up-stream and down-stream trade credit financing by discounted cash flow analysis, European Journal of Operational Research, 243 (2015), 566-575.  doi: 10.1016/j.ejor.2014.12.007.  Google Scholar

[7]

E. P. ChewC. Lee and R. Liu, Joint inventory allocation and pricing decisions for perishable products, International Journal of Production Economics, 120 (2009), 139-150.   Google Scholar

[8]

C.-Y. Dye, The effect of preservation technology investment on a non-instantaneous deteriorating inventory model, Omega, 41 (2013), 872-880.   Google Scholar

[9]

G. FauzaY. AmerS.-H. Lee and H. Prasetyo, An integrated single-vendor multi-buyer production-inventory policy for food products incorporating quality degradation, International Journal of Production Economics, 182 (2016), 409-417.   Google Scholar

[10]

L. FengY.-L. Chan and L. E. C$\acute{a}$rdenas-Barr$\acute{o}$n, Pricing and lot-sizing polices for perishable goods when the demand depends on selling price, displayed stocks, and expiration date, International Journal of Production Economics, 185 (2017), 11-20.   Google Scholar

[11]

P. Ghare and G. Schrader, A model for exponentially decaying inventory, Journal of Industrial Engineering, 14 (1963), 238-243.   Google Scholar

[12]

Y. He and H. Huang, Optimizing inventory and pricing policy for seasonal deteriorating products with preservation technology investment, Journal of Industrial Engineering, 2013 (2013), Article ID 793568, 1-7. Google Scholar

[13]

P. HsuH. Wee and H. Teng, Preservation technology investment for deteriorating inventory, International Journal of Production Economics, 124 (2010), 388-394.   Google Scholar

[14]

V. JeyakumarS. Srisatkunrajah and N. Huy, Kuhn-Tucker sufficiency for global minimum of multi-extremal mathematical programming problems, Journal of Mathematical Analysis and Applications, 335 (2007), 779-788.  doi: 10.1016/j.jmaa.2007.02.013.  Google Scholar

[15]

S. Karlin and C. R. Carr, Prices and optimal inventory policy, Studies in Applied Probability and Management Science, 4 (1962), 159-172.   Google Scholar

[16]

Y. LiS. Zhang and J. Han, Dynamic pricing and periodic ordering for a stochastic inventory system with deteriorating items, Automatica, 76 (2017), 200-213.  doi: 10.1016/j.automatica.2016.11.003.  Google Scholar

[17]

E. S. Mills, Uncertainty and price theory, The Quarterly Journal of Economics, 73 (1959), 116-130.   Google Scholar

[18]

M. $\ddot{O}$nalA. Yenipazarli and O. E. Kundakcioglu, A mathematical model for perishable products with price-and displayed-stock-dependent demand, Computers & Industrial Engineering, 102 (2016), 246-258.   Google Scholar

[19]

S. Priyan and R. Uthayakumar, An integrated production-distribution inventory system for deteriorating products involving fuzzy deterioration and variable setup cost, Journal of Industrial and Production Engineering, 31 (2014), 491-503.   Google Scholar

[20]

J. Qin and W. Liu, The optimal replenishment policy under trade credit financing with ramp type demand and demand dependent production rate, Discrete Dynamics in Nature and Society, 2014 (2014), Art. ID 839418, 18 pp. doi: 10.1155/2014/839418.  Google Scholar

[21]

Y. QinJ. Wang and C. Wei, Joint pricing and inventory control for fresh produce and foods with quality and physical quantity deteriorating simultaneously, International Journal of Production Economics, 152 (2014), 42-48.   Google Scholar

[22]

R. Sachan, On (T, S i) policy inventory model for deteriorating items with time proportional demand, Journal of the Operational Research Society, (1984), 1013-1019.   Google Scholar

[23]

S. SahaI. Nielsen and I. Moon, Optimal retailer investments in green operations and preservation technology for deteriorating items, Journal of Cleaner Production, 140 (2017), 1514-1527.   Google Scholar

[24]

B. Sarkar, An EOQ model with delay in payments and time varying deterioration rate, Mathematical and Computer Modelling, 55 (2012), 367-377.  doi: 10.1016/j.mcm.2011.08.009.  Google Scholar

[25]

B. Sarkar, A production-inventory model with probabilistic deterioration in two-echelon supply chain management, Applied Mathematical Modelling, 37 (2013), 3138-3151.  doi: 10.1016/j.apm.2012.07.026.  Google Scholar

[26]

B. SarkarP. Mandal and S. Sarkar, An EMQ model with price and time dependent demand under the effect of reliability and inflation, Applied Mathematics and Computation, 231 (2014), 414-421.  doi: 10.1016/j.amc.2014.01.004.  Google Scholar

[27]

B. SarkarS. S. Sana and K. Chaudhuri, Optimal reliability, production lotsize and safety stock: An economic manufacturing quantity model, International Journal of Management Science and Engineering Management, 5 (2010), 192-202.   Google Scholar

[28]

B. Sarkar and S. Sarkar, An improved inventory model with partial backlogging, time varying deterioration and stock-dependent demand, Economic Modelling, 30 (2013), 924-932.   Google Scholar

[29]

N. H. ShahU. Chaudhari and M. Y. Jani, Optimal policies for time-varying deteriorating item with preservation technology under selling price and trade credit dependent quadratic demand in a supply chain, International Journal of Applied and Computational Mathematics, 3 (2017), 363-379.  doi: 10.1007/s40819-016-0141-3.  Google Scholar

[30]

N. H. ShahH. N. Soni and K. A. Patel, Optimizing inventory and marketing policy for non-instantaneous deteriorating items with generalized type deterioration and holding cost rates, Omega, 41 (2013), 421-430.   Google Scholar

[31]

K. SkouriI. KonstantarasS. Papachristos and I. Ganas, Inventory models with ramp type demand rate, partial backlogging and Weibull deterioration rate, European Journal of Operational Research, 192 (2009), 79-92.  doi: 10.1016/j.ejor.2007.09.003.  Google Scholar

[32]

J. TangY. LiuR. Y. Fung and X. Luo, Industrial waste recycling strategies optimization problem: mixed integer programming model and heuristics, Engineering Optimization, 40 (2008), 1085-1100.  doi: 10.1080/03052150802294573.  Google Scholar

[33]

J.-T. TengL. E. C$\acute{a}$rdenas-Barr$\acute{o}$nH.-J. ChangJ. Wu and Y. Hu, Inventory lot-size policies for deteriorating items with expiration dates and advance payments, Applied Mathematical Modelling, 40 (2016), 8605-8616.  doi: 10.1016/j.apm.2016.05.022.  Google Scholar

[34]

Y.-C. Tsao, Designing a supply chain network for deteriorating inventory under preservation effort and trade credits, International Journal of Production Research, 54 (2016), 3837-3851.   Google Scholar

[35]

M. Tsiros and C. M. Heilman, The effect of expiration dates and perceived risk on purchasing behavior in grocery store perishable categories, Journal of Marketing, 69 (2005), 114-129.   Google Scholar

[36]

W.-C. WangJ.-T. Teng and K.-R. Lou, Seller's optimal credit period and cycle time in a supply chain for deteriorating items with maximum lifetime, European Journal of Operational Research, 232 (2014), 315-321.  doi: 10.1016/j.ejor.2013.06.027.  Google Scholar

[37]

H. M. Wee and G. A. Widyadana, Economic production quantity models for deteriorating items with rework and stochastic preventive maintenance time, International Journal of Production Research, 50 (2012), 2940-2952.   Google Scholar

[38]

T. M. Whitin, Inventory control and price theory, Management Science, 2 (1955), 61-68.   Google Scholar

[39]

J. WuF. B. Al-KhateebJ.-T. Teng and L. E. Cárdenas-Barrón, Inventory models for deteriorating items with maximum lifetime under downstream partial trade credits to credit-risk customers by discounted cash-flow analysis, International Journal of Production Economics, 171 (2016), 105-115.   Google Scholar

[40]

C.-T. YangC.-Y. Dye and J.-F. Ding, Optimal dynamic trade credit and preservation technology allocation for a deteriorating inventory model, Computers & Industrial Engineering, 87 (2015), 356-369.   Google Scholar

[41]

M. F. Yang and W.-C. Tseng, Deteriorating inventory model for chilled food, Mathematical Problems in Engineering, 2015 (2015), Article ID 816876, 10pp. doi: 10.1155/2015/816876.  Google Scholar

Figure 1.  Process flow
Figure 2.  Product's maximum lifetime versus rate of deterioration
Figure 3.  Inventory behavior during one cycle
Figure 4.  Improvement of profit with application of proposed preservation technology
Figure 5.  Variation in production time by varying the setup cost, material cost, manufacturing cost, inventory holding cost and disposal cost
Figure 6.  Variation in cycle time by varying the setup cost, material cost, manufacturing cost, inventory holding cost and disposal cost
Figure 7.  Variation in cost of preservation by varying the setup cost, material cost, manufacturing cost, inventory holding cost and disposal cost
Figure 8.  Variation in profit by varying the setup cost, material cost, manufacturing cost, inventory holding cost and disposal cost
Table 1.  Authors' contribution to the literature
Reference paper Deterioration Preservation technology MLD selling-price
type formulation
Hsu et al. [13] constant $-$ $\surd$ $-$
Sarkar [24] Time-varying MLD $-$ $-$
Sarkar and Sarkar [28] Time-varying Linear
Dye [8] Time-varying Linear $\surd$ $-$
Qin et al. [21] Time-and temperature Exponential varying $-$ $-$
Wee and Widyadana [37] Constant $-$ $-$ $-$
Chew et al. [7] $-$ $-$ $-$ $\surd$
Sarkar [25] Random Uniform, triangular Beta $-$ $-$
Wang et al. [36] Time-varying MLD $-$ $-$
Priyan and Uthayakumar [19] Fuzzy Triangular $-$ $-$
Shah et al. [29] Time-varying MLD $\surd$ $-$
Tsao [34] Constant $-$ $\surd$
This paper Time-varying MLD $\surd$ $\surd$
Reference paper Deterioration Preservation technology MLD selling-price
type formulation
Hsu et al. [13] constant $-$ $\surd$ $-$
Sarkar [24] Time-varying MLD $-$ $-$
Sarkar and Sarkar [28] Time-varying Linear
Dye [8] Time-varying Linear $\surd$ $-$
Qin et al. [21] Time-and temperature Exponential varying $-$ $-$
Wee and Widyadana [37] Constant $-$ $-$ $-$
Chew et al. [7] $-$ $-$ $-$ $\surd$
Sarkar [25] Random Uniform, triangular Beta $-$ $-$
Wang et al. [36] Time-varying MLD $-$ $-$
Priyan and Uthayakumar [19] Fuzzy Triangular $-$ $-$
Shah et al. [29] Time-varying MLD $\surd$ $-$
Tsao [34] Constant $-$ $\surd$
This paper Time-varying MLD $\surd$ $\surd$
Table 2.  Values of the parameters for numerical experiment
$C_s$ = $500/setup $h$ = $0.8/unit/month $a$ = 1500 units/month $L$ = 4 months
$δ$ = 0.05 $C_{mt}$ = $15/unit $C_d$ = $0.5/unit $b$ = 60 units/month
$k$ = 3.2 $C_m$ = $10/unit $\varepsilon$ = $100/unit $ M$ = $10/unit
$γ$ = 0.005
$C_s$ = $500/setup $h$ = $0.8/unit/month $a$ = 1500 units/month $L$ = 4 months
$δ$ = 0.05 $C_{mt}$ = $15/unit $C_d$ = $0.5/unit $b$ = 60 units/month
$k$ = 3.2 $C_m$ = $10/unit $\varepsilon$ = $100/unit $ M$ = $10/unit
$γ$ = 0.005
Table 3.  Optimal solution for the numerical experiment when preservation technology is applied
$t_1^*$ = 0.21 month $T^*$ = 0.68 month $C_p^*$ = $1.78 /unit/unit time $\pi^*$ = $115955/month
$t_1^*$ = 0.21 month $T^*$ = 0.68 month $C_p^*$ = $1.78 /unit/unit time $\pi^*$ = $115955/month
Table 4.  Optimal solution for the numerical experiment when preservation technology is not applied
$t_1^*$= 0:18 month $T^*$= 0.54 month $\pi^*$= $ \$ $111106/month
$t_1^*$= 0:18 month $T^*$= 0.54 month $\pi^*$= $ \$ $111106/month
Table 5.  Sensitivity analysis
Parameters Changes of parameters (in %) $t_1^*$(in %) $T^*$ $C_p^*$ $\pi^*$
-50% -26.82 -22.62 -19.26 +10.72
-25% -11.73 -9.31 -10.00 +5.03
$C_s$ +25% +9.50 +7.98 +8.15 -8.92
+50% +18.44 +14.86 +15.19 -8.92
-50% +50.84 +38.14 +39.63 +53.65
-25% +19.55 +15.52 +15.93 +26.11
$C_{mt}$ +25% -13.41 -11.09 -11.85 -25.15
+50% -23.46 -19.29 -20.37 -49.63
-50% +12.29 +9.76 +10.00 +17.27
-25% +5.59 +4.66 +4.44 +8.58
$C_m$ +25% -5.03 -3.99 -4.44 -8.47
+50% -9.50 -7.76 -8.15 -16.85
-50% +11.17 +9.09 +9.26 +3.30
-25% +5.03 +4.21 +4.07 +1.60
$h$ +25% -5.03 -3.55 -4.07 -1.52
+50% -8.94 -6.87 -7.41 -2.95
-50% +0.56 +0.67 +0.37 +0.22
-25% +0.00 +0.22 +0.00 +0.10
$C_d$ +25% -0.56 -0.22 -0.37 -0.10
+50% -0.56 -0.44 -0.74 -0.22
Parameters Changes of parameters (in %) $t_1^*$(in %) $T^*$ $C_p^*$ $\pi^*$
-50% -26.82 -22.62 -19.26 +10.72
-25% -11.73 -9.31 -10.00 +5.03
$C_s$ +25% +9.50 +7.98 +8.15 -8.92
+50% +18.44 +14.86 +15.19 -8.92
-50% +50.84 +38.14 +39.63 +53.65
-25% +19.55 +15.52 +15.93 +26.11
$C_{mt}$ +25% -13.41 -11.09 -11.85 -25.15
+50% -23.46 -19.29 -20.37 -49.63
-50% +12.29 +9.76 +10.00 +17.27
-25% +5.59 +4.66 +4.44 +8.58
$C_m$ +25% -5.03 -3.99 -4.44 -8.47
+50% -9.50 -7.76 -8.15 -16.85
-50% +11.17 +9.09 +9.26 +3.30
-25% +5.03 +4.21 +4.07 +1.60
$h$ +25% -5.03 -3.55 -4.07 -1.52
+50% -8.94 -6.87 -7.41 -2.95
-50% +0.56 +0.67 +0.37 +0.22
-25% +0.00 +0.22 +0.00 +0.10
$C_d$ +25% -0.56 -0.22 -0.37 -0.10
+50% -0.56 -0.44 -0.74 -0.22
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