May  2022, 18(3): 1891-1913. doi: 10.3934/jimo.2021048

Optimal lot-sizing policy for a failure prone production system with investment in process quality improvement and lead time variance reduction

Department of Mathematics, Jadavpur University, Kolkata - 700032, India

* Corresponding author: ss.sumonsarkar@gmail.com (Sumon Sarkar)

Received  April 2019 Revised  November 2020 Published  May 2022 Early access  March 2021

To survive in today's competitive market, it is not enough to produce low-cost products but also quality-related issues and lead time needs to be considered in the decision-making process. This paper extends the previous research by developing a stochastic economic manufacturing quantity (EMQ) model for a production system which is subject to process shifts from an in-control state to an out-of-control state at any random time. Moreover, we consider the option of investment to increase the process quality and decrease the lead-time variability. Closed-form solutions of the proposed models are obtained by applying the classical optimization technique. Some lemmas and theorems are developed to determine the optimal solution of the decision variables. Numerical results are obtained for each of these models and compared with those of the basic EMQ model without any investment. From the numerical analysis, it has been observed that our proposed model can significantly reduce the cost of the system compared to the basic model.

Citation: Sumon Sarkar, Bibhas C. Giri. Optimal lot-sizing policy for a failure prone production system with investment in process quality improvement and lead time variance reduction. Journal of Industrial and Management Optimization, 2022, 18 (3) : 1891-1913. doi: 10.3934/jimo.2021048
References:
[1]

M. Al-Salamah, Economic production quantity with the presence of imperfect quality and random machine breakdown and repair based on the artificial bee colony heuristic,, Applied Mathematical Modelling, 63 (2018), 68-83.  doi: 10.1016/j.apm.2018.06.034.

[2]

C. Chandra and J. Grabis, Inventory management with variable lead-time dependent procurement cost, Omega, 36 (2008), 877-887. 

[3]

T. C. E. Cheng, EPQ with process capability and quality assurance considerations, Journal of Operational Research Society, 42 (1991), 713-720. 

[4]

V. ChoudriM. Venkatachalam and S. Panayappan, Production inventory model with deteriorating items, two rates of production cost and taking account of time value of money, Journal of Industrial and Management Optimization, 12 (2016), 1153-1172.  doi: 10.3934/jimo.2016.12.1153.

[5]

M. DeB. Das and M. Maiti, EPL models with fuzzy imperfect production system including carbon emission: a fuzzy differential equation approach, Soft Computing, 24 (2020), 1293-1313.  doi: 10.1007/s00500-019-03967-8.

[6]

E. A. Elsayed and T. O. Boucher, Analysis and Control of Production Systems, Prentice-Hall, , Englewood Cliffs, NJ, 1985.

[7]

L. George and S. Rajagopalan, Process improvement, quality, and learning effects, Management Science, 44 (1998), 1517-1532. 

[8]

B. C. Giri and T. Dohi, Exact formulation of stochastic EMQ model for an unreliable production system, Journal of the Operational Research Society, 56 (2005), 563-575.  doi: 10.1057/palgrave.jors.2601840.

[9]

H. GroeneveltL. Pintelon and A. Seidmann, Production lot sizing with machine breakdown, Management Science, 38 (1992), 104-123.  doi: 10.1287/mnsc.38.1.104.

[10]

H. GroeneveltL. Pintelon and A. Seidmann, Production batching with machine breakdown and safety stocks, Operations Research, 40 (1992), 959-971.  doi: 10.1287/opre.40.5.959.

[11]

D. Gross and A. Soriano, The effect of reducing leadtime on inventory levels-simulation analysis, Management Science, 16 (1969), B61–B67. doi: 10.1287/mnsc.16.2.B61.

[12]

R. Hall, Zero Inventories, , Dow Jones-Irwin, Homewood, IL 1983.

[13]

E. Heard and G. Plossl, Lead times revisited, Production and Inventory Management, 23 (1984), 32-47. 

[14]

A. C. Hax and D. Candea, Production and Inventory Management, Prentice-Hall, Englewood Cliffs, New Jersey, 1984.

[15]

J. C. HayyaT. P. Harrison and X. J. He, The impact of stochastic lead time reduction on inventory cost under order crossover, European Journal of Operational Research, 211 (2011), 274-281.  doi: 10.1016/j.ejor.2010.11.025.

[16]

K. L. Hou and L. C. Lin, Optimal production run length and capital investment in quality improvement with an imperfect production process, International Journal of System Science, 35 (2004), 133-137. 

[17]

K. L. Hou, An EPQ model with setup cost and process quality as functions of capital expenditure, Applied Mathematical Modelling, 31 (2007), 10-17.  doi: 10.1016/j.apm.2006.03.034.

[18]

C. H. Kim CH and Y. Hong, An optimal production run length in deteriorating production processes, International Journal of Production Economics, 58 (1999), 183-189. 

[19]

X. LaiZ. Chen and B. Bidanda, Optimal decision of an economic production quantity model for imperfect manufacturing under hybrid maintenance policy with shortages and partial backlogging, International Journal of Production Research, 57 (2019), 6061-6085.  doi: 10.1080/00207543.2018.1562249.

[20]

L. R. A. CunhaA. P. S. DelfinoK. A. dos Reis and A. Leiras, Economic production quantity (EPQ) model with partial backordering and a discount for imperfect quality batches, International Journal of Production Research, 56 (2018), 6279-6293. 

[21]

G. L. Liao, Joint production and maintenance strategy for economic production quantity model with imperfect production processes, Journal of Intelligent Manufacturing, 24 (2013), 1229-1240.  doi: 10.1007/s10845-012-0658-1.

[22]

H. H. LeeM. J. Chandra and V. J. Deleveaux, Optimal batch size and investment in multistage production systems with scrap, Production Planning & Control, 8 (1997), 586-596. 

[23]

J. Y. Lee and L. B. Schwarz, Lead time management in a periodic-review inventory system: A state-dependent base-stock policy, European Journal of Operational Research, 199 (2009), 122-129.  doi: 10.1016/j.ejor.2008.10.024.

[24]

M. Liberatore, The EOQ model under stochastic lead time, Operations Research, 27 (1979), 391-396.  doi: 10.1287/opre.27.2.391.

[25]

A. K. MannaJ. K. Dey and S. K. Mondal, Two layers supply chain in an imperfect production inventory model with two storage facilities under reliability consideration., Journal of Industrial and Production Engineering, 35 (2018), 57-73.  doi: 10.1080/21681015.2017.1415230.

[26]

F. NasriJ. F. Affisco and M. J. Paknejad, Setup cost reduction in an inventory model with finite-range stochastic lead times, International Journal of Production Research, 28 (1990), 199-212.  doi: 10.1080/00207549008942693.

[27]

A. H. NobilA. H. A. Sedigh and L. E. C$\acute{a}$rdenas-Barr$\acute{o}$n, Multi-machine economic production quantity for items with scrapped and rework with shortages and allocation decisions, Scientia Iranica: Transection E, Industrial Engineering, 25 (2018), 2331-2346.  doi: 10.24200/sci.2017.4453.

[28]

M. J. PaknejadF. Nasri and J. F. Affisco, Lead-time variability reduction in stochastic inventory models, European Journal of Operations Research, 62 (1992), 311-322.  doi: 10.1016/0377-2217(92)90121-O.

[29]

E. L. Porteus, Optimal lot sizing, process quality improvement and setup cost reduction, Operations Research, 34 (1986), 137-144.  doi: 10.1287/opre.34.1.137.

[30]

M. J. Rosenblatt and H. L. Lee, Economic production cycles with imperfect production processes, IIE Transactions, 18 (1986), 48-55.  doi: 10.1080/07408178608975329.

[31]

S. S. Sana, An economic production lot size model in an imperfect production system, European Journal of Operational Research, 201 (2010), 158-170. 

[32]

S. Sarkar and B.C. Giri, Stochastic supply chain model with imperfect production and controllable defective rate, International Journal of System Science: Operations & Logistics, 7 (2020), 133-146.  doi: 10.1080/23302674.2018.1536231.

[33]

A. SofanaC. N. Rosyidi and E. Pujiyanto, Product quality improvement model considering quality investment in rework policies and supply chain proft sharing, Journal of Industrial Engineering International, 15 (2019), 637-649. 

[34]

B. K. SettS. Sarkar and B. Sarkar, Optimal buffer inventory and inspection errors in an imperfect production system with preventive maintenance, The International Journal of Advanced Manufacturing Technology, 90 (2017), 545-560.  doi: 10.1007/s00170-016-9359-9.

[35]

B. SarkarB. K. Sett BK and S. Sarkar, Optimal production run time and inspection errors in an imperfect production system with warranty, Journal of Industrial and Management Optimization, 14 (2018), 267-282.  doi: 10.3934/jimo.2017046.

[36]

B. R. Sarker and E. R. Coates, Manufacturing setup cost reduction under variable lead times and finite opportunities for investment, International Journal of Production Economics, 49 (1997), 237-247.  doi: 10.1016/S0925-5273(97)00010-8.

[37]

E. A. Silver and R. Peterson, Decision Systems for Inventory Management and Production Planning, Wiley, New York, 1985.

[38]

G. P. Sphicas, On the solution of an inventory model with variable lead times, Operations Research, 30 (1982), 404-410. 

[39]

G. P. Sphicas and F. Nasri, An inventory model with finite-range stochastic lead times, Naval Research Logistics Quarterly, 31 (1984), 609-616.  doi: 10.1002/nav.3800310410.

[40]

A. A. TaleizadehaV. R. Soleymanfar and K. Govindan, Sustainable economic production quantity models for inventory systems with shortage, Journal of Cleaner Production, 174 (2018), 1011-1020. 

[41]

J. Taheri-TolgariM. MohammadiB. NaderiA. Arshadi-Khamseh and A. Mirzazadeh, An inventory model with imperfect item, inspection errors, preventive maintenance and partial backlogging in uncertain environment, Journal of Industrial and Management Optimization, 15 (2019), 1317-1344.  doi: 10.3934/jimo.2018097.

[42]

S. TiwariS. S. Sana and S. Sarkar, Joint economic lot sizing model with stochastic demand and controllable lead-time by reducing ordering cost and setup cost, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 112 (2018), 1075-1099.  doi: 10.1007/s13398-017-0410-y.

[43]

C. E. Vinson, The cost of ignoring lead-time unreliability in inventory theory, Decision Sciences, 3 (1972), 87-105.  doi: 10.1111/j.1540-5915.1972.tb00538.x.

[44]

D. WenP. ErshunW. Ying and L. Wenzhu, An economic production quantity model for a deteriorating system integrated with predictive maintenance strategy, Journal of Intelligent Manufacturing, 27 (2016), 1323-1333.  doi: 10.1007/s10845-014-0954-z.

[45]

C. A. Yano and H. L. Lee, Lot sizing with random yields: A review, Operations Research, 43 (1995), 311-334.  doi: 10.1287/opre.43.2.311.

show all references

References:
[1]

M. Al-Salamah, Economic production quantity with the presence of imperfect quality and random machine breakdown and repair based on the artificial bee colony heuristic,, Applied Mathematical Modelling, 63 (2018), 68-83.  doi: 10.1016/j.apm.2018.06.034.

[2]

C. Chandra and J. Grabis, Inventory management with variable lead-time dependent procurement cost, Omega, 36 (2008), 877-887. 

[3]

T. C. E. Cheng, EPQ with process capability and quality assurance considerations, Journal of Operational Research Society, 42 (1991), 713-720. 

[4]

V. ChoudriM. Venkatachalam and S. Panayappan, Production inventory model with deteriorating items, two rates of production cost and taking account of time value of money, Journal of Industrial and Management Optimization, 12 (2016), 1153-1172.  doi: 10.3934/jimo.2016.12.1153.

[5]

M. DeB. Das and M. Maiti, EPL models with fuzzy imperfect production system including carbon emission: a fuzzy differential equation approach, Soft Computing, 24 (2020), 1293-1313.  doi: 10.1007/s00500-019-03967-8.

[6]

E. A. Elsayed and T. O. Boucher, Analysis and Control of Production Systems, Prentice-Hall, , Englewood Cliffs, NJ, 1985.

[7]

L. George and S. Rajagopalan, Process improvement, quality, and learning effects, Management Science, 44 (1998), 1517-1532. 

[8]

B. C. Giri and T. Dohi, Exact formulation of stochastic EMQ model for an unreliable production system, Journal of the Operational Research Society, 56 (2005), 563-575.  doi: 10.1057/palgrave.jors.2601840.

[9]

H. GroeneveltL. Pintelon and A. Seidmann, Production lot sizing with machine breakdown, Management Science, 38 (1992), 104-123.  doi: 10.1287/mnsc.38.1.104.

[10]

H. GroeneveltL. Pintelon and A. Seidmann, Production batching with machine breakdown and safety stocks, Operations Research, 40 (1992), 959-971.  doi: 10.1287/opre.40.5.959.

[11]

D. Gross and A. Soriano, The effect of reducing leadtime on inventory levels-simulation analysis, Management Science, 16 (1969), B61–B67. doi: 10.1287/mnsc.16.2.B61.

[12]

R. Hall, Zero Inventories, , Dow Jones-Irwin, Homewood, IL 1983.

[13]

E. Heard and G. Plossl, Lead times revisited, Production and Inventory Management, 23 (1984), 32-47. 

[14]

A. C. Hax and D. Candea, Production and Inventory Management, Prentice-Hall, Englewood Cliffs, New Jersey, 1984.

[15]

J. C. HayyaT. P. Harrison and X. J. He, The impact of stochastic lead time reduction on inventory cost under order crossover, European Journal of Operational Research, 211 (2011), 274-281.  doi: 10.1016/j.ejor.2010.11.025.

[16]

K. L. Hou and L. C. Lin, Optimal production run length and capital investment in quality improvement with an imperfect production process, International Journal of System Science, 35 (2004), 133-137. 

[17]

K. L. Hou, An EPQ model with setup cost and process quality as functions of capital expenditure, Applied Mathematical Modelling, 31 (2007), 10-17.  doi: 10.1016/j.apm.2006.03.034.

[18]

C. H. Kim CH and Y. Hong, An optimal production run length in deteriorating production processes, International Journal of Production Economics, 58 (1999), 183-189. 

[19]

X. LaiZ. Chen and B. Bidanda, Optimal decision of an economic production quantity model for imperfect manufacturing under hybrid maintenance policy with shortages and partial backlogging, International Journal of Production Research, 57 (2019), 6061-6085.  doi: 10.1080/00207543.2018.1562249.

[20]

L. R. A. CunhaA. P. S. DelfinoK. A. dos Reis and A. Leiras, Economic production quantity (EPQ) model with partial backordering and a discount for imperfect quality batches, International Journal of Production Research, 56 (2018), 6279-6293. 

[21]

G. L. Liao, Joint production and maintenance strategy for economic production quantity model with imperfect production processes, Journal of Intelligent Manufacturing, 24 (2013), 1229-1240.  doi: 10.1007/s10845-012-0658-1.

[22]

H. H. LeeM. J. Chandra and V. J. Deleveaux, Optimal batch size and investment in multistage production systems with scrap, Production Planning & Control, 8 (1997), 586-596. 

[23]

J. Y. Lee and L. B. Schwarz, Lead time management in a periodic-review inventory system: A state-dependent base-stock policy, European Journal of Operational Research, 199 (2009), 122-129.  doi: 10.1016/j.ejor.2008.10.024.

[24]

M. Liberatore, The EOQ model under stochastic lead time, Operations Research, 27 (1979), 391-396.  doi: 10.1287/opre.27.2.391.

[25]

A. K. MannaJ. K. Dey and S. K. Mondal, Two layers supply chain in an imperfect production inventory model with two storage facilities under reliability consideration., Journal of Industrial and Production Engineering, 35 (2018), 57-73.  doi: 10.1080/21681015.2017.1415230.

[26]

F. NasriJ. F. Affisco and M. J. Paknejad, Setup cost reduction in an inventory model with finite-range stochastic lead times, International Journal of Production Research, 28 (1990), 199-212.  doi: 10.1080/00207549008942693.

[27]

A. H. NobilA. H. A. Sedigh and L. E. C$\acute{a}$rdenas-Barr$\acute{o}$n, Multi-machine economic production quantity for items with scrapped and rework with shortages and allocation decisions, Scientia Iranica: Transection E, Industrial Engineering, 25 (2018), 2331-2346.  doi: 10.24200/sci.2017.4453.

[28]

M. J. PaknejadF. Nasri and J. F. Affisco, Lead-time variability reduction in stochastic inventory models, European Journal of Operations Research, 62 (1992), 311-322.  doi: 10.1016/0377-2217(92)90121-O.

[29]

E. L. Porteus, Optimal lot sizing, process quality improvement and setup cost reduction, Operations Research, 34 (1986), 137-144.  doi: 10.1287/opre.34.1.137.

[30]

M. J. Rosenblatt and H. L. Lee, Economic production cycles with imperfect production processes, IIE Transactions, 18 (1986), 48-55.  doi: 10.1080/07408178608975329.

[31]

S. S. Sana, An economic production lot size model in an imperfect production system, European Journal of Operational Research, 201 (2010), 158-170. 

[32]

S. Sarkar and B.C. Giri, Stochastic supply chain model with imperfect production and controllable defective rate, International Journal of System Science: Operations & Logistics, 7 (2020), 133-146.  doi: 10.1080/23302674.2018.1536231.

[33]

A. SofanaC. N. Rosyidi and E. Pujiyanto, Product quality improvement model considering quality investment in rework policies and supply chain proft sharing, Journal of Industrial Engineering International, 15 (2019), 637-649. 

[34]

B. K. SettS. Sarkar and B. Sarkar, Optimal buffer inventory and inspection errors in an imperfect production system with preventive maintenance, The International Journal of Advanced Manufacturing Technology, 90 (2017), 545-560.  doi: 10.1007/s00170-016-9359-9.

[35]

B. SarkarB. K. Sett BK and S. Sarkar, Optimal production run time and inspection errors in an imperfect production system with warranty, Journal of Industrial and Management Optimization, 14 (2018), 267-282.  doi: 10.3934/jimo.2017046.

[36]

B. R. Sarker and E. R. Coates, Manufacturing setup cost reduction under variable lead times and finite opportunities for investment, International Journal of Production Economics, 49 (1997), 237-247.  doi: 10.1016/S0925-5273(97)00010-8.

[37]

E. A. Silver and R. Peterson, Decision Systems for Inventory Management and Production Planning, Wiley, New York, 1985.

[38]

G. P. Sphicas, On the solution of an inventory model with variable lead times, Operations Research, 30 (1982), 404-410. 

[39]

G. P. Sphicas and F. Nasri, An inventory model with finite-range stochastic lead times, Naval Research Logistics Quarterly, 31 (1984), 609-616.  doi: 10.1002/nav.3800310410.

[40]

A. A. TaleizadehaV. R. Soleymanfar and K. Govindan, Sustainable economic production quantity models for inventory systems with shortage, Journal of Cleaner Production, 174 (2018), 1011-1020. 

[41]

J. Taheri-TolgariM. MohammadiB. NaderiA. Arshadi-Khamseh and A. Mirzazadeh, An inventory model with imperfect item, inspection errors, preventive maintenance and partial backlogging in uncertain environment, Journal of Industrial and Management Optimization, 15 (2019), 1317-1344.  doi: 10.3934/jimo.2018097.

[42]

S. TiwariS. S. Sana and S. Sarkar, Joint economic lot sizing model with stochastic demand and controllable lead-time by reducing ordering cost and setup cost, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 112 (2018), 1075-1099.  doi: 10.1007/s13398-017-0410-y.

[43]

C. E. Vinson, The cost of ignoring lead-time unreliability in inventory theory, Decision Sciences, 3 (1972), 87-105.  doi: 10.1111/j.1540-5915.1972.tb00538.x.

[44]

D. WenP. ErshunW. Ying and L. Wenzhu, An economic production quantity model for a deteriorating system integrated with predictive maintenance strategy, Journal of Intelligent Manufacturing, 27 (2016), 1323-1333.  doi: 10.1007/s10845-014-0954-z.

[45]

C. A. Yano and H. L. Lee, Lot sizing with random yields: A review, Operations Research, 43 (1995), 311-334.  doi: 10.1287/opre.43.2.311.

Figure 1.  Logistic diagram of the proposed model
Table 1.  Numerical results for different models
Model type $ \; \; Q $ $ \; \; V $ $ \; \; \theta $ $ E[TC(\cdot)] $ reduction in $ \theta $
%
reduction in $ V $
%
reduction in $ E[TC(\cdot)] $
%
EMQ-SLT Exact $ \; \; 556.4 $ $ - $ $ - $ $ 3369.26 $ $ - $ $ - $ $ - $
Approx. $ \; \; 555.6 $ $ - $ $ - $ $ 3371.76 $ $ - $ $ - $ $ - $
SLT-QI Exact $ \; \; 602.3 $ $ - $ $ 0.02352 $ $ 3203.55 $ $ 84.32 $ $ - $ $ 4.92 $
Approx. $ \; \; 602.2 $ $ - $ $ 0.02331 $ $ 3204.00 $ $ 84.46 $ $ - $ $ 4.98 $
SLT-VR Exact $ \; \; 448.8 $ $ 0.0001660 $ $ - $ $ 3025.87 $ $ - $ $ 78.44 $ $ 10.19 $
Approx $ \; \; 448.2 $ $ 0.0001658 $ $ - $ $ 3027.5 $ $ - $ $ 78.47 $ $ 10.21 $
SLT-SI Exact $ \; \; 487.1 $ $ 0.0001802 $ $ 0.02903 $ $ 2908.06 $ $ 80.64 $ $ 76.60 $ 13.69
Approx. $ \; \; 487.1 $ $ 0.0001801 $ $ 0.02882 $ $ 2908.43 $ $ 80.79 $ $ 76.61 $ 13.68
Model type $ \; \; Q $ $ \; \; V $ $ \; \; \theta $ $ E[TC(\cdot)] $ reduction in $ \theta $
%
reduction in $ V $
%
reduction in $ E[TC(\cdot)] $
%
EMQ-SLT Exact $ \; \; 556.4 $ $ - $ $ - $ $ 3369.26 $ $ - $ $ - $ $ - $
Approx. $ \; \; 555.6 $ $ - $ $ - $ $ 3371.76 $ $ - $ $ - $ $ - $
SLT-QI Exact $ \; \; 602.3 $ $ - $ $ 0.02352 $ $ 3203.55 $ $ 84.32 $ $ - $ $ 4.92 $
Approx. $ \; \; 602.2 $ $ - $ $ 0.02331 $ $ 3204.00 $ $ 84.46 $ $ - $ $ 4.98 $
SLT-VR Exact $ \; \; 448.8 $ $ 0.0001660 $ $ - $ $ 3025.87 $ $ - $ $ 78.44 $ $ 10.19 $
Approx $ \; \; 448.2 $ $ 0.0001658 $ $ - $ $ 3027.5 $ $ - $ $ 78.47 $ $ 10.21 $
SLT-SI Exact $ \; \; 487.1 $ $ 0.0001802 $ $ 0.02903 $ $ 2908.06 $ $ 80.64 $ $ 76.60 $ 13.69
Approx. $ \; \; 487.1 $ $ 0.0001801 $ $ 0.02882 $ $ 2908.43 $ $ 80.79 $ $ 76.61 $ 13.68
Table 2.  Computational results for different lead-time interval
Lead time interval Model type $ \; \; Q $ $ \; \; V $ $ \; \; \theta $ $ E[TC(\cdot)] $ reduction in $ \theta $
(%)
reduction in $ V $
(%)
reduction in $ E[TC(\cdot)] $
(%)
EMQ-SLT $ 420 $ $ - $ $ - $ $ 2552 $ $ - $ $ - $ $ - $
$ 1 $ SLT-QI $ 453 $ $ - $ $ 0.0309 $ $ 2446 $ $ 79.4 $ $ - $ $ 4.15 $
week SLT-VR $ 420 $ $ - $ $ - $ $ 2552 $ $ - $ $ - $ $ - $
SLT-SI $ 453 $ $ - $ $ 0.0309 $ $ 2446 $ $ 79.4 $ $ - $ $ 4.15 $
EMQ-SLT $ \; \; 440 $ $ - $ $ - $ $ 2668 $ $ - $ $ - $ $ - $
$ 2 $ SLT-QI $ \; \; 475 $ $ - $ $ 0.0296 $ $ 2554 $ $ 80.27 $ $ - $ $ 4.27 $
weeks SLT-VR $ \; \; 440 $ $ - $ $ - $ $ 2668 $ $ - $ $ - $ $ - $
SLT-SI $ \; \; 475 $ $ - $ $ 0.0296 $ $ 2554 $ $ 80.27 $ $ - $ $ 4.27 $
EMQ-SLT $ \; \; 470 $ $ - $ $ - $ $ 2852 $ $ - $ $ - $ $ - $
$ 3 $ SLT-QI $ \; \; 508 $ $ - $ $ 0.0276 $ $ 2723 $ $ 81.60 $ $ - $ $ 4.52 $
weeks SLT-VR $ \; \; 448 $ $ 0.000166 $ $ - $ $ 2823 $ $ - $ $ 40.07 $ $ 1.02 $
SLT-SI $ \; \; 487 $ $ 0.000180 $ $ 0.0288 $ $ 2704 $ $ 76.62 $ 35.02 5.19
EMQ-SLT $ \; \; 509 $ $ - $ $ - $ $ 3090 $ $ - $ $ - $ $ - $
$ 4 $ SLT-QI $ \; \; 551 $ $ - $ $ 0.0255 $ $ 2944 $ $ 83.0 $ $ - $ $ 4.72 $
weeks SLT-VR $ \; \; 448 $ $ 0.000166 $ $ - $ $ 2938 $ $ - $ $ 66.33 $ $ 4.91 $
SLT-SI $ \; \; 487 $ $ 0.000180 $ $ 0.0288 $ $ 2819 $ $ 80.8 $ $ 63.49 $ 8.77
EMQ-SLT $ \; \; 556 $ $ - $ $ - $ $ 3372 $ $ - $ $ - $ $ - $
$ 5 $ SLT-QI $ \; \; 602 $ $ - $ $ 0.0233 $ $ 3204 $ $ 84.46 $ $ - $ $ 4.98 $
weeks SLT-VR $ \; \; 448 $ $ 0.000166 $ $ - $ $ 3028 $ $ - $ $ 78.47 $ $ 10.21 $
SLT-SI $ \; \; 487 $ $ 0.000180 $ $ 0.0288 $ $ 2908 $ $ 80.79 $ $ 76.61 $ 13.68
EMQ-SLT $ \; \; 608 $ $ - $ $ - $ $ 3687 $ $ - $ $ - $ $ - $
$ 6 $ SLT-QI $ \; \; 659 $ $ - $ $ 0.0213 $ $ 3495 $ $ 85.8 $ $ - $ $ 5.21 $
weeks SLT-VR $ \; \; 448 $ $ 0.000166 $ $ - $ $ 3100 $ $ - $ $ 85.03 $ $ 15.92 $
SLT-SI $ \; \; 487 $ $ 0.000180 $ $ 0.0288 $ $ 2981 $ $ 80.8 $ $ 83.77 $ 19.15
EMQ-SLT $ \; \; 664 $ $ - $ $ - $ $ 4028 $ $ - $ $ - $ $ - $
$ 7 $ SLT-QI $ \; \; 721 $ $ - $ $ 0.0194 $ $ 3808 $ $ 87.06 $ $ - $ $ 5.46 $
weeks SLT-VR $ \; \; 448 $ $ 0.000166 $ $ - $ $ 3162 $ $ - $ $ 89.01 $ $ 21.50 $
SLT-SI $ \; \; 487 $ $ 0.000180 $ $ 0.0288 $ $ 3043 $ $ 80.8 $ $ 88.08 $ 24.45
Lead time interval Model type $ \; \; Q $ $ \; \; V $ $ \; \; \theta $ $ E[TC(\cdot)] $ reduction in $ \theta $
(%)
reduction in $ V $
(%)
reduction in $ E[TC(\cdot)] $
(%)
EMQ-SLT $ 420 $ $ - $ $ - $ $ 2552 $ $ - $ $ - $ $ - $
$ 1 $ SLT-QI $ 453 $ $ - $ $ 0.0309 $ $ 2446 $ $ 79.4 $ $ - $ $ 4.15 $
week SLT-VR $ 420 $ $ - $ $ - $ $ 2552 $ $ - $ $ - $ $ - $
SLT-SI $ 453 $ $ - $ $ 0.0309 $ $ 2446 $ $ 79.4 $ $ - $ $ 4.15 $
EMQ-SLT $ \; \; 440 $ $ - $ $ - $ $ 2668 $ $ - $ $ - $ $ - $
$ 2 $ SLT-QI $ \; \; 475 $ $ - $ $ 0.0296 $ $ 2554 $ $ 80.27 $ $ - $ $ 4.27 $
weeks SLT-VR $ \; \; 440 $ $ - $ $ - $ $ 2668 $ $ - $ $ - $ $ - $
SLT-SI $ \; \; 475 $ $ - $ $ 0.0296 $ $ 2554 $ $ 80.27 $ $ - $ $ 4.27 $
EMQ-SLT $ \; \; 470 $ $ - $ $ - $ $ 2852 $ $ - $ $ - $ $ - $
$ 3 $ SLT-QI $ \; \; 508 $ $ - $ $ 0.0276 $ $ 2723 $ $ 81.60 $ $ - $ $ 4.52 $
weeks SLT-VR $ \; \; 448 $ $ 0.000166 $ $ - $ $ 2823 $ $ - $ $ 40.07 $ $ 1.02 $
SLT-SI $ \; \; 487 $ $ 0.000180 $ $ 0.0288 $ $ 2704 $ $ 76.62 $ 35.02 5.19
EMQ-SLT $ \; \; 509 $ $ - $ $ - $ $ 3090 $ $ - $ $ - $ $ - $
$ 4 $ SLT-QI $ \; \; 551 $ $ - $ $ 0.0255 $ $ 2944 $ $ 83.0 $ $ - $ $ 4.72 $
weeks SLT-VR $ \; \; 448 $ $ 0.000166 $ $ - $ $ 2938 $ $ - $ $ 66.33 $ $ 4.91 $
SLT-SI $ \; \; 487 $ $ 0.000180 $ $ 0.0288 $ $ 2819 $ $ 80.8 $ $ 63.49 $ 8.77
EMQ-SLT $ \; \; 556 $ $ - $ $ - $ $ 3372 $ $ - $ $ - $ $ - $
$ 5 $ SLT-QI $ \; \; 602 $ $ - $ $ 0.0233 $ $ 3204 $ $ 84.46 $ $ - $ $ 4.98 $
weeks SLT-VR $ \; \; 448 $ $ 0.000166 $ $ - $ $ 3028 $ $ - $ $ 78.47 $ $ 10.21 $
SLT-SI $ \; \; 487 $ $ 0.000180 $ $ 0.0288 $ $ 2908 $ $ 80.79 $ $ 76.61 $ 13.68
EMQ-SLT $ \; \; 608 $ $ - $ $ - $ $ 3687 $ $ - $ $ - $ $ - $
$ 6 $ SLT-QI $ \; \; 659 $ $ - $ $ 0.0213 $ $ 3495 $ $ 85.8 $ $ - $ $ 5.21 $
weeks SLT-VR $ \; \; 448 $ $ 0.000166 $ $ - $ $ 3100 $ $ - $ $ 85.03 $ $ 15.92 $
SLT-SI $ \; \; 487 $ $ 0.000180 $ $ 0.0288 $ $ 2981 $ $ 80.8 $ $ 83.77 $ 19.15
EMQ-SLT $ \; \; 664 $ $ - $ $ - $ $ 4028 $ $ - $ $ - $ $ - $
$ 7 $ SLT-QI $ \; \; 721 $ $ - $ $ 0.0194 $ $ 3808 $ $ 87.06 $ $ - $ $ 5.46 $
weeks SLT-VR $ \; \; 448 $ $ 0.000166 $ $ - $ $ 3162 $ $ - $ $ 89.01 $ $ 21.50 $
SLT-SI $ \; \; 487 $ $ 0.000180 $ $ 0.0288 $ $ 3043 $ $ 80.8 $ $ 88.08 $ 24.45
Table 3.  Critical points
Parameter SLT-QI SLT-VR SLT-SI
$ \Gamma $ $>\frac{i}{R\zeta D\theta_{0}}\sqrt{\frac{4P(PZ+DR\theta_{0}\zeta)(P-D)}{D(2K(P-D)+D(B+H)PV_{0})}} $ $ - $ $>\frac{2(PZ+\theta_{0}R\zeta D)}{\theta_{0}R\zeta D\left(\frac{1}{\beta}+\frac{1}{i}\sqrt{2KD(PZ+\theta_{0}R\zeta D)/P+\frac{i^2}{\beta^2}}\right)} $
$ \Delta $ $ - $ $>\frac{4i(P-D)}{D^2V_{0}(B+H)}\sqrt{\frac{D(2K(P-D)+D(B+H)PV_{0})}{4P(PZ+DR\theta_{0}\zeta)(P-D)}} $ $>\frac{2(2K(1-D/P)+DV_{0}(B+H))}{DV_0(B+H)\left(\frac{1}{\delta}+\frac{1}{i}\sqrt{DZ(2K+V_{0}(B+H)D/(1-D/P))+\frac{i^2}{\delta^2}}\right)} $
$ i $ $<\frac{R\delta\zeta D\theta_{0}}{2}\sqrt{\frac{D(2K(P-D)+D(B+H)PV_{0})}{P(PZ+DR\theta_{0}\zeta)(P-D)}} $ $<\frac{D^2V_{0}(B+H)\beta}{2(P-D)}\sqrt{\frac{P(PZ+DR\theta_{0}\zeta)(P-D)}{D(2K(P-D)+D(B+H)PV_{0})}} $ $<\hbox{min}\; \Bigg\{\frac{R\zeta D\theta_{0}\delta}{2P}\sqrt{\frac{2KPD/\delta}{PZ/\delta+R\zeta D\theta_{0}(\frac{1}{\delta}-\frac{1}{\beta})}} $,
$ \frac{D^2V_{0}Z}{2}\sqrt{\frac{R\zeta(B+H)P\beta}{(P-D)Z(2PK/\beta+DV_{0}(\frac{1}{\beta}-\frac{1}{\delta})R\zeta D)}}\Bigg\} $
Parameter SLT-QI SLT-VR SLT-SI
$ \Gamma $ $>\frac{i}{R\zeta D\theta_{0}}\sqrt{\frac{4P(PZ+DR\theta_{0}\zeta)(P-D)}{D(2K(P-D)+D(B+H)PV_{0})}} $ $ - $ $>\frac{2(PZ+\theta_{0}R\zeta D)}{\theta_{0}R\zeta D\left(\frac{1}{\beta}+\frac{1}{i}\sqrt{2KD(PZ+\theta_{0}R\zeta D)/P+\frac{i^2}{\beta^2}}\right)} $
$ \Delta $ $ - $ $>\frac{4i(P-D)}{D^2V_{0}(B+H)}\sqrt{\frac{D(2K(P-D)+D(B+H)PV_{0})}{4P(PZ+DR\theta_{0}\zeta)(P-D)}} $ $>\frac{2(2K(1-D/P)+DV_{0}(B+H))}{DV_0(B+H)\left(\frac{1}{\delta}+\frac{1}{i}\sqrt{DZ(2K+V_{0}(B+H)D/(1-D/P))+\frac{i^2}{\delta^2}}\right)} $
$ i $ $<\frac{R\delta\zeta D\theta_{0}}{2}\sqrt{\frac{D(2K(P-D)+D(B+H)PV_{0})}{P(PZ+DR\theta_{0}\zeta)(P-D)}} $ $<\frac{D^2V_{0}(B+H)\beta}{2(P-D)}\sqrt{\frac{P(PZ+DR\theta_{0}\zeta)(P-D)}{D(2K(P-D)+D(B+H)PV_{0})}} $ $<\hbox{min}\; \Bigg\{\frac{R\zeta D\theta_{0}\delta}{2P}\sqrt{\frac{2KPD/\delta}{PZ/\delta+R\zeta D\theta_{0}(\frac{1}{\delta}-\frac{1}{\beta})}} $,
$ \frac{D^2V_{0}Z}{2}\sqrt{\frac{R\zeta(B+H)P\beta}{(P-D)Z(2PK/\beta+DV_{0}(\frac{1}{\beta}-\frac{1}{\delta})R\zeta D)}}\Bigg\} $
Table 4.  Critical values for different lead-time intervals for SLT-QI model
Variables Lead time interval
1 2 3 4 5 6 7
$ \theta_{imp} $ $ 0.0309 $ $ 0.0296 $ $ 0.0276 $ $ 0.0255 $ $ 0.0233 $ $ 0.0213 $ $ 0.0194 $
$ i $ $ 0.450 $ $ 0.470 $ $ 0.502 $ 0.544 $ 0.594 $ $ 0.649 $ $ 0.709 $
$ \Gamma $ $ 0.000445 $ $ 0.000425 $ $ 0.000398 $ $ 0.000367 $ $ 0.000337 $ $ 0.000308 $ $ 0.000282 $
Variables Lead time interval
1 2 3 4 5 6 7
$ \theta_{imp} $ $ 0.0309 $ $ 0.0296 $ $ 0.0276 $ $ 0.0255 $ $ 0.0233 $ $ 0.0213 $ $ 0.0194 $
$ i $ $ 0.450 $ $ 0.470 $ $ 0.502 $ 0.544 $ 0.594 $ $ 0.649 $ $ 0.709 $
$ \Gamma $ $ 0.000445 $ $ 0.000425 $ $ 0.000398 $ $ 0.000367 $ $ 0.000337 $ $ 0.000308 $ $ 0.000282 $
Table 5.  Critical values for different lead-time intervals for SLT-VR model
Variables Lead time interval
1 2 3 4 5 6 7
$ V_{imp} $ $ - $ $ - $ $ 0.000166 $ $ 0.000166 $ $ 0.000166 $ $ 0.000166 $ $ 0.000166 $
$ i $ $ - $ $ - $ $ 0.160 $ 0.262 $ 0.375 $ $ 0.494 $ $ 0.615 $
$ \Delta $ $ - $ $ - $ $ 0.000313 $ $ 0.000191 $ $ 0.000133 $ $ 0.000101 $ $ 0.0000813 $
Variables Lead time interval
1 2 3 4 5 6 7
$ V_{imp} $ $ - $ $ - $ $ 0.000166 $ $ 0.000166 $ $ 0.000166 $ $ 0.000166 $ $ 0.000166 $
$ i $ $ - $ $ - $ $ 0.160 $ 0.262 $ 0.375 $ $ 0.494 $ $ 0.615 $
$ \Delta $ $ - $ $ - $ $ 0.000313 $ $ 0.000191 $ $ 0.000133 $ $ 0.000101 $ $ 0.0000813 $
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