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 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 & Management Optimization, 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.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[13]

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

[14]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

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

[38]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[13]

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

[14]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

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

[38]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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