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January  2021, 17(1): 467-484. doi: 10.3934/jimo.2020043

An optimal setup cost reduction and lot size for economic production quantity model with imperfect quality and quantity discounts

1. 

School of Economics and Management, Sanming University, Sanming, Fujian 365004, China

2. 

Department of Mechanical and Industrial Engineering, Louisiana State University, Baton Rouge, LA 70803, USA

3. 

Department of International Business, National Chengchi University, Taipei 11605, Taiwan (R.O.C)

* Corresponding author: Chien-Jui Lin (j698102@gmail.com)

Received  July 2018 Revised  August 2019 Published  January 2021 Early access  March 2020

The purpose of this paper concentrates on an economic production quantity model with the factors of imperfect quality and quantity discounts, in which the inspection action occurs during the production stage. There is specific consideration of there being a finite production rate, and the quantity discounts offered by the supplier serves the purpose of stimulating buying greater quantities. This is in contrast to EPQ models that do not take these added factors into consideration. The objective of this paper is to determine the setup cost reduction, which is a function of capital investment, and inventory lot size. An alternative solution procedure was developed that does not employ the Hessian Matrix concavity in the expected total profit. We develop an algorithm to determine the optimal solution for this model. Theoretical results are discussed and a numerical example is proposed. Managerial insights are also examined.

Citation: Tien-Yu Lin, Bhaba R. Sarker, Chien-Jui Lin. An optimal setup cost reduction and lot size for economic production quantity model with imperfect quality and quantity discounts. Journal of Industrial and Management Optimization, 2021, 17 (1) : 467-484. doi: 10.3934/jimo.2020043
References:
[1]

M. Al-Salamah, Economic production quantity in batch manufacturing with imperfect quality, imperfect inspection, and destructive and non-destructive acceptance sampling in a two-tier market, Comput. Ind. Eng., 93 (2016), 275-285.  doi: 10.1016/j.cie.2015.12.022.

[2]

A. Andijani, Setup cost reduction in a multi-stage order quantity model, Prod. Plan. Control., 9 (1998), 121-126.  doi: 10.1080/095372898234334.

[3]

W. C. Benton and S. Park, A classification of literature on determining the lot size under quantity discounts, Eur. Oper. Res., 92 (1996), 219-238.  doi: 10.1016/0377-2217(95)00315-0.

[4]

T. H. BurwellD. S. DaveK. E. Fitzpatrick and M. R. Roy, Economic lot size model for price-depend demand under quantity and freight discounts, Int. J. Prod. Econ., 48 (1997), 141-155. 

[5]

A. BurnetasS. M. Gilbert and C. E. Smith, Quantity discounts in single-period supply contracts with asymmetric demand information, IIE Trans., 39 (2007), 465-479. 

[6]

Q.-G. Bai and J.-T. Xu, Optimal solutions for the economic lot-sizing problem with multiple suppliers and cost structures, J. Appl. Math. Comput., 37 (2011), 331-345.  doi: 10.1007/s12190-010-0437-0.

[7]

P. J. Billington, The classic economic production quantity model with setup cost as a function of capital expenditure, Decis. sci., 18 (1987), 25-40.  doi: 10.1111/j.1540-5915.1987.tb01501.x.

[8]

H.-C. Chang, A note on an economic lot size model for price-dependent demand under quantity and freight discounts, Int. J. Prod. Econ., 144 (2013), 175-179.  doi: 10.1016/j.ijpe.2013.02.001.

[9]

G. Q. ChengB. H. Zhou and L. Li, Integrated production, quality control and condition-based maintenance for imperfect production systems, Reliab. Eng. Syst. Safe., 175 (2018), 251-264. 

[10]

K. L. CheungJ.-S. Song and Y. Zhang, Cost reduction through operations reversal, Eur. J. Oper. Res., 259 (2017), 100-112.  doi: 10.1016/j.ejor.2016.09.052.

[11]

R. DuA. Banerjee and S.-L. Kim, Coordination of two-echelon supply chains using wholesale price discount and credit option, Int. J. Prod. Econ., 143 (2013), 327-334.  doi: 10.1016/j.ijpe.2011.12.017.

[12]

A. Eroglu and G. Ozdemir, An economic order quantity model with defective items and shortages, Int. J. Prod. Econ., 106 (2007), 544-549.  doi: 10.1016/j.ijpe.2006.06.015.

[13]

R. M. EbrahimJ. Razmy and H. Haleh, Scatter search algorithm for supplier selection and order lot sizing under multiple price discount environment, Adv. Eng. Softw., 40 (2009), 766-776.  doi: 10.1016/j.advengsoft.2009.02.003.

[14]

C. K. Huang, An optimal policy for a single-vendor single buyer integrated production-inventory problem with process unreliability consideration, Int. J. Prod. Econ., 91 (2004), 91-98.  doi: 10.1016/S0925-5273(03)00220-2.

[15]

J. D. Hong and J. C. Hayya, Dynamic lot sizing with setup reduction, Comput. Ind. Eng., 24 (1993), 209-218.  doi: 10.1016/0360-8352(93)90009-M.

[16]

Y. HongC. R. Glassey and D. Seong, The analysis of a production line with unreliable machine and random processing times, IIE Trans., 24 (1992), 77-83. 

[17]

J. D. Hong and J. C. Hayya, Joint investment in quality improvement and setup reduction, Comput. Oper. Res., 22 (1995), 567-574.  doi: 10.1016/0305-0548(94)00054-C.

[18]

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

[19]

C.-K. HuangD. M. TsaiJ. C. Wu and K. J. Chung, An integrated vendor-buyer inventory model with order-processing cost reduction and permissible delay in payments, Appl. Math. Model., 34 (2010), 1352-1359.  doi: 10.1016/j.apm.2009.08.015.

[20]

W. JiaoJ.-L. Zhang and H. Yan, The stochastic lot-sizing problem with quantity discounts, Comput. Oper. Res., 80 (2017), 1-10.  doi: 10.1016/j.cor.2016.11.014.

[21]

T. D. KlastorinK. Moinzadeh and J. Son, Coordinating orders in supply chains through price discounts, IIE Trans., 34 (2002), 679-689.  doi: 10.1080/07408170208928904.

[22]

K. L. KimJ. C. Hayya and J. D. Hong, Setup reduction in economic production quantity model, Decis. Sci., 23 (1992), 500-508.  doi: 10.1111/j.1540-5915.1992.tb00402.x.

[23]

T.-Y. LinM.-T. Chen and K.-L. Hou, An inventory model for items with imperfect quality and quantity discounts under adjusted screening rate and earned interest, J. Ind. Manag. Optim., 12 (2016), 1333-1347.  doi: 10.3934/jimo.2016.12.1333.

[24]

T.-Y. Lin, An economic order quantity with imperfect quality and quantity discounts, Appl. Math. Model., 34 (2010), 3158-3165.  doi: 10.1016/j.apm.2010.02.004.

[25]

T.-Y. Lin, Coordination policy for a two-stage supply chain considering quantity discounts and overlapped delivery with imperfect quality, Comput. Ind. Eng., 66 (2013), 53-62.  doi: 10.1016/j.cie.2013.06.012.

[26]

J. L. Li and L. W. Liu, Supply chain coordination with quantity discount policy, Int. J. Prod. Econ., 101 (2006), 89-98.  doi: 10.1016/j.ijpe.2005.05.008.

[27]

A. H. I. LeeH.-Y. KangC.-M. Lai and W.-Y. Hong, An integrated model for lot sizing with supplier selection and quantity discounts, Appl. Math. Model., 37 (2013), 4733-4746.  doi: 10.1016/j.apm.2012.09.056.

[28]

B. Maddah and M. Y. Jaber, Economic order quantity for items with imperfect quality: Revisited, Int. J. Prod. Econ., 112 (2008), 808-815.  doi: 10.1016/j.ijpe.2007.07.003.

[29]

A. K. MannaJ. K. Dey and S. K. Mondal, Imperfect production inventory model with production rate dependent defective rate and advertisement dependent demand, Comput. Ind. Eng., 104 (2017), 9-22.  doi: 10.1016/j.cie.2016.11.027.

[30]

L. Moussawi-HaidarM. Salameh and W. Nasr, Production lot sizing with quality screening and rework, Appl. Math. Model., 40 (2016), 3242-3256.  doi: 10.1016/j.apm.2015.09.095.

[31]

A. Mendoza and J. A. Ventura, Incorporating quantity discounts to the EOQ model with transportation costs, Int. J. Prod. Econ., 113 (2008), 754-765.  doi: 10.1016/j.ijpe.2007.10.010.

[32]

C. L. Munson and J. Hu, Incorporating quantity discounts and their inventory impacts into the centralized purchasing decision, Eur. Oper. Res., 201 (2010), 581-592.  doi: 10.1016/j.ejor.2009.03.043.

[33]

R. MansiniM. W. P. Savelsbergh and B. Tocchella, The supplier selection problem with quantity discounts and truckload shipping, Omega, 40 (2012), 445-455.  doi: 10.1016/j.omega.2011.09.001.

[34]

P. L. Meena and S. P. Sarmah, Multiple sourcing under supplier failure risk and quantity discount: A genetic algorithm approach, Transport. Res. E-Log., 50 (2013), 84-97.  doi: 10.1016/j.tre.2012.10.001.

[35]

I. MoonB. C. Giri and K. Choi, Economic lot scheduling problem with imperfect production processes and setup times, J. Oper. Res. Soc., 53 (2002), 620-629.  doi: 10.1057/palgrave.jors.2601350.

[36]

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

[37]

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

[38]

J. PaknejadF. Nasri and J. F. Affisco, Quality improvement in an inventory model with finite-range stochastic lead times, J. Appl. Math. and Deci. Sci., 3 (2005), 177-189.  doi: 10.1155/JAMDS.2005.177.

[39]

S. Papachristos and K. Skouri, An inventory model with deteriorating items, quantity discount, pricing and time-dependent partial backlogging, Int. J. Prod. Econ., 83 (2003), 247-256.  doi: 10.1016/S0925-5273(02)00332-8.

[40]

E. L. Porteus, Investing in reduced setups in the EOQ model, Manage. Sci., 31 (1985), 998-1010.  doi: 10.1287/mnsc.31.8.998.

[41]

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

[42]

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

[43]

M. K. Salameh and M. Y. Jaber, Economic production quantity model for items with imperfect quality, Int. J. Prod. Econ., 64 (2000), 59-64.  doi: 10.1016/S0925-5273(99)00044-4.

[44]

S. S. Sana and K. S. Chaudhuri, A deterministic EOQ model with delays in payments and price-discount offers, Eur. Oper. Res., 184 (2008), 509-533.  doi: 10.1016/j.ejor.2006.11.023.

[45]

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

[46]

B. Sarkar and I. Moon, Improved quality, setup cost reduction, and variable backorder costs in an imperfect production process, Int. J. Prod. Econ., 155 (2014), 204-213.  doi: 10.1016/j.ijpe.2013.11.014.

[47]

R. P. Tripati and S. S. Tomar, Optimal order policy for deteriorating items with time-dependent demand in response to temporary price discount linked to order quantity, Int. J. Math. Anal., 9 (2015), 1095-1109.  doi: 10.12988/ijma.2015.5235.

[48]

B. B. Venegas and J. A. Ventura, A two-stage supply chain coordination mechanism considering price sensitive demand and quantity discounts, Eur. J. Oper. Res., 264 (2018), 524-533.  doi: 10.1016/j.ejor.2017.06.030.

[49]

G. Voigt and K. Inderfurth, Supply chain coordination and setup cost reduction in case of asymmetric information, OR Spectrum, 33 (2011), 99-122.  doi: 10.1007/s00291-009-0173-8.

[50]

Q. N. Wang and R. F. Wang, Quantity discount pricing policies for heterogeneous retailers with price sensitive demand, Nav. Res. Log., 52 (2005), 645-658.  doi: 10.1002/nav.20103.

[51]

S.-Y. Wu, Optimal policy for set-up time reduction in a multistage production-inventory system, Int. J. Syst. Sci., 33 (2002), 551-556.  doi: 10.1080/00207720210123724.

[52]

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

show all references

References:
[1]

M. Al-Salamah, Economic production quantity in batch manufacturing with imperfect quality, imperfect inspection, and destructive and non-destructive acceptance sampling in a two-tier market, Comput. Ind. Eng., 93 (2016), 275-285.  doi: 10.1016/j.cie.2015.12.022.

[2]

A. Andijani, Setup cost reduction in a multi-stage order quantity model, Prod. Plan. Control., 9 (1998), 121-126.  doi: 10.1080/095372898234334.

[3]

W. C. Benton and S. Park, A classification of literature on determining the lot size under quantity discounts, Eur. Oper. Res., 92 (1996), 219-238.  doi: 10.1016/0377-2217(95)00315-0.

[4]

T. H. BurwellD. S. DaveK. E. Fitzpatrick and M. R. Roy, Economic lot size model for price-depend demand under quantity and freight discounts, Int. J. Prod. Econ., 48 (1997), 141-155. 

[5]

A. BurnetasS. M. Gilbert and C. E. Smith, Quantity discounts in single-period supply contracts with asymmetric demand information, IIE Trans., 39 (2007), 465-479. 

[6]

Q.-G. Bai and J.-T. Xu, Optimal solutions for the economic lot-sizing problem with multiple suppliers and cost structures, J. Appl. Math. Comput., 37 (2011), 331-345.  doi: 10.1007/s12190-010-0437-0.

[7]

P. J. Billington, The classic economic production quantity model with setup cost as a function of capital expenditure, Decis. sci., 18 (1987), 25-40.  doi: 10.1111/j.1540-5915.1987.tb01501.x.

[8]

H.-C. Chang, A note on an economic lot size model for price-dependent demand under quantity and freight discounts, Int. J. Prod. Econ., 144 (2013), 175-179.  doi: 10.1016/j.ijpe.2013.02.001.

[9]

G. Q. ChengB. H. Zhou and L. Li, Integrated production, quality control and condition-based maintenance for imperfect production systems, Reliab. Eng. Syst. Safe., 175 (2018), 251-264. 

[10]

K. L. CheungJ.-S. Song and Y. Zhang, Cost reduction through operations reversal, Eur. J. Oper. Res., 259 (2017), 100-112.  doi: 10.1016/j.ejor.2016.09.052.

[11]

R. DuA. Banerjee and S.-L. Kim, Coordination of two-echelon supply chains using wholesale price discount and credit option, Int. J. Prod. Econ., 143 (2013), 327-334.  doi: 10.1016/j.ijpe.2011.12.017.

[12]

A. Eroglu and G. Ozdemir, An economic order quantity model with defective items and shortages, Int. J. Prod. Econ., 106 (2007), 544-549.  doi: 10.1016/j.ijpe.2006.06.015.

[13]

R. M. EbrahimJ. Razmy and H. Haleh, Scatter search algorithm for supplier selection and order lot sizing under multiple price discount environment, Adv. Eng. Softw., 40 (2009), 766-776.  doi: 10.1016/j.advengsoft.2009.02.003.

[14]

C. K. Huang, An optimal policy for a single-vendor single buyer integrated production-inventory problem with process unreliability consideration, Int. J. Prod. Econ., 91 (2004), 91-98.  doi: 10.1016/S0925-5273(03)00220-2.

[15]

J. D. Hong and J. C. Hayya, Dynamic lot sizing with setup reduction, Comput. Ind. Eng., 24 (1993), 209-218.  doi: 10.1016/0360-8352(93)90009-M.

[16]

Y. HongC. R. Glassey and D. Seong, The analysis of a production line with unreliable machine and random processing times, IIE Trans., 24 (1992), 77-83. 

[17]

J. D. Hong and J. C. Hayya, Joint investment in quality improvement and setup reduction, Comput. Oper. Res., 22 (1995), 567-574.  doi: 10.1016/0305-0548(94)00054-C.

[18]

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

[19]

C.-K. HuangD. M. TsaiJ. C. Wu and K. J. Chung, An integrated vendor-buyer inventory model with order-processing cost reduction and permissible delay in payments, Appl. Math. Model., 34 (2010), 1352-1359.  doi: 10.1016/j.apm.2009.08.015.

[20]

W. JiaoJ.-L. Zhang and H. Yan, The stochastic lot-sizing problem with quantity discounts, Comput. Oper. Res., 80 (2017), 1-10.  doi: 10.1016/j.cor.2016.11.014.

[21]

T. D. KlastorinK. Moinzadeh and J. Son, Coordinating orders in supply chains through price discounts, IIE Trans., 34 (2002), 679-689.  doi: 10.1080/07408170208928904.

[22]

K. L. KimJ. C. Hayya and J. D. Hong, Setup reduction in economic production quantity model, Decis. Sci., 23 (1992), 500-508.  doi: 10.1111/j.1540-5915.1992.tb00402.x.

[23]

T.-Y. LinM.-T. Chen and K.-L. Hou, An inventory model for items with imperfect quality and quantity discounts under adjusted screening rate and earned interest, J. Ind. Manag. Optim., 12 (2016), 1333-1347.  doi: 10.3934/jimo.2016.12.1333.

[24]

T.-Y. Lin, An economic order quantity with imperfect quality and quantity discounts, Appl. Math. Model., 34 (2010), 3158-3165.  doi: 10.1016/j.apm.2010.02.004.

[25]

T.-Y. Lin, Coordination policy for a two-stage supply chain considering quantity discounts and overlapped delivery with imperfect quality, Comput. Ind. Eng., 66 (2013), 53-62.  doi: 10.1016/j.cie.2013.06.012.

[26]

J. L. Li and L. W. Liu, Supply chain coordination with quantity discount policy, Int. J. Prod. Econ., 101 (2006), 89-98.  doi: 10.1016/j.ijpe.2005.05.008.

[27]

A. H. I. LeeH.-Y. KangC.-M. Lai and W.-Y. Hong, An integrated model for lot sizing with supplier selection and quantity discounts, Appl. Math. Model., 37 (2013), 4733-4746.  doi: 10.1016/j.apm.2012.09.056.

[28]

B. Maddah and M. Y. Jaber, Economic order quantity for items with imperfect quality: Revisited, Int. J. Prod. Econ., 112 (2008), 808-815.  doi: 10.1016/j.ijpe.2007.07.003.

[29]

A. K. MannaJ. K. Dey and S. K. Mondal, Imperfect production inventory model with production rate dependent defective rate and advertisement dependent demand, Comput. Ind. Eng., 104 (2017), 9-22.  doi: 10.1016/j.cie.2016.11.027.

[30]

L. Moussawi-HaidarM. Salameh and W. Nasr, Production lot sizing with quality screening and rework, Appl. Math. Model., 40 (2016), 3242-3256.  doi: 10.1016/j.apm.2015.09.095.

[31]

A. Mendoza and J. A. Ventura, Incorporating quantity discounts to the EOQ model with transportation costs, Int. J. Prod. Econ., 113 (2008), 754-765.  doi: 10.1016/j.ijpe.2007.10.010.

[32]

C. L. Munson and J. Hu, Incorporating quantity discounts and their inventory impacts into the centralized purchasing decision, Eur. Oper. Res., 201 (2010), 581-592.  doi: 10.1016/j.ejor.2009.03.043.

[33]

R. MansiniM. W. P. Savelsbergh and B. Tocchella, The supplier selection problem with quantity discounts and truckload shipping, Omega, 40 (2012), 445-455.  doi: 10.1016/j.omega.2011.09.001.

[34]

P. L. Meena and S. P. Sarmah, Multiple sourcing under supplier failure risk and quantity discount: A genetic algorithm approach, Transport. Res. E-Log., 50 (2013), 84-97.  doi: 10.1016/j.tre.2012.10.001.

[35]

I. MoonB. C. Giri and K. Choi, Economic lot scheduling problem with imperfect production processes and setup times, J. Oper. Res. Soc., 53 (2002), 620-629.  doi: 10.1057/palgrave.jors.2601350.

[36]

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

[37]

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

[38]

J. PaknejadF. Nasri and J. F. Affisco, Quality improvement in an inventory model with finite-range stochastic lead times, J. Appl. Math. and Deci. Sci., 3 (2005), 177-189.  doi: 10.1155/JAMDS.2005.177.

[39]

S. Papachristos and K. Skouri, An inventory model with deteriorating items, quantity discount, pricing and time-dependent partial backlogging, Int. J. Prod. Econ., 83 (2003), 247-256.  doi: 10.1016/S0925-5273(02)00332-8.

[40]

E. L. Porteus, Investing in reduced setups in the EOQ model, Manage. Sci., 31 (1985), 998-1010.  doi: 10.1287/mnsc.31.8.998.

[41]

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

[42]

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

[43]

M. K. Salameh and M. Y. Jaber, Economic production quantity model for items with imperfect quality, Int. J. Prod. Econ., 64 (2000), 59-64.  doi: 10.1016/S0925-5273(99)00044-4.

[44]

S. S. Sana and K. S. Chaudhuri, A deterministic EOQ model with delays in payments and price-discount offers, Eur. Oper. Res., 184 (2008), 509-533.  doi: 10.1016/j.ejor.2006.11.023.

[45]

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

[46]

B. Sarkar and I. Moon, Improved quality, setup cost reduction, and variable backorder costs in an imperfect production process, Int. J. Prod. Econ., 155 (2014), 204-213.  doi: 10.1016/j.ijpe.2013.11.014.

[47]

R. P. Tripati and S. S. Tomar, Optimal order policy for deteriorating items with time-dependent demand in response to temporary price discount linked to order quantity, Int. J. Math. Anal., 9 (2015), 1095-1109.  doi: 10.12988/ijma.2015.5235.

[48]

B. B. Venegas and J. A. Ventura, A two-stage supply chain coordination mechanism considering price sensitive demand and quantity discounts, Eur. J. Oper. Res., 264 (2018), 524-533.  doi: 10.1016/j.ejor.2017.06.030.

[49]

G. Voigt and K. Inderfurth, Supply chain coordination and setup cost reduction in case of asymmetric information, OR Spectrum, 33 (2011), 99-122.  doi: 10.1007/s00291-009-0173-8.

[50]

Q. N. Wang and R. F. Wang, Quantity discount pricing policies for heterogeneous retailers with price sensitive demand, Nav. Res. Log., 52 (2005), 645-658.  doi: 10.1002/nav.20103.

[51]

S.-Y. Wu, Optimal policy for set-up time reduction in a multistage production-inventory system, Int. J. Syst. Sci., 33 (2002), 551-556.  doi: 10.1080/00207720210123724.

[52]

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

Figure 1.  The behavior of the inventory level per cycle
Figure 2.  The expected total profit
Table 1.  Procurement cost structure for the manufacture
$r$$Q_{r-1} \sim Q_{r}$$c_{r}$
1$0 < Q < 150$$c_{1} = 20.05$
2$150 \leq Q < 400$$c_{2} = 20.04$
3$400 \leq Q < 800$$c_{3} = 20.03$
4$800 \leq Q < 1250$$c_{4} = 20.02$
5$Q \geq 800$$c_{5} = 20.01$
$r$$Q_{r-1} \sim Q_{r}$$c_{r}$
1$0 < Q < 150$$c_{1} = 20.05$
2$150 \leq Q < 400$$c_{2} = 20.04$
3$400 \leq Q < 800$$c_{3} = 20.03$
4$800 \leq Q < 1250$$c_{4} = 20.02$
5$Q \geq 800$$c_{5} = 20.01$
Table 2.  The values of $Q^{*}, S^{*}$ and $E T P U^{*}$ corresponding to 32 combinations of $\sigma, f_{g} M, i, U(d)$
$\sigma$$f_{g}$$M$$i$$U(d)$$Q^{*}$$S^{*}$$E T P U^{*}$
96000.051920000.20.04858.9175.3274724.7
0.056866.3175.4275046.5
0.280.04910.9200274036.9
0.056918.8200274358.8
2688000.20.04846.1163.3274694.8
0.056853.5172.8275016.6
0.280.04897.4200274004.6
0.056905.2200274326.4
0.071920000.20.04800163.3274570.9
0.056800162274892.8
0.280.04858.9200273906.7
0.056866.4200274228.7
2688000.20.04800163.3374599.6
0.056800162274862.7
0.280.04846200273872.3
0.056853.5200274194.3
134400.051920000.20.04877.4128385689.5
0.056885128.1386139.7
0.280.04930.5190384839
0.056938.6190.1385289.5
2688000.20.04858.9125.2385646.9
0.056866.3125.3386097
0.280.04910.9186384779.3
0.056918.8186.1385229.8
0.071920000.20.04812.5118.4385535.8
0.056819.6118.5385986.1
0.280.04877.4179.1384674.4
0.056885.1179.2385125.1
2688000.20.04800116.7385493.1
0.056800115.7385943.3
0.280.04858.9175.3384614.7
0.056866.4175.4385065.3
$\sigma$$f_{g}$$M$$i$$U(d)$$Q^{*}$$S^{*}$$E T P U^{*}$
96000.051920000.20.04858.9175.3274724.7
0.056866.3175.4275046.5
0.280.04910.9200274036.9
0.056918.8200274358.8
2688000.20.04846.1163.3274694.8
0.056853.5172.8275016.6
0.280.04897.4200274004.6
0.056905.2200274326.4
0.071920000.20.04800163.3274570.9
0.056800162274892.8
0.280.04858.9200273906.7
0.056866.4200274228.7
2688000.20.04800163.3374599.6
0.056800162274862.7
0.280.04846200273872.3
0.056853.5200274194.3
134400.051920000.20.04877.4128385689.5
0.056885128.1386139.7
0.280.04930.5190384839
0.056938.6190.1385289.5
2688000.20.04858.9125.2385646.9
0.056866.3125.3386097
0.280.04910.9186384779.3
0.056918.8186.1385229.8
0.071920000.20.04812.5118.4385535.8
0.056819.6118.5385986.1
0.280.04877.4179.1384674.4
0.056885.1179.2385125.1
2688000.20.04800116.7385493.1
0.056800115.7385943.3
0.280.04858.9175.3384614.7
0.056866.4175.4385065.3
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