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October  2017, 13(4): 1975-1990. doi: 10.3934/jimo.2017027

An integrated inventory model with variable transportation cost, two-stage inspection, and defective items

1. 

Department of Industrial & Management Engineering, Hanyang University, Ansan Gyeonggi-do, 15588, South Korea

2. 

Department of Mathematics & Statistics, Banasthali Vidyapith, Banasthali, Rajasthan, 304 022, India

3. 

Department of Industrial Engineering, Hanyang University 220 Wangsimni-ro, Seongdong-gu, Seoul, 04763, Korea

* Corresponding author: mitalisarkar.ms@gmail.com(Mitali Sarkar), Phone Number-010-7490-1981, Fax No +82-31-436-8146.

Received  January 2016 Revised  June 2016 Published  April 2017

The paper deals with an integrated inventory model with make-to-order policy from buyer to vendor. A variable transportation cost is used as a power function of the delivery quantity for either tapering or considering proportional rate data to maintain single-setup multi-delivery (SSMD) policy with reduced transportation cost. A two-stage inspection process is introduced by the vendor to ensure the perfect quality of product even though the first inspection process indicates a constant defective rate of imperfect production is present during the long-run production system and all defective items are reworked with some fixed cost. The aim is to minimize the total cost of the integrated inventory model by using classical optimization technique. Two numerical examples, sensitivity analysis, and graphical representations are given to illustrate the model.

Citation: Biswajit Sarkar, Bijoy Kumar Shaw, Taebok Kim, Mitali Sarkar, Dongmin Shin. An integrated inventory model with variable transportation cost, two-stage inspection, and defective items. Journal of Industrial and Management Optimization, 2017, 13 (4) : 1975-1990. doi: 10.3934/jimo.2017027
References:
[1]

A. Banerjee, A joint economic lot size model for purchaser and vendor, Decision Sciences, 17 (1986), 292-311.  doi: 10.1111/j.1540-5915.1986.tb00228.x.

[2]

M. Ben-DayaM. Darwish and K. Ertogral, The joint economic lot sizing problem: Review and extensions, European Journal of Operational Research, 185 (2008), 726-742.  doi: 10.1016/j.ejor.2006.12.026.

[3]

L. E. Cárdenas-BarrónK.-J. Chung and G. T. Garza, Celebrating a century of the economic order quantity model in honor of Ford Whitman Harris, International Journal of Production Economics, 155 (2014), 1-7. 

[4]

L. E. Cárdenas-Barrón and S. S. Sana, A production-inventory model for a two-echelon supply chain when demand is dependent on sales teams' initiatives, International Journal of Production Economics, 155 (2014), 249-258. 

[5]

L. E. Cárdenas-Barrónn and S. S. Sana, Multi-item EOQ inventory model in a two-layer supply chain while demand varies with promotional effort, Applied Mathematical Modeling, (2014), In Press, Corrected Proof, Available online 25 February 2015.

[6]

C. K. Chan and B. G. Kingsman, Coordination in a single-vendor multi-buyer supply chain by synchronizing delivery and production cycles, Transportation Research: Part E, 43 (2007), 90-111.  doi: 10.1016/j.tre.2005.07.008.

[7]

K. ErtogralM. Darwish and M. Ben-Daya, Production and shipment lot sizing in a vendorbuyer supply chain with transportation cost, European Journal of Operational Research, 176 (2007), 1592-1606.  doi: 10.1016/j.ejor.2005.10.036.

[8]

C. H. Glock, The joint economic lot size problem: A review, Int. J. Prod. Econ., 135 (2012), 671-686.  doi: 10.1016/j.ijpe.2011.10.026.

[9]

D. Y. Golhar and B. R. Sarker, Economic manufacturing quantity in a just-in-time delivery system, International Journal of Production Research, 30 (1992), 961-972.  doi: 10.1080/00207549208942936.

[10]

S. K. Goyal, An integrated inventory model for a single supplier-single customer problem, The International Journal of Production Research, 15 (1976), 107-111.  doi: 10.1080/00207547708943107.

[11]

S. K. Goyal, A joint economic lot size model for purchaser and vendor: A comment, Decision Sciences, 19 (1988), 236-241.  doi: 10.1111/j.1540-5915.1988.tb00264.x.

[12]

G. Hadley and T. Whitin, Analysis of Inventory Systems, Prentice Hall, Englewood Cliffs NJ, 1963.

[13]

F. W. Harris, How many parts to make at once, Factory, The Magazine of Management, 38 (1990), 947-950.  doi: 10.1287/opre.38.6.947.

[14]

R. M. Hill, The single-vendor single-buyer integrated production inventory model with a generalized policy, European Journal of Operational Research, 97 (1997), 493-499. 

[15]

R. M. Hill, The optimal production and shipment policy for the single-vendor single-buyer integrated production inventory problem, International Journal of Production Research, 37 (1999), 2463-2475. 

[16]

S. D. Lee and Y. C. Fu, Joint production and delivery lot sizing for a make-to-order producer-buyer supply chain with transportation cost, Transportation Research: Part E, 66 (2014), 23-35. 

[17]

L. Lu, A one-vendor multi-buyer integrated inventory model, European Journal of Operational Research, 81 (1995), 312-323.  doi: 10.1016/0377-2217(93)E0253-T.

[18]

D. MunganJ. Yu and B. R. Sarker, Manufacturing lot-sizing procurement and delivery schedules over a finite planning horizon, International Journal of Production Research, 48 (2009), 3619-3636.  doi: 10.1080/00207540902878228.

[19]

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.

[20]

B. Sarkar, Supply chain coordination with variable backorder, inspections, and discount policy for fixed lifetime products Mathematical Problems in Engineering, 2016 (2016), Art. ID 6318737, 14 pp. doi: 10.1155/2016/6318737.

[21]

B. SarkarL. E. Cárdenas-BarrónM. Sarkar and M. L. Singgihd, An economic production quantity model with random defective rate, rework process and backorders for a single stage production system, Journal of Manufacturing Systems, 33 (2014), 423-435.  doi: 10.1016/j.jmsy.2014.02.001.

[22]

B. SarkarK. S. Chaudhuri and I. K. Moon, Manufacturing setup cost reduction and quality improvement for the distribution free continuous-review inventory model with a service level constraint, Journal of Manufacturing Systems, 34 (2015), 74-82.  doi: 10.1016/j.jmsy.2014.11.003.

[23]

B. SarkarB. GangulyM. Sarkar and S. Pareek, Effect of variable transportation and carbon emission in a three-echelon supply chain model, Effect of variable transportation and carbon emission in a three-echelon supply chain model, Transportation Research: Part E. Log.Tran. Rev., 91 (2016), 112-128.  doi: 10.1016/j.tre.2016.03.018.

[24]

B. Sarkar and A. Majumder, Integrated vendor-buyer supply chain model with vendor's setup cost reduction, Applied Mathematical Modelling, 224 (2013), 362-371.  doi: 10.1016/j.amc.2013.08.072.

[25]

B. SarkarA. MajumderM. SarkarB. K. Dey and G. Roy, Two-echelon supply chain model with manufacturing quality improvement and setup cost reduction, Journal of Industrial Management Optimization, 13 (2016), 1085-1104.  doi: 10.3934/jimo.2016063.

[26]

B. SarkarB. Mandal and S. Sarkar, Preservation of deteriorating seasonal products with stock-dependent consumption rate and shortages, Journal of Industrial Management Optimization, 13 (2017), 187-206.  doi: 10.3934/jimo.2016011.

[27]

B. SarkarB. Mandal and S. Sarkar, Quality improvement and backorder price discount under controllable lead time in an inventory model, Journal of Manufacturing Systems, 35 (2015), 26-36.  doi: 10.1016/j.jmsy.2014.11.012.

[28]

B. R. Sarker and G. R. Parija, An optimal batch size for a production system operating under a fixed-quantity, periodic delivery policy, The Journal of the Operational Research Society, 45 (1994), 891-900. 

[29]

B. R. Sarker and G. R. Parija, Optimal batch size and raw material ordering policy for a production system with a fixed-interval, lumpy demand delivery system, European Journal of Operational Research, 89 (1996), 593-608.  doi: 10.1016/0377-2217(94)00277-0.

[30]

B. Sarkar and S. Saren, Product inspection policy for an imperfect production system with inspection errors and warranty cost, European Journal of Operational Research, 248 (2016), 263-271.  doi: 10.1016/j.ejor.2015.06.021.

[31]

B. SarkarS. Saren and L. E. Cárdenas-Barrón, An inventory model with trade-credit policy and variable deterioration for fixed lifetime products, Ann Operational Research, 229 (2015), 677-702.  doi: 10.1007/s10479-014-1745-9.

[32]

B. Sarkar, S. Saren, D. Sinha and S. Hur, Effect of unequal lot sizes, variable setup cost, and carbon emission cost in a supply chain model, Mathematical Problems in Engineering, 2015 (2015), Art. ID 469486, 13 pp. doi: 10.1155/2015/469486.

[33]

E. A. Silver, D. F. Pyke and R. Peterson, Inventory Management and Production Planning and Scheduling, Third Edition, John Wiley & Sons Inc., New York, 1998.

[34]

A. A. TaleizadehF. Barzinpour and H.-M. Wee, Meta-heuristic algorithms for solving a fuzzy single-period problem, Mathematical and Computer Modelling, 54 (2011), 1273-1285.  doi: 10.1016/j.mcm.2011.03.038.

[35]

A. A. TaleizadehL. E. Cárdenas-BarrónJ. Biabanic and R. Nikousokhan, Multi products single machine EPQ model with immediate rework process, International Journal of Industrial Engineering Computations, 3 (2012), 93-102.  doi: 10.5267/j.ijiec.2011.09.001.

[36]

A. A. TaleizadehS. S. Kalantari and L. E. Cárdenas-Barrón, Determining optimal price, replacement lot size and number of shipments forn an EPQ model with rework and multiple shipments, Journal of Industrial Management Optimization, 11 (2015), 1059-1071. 

[37]

A. A. TaleizadehH. MoghadasiS. T. A. Niaki and A. Eftekhari, An economic order quantity under joint replenishment policy to supply expensive imported raw materials with payment in advance, Journal of Applied Science, 8 (2008), 4263-4273. 

[38]

A. A. TaleizadehB. MohammadiL. E. Cárdenas-Barrón and H. Samimi, An EOQ model for perishable product with special sale and shortage, International Journal of Production Economics, 145 (2013), 318-338.  doi: 10.1016/j.ijpe.2013.05.001.

[39]

A. A. TaleizadehS. T. A. NiakiM.-B. Aryanezhad and N. Shafii, A hybrid method of fuzzy simulation and genetic algorithm to optimize constrained inventory control systems with stochastic replenishments and fuzzy demand, Information Sciences, 220 (2013), 425-441.  doi: 10.1016/j.ins.2012.07.027.

[40]

A. A. TaleizadehS. T. A. NiakiM.-B. Aryanezhad and A. F. Tafti, A genetic algorithm to optimize multi-product multi-constraint inventory control systems with stochastic replenishment intervals and discount, International Journal of Advance Manufacturing Technology, 51 (2010), 311-323. 

[41]

A. A. TaleizadehS. T. A. Niaki and R. G. Meibodi, Replenish-up-to multi chance-constraint inventory control system with stochastic period lengths and total discount under fuzzy purchasing price and holding costs, Knowledge Based Systems, 53 (2013), 147-156. 

[42]

A. A. TaleizadehM. Noori-daryan and L. E. Cárdenas-Barrón, Joint optimization of price, replenishment frequency, replenishment cycle and production rate in vendor managed inventory system with deteriorating items, International Journal of Production Economics, 159 (2015), 285-295.  doi: 10.1016/j.ijpe.2014.09.009.

[43]

A. A. TaleizadehD. W. PenticoM. Aryanezhad and S. M. Ghoreyshi, An economic order quantity model with partial backordering and a special sale price, European Journal of Operational Research, 221 (2012), 571-583.  doi: 10.1016/j.ejor.2012.03.032.

[44]

A. A. TaleizadehD. W. PenticoM. S. Jabalameli and M. Aryanezhada, An economic order quantity model with multiple partial prepayments and partial backordering, Mathematical and Computer Modelling, 57 (2013), 311-323.  doi: 10.1016/j.mcm.2012.07.002.

[45]

A. A. Taleizadeh and D. W. Pentico, An economic order quantity model with a known price increase and partial backordering, European Journal of Operational Research, 228 (2013), 516-525.  doi: 10.1016/j.ejor.2013.02.014.

[46]

A. A. TaleizadehG. A. WidyadanaH. M. Wee and J. Biabani, Multi products single machine economic production quantity model with multiple batch size, International Journal of Industrial Engineering Computations, 2 (2011), 213-224.  doi: 10.5267/j.ijiec.2011.01.002.

[47]

H. M. Wee and C. J. Chung, An note on the economic lot size of the integrated vendor-buyer inventory system derived without derivatives, European Journal of Operational Research, 177 (2007), 1289-1293.  doi: 10.1016/j.ejor.2005.11.035.

[48]

B. SarkarA. MajumderM. SarkarB. K. Dey and G. Roy, Two-echelon supply chain model with manufacturing quality improvement and setup cost reduction, Journal of Industrial and Managment Optimization, 13 (2017), 1085-1104.  doi: 10.9934/j.jimo.2016.06.3.

show all references

References:
[1]

A. Banerjee, A joint economic lot size model for purchaser and vendor, Decision Sciences, 17 (1986), 292-311.  doi: 10.1111/j.1540-5915.1986.tb00228.x.

[2]

M. Ben-DayaM. Darwish and K. Ertogral, The joint economic lot sizing problem: Review and extensions, European Journal of Operational Research, 185 (2008), 726-742.  doi: 10.1016/j.ejor.2006.12.026.

[3]

L. E. Cárdenas-BarrónK.-J. Chung and G. T. Garza, Celebrating a century of the economic order quantity model in honor of Ford Whitman Harris, International Journal of Production Economics, 155 (2014), 1-7. 

[4]

L. E. Cárdenas-Barrón and S. S. Sana, A production-inventory model for a two-echelon supply chain when demand is dependent on sales teams' initiatives, International Journal of Production Economics, 155 (2014), 249-258. 

[5]

L. E. Cárdenas-Barrónn and S. S. Sana, Multi-item EOQ inventory model in a two-layer supply chain while demand varies with promotional effort, Applied Mathematical Modeling, (2014), In Press, Corrected Proof, Available online 25 February 2015.

[6]

C. K. Chan and B. G. Kingsman, Coordination in a single-vendor multi-buyer supply chain by synchronizing delivery and production cycles, Transportation Research: Part E, 43 (2007), 90-111.  doi: 10.1016/j.tre.2005.07.008.

[7]

K. ErtogralM. Darwish and M. Ben-Daya, Production and shipment lot sizing in a vendorbuyer supply chain with transportation cost, European Journal of Operational Research, 176 (2007), 1592-1606.  doi: 10.1016/j.ejor.2005.10.036.

[8]

C. H. Glock, The joint economic lot size problem: A review, Int. J. Prod. Econ., 135 (2012), 671-686.  doi: 10.1016/j.ijpe.2011.10.026.

[9]

D. Y. Golhar and B. R. Sarker, Economic manufacturing quantity in a just-in-time delivery system, International Journal of Production Research, 30 (1992), 961-972.  doi: 10.1080/00207549208942936.

[10]

S. K. Goyal, An integrated inventory model for a single supplier-single customer problem, The International Journal of Production Research, 15 (1976), 107-111.  doi: 10.1080/00207547708943107.

[11]

S. K. Goyal, A joint economic lot size model for purchaser and vendor: A comment, Decision Sciences, 19 (1988), 236-241.  doi: 10.1111/j.1540-5915.1988.tb00264.x.

[12]

G. Hadley and T. Whitin, Analysis of Inventory Systems, Prentice Hall, Englewood Cliffs NJ, 1963.

[13]

F. W. Harris, How many parts to make at once, Factory, The Magazine of Management, 38 (1990), 947-950.  doi: 10.1287/opre.38.6.947.

[14]

R. M. Hill, The single-vendor single-buyer integrated production inventory model with a generalized policy, European Journal of Operational Research, 97 (1997), 493-499. 

[15]

R. M. Hill, The optimal production and shipment policy for the single-vendor single-buyer integrated production inventory problem, International Journal of Production Research, 37 (1999), 2463-2475. 

[16]

S. D. Lee and Y. C. Fu, Joint production and delivery lot sizing for a make-to-order producer-buyer supply chain with transportation cost, Transportation Research: Part E, 66 (2014), 23-35. 

[17]

L. Lu, A one-vendor multi-buyer integrated inventory model, European Journal of Operational Research, 81 (1995), 312-323.  doi: 10.1016/0377-2217(93)E0253-T.

[18]

D. MunganJ. Yu and B. R. Sarker, Manufacturing lot-sizing procurement and delivery schedules over a finite planning horizon, International Journal of Production Research, 48 (2009), 3619-3636.  doi: 10.1080/00207540902878228.

[19]

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.

[20]

B. Sarkar, Supply chain coordination with variable backorder, inspections, and discount policy for fixed lifetime products Mathematical Problems in Engineering, 2016 (2016), Art. ID 6318737, 14 pp. doi: 10.1155/2016/6318737.

[21]

B. SarkarL. E. Cárdenas-BarrónM. Sarkar and M. L. Singgihd, An economic production quantity model with random defective rate, rework process and backorders for a single stage production system, Journal of Manufacturing Systems, 33 (2014), 423-435.  doi: 10.1016/j.jmsy.2014.02.001.

[22]

B. SarkarK. S. Chaudhuri and I. K. Moon, Manufacturing setup cost reduction and quality improvement for the distribution free continuous-review inventory model with a service level constraint, Journal of Manufacturing Systems, 34 (2015), 74-82.  doi: 10.1016/j.jmsy.2014.11.003.

[23]

B. SarkarB. GangulyM. Sarkar and S. Pareek, Effect of variable transportation and carbon emission in a three-echelon supply chain model, Effect of variable transportation and carbon emission in a three-echelon supply chain model, Transportation Research: Part E. Log.Tran. Rev., 91 (2016), 112-128.  doi: 10.1016/j.tre.2016.03.018.

[24]

B. Sarkar and A. Majumder, Integrated vendor-buyer supply chain model with vendor's setup cost reduction, Applied Mathematical Modelling, 224 (2013), 362-371.  doi: 10.1016/j.amc.2013.08.072.

[25]

B. SarkarA. MajumderM. SarkarB. K. Dey and G. Roy, Two-echelon supply chain model with manufacturing quality improvement and setup cost reduction, Journal of Industrial Management Optimization, 13 (2016), 1085-1104.  doi: 10.3934/jimo.2016063.

[26]

B. SarkarB. Mandal and S. Sarkar, Preservation of deteriorating seasonal products with stock-dependent consumption rate and shortages, Journal of Industrial Management Optimization, 13 (2017), 187-206.  doi: 10.3934/jimo.2016011.

[27]

B. SarkarB. Mandal and S. Sarkar, Quality improvement and backorder price discount under controllable lead time in an inventory model, Journal of Manufacturing Systems, 35 (2015), 26-36.  doi: 10.1016/j.jmsy.2014.11.012.

[28]

B. R. Sarker and G. R. Parija, An optimal batch size for a production system operating under a fixed-quantity, periodic delivery policy, The Journal of the Operational Research Society, 45 (1994), 891-900. 

[29]

B. R. Sarker and G. R. Parija, Optimal batch size and raw material ordering policy for a production system with a fixed-interval, lumpy demand delivery system, European Journal of Operational Research, 89 (1996), 593-608.  doi: 10.1016/0377-2217(94)00277-0.

[30]

B. Sarkar and S. Saren, Product inspection policy for an imperfect production system with inspection errors and warranty cost, European Journal of Operational Research, 248 (2016), 263-271.  doi: 10.1016/j.ejor.2015.06.021.

[31]

B. SarkarS. Saren and L. E. Cárdenas-Barrón, An inventory model with trade-credit policy and variable deterioration for fixed lifetime products, Ann Operational Research, 229 (2015), 677-702.  doi: 10.1007/s10479-014-1745-9.

[32]

B. Sarkar, S. Saren, D. Sinha and S. Hur, Effect of unequal lot sizes, variable setup cost, and carbon emission cost in a supply chain model, Mathematical Problems in Engineering, 2015 (2015), Art. ID 469486, 13 pp. doi: 10.1155/2015/469486.

[33]

E. A. Silver, D. F. Pyke and R. Peterson, Inventory Management and Production Planning and Scheduling, Third Edition, John Wiley & Sons Inc., New York, 1998.

[34]

A. A. TaleizadehF. Barzinpour and H.-M. Wee, Meta-heuristic algorithms for solving a fuzzy single-period problem, Mathematical and Computer Modelling, 54 (2011), 1273-1285.  doi: 10.1016/j.mcm.2011.03.038.

[35]

A. A. TaleizadehL. E. Cárdenas-BarrónJ. Biabanic and R. Nikousokhan, Multi products single machine EPQ model with immediate rework process, International Journal of Industrial Engineering Computations, 3 (2012), 93-102.  doi: 10.5267/j.ijiec.2011.09.001.

[36]

A. A. TaleizadehS. S. Kalantari and L. E. Cárdenas-Barrón, Determining optimal price, replacement lot size and number of shipments forn an EPQ model with rework and multiple shipments, Journal of Industrial Management Optimization, 11 (2015), 1059-1071. 

[37]

A. A. TaleizadehH. MoghadasiS. T. A. Niaki and A. Eftekhari, An economic order quantity under joint replenishment policy to supply expensive imported raw materials with payment in advance, Journal of Applied Science, 8 (2008), 4263-4273. 

[38]

A. A. TaleizadehB. MohammadiL. E. Cárdenas-Barrón and H. Samimi, An EOQ model for perishable product with special sale and shortage, International Journal of Production Economics, 145 (2013), 318-338.  doi: 10.1016/j.ijpe.2013.05.001.

[39]

A. A. TaleizadehS. T. A. NiakiM.-B. Aryanezhad and N. Shafii, A hybrid method of fuzzy simulation and genetic algorithm to optimize constrained inventory control systems with stochastic replenishments and fuzzy demand, Information Sciences, 220 (2013), 425-441.  doi: 10.1016/j.ins.2012.07.027.

[40]

A. A. TaleizadehS. T. A. NiakiM.-B. Aryanezhad and A. F. Tafti, A genetic algorithm to optimize multi-product multi-constraint inventory control systems with stochastic replenishment intervals and discount, International Journal of Advance Manufacturing Technology, 51 (2010), 311-323. 

[41]

A. A. TaleizadehS. T. A. Niaki and R. G. Meibodi, Replenish-up-to multi chance-constraint inventory control system with stochastic period lengths and total discount under fuzzy purchasing price and holding costs, Knowledge Based Systems, 53 (2013), 147-156. 

[42]

A. A. TaleizadehM. Noori-daryan and L. E. Cárdenas-Barrón, Joint optimization of price, replenishment frequency, replenishment cycle and production rate in vendor managed inventory system with deteriorating items, International Journal of Production Economics, 159 (2015), 285-295.  doi: 10.1016/j.ijpe.2014.09.009.

[43]

A. A. TaleizadehD. W. PenticoM. Aryanezhad and S. M. Ghoreyshi, An economic order quantity model with partial backordering and a special sale price, European Journal of Operational Research, 221 (2012), 571-583.  doi: 10.1016/j.ejor.2012.03.032.

[44]

A. A. TaleizadehD. W. PenticoM. S. Jabalameli and M. Aryanezhada, An economic order quantity model with multiple partial prepayments and partial backordering, Mathematical and Computer Modelling, 57 (2013), 311-323.  doi: 10.1016/j.mcm.2012.07.002.

[45]

A. A. Taleizadeh and D. W. Pentico, An economic order quantity model with a known price increase and partial backordering, European Journal of Operational Research, 228 (2013), 516-525.  doi: 10.1016/j.ejor.2013.02.014.

[46]

A. A. TaleizadehG. A. WidyadanaH. M. Wee and J. Biabani, Multi products single machine economic production quantity model with multiple batch size, International Journal of Industrial Engineering Computations, 2 (2011), 213-224.  doi: 10.5267/j.ijiec.2011.01.002.

[47]

H. M. Wee and C. J. Chung, An note on the economic lot size of the integrated vendor-buyer inventory system derived without derivatives, European Journal of Operational Research, 177 (2007), 1289-1293.  doi: 10.1016/j.ejor.2005.11.035.

[48]

B. SarkarA. MajumderM. SarkarB. K. Dey and G. Roy, Two-echelon supply chain model with manufacturing quality improvement and setup cost reduction, Journal of Industrial and Managment Optimization, 13 (2017), 1085-1104.  doi: 10.9934/j.jimo.2016.06.3.

Figure 1.  Integrated vendor-buyer production-inventory model.
Figure 2.  Graphical illustration of the unit time cost with respect to joint decision(n, q) of Example 1.
Figure 3.  The curve of total cost function TRC when $a$ varies as on Example 1.
Figure 4.  Graphical illustration of the unit time cost with respect to joint decision(n, q) of Example 2.
Figure 5.  The curve of total cost function TRC when $a$ varies as on Example 2.
Table 1.  A summary of production-delivery lot sizing models.
Author(s) MTO/MTS strategy Production quantity Delivery quantity Delivery costInspection policy
C$\acute{a}$rdenas$-$Barr$\acute{o}$n and Sana [4,5] MTO Variable $-$ $-$ $-$
Taleizadeh et al. [42] MTO Variable $-$ $-$ $-$
Benerjee [1] and Goyal [11] MTO Variable Costant $-$ $-$
Goyal [10] MTO Costant $-$ $-$ $-$
Sarker and Parija [28,29] MTO Variable Costant $-$ $-$
Sarkar and Majumder[24] MTO Costant Variable Considered$-$
Sarkar et al.[32] MTO Costant Variable $-$ $-$
Golhar and Sarker [9] MTO Variable Costant $-$ $-$
Lu [17] MTS Variable Costant $-$ $-$
Hill [14,15] MTS Variable Variable $-$ $-$
Chan and Kingsman [6] MTS Variable Variable $-$ $-$
Ben$-$Daya et al. [2] MTS Variable Variable $-$ $-$
Glock [8] MTS Variable Variable $-$ $-$
Ertogral et al. [7] MTS Variable Variable Considered$-$
Wee and Chung[47] MTO Costant Costant $-$Included
Lee and Fu [16] MTO Variable Variable Considered $-$
This study MTO Variable Variable Considered Included
"$-$" indicates non-availability of the research contribution, $"MTO"$ denotes make$-$to$-$order, and $"MTS"$ denotes make$-$to$-$stock.
Author(s) MTO/MTS strategy Production quantity Delivery quantity Delivery costInspection policy
C$\acute{a}$rdenas$-$Barr$\acute{o}$n and Sana [4,5] MTO Variable $-$ $-$ $-$
Taleizadeh et al. [42] MTO Variable $-$ $-$ $-$
Benerjee [1] and Goyal [11] MTO Variable Costant $-$ $-$
Goyal [10] MTO Costant $-$ $-$ $-$
Sarker and Parija [28,29] MTO Variable Costant $-$ $-$
Sarkar and Majumder[24] MTO Costant Variable Considered$-$
Sarkar et al.[32] MTO Costant Variable $-$ $-$
Golhar and Sarker [9] MTO Variable Costant $-$ $-$
Lu [17] MTS Variable Costant $-$ $-$
Hill [14,15] MTS Variable Variable $-$ $-$
Chan and Kingsman [6] MTS Variable Variable $-$ $-$
Ben$-$Daya et al. [2] MTS Variable Variable $-$ $-$
Glock [8] MTS Variable Variable $-$ $-$
Ertogral et al. [7] MTS Variable Variable Considered$-$
Wee and Chung[47] MTO Costant Costant $-$Included
Lee and Fu [16] MTO Variable Variable Considered $-$
This study MTO Variable Variable Considered Included
"$-$" indicates non-availability of the research contribution, $"MTO"$ denotes make$-$to$-$order, and $"MTS"$ denotes make$-$to$-$stock.
Table 2.  Cost sensitiveness based on the parameter $a$ from Example 1
$a$ $TRC$
$0$ 15722.9
$0.1$ 15816.6
$0.2$ 15935.4
$0.3$ 16085.9
$0.4$16276.8
$0.5$ 16518.7
$0.6$ 16825.3
$0.7$ 17214.0
$0.8$ 17706.6
$0.9$ 18331.1
$1$ 19122.8
$a$ $TRC$
$0$ 15722.9
$0.1$ 15816.6
$0.2$ 15935.4
$0.3$ 16085.9
$0.4$16276.8
$0.5$ 16518.7
$0.6$ 16825.3
$0.7$ 17214.0
$0.8$ 17706.6
$0.9$ 18331.1
$1$ 19122.8
Table 3.  Cost sensitiveness based on the parameter $a$ from Example 2
$a$ $TRC$
$0$ 12126.6
$0.1$ 12193.3
$0.2$ 12275.0
$0.3$ 12375.0
$0.4$ 12497.4
$0.5$ 12647.3
$0.6$ 12830.7
$0.7$ 13055.3
$0.8$ 13330.3
$0.9$ 13666.9
$1$ 14078.9
$a$ $TRC$
$0$ 12126.6
$0.1$ 12193.3
$0.2$ 12275.0
$0.3$ 12375.0
$0.4$ 12497.4
$0.5$ 12647.3
$0.6$ 12830.7
$0.7$ 13055.3
$0.8$ 13330.3
$0.9$ 13666.9
$1$ 14078.9
Table 4.  Sensitivity analysis of key parameters
Parameters Changes(in %) $TRC$
-50% -12.45
-25% -5.04
$A_1$ +25% 6.23
+50% 12.45
-50% -2.31
-25% -1.15
$A_2$ +25% 1.15
+50% 2.31
-50% -12.70
-25% -6.35
$h_1$ +25% 6.35
+50% 12.70
-50% -8.32
-25% -4.16
$h_2$ +25% 4.16
+50% 8.32
Parameters Changes(in %) $TRC$
-50% -12.45
-25% -5.04
$A_1$ +25% 6.23
+50% 12.45
-50% -2.31
-25% -1.15
$A_2$ +25% 1.15
+50% 2.31
-50% -12.70
-25% -6.35
$h_1$ +25% 6.35
+50% 12.70
-50% -8.32
-25% -4.16
$h_2$ +25% 4.16
+50% 8.32
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