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October  2015, 11(4): 1059-1071. doi: 10.3934/jimo.2015.11.1059

Determining optimal price, replenishment lot size and number of shipments for an EPQ model with rework and multiple shipments

Ata Allah Taleizadeh 1, , Solaleh Sadat Kalantari 2, and  Leopoldo Eduardo Cárdenas-Barrón 3,

1. 

School of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran
2. 

School of Industrial and Systems Engineering, College of Engineering, University of Tehran, Tehran, Iran
3. 

School of Engineering and Sciences, Tecnológico de Monterrey, E. Garza Sada 2501 Sur, C.P. 64849, Monterrey, Nuevo León, Mexico

Received  May 2014 Revised  September 2014 Published  March 2015

In a regular manufacturing process at the end of each production cycle defective items are identified. Then some defective items are considered scrap and the others, after a reworking process with a defective rate, could convert in a perfect quality items. In this direction, this research work deals with the problem of the joint determination of selling price, replenishment lot size and the number of shipments for an economic production quantity (EPQ) model with rework of defective items when multi-shipment policy is used. After proving concavity of the long-run average benefit function, a practical algorithm is developed to find the optimal price, replenishment lot size and number of shipments in order to maximize the average long-run benefit function. A special case when there is no scrap is identified and explained. Furthermore, in order to show the practical usage of the proposed algorithm a case study from real world is presented and solved.
Keywords: number of delivery shipments, replenishment policy, EPQ model., Pricing.
Mathematics Subject Classification: 90B0.
Citation: Ata Allah Taleizadeh, Solaleh Sadat Kalantari, Leopoldo Eduardo Cárdenas-Barrón. Determining optimal price, replenishment lot size and number of shipments for an EPQ model with rework and multiple shipments. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1059-1071. doi: 10.3934/jimo.2015.11.1059
References:
[1]

S. Banerjee and A. Sharma, Optimal procurement and pricing policies for inventory models with price and time dependent seasonal demand,, Mathematical and Computer Modeling, 51 (2010), 700.  doi: 10.1016/j.mcm.2009.10.022.  Google Scholar

[2]

W. M. Chan, R. N. Ibrahim and P. B. Lochert, A new EPQ model: Integrating lower pricing, rework and reject situations,, Production Planning & Control, 14 (2003), 588.   Google Scholar

[3]

K. K. Chen, M.-F. Wu, S. W. Chiu and C.-H. Lee, Alternative approach for solving replenishment lot size problem with discontinuous issuing policy and rework,, Expert Systems with Applications, 39 (2012), 2232.  doi: 10.1016/j.eswa.2011.08.030.  Google Scholar

[4]

S. Chiu, H. Lin, M. Wu and J. Yang, Determining replenishment lot size and shipment policy for an extended EPQ model with delivery and quality assurance issues,, ScienticaIranica, 18 (2011), 1537.  doi: 10.1016/j.scient.2011.09.008.  Google Scholar

[5]

S. W. Chiu, D. C. Gong, C. L. Chiu and C. L. Chung, Joint Determination of the production lot size and number of shipments EPQ model with rework,, Mathematical and Computational Applications, 16 (2011), 317.   Google Scholar

[6]

S. W. Chiu, Y. S. P. Chiu and J. C. Yang, Combining an alternative multi delivery policy into economic lot size problem with partial rework,, Expert Systems with Applications, 39 (2012), 2578.  doi: 10.1016/j.eswa.2011.08.112.  Google Scholar

[7]

L. E. Cárdenas-Barrón, Economic production quantity with rework process at a single-stage manufacturing system with planned backorders,, Computers & Industrial Engineering, 57 (2009), 1105.   Google Scholar

[8]

L. E. Cárdenas-Barrón, B. Sarkar and G. Trevino-Garza, An improved solution to the replenishment policy for the EMQ model with rework and multiple shipments,, Applied Mathematical Modelling, 37 (2013), 5549.  doi: 10.1016/j.apm.2012.10.017.  Google Scholar

[9]

L. E. Cárdenas-Barrón, B. Sarkar and G. Trevino-Garza, Easy and improved algorithms to joint determination of the replenishment lot size and number of shipments for an EPQ model with rework,, Mathematical and Computational Applications, 18 (2013), 132.   Google Scholar

[10]

L. E. Cárdenas-Barrón, A. A. Taleizadeh and G. Trevino-Garza, An improved solution to the replenishment lot size problem with discontinuous issuing policy and rework, and the multi-delivery policy into economic production lot size problem with partial rework,, Expert Systems with Applications, 39 (2012), 13540.   Google Scholar

[11]

C. Y. Dye, A finite horizon deteriorating inventory model with two-phase pricing and time varying demand and cost under trade credit financing using particle swarm optimization,, Swarm and Evolutionary Computation, 5 (2012), 37.  doi: 10.1016/j.swevo.2012.03.002.  Google Scholar

[12]

J. T. Hsu and L. F. Hsu, Two EPQ models with imperfect production processes, inspection errors, planned backorders, and sales returns,, Computers & Industrial Engineering, 64 (2013), 389.  doi: 10.1016/j.cie.2012.10.005.  Google Scholar

[13]

F. W. Harris, How many parts to make at once, Factory,, The Magazine of Management, 10 (1913), 135.   Google Scholar

[14]

A. M. M. Jamal and B. R. Sarker, An optimal batch size for a production system operating under a just-in-time delivery system,, International Journal of Production Economics, 32 (1993), 255.  doi: 10.1016/0925-5273(93)90072-S.  Google Scholar

[15]

I. Konstantaras, K. Skouri and M. Y. Jaber, Inventory models for imperfect quality items with shortages and learning in inspection,, Applied Mathematical Modelling, 36 (2012), 5334.  doi: 10.1016/j.apm.2011.12.005.  Google Scholar

[16]

R. Maihami, K. Nakhai and I. Abadi, Joint control of inventory and its pricing for non-instantaneously deteriorating items under permissible delay in payments and partial backlogging,, Mathematical and Computer Modelling, 55 (2012), 1722.  doi: 10.1016/j.mcm.2011.11.017.  Google Scholar

[17]

S. Pal, M. K. Maiti and M. Maiti, An EPQ model with price discounted promotional demand in an imprecise planning horizon via Genetic Algorithm,, Computers & Industrial Engineering, 57 (2009), 181.  doi: 10.1016/j.cie.2008.11.016.  Google Scholar

[18]

B. Pal, S. S. Sana and K. Chaudhuri, Multi-item EOQ model while demand is sales price and price break sensitive,, Economic Modelling, 29 (2012), 2283.  doi: 10.1016/j.econmod.2012.06.039.  Google Scholar

[19]

B. Pal, S. S. Sana and K. Chaudhuri, A multi-echelon supply chain model for reworkable items in multiple-markets with supply disruption,, Economic Modelling, 29 (2012), 1891.  doi: 10.1016/j.econmod.2012.06.005.  Google Scholar

[20]

S. Papachristos and K. Skouri, An inventory model with deteriorating items, quantity discount, pricing and time-dependent partial backlogging,, International Journal of Production Economics, 83 (2003), 247.  doi: 10.1016/S0925-5273(02)00332-8.  Google Scholar

[21]

S. H. R. Pasandideh, S. T. Akhavan Niaki and J. Aryan Yeganeh, A parameter-tuned genetic algorithm for multi-product economic production quantity model with space constraint, discrete delivery orders and shortages,, Advances in Engineering Software, 41 (2010), 306.   Google Scholar

[22]

S. Ray, Y. Gerchak and E. M. Jewkes, Joint pricing and inventory policies for make-to-stock products with deterministic price-sensitive demand,, International Journal of Production Economics, 97 (2005), 143.  doi: 10.1016/j.ijpe.2004.06.054.  Google Scholar

[23]

S. S. Sana, An economic order quantity model for nonconforming quality products,, Service Science, 4 (2012), 331.  doi: 10.1287/serv.1120.0027.  Google Scholar

[24]

S. S. Sana, Sales team's initiatives and stock sensitive demand - A production control policy,, Economic Modelling, 31 (2013), 783.  doi: 10.1016/j.econmod.2013.01.026.  Google Scholar

[25]

S. S. Sana, Optimal production lot size and reorder point of a two-stage supply chain while random demand is sensitive with sales teams' initiatives,, International Journal of Systems Science, (2014).  doi: 10.1080/00207721.2014.886748.  Google Scholar

[26]

S. S. Sana, A production-inventory model of imperfect quality products in a three-layer supply chain,, Decision Support Systems, 50 (2011), 539.  doi: 10.1016/j.dss.2010.11.012.  Google Scholar

[27]

S. S. Sana, A production-inventory model in an imperfect production process,, European Journal of Operational Research, 200 (2010), 451.  doi: 10.1016/j.ejor.2009.01.041.  Google Scholar

[28]

A. A. Taleizadeh, H. M. Wee and S. G. Jalali-Naini, Economic production quantity model with repair failure and limited capacity,, Applied Mathematical Modelling, 37 (2013), 2765.  doi: 10.1016/j.apm.2012.06.006.  Google Scholar

[29]

A. A. Taleizadeh, S. T. AkhavanNiaki and A. A. Najafi, Multi product single-machine production system with stochastic scrapped production rate, partial backordering and service level constraint,, Journal of Computational and Applied Mathematics, 233 (2010), 1834.  doi: 10.1016/j.cam.2009.09.021.  Google Scholar

[30]

A. A. Taleizadeh, L. E. Cárdenas-Barrón, J. Biabani and R. Nikousokh, Multi products single machine EPQ model with immediate rework process,, International Journal of Industrial Engineering Computations, 3 (2012), 93.  doi: 10.5267/j.ijiec.2011.09.001.  Google Scholar

[31]

G. Trevino-Garza, K. Castillo-Villar and L. E. CCárdenas-Barrón, Joint determination of the lot size and number of shipments for a family of integrated vendor-buyer systems considering defective products,, International Journal of Systems Science, (2014).  doi: 10.1080/00207721.2014.886750.  Google Scholar

[32]

D. M. Tsai, Economic production quantity concerning learning and the reworking of imperfect items,, Yugoslav Journal of Operations Research, 22 (2012), 313.  doi: 10.2298/YJOR091109012T.  Google Scholar

[33]

A. Yassine, B. Maddah and M. Salameh, Disaggregation and consolidation of imperfect quality shipments in an extended EPQ model,, International Journal of Production Economics, 135 (2012), 345.  doi: 10.1016/j.ijpe.2011.08.010.  Google Scholar

show all references

References:
[1]

S. Banerjee and A. Sharma, Optimal procurement and pricing policies for inventory models with price and time dependent seasonal demand,, Mathematical and Computer Modeling, 51 (2010), 700.  doi: 10.1016/j.mcm.2009.10.022.  Google Scholar

[2]

W. M. Chan, R. N. Ibrahim and P. B. Lochert, A new EPQ model: Integrating lower pricing, rework and reject situations,, Production Planning & Control, 14 (2003), 588.   Google Scholar

[3]

K. K. Chen, M.-F. Wu, S. W. Chiu and C.-H. Lee, Alternative approach for solving replenishment lot size problem with discontinuous issuing policy and rework,, Expert Systems with Applications, 39 (2012), 2232.  doi: 10.1016/j.eswa.2011.08.030.  Google Scholar

[4]

S. Chiu, H. Lin, M. Wu and J. Yang, Determining replenishment lot size and shipment policy for an extended EPQ model with delivery and quality assurance issues,, ScienticaIranica, 18 (2011), 1537.  doi: 10.1016/j.scient.2011.09.008.  Google Scholar

[5]

S. W. Chiu, D. C. Gong, C. L. Chiu and C. L. Chung, Joint Determination of the production lot size and number of shipments EPQ model with rework,, Mathematical and Computational Applications, 16 (2011), 317.   Google Scholar

[6]

S. W. Chiu, Y. S. P. Chiu and J. C. Yang, Combining an alternative multi delivery policy into economic lot size problem with partial rework,, Expert Systems with Applications, 39 (2012), 2578.  doi: 10.1016/j.eswa.2011.08.112.  Google Scholar

[7]

L. E. Cárdenas-Barrón, Economic production quantity with rework process at a single-stage manufacturing system with planned backorders,, Computers & Industrial Engineering, 57 (2009), 1105.   Google Scholar

[8]

L. E. Cárdenas-Barrón, B. Sarkar and G. Trevino-Garza, An improved solution to the replenishment policy for the EMQ model with rework and multiple shipments,, Applied Mathematical Modelling, 37 (2013), 5549.  doi: 10.1016/j.apm.2012.10.017.  Google Scholar

[9]

L. E. Cárdenas-Barrón, B. Sarkar and G. Trevino-Garza, Easy and improved algorithms to joint determination of the replenishment lot size and number of shipments for an EPQ model with rework,, Mathematical and Computational Applications, 18 (2013), 132.   Google Scholar

[10]

L. E. Cárdenas-Barrón, A. A. Taleizadeh and G. Trevino-Garza, An improved solution to the replenishment lot size problem with discontinuous issuing policy and rework, and the multi-delivery policy into economic production lot size problem with partial rework,, Expert Systems with Applications, 39 (2012), 13540.   Google Scholar

[11]

C. Y. Dye, A finite horizon deteriorating inventory model with two-phase pricing and time varying demand and cost under trade credit financing using particle swarm optimization,, Swarm and Evolutionary Computation, 5 (2012), 37.  doi: 10.1016/j.swevo.2012.03.002.  Google Scholar

[12]

J. T. Hsu and L. F. Hsu, Two EPQ models with imperfect production processes, inspection errors, planned backorders, and sales returns,, Computers & Industrial Engineering, 64 (2013), 389.  doi: 10.1016/j.cie.2012.10.005.  Google Scholar

[13]

F. W. Harris, How many parts to make at once, Factory,, The Magazine of Management, 10 (1913), 135.   Google Scholar

[14]

A. M. M. Jamal and B. R. Sarker, An optimal batch size for a production system operating under a just-in-time delivery system,, International Journal of Production Economics, 32 (1993), 255.  doi: 10.1016/0925-5273(93)90072-S.  Google Scholar

[15]

I. Konstantaras, K. Skouri and M. Y. Jaber, Inventory models for imperfect quality items with shortages and learning in inspection,, Applied Mathematical Modelling, 36 (2012), 5334.  doi: 10.1016/j.apm.2011.12.005.  Google Scholar

[16]

R. Maihami, K. Nakhai and I. Abadi, Joint control of inventory and its pricing for non-instantaneously deteriorating items under permissible delay in payments and partial backlogging,, Mathematical and Computer Modelling, 55 (2012), 1722.  doi: 10.1016/j.mcm.2011.11.017.  Google Scholar

[17]

S. Pal, M. K. Maiti and M. Maiti, An EPQ model with price discounted promotional demand in an imprecise planning horizon via Genetic Algorithm,, Computers & Industrial Engineering, 57 (2009), 181.  doi: 10.1016/j.cie.2008.11.016.  Google Scholar

[18]

B. Pal, S. S. Sana and K. Chaudhuri, Multi-item EOQ model while demand is sales price and price break sensitive,, Economic Modelling, 29 (2012), 2283.  doi: 10.1016/j.econmod.2012.06.039.  Google Scholar

[19]

B. Pal, S. S. Sana and K. Chaudhuri, A multi-echelon supply chain model for reworkable items in multiple-markets with supply disruption,, Economic Modelling, 29 (2012), 1891.  doi: 10.1016/j.econmod.2012.06.005.  Google Scholar

[20]

S. Papachristos and K. Skouri, An inventory model with deteriorating items, quantity discount, pricing and time-dependent partial backlogging,, International Journal of Production Economics, 83 (2003), 247.  doi: 10.1016/S0925-5273(02)00332-8.  Google Scholar

[21]

S. H. R. Pasandideh, S. T. Akhavan Niaki and J. Aryan Yeganeh, A parameter-tuned genetic algorithm for multi-product economic production quantity model with space constraint, discrete delivery orders and shortages,, Advances in Engineering Software, 41 (2010), 306.   Google Scholar

[22]

S. Ray, Y. Gerchak and E. M. Jewkes, Joint pricing and inventory policies for make-to-stock products with deterministic price-sensitive demand,, International Journal of Production Economics, 97 (2005), 143.  doi: 10.1016/j.ijpe.2004.06.054.  Google Scholar

[23]

S. S. Sana, An economic order quantity model for nonconforming quality products,, Service Science, 4 (2012), 331.  doi: 10.1287/serv.1120.0027.  Google Scholar

[24]

S. S. Sana, Sales team's initiatives and stock sensitive demand - A production control policy,, Economic Modelling, 31 (2013), 783.  doi: 10.1016/j.econmod.2013.01.026.  Google Scholar

[25]

S. S. Sana, Optimal production lot size and reorder point of a two-stage supply chain while random demand is sensitive with sales teams' initiatives,, International Journal of Systems Science, (2014).  doi: 10.1080/00207721.2014.886748.  Google Scholar

[26]

S. S. Sana, A production-inventory model of imperfect quality products in a three-layer supply chain,, Decision Support Systems, 50 (2011), 539.  doi: 10.1016/j.dss.2010.11.012.  Google Scholar

[27]

S. S. Sana, A production-inventory model in an imperfect production process,, European Journal of Operational Research, 200 (2010), 451.  doi: 10.1016/j.ejor.2009.01.041.  Google Scholar

[28]

A. A. Taleizadeh, H. M. Wee and S. G. Jalali-Naini, Economic production quantity model with repair failure and limited capacity,, Applied Mathematical Modelling, 37 (2013), 2765.  doi: 10.1016/j.apm.2012.06.006.  Google Scholar

[29]

A. A. Taleizadeh, S. T. AkhavanNiaki and A. A. Najafi, Multi product single-machine production system with stochastic scrapped production rate, partial backordering and service level constraint,, Journal of Computational and Applied Mathematics, 233 (2010), 1834.  doi: 10.1016/j.cam.2009.09.021.  Google Scholar

[30]

A. A. Taleizadeh, L. E. Cárdenas-Barrón, J. Biabani and R. Nikousokh, Multi products single machine EPQ model with immediate rework process,, International Journal of Industrial Engineering Computations, 3 (2012), 93.  doi: 10.5267/j.ijiec.2011.09.001.  Google Scholar

[31]

G. Trevino-Garza, K. Castillo-Villar and L. E. CCárdenas-Barrón, Joint determination of the lot size and number of shipments for a family of integrated vendor-buyer systems considering defective products,, International Journal of Systems Science, (2014).  doi: 10.1080/00207721.2014.886750.  Google Scholar

[32]

D. M. Tsai, Economic production quantity concerning learning and the reworking of imperfect items,, Yugoslav Journal of Operations Research, 22 (2012), 313.  doi: 10.2298/YJOR091109012T.  Google Scholar

[33]

A. Yassine, B. Maddah and M. Salameh, Disaggregation and consolidation of imperfect quality shipments in an extended EPQ model,, International Journal of Production Economics, 135 (2012), 345.  doi: 10.1016/j.ijpe.2011.08.010.  Google Scholar

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