# American Institute of Mathematical Sciences

January  2012, 8(1): 141-162. doi: 10.3934/jimo.2012.8.141

## A common cycle approach for solving the economic lot and inspection scheduling problem

 1 Department of Transportation Technology and Management, National Chiao Tung University, Hsinchu, 30010, Taiwan 2 Department of Industrial Engineering and Management, Hsiuping University of Science and Technology, Taichung, 41280, Taiwan 3 Department of Information Management, Tunghai University, Taichung, 40704, Taiwan

Received  October 2010 Revised  July 2011 Published  November 2011

In this study, we consider an imperfect production system in which the manager not only faces the Economic Lot Scheduling Problem, but also needs to conduct multiple inspections during a production lot of any product. Inspection plays an important role in an imperfect production system since it saves the cost from producing and restoring defective items though it also incurs extra inspection cost at the same time. In this study, we employ the common cycle approach in which all the products share the same replenishment cycle, and adopt a consensus inspection policy. The focus of this study is to determine the optimal cycle time and an optimal production and inspection schedule that minimize the total cost per unit time. We formulate a mathematical model in which we take into accounts both the production capacity and inspection capacity constraints. Also, we conduct full theoretical analysis and propose an effective search algorithm for solving an optimal solution. Our numerical experiments demonstrate the effectiveness of the proposed search algorithm.
Citation: Ming-Jong Yao, Shih-Chieh Chen, Yu-Jen Chang. A common cycle approach for solving the economic lot and inspection scheduling problem. Journal of Industrial and Management Optimization, 2012, 8 (1) : 141-162. doi: 10.3934/jimo.2012.8.141
##### References:
 [1] Y. C. Chen, Optimal inspection and economical production quantity strategy for an imperfect production process, International Journal of Systems Science, 37 (2006), 295-302. doi: 10.1080/00207720600566602. [2] I. Djamaludin, R. J. Wilson, and D. N. P. Murthy, Lot sizing and testing for items with uncertain quality, Stochastic Models in Engineering, Technology and Management (Gold Coast, 1994), Mathematical and Computer Modelling, 22 (1995), 35-44. doi: 10.1016/0895-7177(95)00178-5. [3] G. Dobson, The cyclic lot scheduling problem with sequence-dependent setups, Operations Research, 40 (1992), 736-749. doi: 10.1287/opre.40.4.736. [4] S. E. Elmaghraby, The economic lot scheduling problem (ELSP): Review and extension, Management Science, 24 (1978), 587-597. doi: 10.1287/mnsc.24.6.587. [5] B. Faaland, T. Schmitt and T. Arreola-Risa, Economic lot scheduling with lose sales and setup times, IIE Transactions, 36 (2004), 629-640. doi: 10.1080/07408170490278238. [6] B. C. Giri, T. Dohi, T. Schmitt and T. Arreola-Risa, Inspection scheduling for imperfect production processes under free repair warranty contract, European Journal of Operational Research, 183 (2007), 238-252. doi: 10.1016/j.ejor.2006.09.062. [7] F. Hanssmann, "Operations Research in Production and Inventory Control," John Wiley & Sons, Inc., New York-London, 1962. [8] W. Hsu, On the general feasibility test of scheduling lot sizes for several products on one machine, Management Science, 29 (1983), 93-105. doi: 10.1287/mnsc.29.1.93. [9] F. Hu and Q. Zong, Optimal production run time for a deteriorating production system under an extended inspection policy, European Journal of Operational Research, 196 (2009), 979-986. doi: 10.1016/j.ejor.2008.05.008. [10] C. H. Kim and Y. Hong, An optimal production run length in deteriorating production processes, International Journal of Production Economics, 58 (1999), 183-189. doi: 10.1016/S0925-5273(98)00119-4. [11] C. H. Kim, Y. Hong and S. Y. Chang, Optimal production run length and inspection schedules in a deteriorating production process, IIE Transactions, 33 (2001), 421-426. doi: 10.1080/07408170108936840. [12] H. L. Lee and M. J. Rosenblatt, Simultaneous determination of production cycle and inspection schedules in a production system, Management Science, 33 (1987), 1125-1136. doi: 10.1287/mnsc.33.9.1125. [13] H. L. Lee and M. J. Rosenblatt, A production and maintenance planning model with restoration cost dependent on detection delay, IIE Transactions, 21 (1989), 368-375. doi: 10.1080/07408178908966243. [14] J. S. Lee and K. S. Park, Joint determination of production cycle and inspection intervals in a deteriorating production system, Journal of the Operational Research Society, 42 (1991), 775-783. [15] C. S. Lin, C. H. Chen and D. E. Kroll, Integrated production-inventory models for imperfect production processes under inspection schedules, Computers and Industrial Engineering, 44 (2003), 633-650. doi: 10.1016/S0360-8352(02)00239-5. [16] M. A. Lopez and B. G. Kingsmans, The economic lot scheduling problem: Theory and practice, International Journal of Production Economics, 23 (1991), 147-164. doi: 10.1016/0925-5273(91)90058-2. [17] I. K. Moon, B. C. Cha and H. C. Bae, Hybrid genetic algorithm for group technology economic lot scheduling problem, International Journal of Production Research, 44 (2006), 4551-4568. doi: 10.1080/00207540500534405. [18] J. Neter, W. Wasserman and M. Kutner, "Applied Linear Statistical Models," Irvine, Chicago, IL, 1996. [19] H. Ouyang and X. Zhu, A economic lot scheduling problem for manufacturing and remanufacturing, in "2008 IEEE International Conference on Cybernetics and Intelligent Systems," CIS, 2008. doi: 10.1109/ICCIS.2008.4670892. [20] 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. [21] E. L. Porteus, The impact of inspection delay on process and inspection lot sizing, Management Science, 36 (1990), 999-1007. doi: 10.1287/mnsc.36.8.999. [22] M. A. Rahim, Joint determination of production quantity inspection schedule, and control chart design, International Journal of Production Research, 36 (1994), 277-289. doi: 10.1080/002075498194047. [23] J. Rogers, A computational approach to the economic lot scheduling problem, Management Science, 36 (1958), 264-291. doi: 10.1287/mnsc.4.3.264. [24] M. J. Rosenblatt and H. L. Lee, Economic production cycle with imperfect production process, IIE Transactions, 18 (1986), 48-55. doi: 10.1080/07408178608975329. [25] M. J. Rosenblatt and H. L. Lee, A comparative study of continuous and periodic inspection policies in deteriorating production systems, IIE Transactions, 18 (1986), 2-9. doi: 10.1080/07408178608975323. [26] L. Salvietti and N. R. Smith, A profit-maximizing economic lot scheduling problem with price optimization, European Journal of Operational Research, 184 (2008), 900-914. doi: 10.1016/j.ejor.2006.11.031. [27] C. A. Soman, D. P. Van Donk and G. Gaalman, A basic period approach to the economic lot scheduling problem with shelf life considerations, International Journal of Production Research, 42 (2004), 1677-1689. doi: 10.1080/00207540310001645165. [28] O. Tang and R. H. Teunter, Economic lot scheduling problem with returns, Operations Management, 15 (2006), 488-497. doi: 10.1111/j.1937-5956.2006.tb00158.x. [29] R. Teunter, K. Kaparis and O. Tang, Multi-product economic lot scheduling problem with separate production lines for manufacturing and remanufacturing, European Journal of Operational Research, 191 (2008), 1241-1253. doi: 10.1016/j.ejor.2007.08.003. [30] B. Wagner and D. J. Davis, A search heuristic for the sequence-dependent economic lot scheduling problem, European Journal of Operational Research, 141 (2002), 133-146. doi: 10.1016/S0377-2217(01)00265-X. [31] C. H. Wang and S. H. Sheu, Simultaneous determination of the optimal production-inventory and product inspection policies for a deteriorating production system,, Computers and Operations Research, 28 (2001), 1093-1110. doi: 10.1016/S0305-0548(00)00030-7. [32] C. H. Wang, Integrated production and product inspection policy for a deteriorating production system, International Journal of Production Economics, 95 (2005), 123-134. doi: 10.1016/j.ijpe.2003.11.012. [33] M. J. Yao, "The Economic Lot Scheduling Problem with Extension to Multiple Resource Constraints," Unpublished Ph.D thesis, North Carolina State University, USA, 1999. [34] M. J. Yao, S. E. Elmaghraby and I. C. Chen, On the feasibility testing of the economic lot scheduling problem using the extended basic period approach, Journal of the Chinese Institute of Industrial Engineering, 20 (2003), 435-448. doi: 10.1080/10170660309509249. [35] M. J. Yao and Y. J. Chang, A genetic algorithm for solving the economic lot schedule problem with reworks, Journal of the Chinese Institute of Industrial Engineering, 26 (2009), 411-425. doi: 10.1080/10170660909509155. [36] M. J. Yao and S. C. Chen, On the determination of the optimal replenishment and inspection schedule in an imperfect production-inventory system, Journal of Operations and Logistics, 2 (2009), 1-15. [37] R. H. Yeh and T. H. Chen, Optimal lot size and inspection policy for products sold with warranty, European Journal of Operational Research, 174 (2006), 766-776. doi: 10.1016/j.ejor.2005.02.049.

show all references

##### References:
 [1] Y. C. Chen, Optimal inspection and economical production quantity strategy for an imperfect production process, International Journal of Systems Science, 37 (2006), 295-302. doi: 10.1080/00207720600566602. [2] I. Djamaludin, R. J. Wilson, and D. N. P. Murthy, Lot sizing and testing for items with uncertain quality, Stochastic Models in Engineering, Technology and Management (Gold Coast, 1994), Mathematical and Computer Modelling, 22 (1995), 35-44. doi: 10.1016/0895-7177(95)00178-5. [3] G. Dobson, The cyclic lot scheduling problem with sequence-dependent setups, Operations Research, 40 (1992), 736-749. doi: 10.1287/opre.40.4.736. [4] S. E. Elmaghraby, The economic lot scheduling problem (ELSP): Review and extension, Management Science, 24 (1978), 587-597. doi: 10.1287/mnsc.24.6.587. [5] B. Faaland, T. Schmitt and T. Arreola-Risa, Economic lot scheduling with lose sales and setup times, IIE Transactions, 36 (2004), 629-640. doi: 10.1080/07408170490278238. [6] B. C. Giri, T. Dohi, T. Schmitt and T. Arreola-Risa, Inspection scheduling for imperfect production processes under free repair warranty contract, European Journal of Operational Research, 183 (2007), 238-252. doi: 10.1016/j.ejor.2006.09.062. [7] F. Hanssmann, "Operations Research in Production and Inventory Control," John Wiley & Sons, Inc., New York-London, 1962. [8] W. Hsu, On the general feasibility test of scheduling lot sizes for several products on one machine, Management Science, 29 (1983), 93-105. doi: 10.1287/mnsc.29.1.93. [9] F. Hu and Q. Zong, Optimal production run time for a deteriorating production system under an extended inspection policy, European Journal of Operational Research, 196 (2009), 979-986. doi: 10.1016/j.ejor.2008.05.008. [10] C. H. Kim and Y. Hong, An optimal production run length in deteriorating production processes, International Journal of Production Economics, 58 (1999), 183-189. doi: 10.1016/S0925-5273(98)00119-4. [11] C. H. Kim, Y. Hong and S. Y. Chang, Optimal production run length and inspection schedules in a deteriorating production process, IIE Transactions, 33 (2001), 421-426. doi: 10.1080/07408170108936840. [12] H. L. Lee and M. J. Rosenblatt, Simultaneous determination of production cycle and inspection schedules in a production system, Management Science, 33 (1987), 1125-1136. doi: 10.1287/mnsc.33.9.1125. [13] H. L. Lee and M. J. Rosenblatt, A production and maintenance planning model with restoration cost dependent on detection delay, IIE Transactions, 21 (1989), 368-375. doi: 10.1080/07408178908966243. [14] J. S. Lee and K. S. Park, Joint determination of production cycle and inspection intervals in a deteriorating production system, Journal of the Operational Research Society, 42 (1991), 775-783. [15] C. S. Lin, C. H. Chen and D. E. Kroll, Integrated production-inventory models for imperfect production processes under inspection schedules, Computers and Industrial Engineering, 44 (2003), 633-650. doi: 10.1016/S0360-8352(02)00239-5. [16] M. A. Lopez and B. G. Kingsmans, The economic lot scheduling problem: Theory and practice, International Journal of Production Economics, 23 (1991), 147-164. doi: 10.1016/0925-5273(91)90058-2. [17] I. K. Moon, B. C. Cha and H. C. Bae, Hybrid genetic algorithm for group technology economic lot scheduling problem, International Journal of Production Research, 44 (2006), 4551-4568. doi: 10.1080/00207540500534405. [18] J. Neter, W. Wasserman and M. Kutner, "Applied Linear Statistical Models," Irvine, Chicago, IL, 1996. [19] H. Ouyang and X. Zhu, A economic lot scheduling problem for manufacturing and remanufacturing, in "2008 IEEE International Conference on Cybernetics and Intelligent Systems," CIS, 2008. doi: 10.1109/ICCIS.2008.4670892. [20] 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. [21] E. L. Porteus, The impact of inspection delay on process and inspection lot sizing, Management Science, 36 (1990), 999-1007. doi: 10.1287/mnsc.36.8.999. [22] M. A. Rahim, Joint determination of production quantity inspection schedule, and control chart design, International Journal of Production Research, 36 (1994), 277-289. doi: 10.1080/002075498194047. [23] J. Rogers, A computational approach to the economic lot scheduling problem, Management Science, 36 (1958), 264-291. doi: 10.1287/mnsc.4.3.264. [24] M. J. Rosenblatt and H. L. Lee, Economic production cycle with imperfect production process, IIE Transactions, 18 (1986), 48-55. doi: 10.1080/07408178608975329. [25] M. J. Rosenblatt and H. L. Lee, A comparative study of continuous and periodic inspection policies in deteriorating production systems, IIE Transactions, 18 (1986), 2-9. doi: 10.1080/07408178608975323. [26] L. Salvietti and N. R. Smith, A profit-maximizing economic lot scheduling problem with price optimization, European Journal of Operational Research, 184 (2008), 900-914. doi: 10.1016/j.ejor.2006.11.031. [27] C. A. Soman, D. P. Van Donk and G. Gaalman, A basic period approach to the economic lot scheduling problem with shelf life considerations, International Journal of Production Research, 42 (2004), 1677-1689. doi: 10.1080/00207540310001645165. [28] O. Tang and R. H. Teunter, Economic lot scheduling problem with returns, Operations Management, 15 (2006), 488-497. doi: 10.1111/j.1937-5956.2006.tb00158.x. [29] R. Teunter, K. Kaparis and O. Tang, Multi-product economic lot scheduling problem with separate production lines for manufacturing and remanufacturing, European Journal of Operational Research, 191 (2008), 1241-1253. doi: 10.1016/j.ejor.2007.08.003. [30] B. Wagner and D. J. Davis, A search heuristic for the sequence-dependent economic lot scheduling problem, European Journal of Operational Research, 141 (2002), 133-146. doi: 10.1016/S0377-2217(01)00265-X. [31] C. H. Wang and S. H. Sheu, Simultaneous determination of the optimal production-inventory and product inspection policies for a deteriorating production system,, Computers and Operations Research, 28 (2001), 1093-1110. doi: 10.1016/S0305-0548(00)00030-7. [32] C. H. Wang, Integrated production and product inspection policy for a deteriorating production system, International Journal of Production Economics, 95 (2005), 123-134. doi: 10.1016/j.ijpe.2003.11.012. [33] M. J. Yao, "The Economic Lot Scheduling Problem with Extension to Multiple Resource Constraints," Unpublished Ph.D thesis, North Carolina State University, USA, 1999. [34] M. J. Yao, S. E. Elmaghraby and I. C. Chen, On the feasibility testing of the economic lot scheduling problem using the extended basic period approach, Journal of the Chinese Institute of Industrial Engineering, 20 (2003), 435-448. doi: 10.1080/10170660309509249. [35] M. J. Yao and Y. J. Chang, A genetic algorithm for solving the economic lot schedule problem with reworks, Journal of the Chinese Institute of Industrial Engineering, 26 (2009), 411-425. doi: 10.1080/10170660909509155. [36] M. J. Yao and S. C. Chen, On the determination of the optimal replenishment and inspection schedule in an imperfect production-inventory system, Journal of Operations and Logistics, 2 (2009), 1-15. [37] R. H. Yeh and T. H. Chen, Optimal lot size and inspection policy for products sold with warranty, European Journal of Operational Research, 174 (2006), 766-776. doi: 10.1016/j.ejor.2005.02.049.
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