Optimal Batch Production for a Single Machine System with Accumulated Defectives and Random Rate of Rework

Document Type : Research Paper


Industrial Engineering, Sharif unversity of technology, Teharan, Tehran, Iran


In this paper we consider an imperfect production system which produces good and defective items and assume that defective items can be reworked. Due to the nature of rework process we do not restrict the rework rate to be equal to normal production rate or constant and assume that it is a random variable with an arbitrary distribution function. We also assume as it is true in most real world situation that the unit rework processing cost need not to be equal to unit normal processing cost. We consider the case in which during a specified period of time consisting of several cycles the rework process takes place only in its last cycle. For practical reasons such as the ease of production scheduling and resource planning we require that the length of the last cycle must have the same length as the other cycles of this period. For this purpose we assume in the last cycle of this period first the normal production starts after which the rework of all defective units produced in the period begins. After the end of the last cycle of the period the whole process will start all over again. Further we assume that, for reduction of set up cost for rework per unit time, the total number of setups for rework must be as small as possible. For this system we derive the expected total cost function consisting of inventory holding, waiting time of defectives for rework, setup, and processing costs per unit time. Then we obtain the economic batch quantity which minimizes this expected cost.


Main Subjects

[1] Buscher U., Lindner G. (2007), Optimizing a production system with rework and equal sized batch
shipments; Computers and Operations Research 34(2); 515-535.
[2] Chakrabarty S., Rao N.V. (1988), EBQ for a multi stage production system considering rework;
Opsearch 25(2); 75-88.
[3] Cheng T.C.E. (1991), An economic order quantity model with production processes; IIE Transactions
23(1); 23-28.
[4] Cárdenas-Barrón L.E. (2009), Economic production quantity with rework process at a single-stage
manufacturing system with planned backorders; Computers & Industrial Engineering 57; 1105–1113.
[5] Gupta T., Chakrabarty S. (1984), Looping in a multistage production system; International Journal of
Production Research 22(2); 299-311.
[6] Haji B., Haji A., Rahmati Tavakol A. (2008), Scheduling Accumulated Rework in a Normal Cycle:
Optimal Batch Production with Minimum Rework Cycles; Journal of Industrial and Systems
Engineering 2(3); 236-249.
[7] Haji R., Haji A., Sajadifar M., Zolfaghari S. (2008), Lot sizing with non-zero setup times for rework;
Journal of Systems Science and Systems Engineering 17(2); 230-240.
[8] Hayek P.A., Salameh M.K. (2001), Production lot sizing with the reworking of imperfect quality item
produced; Production Planning and Control 12(6); 584-590.
[9] Jamal A.M.M., Sarker B.R., Mondal S. (2004), Optimal manufacturing batch size with network
process at a single-stage production system; Computers & Industrial Engineering 47; 77-89.
[10] Johnson L.A., Montgomery D.C. (1974), Operations research in production planning and inventory
control; New York, John Wiley and Sons.
[11] Kalro A.H., Gohil M.M. (1982), A lot size model with backlogging when the amount received is
uncertain; International Journal of Production Research 20(6); 775–786.
[12] Lee H.L., Rosenblatt M.J. (1987), Simultaneous determination of production cycle and inspection
schedules in a production system; Management Science 33(9); 1125-1136.
[13] Lee H.L. (1992), Lot sizing to reduce capacity utilization in production process with defective item,
process corrections and rework; Management Science 38(9); 1314-1328.
[14] Lee H.L., Chandra M.J., Deleveaux V.J. (1997), Optimal batch size and investment in multistage
production systems with scrap; Production Planning and Control 8(6); 586-596.
[15] Love S. (1979), Inventory Control; McGraw-Hill, New York.
[16] Nahmias S. (2005), Production and Operations Analysis; McGraw-Hill, 5th ed; new York.
[17] Porteus E.L. (1986), Optimal lot sizing, process quality improvement and setup cost reduction;
Operations Research 34(1); 137–144.
[18] Rosenblatt M.J., Lee H.L.(1986), Economic production cycles with imperfect production processes;
IIE Transactions 18(1); 48-55.
[19] Salameh M.K., Jaber M.Y.(2000), Economic production quantity model for items with imperfect
quality; International Journal of Production Economics 64; 59–64.
[20] Sarker B.R., Jamal A.M.M., Mondal S.(2008), Optimal batch sizing in a process at multi-stage
production system with network consideration; European Journal of Operational research 184(3);
[21] Schwaller R.L. (1988), EOQ under inspection costs; Production and Inventory Management Journal
29(3); 22-24.
[22] Zhang X., Gerchak Y. (1990), Joint lot sizing and inspection policy in an EOQ model with random
yield; IIE Transactions 22(1); 41-47.
Volume 3, Issue 4 - Serial Number 4
February 2010
Pages 243-256
  • Receive Date: 01 September 2008
  • Revise Date: 01 February 2009
  • Accept Date: 01 October 2009
  • First Publish Date: 01 February 2010