A proposed procedure for variable sampling plans design based on process capability index

  • Muhammad Aslam
  • Chien-Wei Wu
  • Muhammad Azam
  • Chi-Hyuck Jun


The article presents the tightened-normal-tightened acceptance sampling plan using the process capability index Cpk. The plan parameters of the proposed plan are obtained by solving an optimization problem such that the producer’s risk and the consumer’s risk are satisfied simultaneously for several specified combinations of acceptable quality level and limiting quality level. The plan parameters are tabulated and discussed for the industrial use when the fractions non-conforming beyond the lower and the upper specification limits are symmetric or asymmetric. We illustrate the proposed plan with the help of a real example. We compare the efficiency of the proposed plan with the existing sampling plans.


Aslam, M., Azam, M. & Jun, C.H. 2015d. Improved acceptance sampling plan based on EWMA Statistic. Sequential Analysis, 34:406-422.

Aslam, M., Balamurali, S., Azam, M., Rao, G.S. & Jun, C.H. 2015a. Mixed multiple dependent state sampling plans based on process capability index. Journal of Testing and Evaluation, 43(1):171-178.

Aslam, M., Jun, C.H. & Arshad, A. 2015c. SKSP-V Sampling plan for accelerated life tests. Journal of Risk and Reliability, 229(3):193-199.

Aslam, M., Khan, N. & Khan, H. 2015b. SkSP-V Acceptance sampling plan based on process capability index. Chiang Mai Journal of Science, 42(1):258-267.

Aslam, M., Wu, C.W., Azam, M. & Jun C.H. 2013. Sampling inspection by variables for resubmitted lots based on process capability index Cpk for normally distributed items. Applied Mathematical Modeling, 37(3):667-675.

Aslam, M., Wu, C.W., Azam, M. & Jun, C.H. 2014. Mixed acceptance sampling plans for product inspection using process capability index. Quality Engineering, 26(4):450-459.

Aslam, M., Balamurali, S., Jun, C.H. & Ahmad, M. 2010. A two-plan sampling system for life testing under Weibull distribution. Industrial Engineering & Management Systems, 9(1):54-59.

Calvin, T. W. 1977. TNT zero acceptance number sampling, American Society for Quality Control, Annual Technical Conference Transaction, Philadelphia, Pennsylvania, 35–39.

Govindaraju, K. & Kissling, R. 2015. A combined attributes–variables plan. Applied Stochastic Models in Business and Industry, 31:575–583

Kane, V.E. 1986. Process capability indices. Journal of Quality Technology, 18(1):41-52.

Kotz, S. & Johnson, N.L. 2002. Process capability indices – a review, 1992-2000.

Liu, S.W., Lin, S.W. & Wu, C.W. 2014. A resubmitted sampling scheme by variables inspection for controlling lot fraction nonconforming. International Journal of Production Research, 52(12):3744-3754.

Montgomery, D.C. 2009. Introduction to Statistical Quality Control (6th Edition), Wiley, New York.

Muthuraj, D. & Senthilkumar, D. 2006. Designing and construction of tightened–normal–tightened variables sampling scheme. Journal of Applied Statistics, 33:101–111.

Negrin, I., Parmet, Y. & Schechtman, E. 2009. Developing a sampling plan based on Cpk. Quality Engineering, 21:306-318.

Negrin, I., Parmet, Y. & Schechtman, E. 2011. Developing a sampling plan based on Cpk-unknown variance. Quality and Reliability Engineering International, 27:3-14.

Pearn, W.L. & Wu, C.W. 2006a. Critical acceptance values and sample sizes of a variables sampling plan for very low fraction of defectives. Omega – International Journal of Management Science,


Pearn, W.L. & Wu, C.W. 2006b. Variables sampling plans with ppm fraction of defectives and process loss consideration. Journal of the Operational Research Society, 57(4):450-459.

Pearn, W.L. & Wu, C.W. 2007. An effective decision making method for product acceptance. Omega - International Journal of Management Science, 35(1):12-21.

Senthilkumar, D. & Muthuraj, D. 2010. Construction and selection of tightened- normal- tightened variables sampling schemes of type TNTVSS . Journal of Applied Statistics, 37 (3):375-

Soundararajan, V. & Vijayaraghavan, R. 1992. Construction and selection of tightened-normal- tightened sampling inspection scheme of type TNT-(n1, n2; c). Journal of Applied Statistics, 19(3):339- 349.

Vijayaraghavan, R. & Soundararajan, V. 1996. Procedures and tables for the selection of tightenednormal-tightened (TNT- ) sampling schemes. Journal of Applied Statistics, 23(1):69-79.

Wu, C.W. & Pearn, W.L. 2008. A variables sampling plan based on for product acceptance determination. European Journal of Operational Research, 184(2): 549-560.

Wu, C.W. 2012. An efficient inspection scheme for variables based on Taguchi capability index. European Journal of Operational Research, 223(1):116-122.

Wu, C.W., Aslam, M. & Jun, C.H. 2012. Variables sampling inspection scheme for resubmitted lots based on the process capability index Cpk. European Journal of Operational Research, 217:560-566.

Yen, C.H & Chang, C.H. 2009. Designing variables sampling plans with process loss consideration. Communications in Statistics: Simulation and Computation, 38:1579-1591.

Yen, C.H., Chang, C.H, & Aslam, M. 2015. Repetitive variable acceptance sampling plan for one-sided specification. Journal of Statistical Computation and Simulation, 85(6):1102-1116.

Yen, C.H., Ma, H., Yeh, C.H. & Chang, C.H. 2015. The economic design of variable acceptance sampling plan with rectifying inspection. Kybernetes, 44(3):440-450.

Industrial Engineering