Designing Tolerance of Assembled Components Using Weibull Distribution

Document Type : Original Manuscript


1 Department of Industrial Management, Firoozkooh Branch, Islamic Azad University, Firoozkooh, Iran.

2 Department of Industrial Management, Science and Technology Branch, Islamic Azad University, Tehran, Iran.



Tolerancing is one of the most important tools for planning, controlling, and improving quality in the industry. Tolerancing conducted by design engineers to meet customers’ needs is a prerequisite for producing high-quality products. Engineers use handbooks to conduct tolerancing. While use of statistical methods for tolerancing is not a new concept, engineers often use known distributions, including the normal distribution. However, if the statistical distribution of the given variable is unknown, a new statistical method will be employed to design tolerance. Therefore, in this study we want to offer a proper statistical method for determining tolerance. The use of statistical methods to design tolerance is not a new concept; however, flexible use of statistical distributions can enhance its performance. In this regard, Weibull distribution is proposed. To illustrate the proposed method first technical characteristics of production parts were selected randomly, and then manufacturing parameters were determined using maximum likelihood method.  Finally, the Goodness of Fit test was used to ensure the accuracy of the obtained results.

Graphical Abstract

Designing Tolerance of Assembled Components Using Weibull Distribution


  • Nowadays, tolerance of components is determined by using engineering handbooks. Since the process capability is not taken into account for this purpose; tolerance is determined sometimes less and sometimes more than process capability. In the first case, many components are produced as waste or correction is needed; in the second case, components are manufactured with greater precision than needed. As a result, the product cost rises. In any case, the total cost of production will rise.
  • In practice, produced components tend to be assembled together. Therefore, if probabilistic relationships, process capability, and products’ functional needs are simultaneously and properly considered for determining the tolerance of assembled components, we can determine the tolerance of each of assembled components more judiciously and greater than its current value, so that in practice they can be produced more easily whereas they still serve the intended function.
  • Another point that can be stated here is related to the nature of probability distribution function of produced components. For example, if the distribution function of components is normal or at least symmetrical, or if several samples can be picked up from the production line simultaneously, and their average quality characteristics can be considered as a random variable, fewer problems will arise. But if the distribution of quality characteristics is neither normal nor at least symmetrical, or we have individual measurement of variable characteristics, using a flexible distribution such as Weibull distribution, which can embrace several distributions including the normal distribution, would seem appropriate.


Main Subjects

Armillotta, A., (2016), Tolerance analysis considering form errors in planar datum features, Procedia CIRP 43, 64 – 69.
Asadi, M. (2013), Introduction to reliability theory, University Press Center,  Tehran. pp. 144-147.
Beaucaire, P., Gayton, N., Duc, E., and Dantan, J.,Y., (2013), Statistical tolerance analysis of a mechanism with gaps based on system reliability methods, 12th CIRP Conference on Computer Aided Tolerancing, Procedia CIRP 10, 2 – 8.
Chandra,  M. J. (2001), Statistical quality control. CRC Press LLC, 5–53.
Devor, R. E., Tsong-How Chang, and Sutherland, J. W.,  (2007), Statistical quality design and control. Pearson Prentice hall, Upper Saddle River, pp 366–404.
Fathi, P., Saeedi, M,. Hamidi, M. (2005), providing a model of tolerancing parts to reduce loss of quality of production costs, Fourth International Conference on Industrial Engineering, Tarbiat Modarres University.
George J. Kaisarlis, (2012), A Systematic Approach for Geometrical and Dimensional Tolerancing in Reverse Engineering, Reverse Engineering – Recent Advances and Applications.
Ginsberg, Robert H., (1981), Outline of tolerancing from performance specification to toleranced drawings, Hughes Aircraft Company, Optical Engineering 20(2), 175-180 March/April.
Heling, B., Aschenbrenner, A., Walter, M.S.J., and Wartzack, S., (2016), On Connected Tolerances in Statistical Tolerance-Cost-Optimization of Assemblies with Interrelated Dimension Chains, 14th CIRP Conference on Computer Aided Tolerancing (CAT), Procedia CIRP 43, 262 – 267.
Hoecke. A.V., (2016), Tool risk setting in statistical tolerancing and its management in verification, in order to optimize customer’s and supplier’s risks, 14th CIRP Conference on Computer Aided Tolerancing (CAT), Procedia CIRP 43, 250 – 255.
Huang, W., Ceglarex, D., and Zhou, Z., (2004) Tolerance Analysis for Design of Multistage Manufacturing Processes Using Number-Theoretical Net Method (NT-net), The International Journal of Flexible Manufacturing Systems, 16, 65–90.
JUDIC Jean-Marc, (2016), Process Tolerancing: a new approach to better integrate the truth of the processes in tolerance analysis and synthesis, 14th CIRP Conference on Computer Aided Tolerancing (CAT), Procedia CIRP 43, 244 – 249.
Mansuy, M., Giordano, M., and Hernandez, P., (2013), A generic method for the worst case and statistical tridimensional tolerancing analysis, 12th CIRP conference on computer aided tolerancing, Procedia CIRP 10, 276-282.
Macko, M, Ilic´ S, Jezdimirovic´ M., (2012), The influence of part dimensions and tolerance size to trigger characteristics. Strojnis ˇki vestnik J Mech Eng 58(6): 411–415.
Movahedi, M.M., Khounsiavash, M., Otadi, M., Mosleh M., (2016), A new statistical method for design and analyses of component tolerance, J Ind Eng Int.
Noorul Haq A., Sivakumar K., Saravanan R., Muthiah V., (2005) Tolerance design optimization of machine elements using genetic algorithm, Int J Adv Manuf Technol, 25: 385–391.
Rausand, M., and Hoyland, A., (2004), System Reliability Theory Models, Statistical Methods, and Applications, Second Edition, A Join Wiley & Sons, INC., publication.
Rout B.K., & Mittal R.K., (2007) Tolerance design of manipulator parameters using design of experiment approach, Struct Multidisc Optim, 34:445–462.
Sampath Kumar R., Alagumurthi, N., Ramesh, R., (2009) Optimization of design tolerance and asymmetric quality loss cost using pattern search algorithm. Int J Phys Sci 4(11):629–637.
Weibull, W., (1951), A statistical distribution function of wide applicability, Journal of applied mechanics, 18: 280-281.
Yan, H., Wu, X., and Yang, J., (2015), Application of Monte Carlo Method in Tolerance Analysis, 13th CIRP conference on Computer Aided Tolerancing, Procedia CIRP 27, 281 – 285.
Yan, H., Cao, Y., and Yang J., (2016), Statistical tolerance analysis based on good point set and homogeneous transform matrix, 14th CIRP Conference on Computer Aided Tolerancing (CAT), Procedia CIRP 43, 178 – 183.
Yang, J., Wang, J., Wu., Z., and Anwer, N., (2013), Statistical tolerancing based on variation of point-set, 12th CIRP Conference on Computer Aided Tolerancing, Procedia CIRP 10, 9 – 16.
Young Lee G., Shim J.S., Cho B., Joo B.C., Jung J.Y.,  Lee D.S., Oh H.H., (2011) Stochastic acquisition of a stem cell-like state and drug tolerance in leukemia cells stressed by radiation, Int J Hematol, 93:27–35.
Zhang, Y., and Lu, W., (2016), Development and Standardization of Quality-oriented Statistical Tolerancing in China, 14th CIRP Conference on Computer Aided Tolerancing (CAT), Procedia CIRP 43, 268 – 273.
Zhang, H.C., and Huq, M.E., (1992), Tolerancing technique: the state-of-the art, Int. J. Prod. Res., 30, 2111.