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Concurrent optimal allocation of geometric and process tolerances based on the present worth of quality loss using evolutionary optimisation techniques

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Abstract

Concurrent tolerancing becomes an optimisation problem to find out the optimum allocation of the process tolerances in the given design function constraints. In traditional optimisation methods, finding out the optimum solution for this advanced tolerance design problem is complex. The proposed algorithms (elitist non-dominated sorting genetic algorithm) and differential evolution extensively do better than the previous algorithms for attaining the optimum result. The aim of this paper is to suggest a model for optimal tolerance allocation by considering both tolerance cost and the present worth of quality loss such that the total manufacturing cost/loss is minimised. The suggested model takes into account the time value of money for quality loss and product degradation over time and consists of two new parameters: the planning horizon and the product user’s discount rate. From the outcome of this study, a longer planning horizon results in an increase in both tolerance cost and quality loss; however, a larger value of discount rate gives up a decrease in both tolerance cost and quality loss. Finally, a practical example is brought into reveal the effectiveness of the suggested method.

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References

  • Chou C-Y, Chang C-L (2001) Minimum-loss assembly tolerance allocation by considering product degradation and time value of money. Int J Adv Manuf Technol 17:139–146

    Article  Google Scholar 

  • Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6(2):182–197

    Article  Google Scholar 

  • Huang M, Zhong Y (2008) Dimensional and geometrical tolerance balancing in concurrent design. Int J Adv Manuf Technol 35(7–8):723–735

    Article  Google Scholar 

  • Huang MF, Zhong YR, Xu ZG (2005) Concurrent process tolerance design based on minimum product manufacturing cost and quality loss. Int J Adv Manuf Technol 25:714–722

    Article  Google Scholar 

  • Janakiraman V, Saravanan R (2010) Concurrent optimization of machining process parameters and tolerance allocation. Int J Adv Manuf Technol 51:357–369

    Article  Google Scholar 

  • Jeang A (2001) Combined parameter and tolerance design optimization with quality and cost. Int J Prod Res 39(5):923–952

    Article  MATH  Google Scholar 

  • Jeang A, Chung C-P, Hsieh C-K (2007) Simultaneous process mean and process tolerance determination with asymmetrical loss function. Int J Adv Manuf Technol 31:694–704

    Article  Google Scholar 

  • Peng HP (2012) Concurrent tolerancing for design and manufacturing based on the present worth of quality loss. Int J Adv Manuf Technol 59:929–937

    Article  Google Scholar 

  • Peng HP, Jiang XQ, Liu XJ (2008a) Concurrent optimal allocation of design and process tolerances for mechanical assemblies with interrelated dimension chains. Int J Prod Res 46(24):6963–6979

    Article  Google Scholar 

  • Peng HP, Jiang XQ, Xu ZG, Liu XJ (2008b) Optimal tolerance design for products with correlated characteristics by considering the present worth of quality loss. Int J Adv Manuf Technol 39:1–8

    Article  Google Scholar 

  • Price K, Storn R (1997) Differential evolution—a simple evolution strategy for fast optimization. Dr. Dobb’s J 22(4):18–24 & 78

    MATH  Google Scholar 

  • Taguchi G, Elsayed EA, Hsiang TC (1989) Quality engineering in production systems. McGraw-Hill, New York

    Google Scholar 

  • Tanchoco JMA, Buck JR, Leung LC (1981) Modeling and discounting of continuous cash flows under risk. Eng Costs Prod Econ 5:205–216

    Article  Google Scholar 

  • Teran A, Pratt DB, Case KE (1996) Present worth of external quality losses for symmetric nominal-is-better quality characteristics. Eng Econ 42(1):39–52

    Article  Google Scholar 

  • Ye B, Salustri FA (2003) Simultaneous tolerance synthesis for manufacturing and quality. Res Eng Des 14:98–106

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, College of Engineering, Anna University, Guindy Campus, Chennai, Tamil Nadu, 600025, India

    C. Balamurugan

  2. Department of Mechanical Engineering, Saranathan College of Engineering, Tiruchirappalli, Tamil Nadu, 620012, India

    A. Saravanan

  3. School of Mechanical Engineering, SASTRA University, Thanjavur, Tamil Nadu, 613401, India

    P. Dinesh Babu, S. Jagan & S. Ranga Narasimman

Authors
  1. C. Balamurugan
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  2. A. Saravanan
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  3. P. Dinesh Babu
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  4. S. Jagan
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  5. S. Ranga Narasimman
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Corresponding author

Correspondence to C. Balamurugan.

Appendix

Appendix

The pseudo-code of DE algorithm is presented below.

The following assumes that we are minimising all the objective functions, F c :

  1. 1.

    Generate box, P, of N p parent vectors using a random number code to generate several real variables. These vectors are given a sequence (position) number as generated.

  2. 2.

    Classify these vectors into fronts based on non-domination as follows:

    1. (a)

      Create new (empty) box, P′, of size, N p.

    2. (b)

      Transfer ith vector from P to P′, starting with i = 1.

    3. (c)

      Compare vector I with each member, say j, already present in P′, one at a time.

    4. (d)

      If i dominates over j (i.e. it is superior to or better than j in terms of all objective functions), remove the jth vector from P′ and put it back in its original location in P.

    5. (e)

      If i is dominated over by j, remove i from P′ and put it back in its position in P.

    6. (f)

      If i and j are non-dominating (i.e. there is at least one objective function associated with i that is superior to/better than that of j), keep both i and j in P′ (in sequence). Test for all j present in P′.

    7. (g)

      Repeat for next vector (in the sequence, without going back) in P till all N p are tested. P′ now contains a sub-box (of size ≤N p) of non-dominated vectors (a subset of P), referred to as the first front or sub-box. Assign it a rank number, I rank, of I.

    8. (h)

      Create subsequent fronts in (lower) sub-boxes of P′, using step 2b above (with the vectors remaining in P). Compare these members only with the members present in the current sub-box, and not with those in earlier (better) sub-boxes. Assign these I rank = 2, 3…. Finally, we have all N p vectors in P′, boxed into one or more fronts.

  3. 3.

    Spreading out: Evaluate the crowding distance, I i,dist, for the ith vector in any front, j, of P′ using the following procedure:

    1. (a)

      Rearrange all vectors in front j in ascending order of the values of any one (say, the qth) of their several objective functions (fitness functions). This provides a sequence, and, thus, defines the nearest neighbours of any vector in front j.

    2. (b)

      Find the largest cuboid (rectangle for two fitness functions) enclosing vector i that just touches its nearest neighbours in the f-space.

    3. (c)

      I i,dist = ½ × (sum of all sides of this cuboid).

    4. (d)

      Assign large values of I i,dist to solutions at the boundaries (the convergence characteristics would be influenced by this choice).

  4. 4.

    Perform DE operation over the NP target vectors in P′ to generate NP trial vectors and store it in P″.

    1. (a)

      Create new (empty) box, P″, of size, N p.

    2. (b)

      Select a target vector, i in P′, starting with i = 1.

    3. (c)

      Choose two vectors r1 and r2 at random from the NP vectors in P′ and find the weighted difference. This is carried out by the following steps: (1) generate two random numbers, (2) decide which two population members are to be selected, (3) find the vector difference between the two vectors. Multiply this difference with F to obtain the weighted difference.

    4. (d)

      Find the noisy random vector. This is done by (1) generate a random number, (2) choose a third random vector, r3, from the NP vectors in P′, (3) add this vector to the weighted difference to obtain the noisy random vector.

    5. (e)

      Perform crossover between the target vector and noisy random vector to find the trial vector and put it in box P″. This is carried out by (1) generate random numbers equal to the dimension of the problem, (2) for each of the dimensions: if random no. > CR, copy the value from the target vector, else copy the value from the noisy random vector into the trial vector and put it in box P″.

  5. 5.

    Elitism: copy all the N p parent vectors (P′) and all the N p trial vectors (P″) into box PT. Box PT has 2N p vectors

    1. (a)

      Reclassify these 2N p vectors into fronts (box PT′) using only non-domination (as described in step 2 above).

    2. (b)

      Take the best N p from box PT′ and put into box P‴. The following procedure is adopted to identify the better of the two chromosomes. Chromosome i is better than chromosome j if

$$\begin{aligned} I_{{i,{\text{rank}}}} \ne I_{{j,{\text{rank}}}} :I_{{i,{\text{rank}}}} < I_{{j,{\text{rank}}}} \hfill \\ I_{{i,{\text{rank}}}} = I_{{j,{\text{rank}}}} :I_{{i,{\text{dist}}}} > I_{{j,{\text{dist}}}} \hfill \\ \end{aligned}$$

This completes one generation. Stop if appropriate criteria are met, e.g. the generation number > maximum number of generations (user specified). Else, copy P′′′ into starting box P. Go to step 2 above.

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Balamurugan, C., Saravanan, A., Dinesh Babu, P. et al. Concurrent optimal allocation of geometric and process tolerances based on the present worth of quality loss using evolutionary optimisation techniques. Res Eng Design 28, 185–202 (2017). https://doi.org/10.1007/s00163-016-0230-7

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  • DOI: https://doi.org/10.1007/s00163-016-0230-7

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