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A genetic algorithm approach for simultaneous tolerance synthesis for manufacturing and quality with different stack-up conditions.

Introduction

Tolerance design is one of the important aspects of mechanical design, because it affects both the functional performance and manufacturing cost of a product. Dimensional tolerances are of two types: design tolerances and manufacturing tolerances. Design tolerances are related to the functional requirements of a mechanical assembly or of a component whereas the manufacturing tolerances are related to the process plan of manufacturing a part. Unnecessarily tight tolerances lead to higher manufacturing cost while loose tolerances may lead to malfunctioning of the product. Tolerance is a bridge between design, manufacturing, and quality engineers, and as such it plays a key role in concurrent engineering. Ideally, one can imagine that the best technique for tolerance synthesis takes into account the coupling between design, manufacturing, and quality, for the sake of achieving a minimal total cost and reducing lead-time.

Conventionally, tolerance synthesis is carried out in two stages: design and process planning. However, design engineers allocating design tolerances are often unaware of manufacturing processes and their production capabilities. This may be due to either a lack of communication between design engineers and process engineers, or a lack of knowledge of the manufacturing processes by the design engineers. The resulting process plans often cannot be executed effectively, or can only be executed at undesirably high manufacturing cost. When this happens, process engineers must modify the design tolerances. Furthermore, manufacturing engineers who allocate processes must typically work within the tolerance limits set by design engineers; to do otherwise requires cycling through the design/process-planning loop and results in longer lead times. But in accepting tolerances set by design engineers, the process planners also limit the range within which they can set process tolerances. This, in turn, leads to tight process tolerances and higher manufacturing cost. One well-used method for measuring quality is quality loss as introduced by Taguchi [16]. He proposed that performance degradation can be measured as a deviation from some target value, and asserted that the degradation can be related to a loss in value to the consumer called a quality loss. Taguchi emphasized that the level of a product's quality is not the same as the number of defective products; rather, it refers to the magnitude of societal losses. Even if a product is well within its specifications, it has a quality loss if its quality characteristic value is not at the ideal performance target. This loss is defined in monetary terms so that it can be compared to the product's manufacturing cost. Tight tolerances are preferred to ensure product performance, which degrades as parts deviate from nominal values. However, tight tolerances imply higher manufacturing cost, so loose tolerances are preferred from a manufacturing perspective. This conflicting relation between the effect of tolerances on quality loss and on manufacturing cost make it very difficult to establish near-optimal tolerance specifications.

Formulations of tolerance assignment as a non-linear optimization problem were first introduced by Speckhart (1972), Spotts (1973) and Sutherland and Roth (1975). Based upon the general characteristics of a production cost-tolerance data curve, they proposed an exponential cost model, reciprocal squared cost model, reciprocal power model respectively. These researches focused on the mathematical modeling of cost-tolerance relations and optimization of related tolerances, considering discrete cost functions for minimum production costs. The above works pioneered the area of tolerance synthesis (Dong et al. 1994).

At the manufacturing stage tolerances are simply allocated to working dimensions depending upon the process capability of the manufacturing process, stock-removal for each successive operation and the design tolerance level of the nominal dimension. Ostwald and Huang (1977) first formulated a technique for optimal tolerance allocation choosing one of many possible process alternatives. They used linear integer programming, with cost as the objective function and design requirements as constraints. This technique is suitable where sequences and tolerances of operations are fixed. Lee and Woo (1989) presented a different approach by considering tolerance as process specific and optimized by introducing a discrete cost-tolerance model, using a reliability index and integer programming. This method directly utilizes empirical data, thus having no curve fitting error. Chase et al. (1990) presented three methods--exhaustive search, univariate search, and sequential quadratic programming--to solve the models originally proposed by Huang (1977). Nagarwala et al. (1994) proposed a new slope-based method that took into account process selection. This method eliminates component-wise process selection, hence eliminating the generation of process combinations and improving efficiency. All these models assume each component dimension is produced by only one process (discrete cost function). The tolerance obtained from the process has a single fixed value.

However, due to the simple forms of these models (discrete), relatively large model fitting errors were introduced by Wu et al. (1988). Michael and Siddall (1981) solved optimal design problems with both design parameters and their tolerances as design variables, and introduced the Powers and Exponential Hybrid Model, considering continuous cost function. Chase and Greenwood (1988) introduced the Reciprocal Model with better empirical data fitting capability, considering both the continuous and discrete cost functions. Zhang and Wang (1993), proposed mathematical models with the consideration of manufacturing process for continuous cost function and solved it first time by using non-traditional optimization technique called simulated annealing. Zhang (1996) approached the problem in a totally different manner by introducing a new concept of interim tolerances, which help to determine appropriate manufacturing process and solved the problem using a non-linear programming technique (mixed penalty function approach). Iannuzzi and Sandgren (1995) presented a computational design methodology using a non-traditional genetic optimization method coupled with a Monte Carlo-based tolerance analysis to determine the maximum tolerance zone value for each nominal feature while simultaneously meeting all critical dimensional and functional constraints imposed upon the design. Al-Ansary and Deiab (1997) adopted a model similar to that proposed by Zhang and Wang (1993) and solved the model by considering worst-case stack-up criteria using a genetic algorithm. Singh et al. (2003) solved the same model for four different stack-up conditions using a genetic algorithm.

When quality is important for tolerance design, the quality loss is measured as the loss to society that occurs when a product deviates from the optimum set of design parameters. Taguchi (1989) proposed a quality loss function which estimates the cost of quality value versus target value and the variability of the product characteristic in terms of the monetary loss due to product failure in the eyes of consumer. Kapur et al. (1990) proposed a general optimization model in terms of costs associated with variances of the components and losses associated with the variability from the quality characteristic target. Vasseur et al. (1993) proposed a model that allocates tolerances based on profit maximization. The quality loss function is used to determine the reduction in value due to an off target product, which is then balanced against reductions in manufacturing cost. Soderberg (1993) developed a quality loss function based on component lifetime which represents the customer's objective. The function is developed from physical relations between critical dimensions and lifetime.

Clearly, the quality loss function is a way to show the economic value of reducing variability and staying closer to the target value. On the other hand, the manufacturing cost for a product usually increases as the tolerance of the quality characteristic close to the ideal value is reduced. Therefore, a trade-off analysis between manufacturing cost and quality loss should be considered in tolerance design for product improvement and cost reduction Jeang (1995, 1997).

Feng and Kusiak (1997) considered quality loss in addition to manufacturing cost in a discrete cost function. Jeang (1995, 1997) developed a few general mathematical models to determine product tolerances minimizing the combined manufacturing costs and quality losses without considering the manufacturing processes, in a continuous cost function using quadratic and geometrical decay functions. Ye and Salustri (2003) developed a simultaneous tolerance synthesis (STS) model with quality loss considering continuous cost function.

Purpose of this work is to extend the STS optimization method to minimize manufacturing cost with quality loss function in a continuous cost function to achieve near optimal tolerance allocation using total cost minimization as the criteria. The nonlinear multivariable optimization problem formulated in this manner can be solved effectively with the help of global optimization techniques. The solution methodology for the optimization of the above problem using Genetic algorithm a strategic design algorithm is shown. A case study involving piston cylinder assembly for concurrent allocation of design and machining tolerances with quality loss function is presented to demonstrate the effectiveness of the method with four different stack up conditions like WCS, RSS, Spott's and EMS.

The STS Method

Both the traditional tolerance synthesis methods and methods based on quality loss tend to allocate tight tolerances, leading to higher manufacturing costs as a result of a lack of concurrency or a lack of consideration of both manufacturing processes and process tolerance allocation. Some methods (Zhang and Wang (1993)) allow for loosening of process tolerances, but with the result of a relatively uncontrolled quality loss. The STS method simultaneously allocate tolerances to balance manufacturing cost and quality loss and thus optimize cost over the product's life.

Tolerance synthesis can be formulated as an optimization problem. In order to implement this method, an optimization model must be developed. In the model, the objective function is chosen as a combination of manufacturing cost and quality loss. By combining the two measures, the method seeks to balance them and achieve an overall minimum total cost over the product's lifetime. Design tolerance and process tolerance are taken as the decision variables in the model; manufacturing cost is a function of process tolerances, and quality loss is a function of design tolerances. For simultaneous selection of design and manufacturing tolerances, an optimization problem based on minimization of assembly manufacturing cost is formulated as follows.

Objective function

The objective function minimizes the total cost of the simultaneous manufacturing cost and quality loss. The weights W1 and W2 are introduced to represent the relative importance of the two components of the objective function.

[C.sub.T] = [n.summation over(i=1)] [pi.summation over (j=1)] [W.sub.1]C(tpij) + [W.sub.2]K[[sigma].sup.2.sub.y], (1)

The total manufacturing cost is then the sum of the manufacturing costs of each machining process of each component's dimension

[C.sub.m]([tp.sub.ij]) = [n.summation over (i=1)] [[p.sub.i].summation over (j=1)] C([tp.sub.ij]), (2)

where n is the number of dimensions in the dimension chain, pi is the number of processes to produce dimension i, and C(tpij) is the cost-tolerance function of the machining process. In this work, an exponential function is used to model cost-process tolerances. For a particular manufacturing process, we have

[C.sub.m]([tp.sub.ij]) = [A.sub.ij][e.sup.-[B.sub.ij]([tp.sub.ij]-[C.sub.ij]) + [D.sub.ij], (3)

Quality loss is quantified, per Taguchi, as a quadratic expression relating the loss to the variation of a product characteristic

L(Y) = k[(y-m).sup.2], (4)

where k=A/[T.sub.f], A is the cost of replacement or repair if the dimension does not meet the tolerance requirements, [T.sub.f] is the functional tolerance requirement, m is the target value of the functional dimension, and y is the design characteristic.

We assume the functional dimension has a normal distribution and a mean at the target value. The quality loss can then be represented by the standard deviation of the functional dimension. Then, the expected value of the loss function can be written as

QL = E(L(Y)) = k([([mu]-m).sup.2] + [[sigma].sub.y.sup.2]), (5)

where [mu] and [[sigma].sup.2.sub.y.] are the mean and the variance of Y, respectively. The equation combines linearly the variance of Y and distance of the mean of Y, that is, [mu], from the target value m. To lower the quality loss (and hence its associated cost), a quality engineer adjusts [mu] during parameter design. These adjustments do not affect the value of process variability [[sigma].sub.y]. From this point of view, then, the quality loss can be written

QL = E(L(Y))= k[[sigma].sub.y.sup.2], (6)

Based on the design function, the resultant overall quality characteristic can be estimated from the set of individual quality characteristics in the design function. These approximation functions can be found by using Taylor series expansions. The resultant variance can then be expressed in terms of the variances [[sigma].sup.2.sub.x]i, of the individual quality characteristics

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

Tolerances are always related to manufacturing processes, and they must be designed in conjunction with the application of a specific manufacturing process. If a tolerance is determined without considering a specific process, one risks creating a mismatch between the required tolerance and the capability of a given process. One way to express this relationship is with a process capability index Cp, which is the ratio of design tolerance boundaries to the measured variability of the output response of the manufacturing process. The process capability index Cp is a measure of the ability of a process to manufacture a product that meets its specification, and is defined by

Cp = [t.sub.d]/3[sigma], (8)

we can now write

[[sigma].sub.xi] = [t.sub.id]/[3C.sub.pi], (9)

Substituting into above (Eq.7) gives

[[sigma].sup.2.sub.y] = [n.summation over (i=1)] [([partial derivative]f/[partial derivative][x.sub.i]).sup.2] [([t.sub.id]/[3C.sub.p]).sup.2], (10)

and the total quality loss is

QL([t.sub.d]) = k[[sigma].sup.2.sub.y], (11)

Constraints

The aforementioned optimization problem is subjected to the constraints related to both the design and manufacturing tolerances.

Design Tolerance Constraints

The design tolerances on different dimensions are based on the functionality considerations of the assembly. The tolerance on the assembly dimension(s) usually depends on other related dimensions that constitute the dimensional chain(s). A design tolerance constraint is formulated to ensure that the accumulated tolerance in the dimensional chain does not exceed the specified tolerance on the assembly dimension. Accumulated tolerance can be estimated by using different approaches, which are applicable under different conditions. Apart from the commonly used worst case and RSS stack up approaches, a few non-traditional tolerance accumulation formulae have also been used. These formulae lead to a set of design tolerance constraints for different assembly dimensions.

[n.summation over (i=1)] [t.sub.id] [less than or equal to] [T.sub.f], worst case criteria (12)

[square root of ([n.summation over (i=1)] [t.sup.2.sub.id])] [less than or equal to] [T.sub.f], RSS criteria (13)

1/2 [[n.summation over (i=1)] [t.sub.id] + [square root of ([n.summation over (i=1)] [t.sup.2.sub.id])]] [less than or equal to] [T.sub.f], Spotts' criteria (14)

[n.summation over (i=1)] [m.sub.i][t.sub.id] + Z/3 [square root of ([n.summation over (i=1)] [(1-[m.sub.i]).sup.2] [t.sup.2.sub.id]]) [less than or equal to] [T.sub.f], Estimated Mean Shift criteria (15)

where [t.sub.id] are the design tolerances on the constituent dimensions and n is the number of constituent dimensions associated with the dimensional chain, mi are the mean shift factors of process distribution, and [T.sub.f] is the permissible variation in the assembly dimension. Z = 3.00 corresponds to 99.73% yield; this value is most commonly considered in analytical treatment [Singh et al. (2003)].

Manufacturing Tolerance Constraints

In allocating machining tolerances, consideration should be given not only to process capability, but also to the amount of machining allowance for each operation. The machining allowance is the size of the layer of material that is to be removed from the surface of a work piece in order to obtain the required accuracy and surface quality. It influences greatly the quality and production efficiency of the machined part. Excessive machining allowance increases the consumption of material, machining time, tools, and power, thereby increasing manufacturing cost. On the other hand, insufficient machining allowance fails to remove any roughness or surface defects of a previous operation, thus lowering part quality. The amount of machining allowance is the difference between the machining dimension obtained from the preceding operation and that in the current operation. Because of operation errors, the actual amount of material removed varies within some range; this variation is a cumulative sum of manufacturing tolerances. In practice, typical levels of material removal are set on a per-process basis and are defined in various handbooks.

The manufacturing tolerance constraints can be formulated as follows:

[t.sub.ij] + [t.sub.i(j-1)] [less than or equal to] [Tp.sub.ij], (16)

where [t.sub.ij] and [t.sub.i(j-1)] are the manufacturing tolerance obtainable in the jth and j-1th operations respectively, in production of the ith dimension. [Tp.sub.ij] is the difference between the nominal and the minimum stock removal/deformation allowance for manufacturing operation j.

Process Tolerance Constraint.

Each process operation has its own accuracy (again, usually available from reference handbooks) and must be performed within its process capability. Thus

[tp.sup.min.sub.ij] [less than or equal to] [tp.sub.ij] [less than or equal to] [tp.sup.max.sub.ij], (17)

In the STS model, manufacturing cost is a function of process tolerances, while quality loss is a function of design tolerances, acting in combination. Since the intermediate process tolerances are not final tolerances on a manufactured dimension, they affect neither functional performance nor quality, so no quality loss is associated with them. It is instead the tolerances of last processes (i.e. design tolerances) that constitute the final tolerances for a manufactured dimension. Quality loss is associated with design tolerances, which are the tolerances of the last processes. That the last process tolerances equal the design tolerances links the manufacturing cost and the quality loss; that is,

[tp.sub.ipi] = [t.sub.id], (18)

GA-over view

GAs are a very simple, straight forward, yet a powerful approach for global search and optimization of multimodal functions. The approach combines the characteristics of direct search methods and probabilistic selection and is based on the mechanics of natural genetics and natural selection. It makes use of the robustness, efficiency and flexibility of biological systems into artificial ones. Different authors have explained useful schemes of the application of this approach (Goldberg1989, Michalewicz 1996, Deb 1999, 2001). The working principle of GAs is very different from that of the most of the classical optimization techniques.

The distinguishing characteristics of the GAs are as follows.

* Work with a coding of the parameter set, not the parameters themselves.

* Search from a population of points, not a single point.

* Use payoff. (Objective function) information, not derivative or other auxiliary knowledge.

* Use probabilistic transition rules, not deterministic rules.

The general steps to be followed for implementing simple GAs for optimization problem are listed below.

1. The algorithm begins by creating a random initial population.

2. The algorithm then creates a sequence of new populations. At each step, the algorithm uses the individuals in the current generation to create the next population. To create the new population, the algorithm performs the following steps:

a. Scores each member of the current population by computing its fitness value.

b. Scales the raw fitness scores to convert them into a more usable range of values.

c. Selects members, called parents, based on their fitness.

d. Some of the individuals in the current population that have lower fitness are chosen as elite. These elite individuals are passed to the next population.

e. Produces children from the parents. Children are produced either by making random changes to a single parent--mutation--or by combining the vector entries of a pair of parents--crossover.

f. Replaces the current population with the children to form the next generation.

3. The algorithm stops when one of the stopping criteria is met.

A scheme for the application of a GA for the solution of an optimization problem is presented in figure1.

[FIGURE 1 OMITTED]

Case study

An example case study of a simple linear mechanical assembly as shown in fig. 2. is taken to explain the proposed methodology.

[FIGURE 2 OMITTED]

Problem description

For simultaneous determination of optimal design and manufacturing tolerances for the taken cylinder-piston assembly (Zhang and Wang 1993, Al-Ansary and Deiab 1997), the details are as follows:

* Dimensions

* Piston diameter: 50.8 mm.

* Cylinder bore diameter: 50.856 mm.

* Clearance (assembly dimension): 0.056 [+ or -] 0.0005 mm.

* Machining process plan for the piston: rough turning, finish turning, rough grinding, and finish grinding.

* Machining process plan for the cylinder bore: drilling, boring, finish boring, and grinding.

The ranges (mm) of the principal machining tolerances for the piston and cylinder bore have been taken from the aforementioned references and are shown in table 1.

In the present case study there is only one assembly dimension, constituted by only two component dimensions, namely the piston and cylinder (i.e. n = 2). Thus, there are only two design tolerance parameters, one for the piston diameter, the other for the cylinder bore diameter. Also, there are four machining tolerance parameters for machining each of the piston diameter and the cylinder bore diameter corresponding to the given process plans (table 1). Thus, in all there are 10 tolerances or design variables for the assembly. The design tolerance parameters are: tid, i = 1 for the piston and 2 for the cylinder bore. The machining tolerance parameters are: tij, i = 1 for the piston and 2 for the cylinder bore, j = 1 to 4 for the four machining processes for piston, and also the same for the four machining processes for the cylinder bore. The design tolerance for a given feature of a component is equal to the final machining tolerance for that feature. This gives t1d = t14 for the piston and t2d = t24 for the cylinder. The quality loss coefficient A is set at $100 (from Taguchi (1989)).

In the piston-cylinder bore assembly, there is only one resultant dimension (the clearance between the two parts) and two dimensions that form the chain (the diameters of the piston and cylinder bore). So, the design function is

X = [delta]c - [delta]p

Formulation of Optimization Problem

Objective function

An objective function based on the minimization of simultaneous manufacturing cost and quality loss is formulated. The manufacturing cost of an individual process is represented as a monotonically decreasing mathematical relationship between tolerance and the associated manufacturing cost. Several formulations for modeling the cost-tolerance relationship have been evaluated and/or reviewed by different authors (Wu et al. 1988, Singh et al. 2002b). Although non-traditional cost functions model the characteristics of the manufacturing processes more accurately (Dong et al. 1994), for a balance between modeling accuracy and computational simplicity, the exponential cost function is considered the best. In this example, a modified form of the exponential cost function (Zhang et al. 1992, Zhang and Wang 1993, Al-Ansary and Deiab 1997), as expressed in equation (3), has been used. This cost function offers easier manipulation with regard to location of the curve.

[C.sub.m]([tp.sub.ij]) = [A.sub.ij][e.sup.-[B.sub.ij]([tp.sub.ij]-[C.sub.ij]) + [D.sub.ij], (19)

where the constant parameters are determined from the test data. The parameters A, C and D control the position, while B governs the curvature of the cost function.

The total quality loss is

QL([t.sub.d]) = k[[sigma].sup.2.sub.y], (20)

where k=A/Tf, A is the cost of replacement or repair if the dimension does not meet the tolerance requirements, Tf is the functional tolerance requirement and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

Thus, the total cost of the assembly can be expressed as:

Min. [C.sub.T] = [n.summation over (i=1)] [pi.summation over (j=1)] [W.sub.1]C(tpij) + [W.sub.2]K[[sigma].sup.2.sub.y], (22)

Values of the constant parameters for the cost functions of different manufacturing processes are given in table 2 (Al-Ansary and Deiab 1997).

Since we are only interested in determining the near optimal tolerances for these parts with respect to the clearance between them, for representing four possible cases:

W1= W2= 1; Cp1= Cp2= 1;

W1= W2= 2; Cp1= Cp2= 1;

W1= W2= 1; Cp1= Cp2= 0.5;

W1= W2= 1; Cp1= Cp2= 1.5:

In the first case, manufacturing cost and quality loss have the same weight that is, they are equally important and Cp=1 in accordance with typical North American practice for quality standards. In the second case, quality loss is considered twice as important as manufacturing cost. In the third case, quality is set at a low level. Finally, in the fourth case, quality is set at a high level. Our use of weights Wi allows us to separate determining the relative importance of quality loss and manufacturing cost on the one hand from modeling the quality loss and manufacturing cost themselves. The weights may be perform worse than the competing models; in many cases, STS far exceeded the competing models.

Constraints

The constraints on the design tolerances are formulated based on the four stackup conditions, namely worst case, RSS, Spotts' modified method and estimated mean shift criteria. These stack-up conditions yield a set of design constraints as follows:

[t.sub.14] + [t.sub.24] [less than or equal to] 0.001 worst case criteria, (23)

[t.sup.2.sub.14] + [t.sup.2.sub.24] [less than or equal to] [(0.001).sup.2] RSS criteria, (24)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

The sum of the machining tolerance for a process and that for its preceding process should be less than or equal to the difference of the nominal and minimum machining allowances for the process. The nominal and minimum machining allowances are normally listed in the machining manuals or handbooks. This yields the following constraints:

For piston: [t.sub.11] + [t.sub.12] [less than or equal to] 0.02, [t.sub.12] + [t.sub.13] [less than or equal to] 0.005, [t.sub.13] + [t.sub.14] [less than or equal to] 0.0018, (27)

For cylinder bore: [t.sub.21] + [t.sub.22] [less than or equal to] 0.02, [t.sub.22] + [t.sub.23] [less than or equal to] 0.005, [t.sub.23] + [t.sub.24] [less than or equal to] 0.0018 (28)

Each process operation has its own accuracy and must be performed within its process capability. Thus

For piston

0.005 [less than or equal to] t11 [less than or equal to] 0.02, 0.002 [less than or equal to] t12 [less than or equal to] 0.012, 0.0005 [less than or equal to] t13 [less than or equal to] 0.003, 0.0002 [less than or equal to] t14 [less than or equal to] 0.001 (29)

For Cylinder bore

0.005 [less than or equal to] [t.sub.21] [less than or equal to] 0.02, 0.0002 [less than or equal to] [t.sub.24] [less than or equal to] 0.001, 0.0005 [less than or equal to] [t.sub.23] [less than or equal to] 0.003, 0.002 [less than or equal to] [t.sub.22] [less than or equal to] 0.012 (30)

The last process tolerances equal the design tolerances links the manufacturing cost and the quality loss; that is,

[t.sub.1d] = [t.sub.14], [t.sub.2d] = [t.sub.24], (31)

Optimization

Finally the total cost of the assembly as represented by equation (22) is optimized subject to the constraints and ranges of the tolerances (process limits). The scheme of GAs as explained above is used as the optimization strategy. The simulation for each case of the design constraint was carried out on P-IV personal computer using MATLAB. The results are shown in tables 3-18.

Results and discussion

In case study the manufacturing cost and quality loss have been given with four possible cases for quality standards. The manufacturing cost and quality loss cost is solved for stack-up conditions namely WCS,RSS,SPOTT'S and EMS.

The resulting value of the manufacturing cost and quality loss cost throw some light regarding the relationship between manufacturing cost and Taguchi quality loss.

The variation of the minimum of the assembly manufacturing cost from amongst the feasible population in a given number of generations has been plotted for all four stack-up conditions of first case in figure 3. The plot indicates that the rate of reduction in the cost is fast up to first 100 generations, and slows down beyond. The reduction is nominal beyond 200 generations, and hence the curves become almost flat. This flat portion is significant from the optimal design point of view.

[FIGURE 3 OMITTED]

The flat portions of the curves indicate that the minimum total cost of the assembly is lowest with RSS criteria and highest with the worst case criteria. The minimum total costs for the other two stack-up conditions lie in between the aforementioned two extreme cases. Up to 100 generations, the minimum costs obtained with different criteria are 72.23662(RSS), 72.23749 (Spotts' criteria), 72.23798(Greenwood and Chase's unified criteria with mean shift factors 0.25 each) and 72.30495 (worst case) units. The relative location of curve corresponding to the Greenwood and Chase's unified criteria is governed by the values of the mean shift factor; the other curves are fixed. The tolerances associated with the corresponding generations are considered as the solution of the optimal tolerance design problem. These results are shown in tables 3-18.

Conclusion

The method of synthesizing tolerances simultaneously for both manufacturing cost and quality has been presented. The best results were attained in cases using GA where quality was either a high concern or was deliberately set at low levels. The approach of the GAs has been briefly explained and proposed for global optimization.

The methodology was demonstrated with the help of a simple linear assembly considering different tolerance stack-up conditions. Apart from the traditional worst case and RSS criteria, two non-traditional approaches, Spotts' modified and Greenwood and Chase's estimated mean shift stack-up conditions, were explored for allocation.

The method is then well suited to engineering environments where either high quality or low-cost products are designed and manufactured. Criteria for evaluating STS with respect to other models involve trading off near-term manufacturing costs against the losses in quality that adversely affect the long-term operational life of the product. Arguably, this criterion is more realistic than others focusing only on either manufacturing costs or quality losses. As such, the STS method is inherently suited for use in concurrent engineering environments. Weights built into the method allow product- and enterprise-specific factors to be taken into account. The STS method is comparatively simple. It eliminates the need for various intermediate results (e.g. cost-design tolerance functions), thus improving computability and making the model easier to understand by design and manufacturing engineers.

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[18] Ye, B. and Salustri, F.A., 2003, "Simultaneous tolerance synthesis for manufacturing and quality." Res. Eng. Des., 14(2), 98-106.

[19] Zhang, C. and Wang, H.P., 1993, "Integrated tolerance optimization with simulated annealing." Int. J. Adv. Manuf. Tech., 8, 167-174.

[20] Zhang, G., 1996, "Simultaneous tolerancing for design and manufacturing." Int. J. Prod. Res., 34, 3361-3382.

Y.S. Rao (1) and C.S.P. Rao (2)

(1) Associate Professor, Department of Mechanical Engineering QIS College of Engg. & Technology, Ongole-523 001(AP) Email: raoseshuy2@yahoo.co.in

(2) Professor, Department of Mechanical Engineering National Institute of Technology, Warangal-506 004(AP) Email: csp_rao@rediffmail.com
Table 1: Ranges of the principal machining tolerances for the
Piston and cylinder bore diameter

   Piston diameter (mm)         Cylinder bore diameter (mm)

Notation     Lower     Upper    Notation     Lower     Upper
             limit     limit                 limit     limit

   t11      0.005      0.02        t21      0.007      0.02
   t12      0.002      0.012       t22      0.003      0.012
   t13      0.0005     0.003       t23      0.0006     0.005
   t14      0.0002     0.001       t24      0.0003     0.005

Table 2: Values of the constant parameters for the cost functions
of different manufacturing processes

        C11(t11)           C12(t12)           C13(t13)
A          5                  9                  13
B         309                790                3196
C    5x[10.sup.-3]     2.04x[10.sup.-3]   5.3x[10.sup.-4]
D         1.51               4.36               7.48

        C14(t14)           C21(t21)           C22(t22)
A          18                 4                  8
B         8353               299                986
C   2.19x[10.sup.-4]   7.02x[10.sup.-3]   2.97x[10.sup.-3]
D        11.99               2.35               5.29

        C23(t23)           C24(t24)
A          10                 2
B         3206               9428
C    6x[10.sup.-4]      6x[10.sup.-4]
D         9.67              13.12

Table 3: Near-optimal tolerances & costs for example, with W1=W2=1 and
Cp1=Cp2=1 (based on the RSS criteria).

       Piston tolerances                    Cylinder bore
                                             tolerances

[t.sub.11]              0.01629   [t.sub.21]              0.01627
[t.sub.12]              0.00043   [t.sub.22]              0.00373
[t.sub.13]              0.00129   [t.sub.23]              0.00127
[t.sub.14] = [t.sub.1d] 0.00051   [t.sub.24] = [t.sub.2d] 0.00371

   Minimum      Minimum     total
manufacturing   quality     cost
    cost        loss cost

67.2918         4.9444      72.23662

Table 4: Near-optimal tolerances & costs for example, with W1=W2=2 and
Cp1=Cp2=1 (based on the RSS criteria).

       Piston tolerances                    Cylinder bore
                                             tolerances

[t.sub.11]              0.0163    [t.sub.21]              0.01628
[t.sub.12]              0.0037    [t.sub.22]              0.00372
[t.sub.13]              0.0013    [t.sub.23]              0.00128
[t.sub.14] = [t.sub.1d] 0.0005    [t.sub.24] = [t.sub.2d] 0.00037

   Minimum      Minimum     total
manufacturing   quality     cost
    cost        loss cost

98.9468         8.5978      107.54945

Table 5: Near-optimal tolerances & costs for example, with W1=W2=1 and
Cp1=Cp2=1.5 (based on the RSS criteria).

       Piston tolerances                    Cylinder bore
                                             tolerances

[t.sub.11]              0.01641   [t.sub.21]              0.01628
[t.sub.12]              0.00359   [t.sub.22]              0.00372
[t.sub.13]              0.00141   [t.sub.23]              0.00128
[t.sub.14] = [t.sub.1d] 0.00039   [t.sub.24] = [t.sub.2d] 0.00032

   Minimum      Minimum     total
manufacturing   quality     cost
    cost        loss cost

71.7739         11.3111     83.08277

Table 6: Near-optimal tolerances & costs for example, with W1=W2=1 and
Cp1=Cp2=0.5 (based on the RSS criteria).

       Piston tolerances                    Cylinder bore
                                             tolerances

[t.sub.11]              0.01622   [t.sub.21]              0.01626
[t.sub.12]              0.00377   [t.sub.22]              0.00374
[t.sub.13]              0.00123   [t.sub.23]              0.00126
[t.sub.14] = [t.sub.1d] 0.00057   [t.sub.24] = [t.sub.2d] 0.0005

   Minimum      Minimum     total
manufacturing   quality     cost
    cost        loss cost

66.3039         2.8390      69.14427

Table 7: Near-optimal tolerances & costs for example, with W1=W2=1 and
Cp1=Cp2=1 (based on the WC criteria).

        Piston tolerances                   Cylinder bore
                                             tolerances

[t.sub.11]              0.01624     [t.sub.21]              0.01628
[t.sub.12]              0.00371     [t.sub.22]              0.0037
[t.sub.13]              0.00128     [t.sub.23]              0.00129
[t.sub.14] = [t.sub.1d] 0.0005      [t.sub.24] = [t.sub.2d] 0.00044

   Minimum      Minimum     total
manufacturing   quality     cost
    cost        loss cost

67.4166         4.9289      72.30495

Table 8: Near-optimal tolerances & costs for example, with W1=W2=2 and
Cp1=Cp2=1 (based on the WC criteria).

        Piston tolerances                   Cylinder bore
                                             tolerances

[t.sub.11]              0.01607     [t.sub.21]              0.01658
[t.sub.12]              0.00364     [t.sub.22]              0.00339
[t.sub.13]              0.00125     [t.sub.23]              0.00136
[t.sub.14] = [t.sub.1d] 0.00052     [t.sub.24] = [t.sub.2d] 0.00037

   Minimum      Minimum     total
manufacturing   quality     cost
    cost        loss cost

100.2508        9.0511      109.27202

Table 9: Near-optimal tolerances & costs for example, with W1=W2=1 and
Cp1=Cp2=1.5 (based on the WC criteria).

        Piston tolerances                   Cylinder bore
                                             tolerances

[t.sub.11]              0.01634     [t.sub.21]              0.01624
[t.sub.12]              0.00363     [t.sub.22]              0.00376
[t.sub.13]              0.00136     [t.sub.23]              0.00124
[t.sub.14] = [t.sub.1d] 0.0004      [t.sub.24] = [t.sub.2d] 0.00031

   Minimum      Minimum     total
manufacturing   quality     cost
    cost        loss cost

71.7827         11.3822     83.16795

Table 10: Near-optimal tolerances & costs for example, with W1=W2=1 and
Cp1=Cp2=0.5 (based on the WC criteria).

        Piston tolerances                   Cylinder bore
                                             tolerances

[t.sub.11]              0.01624     [t.sub.21]              0.01629
[t.sub.12]              0.00371     [t.sub.22]              0.00364
[t.sub.13]              0.00125     [t.sub.23]              0.00136
[t.sub.14] = [t.sub.1d] 0.00055     [t.sub.24] = [t.sub.2d] 0.00044

   Minimum      Minimum     total
manufacturing   quality     cost
    cost        loss cost

66.9645         2.4499      69.42320

Table 11: Near-optimal tolerances & costs for example, with W1=W2=1 and
Cp1=Cp2=1 (based on the SPOTT'S criteria).

         Piston tolerances                     Cylinder bore
                                                 tolerances

[t.sub.11]               0.01629     [t.sub.21]               0.01627
[t.sub.12]               0.00371     [t.sub.22]               0.00373
[t.sub.13]               0.00129     [t.sub.23]               0.00127
[t.sub.14] = [t.sub.1d]  0.00051     [t.sub.24] = [t.sub.2d]  0.00043

   Minimum      Minimum     total
manufacturing   quality     cost
    cost        loss cost

67.2918         4.9444      72.23749

Table 12: Near-optimal tolerances & costs for example, with W1=W2=2 and
Cp1=Cp2=1 (based on the SPOTT'S criteria).

         Piston tolerances                     Cylinder bore
                                                 tolerances

[t.sub.11]               0.0163      [t.sub.21]               0.01627
[t.sub.12]               0.0037      [t.sub.22]               0.00373
[t.sub.13]               0.0013      [t.sub.23]               0.00127
[t.sub.14] = [t.sub.1d]  0.0005      [t.sub.24] = [t.sub.2d]  0.00037

   Minimum      Minimum     total
manufacturing   quality     cost
    cost        loss cost

98.9469         8.5978      107.53211

Table 13: Near-optimal tolerances & costs for example, with W1=W2=1 and
Cp1=Cp2=1.5 (based on the SPOTT'S criteria).

         Piston tolerances                     Cylinder bore
                                                 tolerances

[t.sub.11]               0.01641     [t.sub.21]               0.01627
[t.sub.12]               0.00359     [t.sub.22]               0.00373
[t.sub.13]               0.00141     [t.sub.23]               0.00127
[t.sub.14] = [t.sub.1d]  0.00039     [t.sub.24] = [t.sub.2d]  0.00032

   Minimum      Minimum     total
manufacturing   quality     cost
    cost        loss cost

71.7740         11.3111     83.08379

Table 14: Near-optimal tolerances & costs for example, with W1=W2=1 and
Cp1=Cp2=0.5 (based on the SPOTT'S criteria).

         Piston tolerances                     Cylinder bore
                                                 tolerances

[t.sub.11]               0.01622     [t.sub.21]               0.01628
[t.sub.11]               0.00378     [t.sub.22]               0.00372
[t.sub.12]               0.00122     [t.sub.23]               0.00128
[t.sub.13]               0.00058     [t.sub.24] = [t.sub.2d]  0.0005
[t.sub.14] = [t.sub.1d]

   Minimum      Minimum     total
manufacturing   quality     cost
    cost        loss cost

66.2523         2.8958      69.1467

Table 15: Near-optimal tolerances & costs for example, with W1=W2=1 and
Cp1=Cp2=1 (based on the EMS criteria).

         Piston tolerances                     Cylinder bore
                                                tolerances

[t.sub.11]               0.01628     [t.sub.21]               0.01627
[t.sub.12]               0.00372     [t.sub.22]               0.00373
[t.sub.13]               0.00128     [t.sub.23]               0.00127
[t.sub.14] = [t.sub.1d]  0.00052     [t.sub.24] = [t.sub.2d]  0.00043

   Minimum      Minimum     total
manufacturing   quality     cost
    cost        loss cost

67.1836         5.0589      72.23798

Table 16: Near-optimal tolerances & costs for example, with W1=W2=2 and
Cp1=Cp2=1 (based on the EMS criteria).

         Piston tolerances                     Cylinder bore
                                                tolerances

[t.sub.11]               0.01628     [t.sub.21]               0.01627
[t.sub.12]               0.00372     [t.sub.22]               0.00373
[t.sub.13]               0.00128     [t.sub.23]               0.00127
[t.sub.14] = [t.sub.1d]  0.00052     [t.sub.24] = [t.sub.2d]  0.00037

   Minimum      Minimum     total
manufacturing   quality     cost
    cost        loss cost

98.4896         9.0511      107.53595

Table 17: Near-optimal tolerances & costs for example, with W1=W2=1 and
Cp1=Cp2=1.5 (based on the EMS criteria).

         Piston tolerances                     Cylinder bore
                                                tolerances

[t.sub.11]               0.01641     [t.sub.21]               0.01628
[t.sub.12]               0.00359     [t.sub.22]               0.00372
[t.sub.13]               0.00141     [t.sub.23]               0.00128
[t.sub.14] = [t.sub.1d]  0.00039     [t.sub.24] = [t.sub.2d]  0.00032

   Minimum      Minimum     total
manufacturing   quality     cost
    cost        loss cost

71.7739         11.3111     83.08311

Table 18: Near-optimal tolerances & costs for example, with W1=W2=1 and
Cp1=Cp2=0.5(based on the EMS criteria).

         Piston tolerances                     Cylinder bore
                                                tolerances

[t.sub.11]               0.01622     [t.sub.21]               0.01627
[t.sub.12]               0.00378     [t.sub.22]               0.00373
[t.sub.13]               0.00122     [t.sub.23]               0.00127
[t.sub.14] = [t.sub.1d]  0.00058     [t.sub.24] = [t.sub.2d]  0.0005

   Minimum      Minimum     total
manufacturing   quality     cost
    cost        loss cost

66.2524         2.8958      69.1444
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Author:Rao, Y.S.; Rao, C.S.P.
Publication:International Journal of Applied Engineering Research
Article Type:Report
Date:Sep 1, 2008
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