CHAPTER 5 OPTIMAL TOLERANCE DESIGN WITH ALTERNATIVE MANUFACTURING PROCESS SELECTION

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1 9 CHAPTER 5 OPTIMAL TOLERANCE DESIGN WITH ALTERNATIVE MANUFACTURING PROCESS SELECTION 5. STAGE 4 COMBINED OBJECTIVE ( OBJECTIVES) CONCURRENT OPTIMAL TOLERANCE DESIGN WITH ALTERNATIVE MANUFACTURING PROCESS SELECTION 5.. Introduction Sivakumar et al (9) presented a method to calculate manufacturing cost based on Tabu search approach, for mechanical assemblies (Knuckle joint assembly and Wheel mounting assembly). In this stage, an optimization procedure based on Differential Evolution (DE) algorithm for the problem dealt by Sivakumar et al (9) is proposed. A new optimization model which considers objective functions (minimization of manufacturing cost and quality loss), constraints (Tolerance stack up) and 8 variables (Process tolerance) is proposed. Also a concurrent optimal tolerance design with alternative manufacturing process selection is considered. 5.. Problem Formulation The selection of machining process including equipment accuracy, set-up mode, machining sequence and cutting parameters is strongly affected by the tolerance of the part to be machined. So it is important to do a

2 3 simultaneous selection of the best machining process while allocating the tolerance. Manufacturing cost usually increases as the tolerance of quality characteristics in relation to the ideal value is reduced, due to the need for more refined and precise operations, while the acceptable ranges of output are reduced. Conversely, large tolerances are less costly to achieve as they require less precise manufacturing processes; but they usually result in poor performance, premature wear and part rejection. Total manufacturing cost of an assembly can be represented by the following equation: C ASM n i C X i ( t ) (5.) ij The following exponential tolerance-cost function is used to find ct ij the manufacturing cost of a part: C X ( tij ) ce c. Where, C ASM is the i total manufacturing cost of the assembly, t ij is the tolerance allocated to the part dimension x i. c, c and c are the coefficients of the function C xi. Variability in the production process is unavoidable due to inconsistency in tool work piece, material and process parameters. Kun-Lin Hsieh (6) suggests that given an ideal target value, an evaluation function associated with deviations from the target can be developed. In this study it is referred to as quality loss function. This loss function is a quadratic expression for measuring the cost of the average value versus the target value and the variability of product characteristics in terms of monetary loss due to product failure in the eyes of consumers. The expected quality loss function (E(L)) is: n E (L)= k i itij i (5.) i

3 3 where k i =A ci /t ij, A ci is the loss at either specification limit, i represents the part dimension number, j represents the manufacturing process number, is a constant ( < i <) and i = y i m i, y i denotes the measurement of the quality characteristic, m i denotes the desired target value. i The improved optimization model opens up possibilities for tolerance control in order to achieve the selection of the best manufacturing processes, economical assembly manufacturing cost and the minimum quality loss of the product. Formulation of the simple optimal tolerance synthesis involves framing of the objective functions and constraints. The objective functions considered are: minimum total manufacturing cost of the assembly (summation of the manufacturing cost involved in the manufacture of all the toleranced dimensions) and minimum quality loss function. The functional requirements of the assembly (stack-up conditions), process precision limits and process selection conditions form the constraints of the problem. The variables are tolerances of all part dimensions. Optimization model is as follows: Combined objective function Minimize Fc W CASM / N W E( L) / N (5.3) where, n C ASM = Total manufacturing cost of the assembly = i n E (L ) = Expected Quality Loss= k i itij i C ij ( t ij ) (5.4) (5.5) i Constraints Stack-up conditions - t n ij i T k asm (5.6)

4 3 Variables Process precision limits: t min ij t ij t max ij i= to n, j= to m i where, W, W = Weightage values given to the respective objective functions (C ASM, E(L)), N, N = Average value of the respective objective functions (C ASM, E(L)), C ij (t ij ) = Cost of producing dimension x i by the process j maintaining tolerance t ij, m i n = Number of the available alternative processes for producing dimension x i, = Number of the component dimensions, T k asm = Permissible variation in the k th assembly dimension, known as assembly tolerance, t ij = Tolerance on the dimension x i produced by the j th process 5..3 Numerical Examples Two complex mechanical assemblies namely knuckle joint and wheel mounting assembly are taken as case studies Knuckle joint assembly (KJA) There are five components (rod, fork, pin, collar and taper pin) and five important dimensions in the knuckle joint assembly (Figure 5.). These dimensions constitute two interrelated dimension chains with X a being

5 33 common to both. Total assembly manufacturing cost is represented by the Equations (5.7) (5.9). Two interrelated dimension chains give rise to two stack-up constraints (Equation (5.) and (5.3)). The constraints formulated here are based on the worst case (WC) criteria. The permissible variations T and T associated with the respective assembly dimensions Y and Y have been assumed to be equal to.3. The manufacturing details (part dimensions and process number), constants of cost models and precision limits (tolerance limits) are given in table 5. (Singh et al 4a). Tables 5. and 5.3 give the value of parameters to find quality loss with corresponding part dimensions of knuckle joint assembly. Figure 5. Knuckle joint assembly Minimize FcKJA W CASM / N W E( L) / N (5.7) W n CX CX C C C / N W k t X X a b 3 X 4 i i ij i / i N (5.8) C X c t ij ( t ) ce c i ij (5.9) k i =A ci /t ij where W, W are weightage values given to the respective objective functions (C ASM, E(L)), N, N are average value of respective objective functions (C ASM,

6 34 E(L)), i represents part dimension number (i =,a,b,3,4) and j represents manufacturing process number (j =,,, 4). W = W =.5, N =375, N =56. Assembly response function: Y Y X a X (5.) 3 ( a b X 4 X X X ) (5.) Tolerance stack-up constraints t t.3 (5.) x x 3t t t.3 (5.3) 3 4 x x x 3 4 Table 5. Exponential cost function constants for Knuckle joint assembly (Singh et al 4a) Parameters of cost functions Minimum Maximum Part Process Tolerance Tolerance Dimension No. C o C C (mm) (mm) X X a,x b X X

7 35 Table 5. Knuckle joint assembly parameters of each component Part Dimensions i (mm) i (mm) z i X X a X b X X Table 5.3 Quality loss data for knuckle joint assembly Part Dimensions A ci ($) t ij (mm) i i (mm) k i ($/mm) X X a X b X X Wheel mounting assembly (WMA) There are five important dimensions considered in this assembly as shown in Figure 5. (Jeang and Chang ). These dimensions constitute two interrelated dimension chains with X being common to both. Two interrelated dimension chains give rise to two stack-up constraints (Equation 5.9 and 5.). The total assembly manufacturing cost is represented by the Equations The constraints formulated here are based on the worst case criteria. The permissible variations T and T associated with the assembly dimensions Y and Y have been assumed to be equal to. and

8 36.4 respectively. The manufacturing details (part dimensions and process number), constants of cost models and precision limits (tolerance limits) are given in Table 5.4 (Singh et al 4a). Tables 5.5 and 5.6 give the value of parameters to find quality loss with corresponding part dimensions of wheel mounting assembly. Figure 5. Wheel mounting assembly Minimize FcWMA W CASM / N W E( L) / N (5.4) W n CX CX C C C / N W k t X 3 X 4 X 5 i i ij i / i N (5.5) C X c t ij ( t ) ce c i ij (5.6) k i =A ci /t ij where W, W are the weightage values given to the respective objective functions (C ASM, E(L)), N, N are the average value of respective objective functions (C ASM, E(L)), i represents the part dimension number (i =

9 37,,3,4,5) and j represents the manufacturing process number (j =,,, 4). W = W =.5, N =56, N =5. Assembly response function: Y X X 4 (5.7) Y X X X 3 X 5 (5.8) Tolerance stack-up constraints: t t. (5.9) x 4 x 4 t t t t.4 (5.) 3 5 x x x x 3 5 Table 5.4 Exponential cost function constants for wheel mounting assembly (Singh et al 4a) Part Dimensions Process No. Parameters of cost functions C o C C Minimum Tolerance (mm) Maximum Tolerance (mm) X, X,X X X

10 38 Table 5.5 Wheel mounting assembly parameters of each component Part dimensions i (mm) i (mm) z i X X X X X Table 5.6 Quality Loss data for wheel mounting assembly Part dimensions A ci ($) t ij (mm) i i (mm) k i ($/mm) X X X X X Implementation of DE Algorithm The combined objective function comprising of assembly manufacturing cost and quality loss function (Equations 5.7 and 5.4) is minimized considering the design constraints based on the assumed stack-up criteria (Equations 5., 5.3, 5.9 and 5.).

11 39 The steps for running DE algorithm are summarized below: Step : The following are inputs to software program of DE algorithm:. Details of Mechanical assemblies: a. Parameters of cost functions and tolerance limits for Knuckle joint assembly (Given in Tables ). b. Parameters of cost functions and tolerance limits for Wheel mounting assembly (Given in Tables ). Formulae to find the total manufacturing cost and the quality loss: a. Knuckle joint assembly (Equations ) b. Wheel mounting assembly (Equations ) 3. The parameters of DE algorithm. (Given in section 5..4.) 4. Formulae to check all constraints. (Equations 5., 5.3, 5.9 and 5. ) Step : The software program of DE algorithm will find the optimal tolerances values (t ij ) in such a way that. The combined objective function (Fc KJA or Fc WMA ) is minimum.. All constraints (tolerance stack-up) are satisfied. Step 3: Step 4: Step will be repeated up to maximum number of iterations. The following are the outputs from DE algorithm:

12 4. Optimal solutions are obtained from DE. Each solution shall have optimal objective function value, optimal value of variables and the constraints value.. The best optimal solution gives the minimum total manufacturing cost, minimum quality loss and optimal tolerances value DE Parameters The following are the values of the parameters of DE technique that have been used to obtain the best optimal results: Strategy = DE/rand//bin, crossover constant CR=.9, number of population NP=5, F=.5 and total number of generations= Results and Discussion Here, two mechanical assemblies namely Knuckle joint assembly and Wheel mounting assembly have been considered. Valid process precision limits have been considered in both case studies. The constraints are based on stack-up conditions and those resulting from stack removal considerations. This problem has been attempted in a Pentium IV personal computer with GB RAM, GB Hard Disk using VC++ computer programs. Table 5.7 shows optimally allocated tolerance value and the corresponding manufacturing process of knuckle joint assembly components. Similar results for wheel mounting assembly are shown in Table 5.8. Also both the tables show a comparison between the results of DE, Singh et al (4a) and Sivakumar et al (9) methods. From Tables 5.7 and 5.8, it is concluded that DE algorithm gives better results than Singh et al (4a) and Sivakumar et al (9) methods. The cost values for different processes obtained from DE technique for knuckle joint and wheel mounting assemblies are given in Tables 5.9 and 5.. In Tables 5.9 and 5., the minimum cost values for different processes obtained from DE technique are given in bold.

13 4 Table 5.7 Result of optimal tolerance synthesis for Knuckle joint assembly Techniques Dimensions (mm) Accumulated tolerance (mm) Cost($) X X a X b X 3 X 4 Process 3 3 Y Y Singh et al. (4a) Process 3 3 Sivakumar et al (9) Process 3 3 Proposed DE Table 5.8 Result of optimal tolerance synthesis for wheel-mounting assembly Accumulated Dimensions (mm) Techniques tolerance (mm) Cost X X X 3 X 4 X 5 ($) Y Y Process Singh et al (4a) Process Sivakumar et al (9) Process Proposed DE Table 5.9 Cost values for different processes obtained from DE technique for Knuckle joint assembly Process Dimensions X X a X b X 3 X Minimum manufacturing cost ($)

14 4 Table 5. Cost values for different processes obtained from DE technique for wheel-mounting assembly Dimensions Process X X X 3 X 4 X Minimum manufacturing cost ($) The Table 5. compares the combination of processes, assembly manufacturing cost ($), CPU running time (seconds) to find the optimal results given by different methods for both knuckle joint and wheel mounting assemblies. From the Table 5., the following conclusion is made: DE method gives better results (less manufacturing cost ($) and CPU time) than the ones of both Singh et al (4a) and Sivakumar et al (9) for both the assemblies. Table 5. Comparison of Assembly manufacturing cost and CPU time from different methods Knuckle joint assembly Wheel mounting assembly Method Combination of processes Assembly manufacturing cost ($) CPU time (s) Combination of processes Assembly manufacturing cost($) CPU time (s) Singh et al (4a) Sivakumar et al (9) Proposed DE

15 Cost saving calculation The manufacturing cost saving (CS ASM ) in both assemblies is calculated for the following manufacturing conditions: products per hour, eight hour per shift, 3 shifts per day and 3 days per year. Cost saving by DE compared to Singh et al (4) CS KJA = ( )**8*3*3 = $46, 3,76/year CS WMA = ( )**8*3*3= $4, 79,5/year Cost saving by DE compared to Sivakumar et al (9) CS KJA = ( )**8*3*3 = $45, 38,88/year The Table 5. gives the saving in assembly manufacturing cost ($) and CPU running time (seconds) to find the optimal results given by different methods for both the knuckle joint and wheel mounting assemblies. From Table 5., the following conclusions are made:. DE method saves nearly $46,3,76 and $4,79,5 per year in manufacturing cost of knuckle joint and wheel mounting assembly respectively than that of Singh et al (4a),. Also DE saves nearly $45,38,88 per year in manufacturing cost of knuckle joint than Sivakumar et al (9), 3. DE saves nearly 3. seconds and 3.9 seconds in CPU running time for knuckle joint and wheel mounting assembly respectively than Singh et al (4a), and 4. Also DE method saves nearly 3. seconds and.98 seconds in CPU running time for knuckle joint and wheel mounting assembly respectively than Sivakumar et al (9). Table 5.3 shows the results (Expected quality loss, total manufacturing cost and combined objective function) obtained from DE algorithm for both the assemblies.

16 44 Table 5. Comparison of cost saving and CPU time saving Knuckle joint assembly Wheel mounting Proposed assembly DE with literature CPU time Cost saving CPU Cost saving (s) ($/year) time (s) ($/year) Singh et al (4a) 3. 46, 3, ,79,5. Sivakumar et al (9) 3. 45, 38, Table 5.3 Results obtained from DE for both assemblies Knuckle joint assembly Part Dimensions Quality loss ($) Wheel mounting assembly Part Dimensions Quality loss ($) X X X a.79 X X b X X X 4.67 X X E(L) E(L) C ASM C ASM Fc.878 Fc.948 Figures 5.3 and 5.4 show the manufacturing cost and quality loss for each dimension of both assemblies (knukle joint assembly and wheel mounting assembly) obtained from DE. Figures 5.5 and 5.6 show a comparison of manufacturing cost obtained from DE, Singh et al (4a) and Sivakumar et al (9) methods for both assemblies (knukle joint assembly and wheel mounting assembly). From Figures 5.5 and 5.6, it is concluded that DE algorithm gives better results than the methods of Singh et al (4a) and

17 45 Sivakumar et al (9). The result histories of DE algorithm for knuckle joint and wheel mounting assembly cases are shown in Figures 5.7 and 5.8 respectively. It is concluded (from Figures 5.7 and 5.8) that DE algorithm converges within 6 iterations in both the cases. Cost ($) Manufacturing cost and quality loss of each dimensions for knukle joint assembly X Xa Xb X3 X4 TOTAL CASM E(L) Part dimensions Figure 5.3 Manufacturing cost and quality loss of each dimension for knuckle joint assembly Cost ($) Manufacturing cost and quality loss of each dimensions for wheel mounting assembly X X X3 X4 X5 TOTAL CASM E(L) Part dimensions Figure 5.4 Manufacturing cost and quality loss of each dimension for wheel mounting assembly

18 46 Manufacturing cost ($) comparison of manufacturing cost-knuckle joint assembly X Xa Xb X3 X4 Singh MS Kumar Proposed Part dimensions Figure 5.5 Comparison of assembly from different methods manufacturing cost for knuckle joint Manufacturing cost ($) Comparison of manufacturing cost - Wheel mounting assembly X X X3 X4 X5 singh MS Kumar proposed Part dimensions Figure 5.6 Comparison of manufacturing cost for wheel mounting assembly from different methods Minimum manufacturing cost($) Generations Figure 5.7 Result history for Knuckle joint assembly from DE

19 47 Minimum manufacturing cost($) Generations Figure 5.8 Result history for wheel-mounting assembly from DE 5..6 Limitations The following are the limitations of this stage:. The problem is approached as a single objective optimisation problem (using normalised weighted objective function method, all objective functions are combined as a single objective function),. The proposed approach is not applicable, if the user does not know what weightage is to be given for each objective function, 3. The method used in this stage cannot be directly used for treating all objectives individually, and 4. Only one optimal solution is obtained in this stage. But for a real world problem, a Pareto optimal front that offers more number of optimal solutions for user s choice is most desirable. Many real-world problems involve simultaneous optimization of several objective functions. Generally, these functions are conflicting objectives. Multi-objective optimization with such conflicting objective functions gives rise to a set of optimal solutions, instead of one optimal solution. The reason for the optimality of many solutions is that no one can be considered to be better than any other with respect to all objective functions. These optimal solutions are known as pareto-optimal solutions. A general muti-objetive optimization problem (MOOP) consists of a number of

20 48 objectives to be optimized simultaneously and is associated with a number of equality and inequality constraints. The main objective of the Multi-Objective (MO) problem is to find the set of acceptable (trade-off) Optimal Solutions. This set of accepted solutions is called Pareto front. These acceptable trade-off solutions give more ability to the user to make an informed decision by seeing a wide range of near optimal solutions that are near optimum from an overall standpoint. Single Objective (SO) optimization may ignore this trade-off viewpoint, which is crucial. The conventional optimization methods such as dynamic programming (DP), linear programming (LP) and non-linear programming (NLP) are not suitable to solve multi-objective optimization problems (MOOP), because these methods use a point-by-point approach, and the outcome of these classical optimization methods is a single optimal solution. For example, the weighted sum method will convert the MOOP into a single objective optimization. By using a single pair of fixed weights, only one point on the Pareto front can be obtained. Therefore, if one would like to obtain the global Pareto optimum, all possible Pareto fronts must first be derived. This requires the algorithms to be executed iteratively, so as to ensure that every weight combination has been used. Obviously, it is impractical to reiterate the algorithms continually to exhaust all the weight combinations. Hence the algorithms should have an ability to learn from previous performance to direct the proper selection of weights in further evolutions. Also conventional methods may face problems, if the optimal solution lies on nonconvex or disconnected regions of function space (Deb, ). So, in the next stage a method based on multi objective optimization algorithms is proposed.

21 49 5. STAGE 5: MULTI-OBJECTIVE CONCURRENT OPTIMAL TOLERANCE DESIGN USING NSGA-II AND MODE 5.. Introduction This stage presents a novel general method for the multi-objective optimal tolerance design of engine assembly and helical spring. The problem considered has 3 objective functions, 6 variables and 4 constraints at the maximum. This stage addresses the tolerance design and development of a method to find optimal tolerances. This stage considers the important decision criteria for the optimal tolerance design of a mechanical assembly (Minimization of stack-up tolerance, total manufacturing cost and quality loss). In this stage, two evolutionary algorithms namely Elitist Nondominated Sorting Genetic Algorithm (NSGA-II) and Multi Objective Differential Evolution (MODE) are proposed to obtain optimal tolerance for engine assembly and helical spring. Two methods (normalized weighting objective functions and average fitness factor) are combinedly used to select the best optimal solution from Pareto optimal fronts. Two multi objective performance measures namely solution spread measure and ratio of nondominated individuals are used to evaluate the strength of Pareto optimal fronts. Two multi objective performance measures namely optimiser overhead and algorithm effort are used to find the computational effort of NSGA-II and MODE algorithms. 5.. Problem Formulation The improved optimization model opens up the possibilities for tolerance control in order to achieve the selection of the best manufacturing processes, economical assembly manufacturing cost and minimum quality loss of the product. Formulation of simple optimal tolerance synthesis

22 5 involves framing of the objective functions and constraints. The objective functions considered are: minimum tolerance stack up (Z for example A and Z, Z for example B), minimum total manufacturing cost of the assembly (Z for example A and Z 3 for example B) (summation of the manufacturing cost involved in the manufacture of all the toleranced dimensions) and minimum quality loss function (Z 3 for example A and Z 4 for example B). The functional requirements of the assembly (stack-up conditions), process precision limits and process selection conditions form the constraints of the problem. The variables are tolerances of all part dimensions. The multicriterion optimisation problem is defined as follows: 5... Example A (Engine assembly) Minimize: Z = Y t4 t4 (5.) x4 x4 Z = 4 c ( t d ) C t a b e (5.) ij ij ij ij ( ij ) ( ij ij ) i j x ij Z 3 = Q A I L t ij 9T i (5.3) Constraints: Tolerance stack-up constraints: t 4 t4. x4 x4 (5.4) x4 4 x4. t 4 t (5.5)

23 5 Machining tolerance constraints: For the piston t x t x t 3 x3 t t t x 3 x3 4 x (5.6) and for the cylinder bore t x t x t 3 x3 t t t x 3 x3 4 x (5.7) Variables: Process precision limits: t min ij t ij t max ij i= to n, j= to m i 5... Example B (Helical spring) Minimize: Z 8 K 8 ( G( d G( d 3 3 ( d t ) ( d t ) ( N t ) i ij i w w ij t wj wj ) 4 w d t ) ( d w t wj t wj ) 4 ) Nj ( N t Nj ) (5.8) Z = d t ij t (5.9) wj Z 3 = C 3 t c e 4 ij ij ( ij ) ( ) i j c t c x ij C C C (5.3) dw di N

24 5 Z 4 = Q A I L t ij 9T i (5.3) Tolerance stack-up Constraints: 8 ( d 8 i ( t d i ij t G( d ) ( d ij w w G( d ) ( d t t w w wj wj t ) t 4 3 ) wj wj ) ( N t 4 3 ) Nj ( N t ) Nj.477 ) (5.3) t t.58 (5.33) ij wj Variables: Process precision limits: t min ij t ij t max ij i= to n, j= to m i where, Y = Assembly response function (Example A), Q L = Quality loss function, C ij (t ij ) = Cost of producing dimension x i maintaining tolerance t ij, by the process j m i n t ij = Number of the available alternative processes for producing dimension x i, = Number of the component dimensions, = Tolerance on the dimension x i produced by the j th process.

25 Numerical Example Two complex mechanical elements namely engine assembly and helical spring are taken as case studies Example A engine assembly Design and manufacturing details of the engine assembly (Figure 5.9) are given below (Singh et al 5b): Piston diameter, X =5.8mm; cylinder bore diameter, X =5.856mm; and clearance, Y=X -X =.56 ±.5mm. The process plans for the piston are rough turning, finish turning, rough grinding, and finish grinding, and for the cylinder bore are drilling, boring, finish boring, and grinding. There are only two design tolerance parameters, t d and t d, for the piston and the cylinder bore diameter respectively. These two design tolerances are met by the final finishing operation in the respective process plan. Also, there are four machining tolerance parameters for the manufacturing of each of the piston and of the cylinder t ij (i = for piston, and for cylinder; j = to 4 for the four manufacturing operations in the respective process plan). For each manufacturing operation alternative machines are available with the details given in Table 5.4 and the only one that yields economical results is selected. For machine selection there is one machine selection variable corresponding to each manufacturing operation, x ij (suffixes as explained above). Thus in all there are 8 decision variables. Since the design tolerances are supposed to be identical with the tolerance achieved in the final finishing operation (i.e.t d = t 4 and t d = t 4 ), the effective number of decision variables becomes 6. Objective functions based on the minimization of the stackup tolerance, assembly manufacturing cost and quality loss are formulated. Total assembly manufacturing cost is the

26 54 summation of all the costs associated with all the operations involved in manufacturing of all the individual dimensions. Constraints on the design tolerances are formulated based on the assumed stack-up criteria. Here the design tolerance for a given feature is same as the final machining tolerance. These conditions yield design constraints represented through Equations (5.4) and (5.5) corresponding to the worst case and the root sum square criteria respectively The constraints on the machining tolerance dictate the sum of machining tolerance for a machine in a particular operation and that for a machine in the preceding operation should be less than or equal to the difference of the nominal and minimum machining allowances for the operation. The nominal and minimum machining allowances are normally listed in the machining manuals or handbooks. The machining tolerance constraints are given in Equations (5.6) and (5.7). Figure 5.9 Engine (Cylinder- Piston) assembly

27 Table 5.4 Details of manufacture for engine assembly (Singh et al 5b) Dimension Piston diameter Manufacturing operation Rough turning Finish turning Rough grinding Finish grinding Machine Parameters of cost function a b c d M M M M M M M M M M M M M M M M Minimum Tolerance (mm).5 Maximum Tolerance(mm)

28 Table 5.4 (Continued) Dimension Cylinder diameter Manufacturing operation Drilling Boring Finish boring Grinding Machine Parameters of cost function a b c d M M M M M M M M M M M M M M M M Minimum Tolerance (mm) Maximum Tolerance(mm)

29 Example B-helical spring For the helical spring shown in Figure 5., there are three decision variables. There are two interrelated dimension chains corresponding to respective design functions; one linear and other nonlinear giving rise to two constraints given below. The constraints formulated here are based on the worst-case criteria. The nonlinear constraints are to be linearized in general. However, with the application of optimization algorithms it is not so with the worst-case criteria. Nevertheless, the constraints corresponding to those based on criteria other than the worst-case, are required to be linearized. Instead of X, Y and i notations, the usual notations as applicable in spring terminology have been used here. These notations have already been explained. The second suffix j associated with tolerance indicates the selected manufacturing process. For each manufacturing operation alternative machines are available with the details given in Table 5.5 and the only one that yields economical results is selected. For machine selection there is one machine selection variable corresponding to each manufacturing operation, x ij (suffixes as explained above). Design functions: 4 Gd K w 3 8( d d ) N, (5.34) i w d d i d w (5.35)

30 58 Figure 5. Helical spring (all length dimensions in mm, force in N) Table 5.5 Details of manufacturing of assemblies for helical spring (Singh et al 5a) Parameters of cost function Minimum Maximum Dimension Process Tolerance Tolerance C C C (mm) (mm) d w d i N

31 NSGA-II Parameters The following are the values of the parameters of NSGA-II technique that have been used to obtain the best optimal results: Variable type = Real variable, Population size =, Crossover probability =.9, Real-parameter mutation probability =., Real-parameter SBX parameter =, Real-parameter mutation parameter =, Total number of generations = MODE Parameters The following are the values of the parameters of MODE technique that have been used to obtain the best optimal results: Strategy = MODE/rand//bin, crossover constant CR=.9, number of population NP=, F s =.5 and total number of generations= 5..4 Implementation of NSGA-II and MODE Algorithms The steps for running the algorithms are summarized below: Step : The following are the inputs to NSGA-II and MODE algorithms:. Details of Mechanical assemblies: Parameters cost functions and tolerance limits for engine assembly (Given in Tables 5.4) Parameters cost functions and tolerance limits for helical spring (Given in Tables 5.5). Details of total Manufacturing cost and quality loss formulae : a) Engine assembly (Equations 5. and 5.3)

32 6 b) Helical spring (Equations 5.3 and 5.3) 3. Formulae to find objective functions (z, z, z 3 and z 4 ) (Equations (5.) - (5.3) for Example A, Equations (5.8) - (5.3) for Example B), Formulae for checking constraints (Equations (5.4) - (5.7) for Example A, Equations (5.3) - (5.33) for Example B)). 4. Formulae to find the multi-objective performance measures namely solution spread measure, ratio of non-dominated individuals, optimiser overhead and algorithm effort. (Equations (3.5) - (3.8)). 5. Formulae to find the combined objective function (f c ) and average fitness factor (F avg ) (Equation 3.3). Based on these values the best optimal solution from Pareto optimal front shall be selected. For this problem, the combined objective function (f c ) is defined as follows: Example A (Engine assembly) Minimize f c =W Z /N +W Z /N +W 3 Z 3 /N 3 (5.36) Example B (Helical spring) Minimize f c =W Z /N +W Z /N +W 3 Z 3 /N 3 +W 4 Z 4 /N 4 (5.37) W, W, W 3 and W 4 are the weightages given to the objective functions,, 3 and 4 respectively. Here normalized weighting objective functions method is used only to select the best optimal solution from Pareto optimal fronts obtained from NSGA-II and MODE. So one can give any weightage to each objective function. But the condition is W +W +W 3 +W 4 =. It means the total weightage should be %. The Value

33 6 of W =W =W 3 =.333 for example A (Engine assembly) and the values of W =W =W 3 =W 4 =.5 for example B (Helical spring). It means that equal weightage is given to all objective functions. N, N, N 3 and N 4 are normalizing parameters of objective functions (Average value of individual objective functions). The values of N =., N = and N 3 = for example A (Engine assembly) and the values of N =N =N 3 =N 4 =. for example B (Helical spring). 6. The variables limits are as follows: a. Engine assembly (Given in Tables 5.4) b. Helical spring (Given in Tables 5.5) 7. The parameters of NSGA-II and MODE algorithms. (Given in sections and ) Step : The software programs of NSGA-II and MODE algorithms find the optimal variables in such a way that i. All objective functions (tolerance stackup, assembly manufacturing cost and quality loss function) are minimum. ii. All constraints (tolerance stack up and design) are satisfied. Step 3: Step is repeated up to maximum number of iterations. Step 4: The following are the outputs from NSGA-II and MODE algorithms: (i) Pareto optimal fronts obtained from NSGA-II and MODE. The Pareto optimal front gives a number of trade-off solutions. Each solution has the optimal objective function value, optimal

34 6 value of variables and constraints value. All constraints shall be satisfied by any solution in the Pareto optimal front. (ii) The best optimal solution from Pareto optimal fronts are selected by normalized weighting objective functions and average fitness factor methods in a combined manner. The best optimal solution is the one that gives a minimum stackup tolerance, economical manufacturing cost and minimum quality loss and optimal tolerances values. (iii) The strength of Pareto optimal fronts evaluated by the two multi-objective performance measures - solution spread measure and ratio of non-dominated individuals. (iv) The computational efforts of NSGA-II and MODE algorithms found by the two multi-objective performance measures (optimiser overhead and algorithm effort) Results and Discussion Tables 5.6 and 5.7 compare optimization results (selected manufacturing process (or machine), manufacturing tolerance, stackup tolerance, manufacturing cost, quality loss) obtained from NSGA-II and MODE for engine assembly and helical spring respectively. The results of average fitness factor value (F avg ), solution spread measure (SSM), ratio of non-dominated individuals (RNI), optimiser overhead (OO) and algorithm effort obtained from NSGA-II and MODE are listed in Tables for both Examples A and B. From Tables , it is observed that NSGA- II technique gives minimum solution spread measure (SSM), maximum average fitness factor (F avg ), maximum optimiser overhead (OO), maximum ratio of non-dominated individuals (RNI) and maximum algorithm effort than those of MODE. So NSGA-II is the best technique for this problem.

35 Table 5.6 Optimization results for engine assembly Technique NSGA-II MODE Machine Manufacture of piston Tolerance notation Optimum variable Tolerance value(mm) Machine Manufacture of cylinder Tolerance notation Tolerance value(mm) M t.5849 M 9 t.35 M 8 t.3478 M 4 t.35 M t M 5 t 3.5 M 4 t 4.33 M 9 t M t.54 M t.76 M 8 t.37 M 4 t.334 M 9 t 3.97 M 5 t 3.33 M 4 t 4. M 9 t 4.3 Objective Functions Z Z Z

36 64 Table 5.7 Optimization results for helical spring Technique NSGA-II MODE Dimension Process Individual tolerance (mm) d w.3683 d i.968 N.7593 d w.738 d i N Objective Functions Z Z Z 3 Z Table 5.8 Results obtained from NSGA-II and MODE algorithms Z Z Z 3 Z 4 avg Example A ( Engine Assembly) Z max Z min NSGA-II MODE Example B ( Helical Spring) Z max Z min NSGA-II MODE

37 65 Table 5.9 Combined objective function and Algorithm effort obtained from NSGA-II and MODE algorithms Proposed algorithm Combined objective function (fc) Example A ( Engine Assembly) Simulation time T run (sec) No. of function evolution (N eval ) Algorithm effort NSGA-II MODE Example B ( Helical Spring) NSGA-II MODE Table 5. SSM, RNI and OO obtained from NSGA-II and MODE algorithms Example No. Techniques SSM RNI OO A NSGA-II MODE B NSGA-II MODE The best solution is selected by average fitness factor method from optimal solution trade-offs obtained from NSGA-II and MODE. The optimal solution trade-offs (Pareto optimal fronts) obtained from NSGA-II and MODE are given in Figures 5. and 5. respectively for engine assembly and Figures 5.4 and 5.5 respectively for helical spring. From Figures 5., 5., 5.4 and 5.5, it is noted that NSGA-II gives more number of optimal solution trade-offs than MODE. So NSGA-II technique is the best one for this problem, if the user wants more number of solutions for his choice. The best solution trade-offs selected by average fitness factor method from the optimal solution trade-offs obtained from NSGA-II and

38 66 MODE are shown in Figures 5.3 (for engine assembly) and 5.6 (for helical spring) respectively. From Figure 5.3, it is noted that MODE gives best results for two objective functions (Minimum values for z, and z 3 ). But NSGA-II gives the best result for z and From Figure 5.6, it is noted that NSGA-II gives best results for three objective functions (Minimum values for z, z and z 4 ). But MODE gives the best result for z 3. So both NSGA-II and MODE are best techniques for this problem. ENGINE ASSEMBLY (NSGA-II) F/F* solutions 3 OBJECTIVE FUNCTION NUMBER Figure 5. Optimal solution trade-offs obtained from NSGA-II for Engine assembly ENGINE ASSEMBLY (MODE) 4 solutions F/F* OBJECTIVE FUNCTION NUMBER Figure 5. Optimal solution trade-offs obtained from MODE for engine assembly

39 67 F ENGINE ASSEMBLY NSGA-II MODE 3 OBJECTIVE FUNCTION NUMBER Figure 5.3 Best solution trade-offs obtained from NSGA-II and MODE for Engine assembly F/F* HELICAL SPRING (NSGA-II) solutions 3 4 OBJECTIVE FUNCTION NUMBER Figure 5.4 Optimal solution trade-offs obtained from NSGA-II for helical spring

40 68 F/F* solutions HELICAL SPRING (MODE) 3 4 OBJECTIVE FUNCTION NUMBER Figure 5.5 Optimal solution trade-offs obtained from MODE for helical spring HELICAL ASSEMBLY NSGA-II MODE 8 6 F OBJECTIVE FUNCTION NUMBER Figure 5.6 Best solution trade-offs obtained from NSGA-II and MODE for helical spring 5..6 Limitations This stage deals the optimal tolerance allocation using NSGA-II and MODE. The same problem dealt in this stage is extended in next stage for some other assemblies and solved by using NSGA-II and Multi Objective Particle Swarm Optimization (MOPSO). Because, the most striking difference

41 69 between MOPSO and the other evolutionary algorithms is that MOPSO chooses the path of cooperation over competition. The other optimization algorithms commonly use some form of decimation, survival of the fittest. In contrast, the MOPSO population is stable and individuals are not destroyed or recreated. Individuals are influenced by the best performance of their neighbors. Individuals eventually converge on optimal points in the problem domain. In addition, the MOPSO traditionally does not have genetic operators like crossover between individuals and mutation, and other individuals never substitute particles during the run. So, in MOPSO all the particles tend to converge to the best solution quickly, as compared to the other optimization algorithms. The main advantages of MOPSO method are:. It works simultaneously with a set of possible solutions, the so-called population, and several nondominated solutions may be found in a single run of the algorithm.. It gives optimal solution trade-offs with more number of nondominated solutions for user s choice than MODE and NSGA-II. 3. It does not require prior knowledge of the relative importance of the objectives. 4. There is a set of acceptable trade-off near optimal solutions. This set is called Pareto front or optimality trade-off surfaces. 5. The algorithm handles constraints in a very simple and efficient way, as comparing to different solutions. 6. It is less sensitive to the shape or continuity of the Pareto surface.

42 7 5.3 STAGE 6: MULTI-OBJECTIVE CONCURRENT OPTIMAL TOLERANCE DESIGN USING NSGA-II AND MOPSO 5.3. Introduction This stage presents a novel general method for computing the optimal tolerances of mechanical assemblies (Assembly A, Assembly B, Gearbox assembly, Shaft and Housing assembly, One-Way Clutch assembly, Knuckle Joint with Three Arms). The optimisation model considers the alternative manufacturing process selection, stack up constraints and Process precision limits. The problem considered has 5 objective functions, 5 variables and 3 constraints at the maximum. This stage addresses the tolerance design and development of a method to find optimal tolerances. This stage considers the important decision criteria for the optimal tolerance design of mechanical assemblies (Minimization of stack-up tolerance, total cost and quality loss). In this stage, two intelligent algorithms namely Elitist Non-dominated Sorting Genetic Algorithm (NSGA-II) and Multi Objective Particle Swarm Optimization (MOPSO) are proposed to obtain optimal tolerance for mechanical assemblies (Assembly A, Assembly B, Gearbox assembly, Shaft and Housing assembly, One-Way Clutch assembly, Knuckle Joint With Three Arms). Two methods (normalized weighting objective functions and average fitness factor) are combinedly used to select the best optimal solution from Pareto optimal fronts. Two multi objective performance measures namely solution spread measure and ratio of non-dominated individuals are used to evaluate the strength of Pareto optimal fronts. Two multi objective performance measures namely optimiser overhead and algorithm effort are used to find the computational effort of NSGA-II and MOPSO algorithms.

43 Problem Formulation The same optimization model used in the previous stage is considered here. The multicriterion optimisation problem is defined as follows: Objective functions Example A (Assembly A) Minimize: Z = Y = t +t +t 3 (5.38) n c t ixi ixi Z = C a b e asm i ixi ixi (5.39) Z 3 = Q A I L t ij 9T i (5.4) Example B (Assembly B) Minimize: Z = Y = t +t +t 3 + t 4 +t 5 +t 6 (5.4) n c t ixi ixi Z = C a b e asm i ixi ixi (5.4) Z 3 = Q A I L t ij 9T i (5.43) Example C (Gear box assembly) Minimize: Z = Y = t +t +t 3 + t 4 +t 5 (5.44) n c t ixi ixi Z = C a b e asm i ixi ixi (5.45) Z 3 = Q A I L t ij 9T i (5.46)

44 7 Example D (Shaft and housing assembly) Minimize: Z = Y = t +t +t 3 + t 4 +t 5 +t 6 +t 7 (5.47) n c t ixi ixi Z = C a b e Z 3 = Q asm i A ixi I L t ij 9T i ixi (5.48) (5.49) Example E (One-way clutch assembly) Minimize: Z = Y = Y t Y t Y t X X X (5.5) 3 3 n c t ixi ixi Z = C a b e Z 3 = Q asm i A ixi I L t ij 9T i ixi (5.5) (5.5) Example F (knuckle joint with three arms) Minimize: Z = t t (5.53) Y j j Z = Y 3t j t3 j (5.54) Z 3 = Y3 3t3 j t4 j t5 j (5.55) Z 4 = Casm CX C X C X C X C 3 X C 3 X C a b a b 4 X (5.56) 5 C X c t ij ( t ) ce c i ij (5.57) Z 5 = Q A I L t ij 9T i (5.58)

45 Constraints Example A (Assembly A) Assembly function: Y=X +X +X 3 (5.59) Stack-up condition: t +t +t 3.5 (5.6) Example B (Assembly B) Assembly function: Y=X +X +X 3 +X 4 +X 5 +X 6 (5.6) Stack-up condition: t +t +t 3 + t 4 +t 5 +t 6. (5.6) Example C (Gear box assembly) Assembly function: Y=X +X -X 3 -X 4 -X 5 (5.63) Stack-up condition: t +t +t 3 + t 4 +t 5.6 (5.64) Set-up reduction condition(s): t = t and t 4 = t 5

46 74 Example D (Shaft and housing assembly) Assembly function: Y= -X +X -X 3 +X 4 -X 5 +X 6 -X 7 (5.65) Stack-up condition: t +t +t 3 + t 4 +t 5 +t 6 +t (5.66) Set-up reduction condition: t 4 = t 6 Example E (One-way clutch assembly) Assembly function: Y= a cos [(X + X )/(X 3 - X )] (5.67) Stack-up condition: Y t Y t Y t3.349 X X X (5.68) 3 which further reduces to n t ij i T kasm (5.69) For a linear assembly Example F (Knuckle joint with three arms) Design functions: Y X b X (5.7) Y X 3b X a X b (5.7)

47 Y 3 X 4 X 3a X 3b X 5 75 (5.7) Tolerance stack up conditions: t t. (5.73) j j 3 j 3 j t t. (5.74) 3 3 j 4 j 5 j t t t. (5.75) Variables Process precision limits: t min ij t ij t max ij i= to n, j= to m i where, Y C asm QL = Assembly response function, = Total assembly manufacturing cost, = Quality loss function, i = Represents part dimension number (i =,a,b,3a,3b,4,5) j = Represents manufacturing process number (j =,,, 4). C ij (t ij ) = Cost of producing dimension x i maintaining tolerance t ij, by the process j m i = Number of the available alternative processes for producing dimension x i, n = Number of the component dimensions, T k asm = Permissible variation in the k th assembly dimension, known as assembly tolerance, t ij = Tolerance on the dimension x i produced by the j th process.

48 Numerical Example A few case studies involving both the linear and non-linear assemblies have been presented to explain the proposed methodology. The assemblies involving a simple dimension chain (for assemblies A, B, C, D and E) and three interrelated dimension chains (for assembly F) have been considered to make a comparison of the results obtained by the two methods possible. Details of example assemblies (number of dimensions, number of processes available for the manufacturing of each dimension, revised number of processes for the manufacturing of each dimension as applicable to the algorithm, parameters of cost functions, stack-up conditions, etc.) have been given in Tables Example assembly B, which is simply assembly A doubled, has been considered to study the effect of the problem size. Assemblies C, D, E and F have also been represented through figures. Assemblies A, B, C, D and F are linear assemblies, while assembly E is a non-linear assembly. Processes marked with an asterisk are fictitious processes, added to fulfil the requirements of the algorithm. Assemblies C, D and F involve a few dimensions that can be produced on the same machine and hence it is desirable to have the same value of associated tolerances for reducing the number of set-ups. The impact of all the toleranced dimensions must be considered in the formulation of the assembly manufacturing cost and the stack-up condition. Assemblies D and E involve a few vendor supplied components, the tolerances of which are considered fixed and do not count as decision variables.

49 77 Table 5. Assembly A (Singh et al 4b) Dimensions X X X 3 Process Parameters of cost function a b c * * * Table 5. Assembly B (Singh et al 4b) Dimensions Process Parameters of cost function a b c X,X * X,X * X 3,X *

50 78 Figure 5.7 Gearbox assembly Table 5.3 Assembly C: Gear box assembly (Singh et al 4b) Dimensions X, X X 3 X 4, X 5 Parameters of cost function Process a b c *

51 79 Figure 5.8 Shaft and housing assembly Table 5.4 Assembly D: Shaft and housing assembly (Singh et al 4b) Dimensions Process Parameters of cost function a b c X Vendor supplied (fixed tolerance t =.38; C = 5.) X X 3,X 7 Vendor supplied (fixed tolerance t 3 =.635; C 3 =5.) X 4,X * X *

52 8 Figure 5.9 One-way clutch assembly Table 5.5 Assembly E: One-way clutch assembly (Singh et al 4b) Parameters of cost function Dimensions Process a b c X (hub) = X (roll) =.86 Vendor supplied (fixed tolerance t =.635; C = 3.) X 3 (cage) =