Optimal Process Tolerance Balancing Based on Process

Int J Adv Manuf Technol(2003)21:501–507 Ownership and Copyright2003Springer-Verlag LondonLimitedOptimal Process Tolerance Balancing Based on Process CapabilitiesY.Gao1and M.Huang21Department of Mechanical Engineering,Hong Kong University of Science and Technology,Clear Water Bay,Kowloon,Hong Kong 2School of Mechanical Science and Engineering,Huazhong University of Science and Technology,Wuhan,Hubei,ChinaAn optimal approach for process tolerance balancing is presented.The new approach is based on process capabilities and is to be used in the stage of process planning.A nonlinear programming model is used to simultaneously optimise process tolerances of required operations.In the optimisation model, the objective function is to minimise the total manufacturing cost with different weighting ing the estimated stan-dard deviations of the dimensions and the manufacturing cost-tolerance functions,the constraint equations for the process tolerance chains and the manufacturing capability indices are established,together with a model for the economical tolerance bounds of machine tools.A practical example was used to verify the usefulness of the proposed approach.The results of the comparative study show that the proposed approach is more advantageous in relaxing tolerance requirements and in reducing scrap rates generally,compared with the existing methods.Keywords:Optimisation;Process capability;Tolerance balancing;Tolerance modelling1.IntroductionTolerancing is of significant importance for the functionality of mechanical products and the manufacturing cost of the parts. It is one of the key techniques in the integration of computer-aided design(CAD)and computer-aided manufacturing(CAM). In general,two types of tolerance are involved in conventional tolerance design,product tolerance and process tolerance.Pro-duct tolerance is designed to meet the functional requirements for a product[1–4].Process tolerance,however,is designed in terms of the available information of the parts to be pro-cessed and the production resources[5–15].The information Correspondence and offprint requests to:Dr Y.Gao,Department of Mechanical Engineering,Hong Kong University of Science andTechnology,Clear Water Bay,Kowloon,Hong Kong.E-mail:meygao@ust.hk of the parts to be processed typically includes size,shape feature,surface roughness,heat treatment requirements,etc. Manually determining the process tolerances is a tedious and time-consuming task that requires experience and skill. Furthermore,the process tolerances allocated to the parts to be machined can be quite different[9].Therefore,a systematic and rational approach for tolerance allocation is necessary.A number of techniques of dimensioning and tolerancing have been investigated[1–22].Optimal solutions to tolerance charts have been the most effective.They were developed to ensure that process dimensions and tolerances meet the assembly functional requirements or the blueprint(BP)specifi-cations[4–10].Linear programming[2,4,6–9,11,13,14]and nonlinear programming[1,12,18,19]have been used.Total manufacturing costs or the total blueprint or process tolerances have been included in the objective functions.Swift et al.[1]and Ngoi and Ong[2,4]introduced techniques for tolerancing for assembly.Zhang[18],Huang et al.[19,20], and Ngoi and Ong[21]presented methods for tolerancing in a concurrent engineering environment.Approaches for the establishment of dimension and tolerance chains have been discussed.He and Gibson[15],Ngoi and Tan[16],and Huang et al.[17]described the models and methods,by which geometrical dimensioning and tolerancing were dealt with sim-ultaneously.Ji[7,8]and Ngoi and Ong[9]proposed linear programming methods for allocating process tolerances based on tolerance charting techniques.Algorithms to determine the unknown working dimensions and process tolerances have been proposed,in which special tree representations were utilised.The dimensioning and toler-ancing chains can be generated by use of the path tracing techniques.Ji[10]proposed an algebraic approach for dimen-sion chain identification for process planning,where reverse dimensions can be identified by the use of the original tolerance chart.The forward dimensional chain can be obtained directly from the reverse dimensional chain through the use of the nominal dimensions.Wei and Lee[11]proposed a linear programming model to determine the process tolerance based on the manufacturing process capability,which takes into account the standard devi-ation of the machine that actually performs the operation.This502Y.Gao and M.Huangapproach can be used to simplify the typical procedures of tolerance chart balancing.A model based on the modified root-tree chart has been proposed.A nonlinear programming model has been examined[12],which can be used to design the process tolerances to minimise the loss of defective parts.This would permit prediction of the production loss at the process planning stage.To maximise the process capability indices of an operation, a fuzzy method for designing the process tolerances has been investigated[13].A method using a linear programming model has been presented to convert multi-objective formulation into a single objective.Chang et al.[14]proposed a method to design the process tolerances through the use of the grey theory.A standardised process tolerance is viewed as an inter-val of a grey number with lower and upper bounds,and the whitening of the grey number is equivalent to the allocation of the process tolerances.A linear programming model was established to maximise concurrently the process tolerance of each operation.It should be noted that the above research has been mainly to design the process tolerances through the use of linear or nonlinear programming models in which the objective functions were often related to the total manufacturing cost or the total process tolerances.Furthermore,the constraints were often the economical process tolerance intervals or the machining accu-racy of an operation.The advantages of the above methods are that they are easy to use as the manufacturing processes are simplified in linear models.The solutions,however,are approximate and,in many cases,would have significant errors, as the real production processes can be nonlinear and are more complex.To deal with the above nonlinear problem,a new optimal approach is proposed for the allocation of process tolerances, where the nonlinear behaviour of processes are significant.In the proposed approach,the whole manufacturing process will be considered in order to be more comprehensive,and a nonlinear programming optimal model will be used to include a new type of constraint,which should be more appropriate, compared to the models of the existing methods.This would improve the rationality of the process tolerances of the opti-mal solution.The process probability index of an operation will be presented as a new type of constraint.The objective function is to minimise the total manufacturing costs with different weight factors to take into account the differences in the costs of different processes.The standard deviation of an operation to produce the actual dimension is related to the process capability of the operation.The distribution of the actual dimension,and the actual dimension deviation,is affected by a number of factors related to the operation.It is necessary that the variation zone of the actual dimension of an operation be confined to a range.Overestimation or underestimation of process tolerances should be avoided.If the process tolerances are loose,the blueprint tolerances would be relaxed,and as a result,the functional specifications of the product could be reduced.Conversely,if the process tolerances are unnecessarily tight,the manufacturing costs could be increased.This requires that the process tolerances are properly balanced during the process planning.Process planning develops a suitable production sequence with specified tolerances,to process the parts economically. Process tolerances are often designed to minimise the total manufacturing costs based on the fact that,for many machine tools and manufacturing processes under normal operation con-ditions,a larger tolerance would help to reduce the manufactur-ing costs.The process tolerance assigned to an operation may be different from planner to planner.In many actual manufacturing situations,two cases are often encountered.Firstly,tolerances allocated to some operations may be too tight.This could result in demanding stricter operations,and so,the manufacturing costs would increase. Secondly,process tolerances for some operations may be too loose.In this case,the functional requirements of the blueprint dimensions would be lost,causing the reduction of values to be greater than the one due to the reduction in the manufactur-ing cost.It can be seen that a systematic tolerance assignment method is needed to take into account of the whole production process and to give deep insight into the tolerance techniques. In general,the actual deviation of a process dimension could be affected by factors such as set up,vibration,wear of the cutting tools,cutting force,change in temperature,and skill of the operators,etc.In the stage of the process planning,the information of dimension,shape,material,surface roughness, heat treatment,and other functional requirements of the parts to be manufactured,should be taken into accounted.Their effects on the distributions of the actual dimensions have to be estimated.This could be realised through experiments or through experience.2.Distribution of Process DimensionsUnder normal operation conditions,the distribution of a process dimension can be regarded as conforming to a Gaussian distri-bution,due to the contributions by many factors.For simplicity, assume that the distribution centre of the dimension coincides with the centre of the tolerance zone.So,the probability density distribution function of the dimension can be expressed as: f(x i)=1ͱ2␲␴i eϪ12(x iϪµi)2␴2i(1)where x i and␮i are the actual and the nominal value of thei th process dimension,respectively,␴iis the standard deviation of the i th process dimension,i=1,2,…,n,and n is the total number of process dimensions.The standard deviation␴i represents the precision of the process and the machine tool in realising the dimension,and hence the process capability.To reflect the relationship between the design expectation in terms of tolerance T i and the capa-bility of the process in terms of standard deviation␴i,process capabilities indices can be used,which are given as:C p i=T i6␴i(2) where C p i and T i are the process capability indeces,and the total process tolerance of the i th process dimension,respect-ively.For simplification in numerical assessment,a normalis-ation process may be used as:Optimal Process Tolerance Balancing503z i=x iϪµi␴i(3)to allow the standardised normal distribution be used for the data processing,where z i is the standardised transform of the dimension x ing Eq.(3),Eq.(1)can be rewritten as:f(z i)=1ͱ2␲eϪ12z2i(4)The probability of the dimension x i being within the interval [Ϫz i␴i,z i␴i]can be obtained as:P(Ϫz i␴iՅx iՅ+z i␴i)=͵+z iϪz i1ͱ2␲eϪ1u2du(5)To ensure a given probability level P(z i),the width of the distribution zone of the process dimension⍀i must satisfy ⍀i=2z i␴iՅT i=2t i(6) where t i is the half width of the bilateral process tolerance for the i th process dimension x i,and T i=2t i.From Eqs(2)and(6),z iՅ3C p i(7) It can be seen that the process capability index C p i will affect the probability P(z i)and hence the scrap rate␥i of the process dimension x i or of the i th process.3.Optimal Tolerance BalancingThe task of the process planning is to determine the best manufacturing tolerances to ensure minimum costs.For this purpose,the objective function of the optimisation problem can be given as:min C=͸n i=1w i c i(8)where w i and c i are the relative cost weighting factor and the manufacturing cost of the i th process dimension,respectively, i=1,2,…,n,and n is the total number of process dimension or operations.For the manufacturing cost,the cost–tolerance relationships are applied.To accommodate the more complex situations,a nonlinear cost–tolerance relationship may be used as:C i=a+b/T e i(9) where C i and T i are the manufacturing cost and the process tolerance of the i th operation,respectively,a and b are the coefficients for C i and e is an exponent.For different processes and design features,the coefficients and the exponent may have different values.For example,it may be assumed,for typical internal surfaces,a=7.2, b=0.015,and e=1.727[18].For external surfaces,a=3.5, b=0.006,and e=1.87[18].For carburising,a=9.5,b=0.02, and e=1.65[18].It should be noted that the relative cost factor w i represents differences in the cost of different machine tools or processes.Generally,the cost-tolerance model can be obtained through a regression of the actual data between manufacturing costs and process tolerances[18].There are several kinds of cost-tolerance models,such as the exponential model,the reciprocal power model,and combinations of them.The best solution to the cost-tolerance model,however,is to use the actual data from a database acquired through practical tests.The model used in this project(Eq.(8))is advantageous in terms of simplicity,as the results can be obtained more con-veniently.The use of the model should not affect the investi-gation into the characteristics of the probability density distri-bution of the process tolerances in this project.In addition, the determination of the relative cost weighting factor w i is a difficult task and the value will have certain effects on the final optimal solution of the process tolerances[11].In this project,it was found,based on the results of the testing,that the effects were limited.In order to generate the constraint Eq.set,the technical issue of tolerance accumulation has to be solved.This is due to the misalignment between different kinds of references,such as those used in design,setup,operation,and measurement. For simplification,it is assumed that realignment is com-pleted properly in the case of a change of setup.So the setup tolerance is not considered in the investigation.Using the worst-case model,the tolerance stackup can be used as the tolerance chain constraints and can be expressed as:͸ni=1␭i T iՅT BP j j=1,2,...,k(10)where T BP j is the blueprint tolerance or the design tolerance of the j th working dimension or operation,k is the number of blueprint tolerances of the workpiece,␭is the tolerance selec-tion coefficient.If a tolerance is involved,␭=1,otherwise,␭=0.n is the total number of operations.The machining accuracy constraints can be expressed as T iՅT max i,where T max i is the maximum tolerance value related to the process or the machine tool.For different operations,the importance and the functional role may be different from the part to be processed.Therefore, the process capability index C p i may be used at different levels. For example,for a rough cut,C p i can be set at a lower level. For thefinish cut,however,C p i can be set at a higher level. Based on this,the process capability constraint Eq.can be expressed as C pmin iՅC p iՅC pmax i,where C pmin i and C pmax i are the lower and upper bounds of the index C p i,respectively.4.Testing and DiscussionIn order to validate the proposed nonlinear optimal model and to facilitate a direct comparison,an example developed by Eary and Johnson[5]and Ngoi[6]will be used[11,13,14]. The workpiece is to be processed with10operations including turning,milling,carburising,and grinding.Figure1[13]shows the workpiece,and Table1[13]lists the operations with refer-ence surfaces and processed surfaces indicated.It is to be noted that all capital letters represent the workpiece surfaces.In addition,B c and G c are used to indicate the504Y.Gao and M.HuangFig.1.Tolerances of the workpiece for testing.Table 1.Process planning for the workpiece.Operation Code Reference Processed Process or Working number surface surface machine tool dimension (mm)110A G Lathe 1.970220a G B Lathe 1.940320b G C Lathe 1.780430a B F Lathe 1.730530b B D Milling 0.260640B G 1Grinding 1.930750a G 1B 1Grinding 1.920850b B B c Carburising and 0.025hardening960G G c Carburising and 0.025hardening 1070FELathe 0.100carburised surfaces of B and G,where B 1and G 1are thesurfaces generated on B and G [13,14],respectively.To derive the process tolerance chains,the information given in Table 1can be used to draw a rooted tree representation,as shown in Fig.2[13].The arrows represent the relevant machining operations.The surfaces that the plain ends of the arrows are connected to are reference surfaces.The surfaces that the arrowheads connected to are surfaces to be ing Fig.2[13],the tolerance chains of the processes can be conveniently obtained.The tolerance chains were obtained based on the fact that the resultant tolerances should be less than,or equal to,the ones in the blueprint tolerance speci fi-cations [13].Assume that the machining accuracy of the operation is set at 0.005mm for the lathe and the milling operation,0.002mm for the grinder,and 0.01mm for the carburising process [6].Based on the figures,the bilateral maximum tolerance bonds of the processes and the machine tools were established.The standard deviations of the processes depend on the type of surface being processed,the dimension,and the machine tool being used.For example,the standard deviations of the operations involving grinding,␴6and ␴7,should besmallerFig.2.Rooted tree representation of the process tolerences.D i –dimension of the process i ;T i –process tolerance;DA,DB,DC,DE,DF,DG –blueprint dimensions;T BP j –blueprint tolerances.than the ones for the other operations (Fig.1and Table 1[13]).Based on these considerations,the standard deviations were obtained as ␴1=0.002mm,␴2=␴3=0.001mm,␴4=␴5=0.0008mm,␴6=␴7=0.0003mm,␴8=␴9=0.003mm,␴10=0.0008mm [11].The upper and the lower bounds of the process capability indices are set at C pmin =2/3and C pmax =1.0,respectively.In order to minimise the total manufacturing cost,the more critical processes should be given higher weightings.In addition,the operation of the same process should have the same weighting factor.Based on the above considerations,the weighting factors are assumed as w 1=w 2=w 3=w 4=w 10=1.0for the turning operation,w 5=1.5for the milling operation,w 8=w 9=3.0for the carburising operation,and w 6=w 7=2.0for the grinding operation.For the operations,Eq.(9)can be used to represent the cost –process-tolerance relationships.Having the above results,the complete optimal mathematical model can be derived as:min C =͸10i =1w i c iwhere w 1=w 2=w 3=w 4=w 10=1.0,w 5=1.5,w 8=w 9=3.0,and w 6=w 7=2.0,as analysed previously.Based on Eq.(9),the cost –tolerance relationships were given as:c 1=3.5+0.006/T 11.87c 2=3.5+0.006/T 21.87c 3=7.2+0.015/T 31.727c 4=3.5+0.006/T 41.87c 5=7.2+0.015/T 51.727c 6=3.5+0.006/T 61.87c 7=3.5+0.006/T 71.87c 8=9.5+0.020/T 81.65Optimal Process Tolerance Balancing505c9=9.5+0.020/T91.65c10=3.5+0.006/T101.87Based on the standard deviations obtained previously, 6ϫ0.002C p1=T16ϫ0.001C p2=T26ϫ0.001C p3=T36ϫ0.0008C p4=T46ϫ0.0008C p5=T56ϫ0.0003C p6=T66ϫ0.0003C p7=T76ϫ0.003C p8=T86ϫ0.003C p9=T96ϫ0.0008C p10=T10The optimisation is subject to the following constraints:1.Process tolerance constraints,3(0.003C p9+0.001C p2+0.0003C p6)Յ0.01(T BP1/2) 3(0.0008C p5+0.0003C p6+0.0003C p7)Յ0.005(T BP2/2) 3(0.001C p3+0.001C p2+0.0003C p6)Յ0.005(T BP3/2) 3(0.0008C p4+0.0003C p6)Յ0.005(T BP4/2) 3(0.0003C p7)Յ0.001(T BP5/2) 3(0.003C p8+0.0003C p6+0.0003C p7)Յ0.01(T BP6/2) 3(0.0008C p10)Յ0.002(T BP7/2) 2.Machining precision constraints,3ϫ0.002C p1Յ0.005(lathe)3ϫ0.001C p2Յ0.005(lathe)3ϫ0.001C p3Յ0.005(lathe)3ϫ0.0008C p4Յ0.005(lathe)3ϫ0.0008C p5Յ0.005(milling)3ϫ0.0003C p6Յ0.002(grinding)3ϫ0.0003C p7Յ0.002(grinding)3ϫ0.003C p8Յ0.01(carburising)3ϫ0.003C p9Յ0.01(carburising)3ϫ0.0008C p10Յ0.005(lathe)3.Process capability index constraints,2/3ՅC p iՅ1.0,i=1,2,…,10The objective function is obtained as C=4427.46,with t1= 0.005mm,t2=0.002mm,t3=0.0021mm,t4=0.0024mm, t5=0.0024mm,t6=0.0009mm,t7=0.0009mm,t8= 0.00819mm,t9=0.00710mm,and t10=0.002mm (Tables2–4).Table2.Final process tolerances and scrap rates.No.Code C p i z i␴i(␮m)t i(␮m)␥i(%)1100.833 2.525 1.24 220a0.667 2.01 2.0 4.56 320b0.700 2.11 2.1 3.58 430a 1.00030.8 2.40.27 530b 1.00030.8 2.40.27 640 1.00030.30.90.27 750a 1.00030.30.90.27 850b0.911 2.7338.190.64 9600.789 2.3737.10 1.78 10700.833 2.50.82 1.24Table2summarises thefinal process tolerances t i and the related scrap rates␥i.It shows that the scrap rates for all of the operations are less than 4.56%(Table2).In addition, 7out of the10operations are estimated to yield a scrap rate␥iՅ1.24%.To demonstrate the effectiveness of the proposed method,a comparison with the existing methods has been made and the results are as shown in Table3,wherefive other existing methods,by Eary and Johnson[5],Ngoi[6],Wei and Lee [11],Lee et al.[13],and Chang et al.[14],respectively,have been used to generate the results for comparison(Table3).It can be seen that the proposed approach is advantageous compared with the existing methods.It is able to relax the process tolerance requirements(Table3).Although three toler-ances in Eary and Johnson’s method[5],four tolerances in Ngoi’s method[6],five tolerances in Wei and Lee’s method [11],five tolerances in Lee et al.’s method[13],and four tolerances in Chang et al.’s method[14]are slightly larger (Table3(a)),the total acceptable probability of the workpiece has been improved or at least maintained at the similar levels (Table3(b)).It is noted that although the scrap rate of Lee et al.’s method[13]is slightly lower(Table3(b)),the sum of the process tolerances is larger than that of the proposed method(Table3(a)).The above results demonstrate that the proposed method has, in general,provided improvements over the existing methods. In particular,the proposed method is able to increase generally the overall acceptable process probability of the workpiece due to the generally reduced scrap rates.In order to establish the effects of the relative cost weighting factor on thefinal process tolerances,five cases of weighting factors were investigated(Table4).Larger weightings were in turn given to a particular type of machining process(Table4). It can be seen that,although the weighting factors were varied over a wide range infive cases to emphasise the processes differently(Table4),most of thefinal process tolerances did not change,and only three process tolerances,t2,t3,and t9, had small changes.The results demonstrated that the process tolerances obtained were quite consistent and the proposed approach is quite stable.5.ConclusionsProcess tolerances have influences not only on manufacturing costs,but also on the achievement of the required specifications of a product.The existing research work has not dealt with the more complex nonlinear situations of manufacturing pro-cesses.For this,skills in process planning and extensive experi-ence of manufacturing processes are required in order to make process tolerances assigned to each process dimension more economical and rational.To deal with this issue,a nonlinear, and thus more rational,optimal process tolerance allocation approach is proposed.The new approach is to optimise process tolerances based on manufacturing capability indices.In addition,the economical tolerance bounds of machine tools are utilised to permit improved tolerance balancing.The results of the testing and the results of a comparison with the existing methods show that the proposed approach is quite stable and506Y.Gao and M.HuangTable3.Optimal solutions compared with other methods.(a)Process tolerances(␮m).No.Code Reference Processed[5][6][11][13][14]Proposed surface surface approach110A G255555220a G B22 2.17 2.2 2.3 2.0320b G C22 2.17 2.2 1.9 2.1430a B F44 3.12 3.12 2.5 2.4530b B D23 2.48 2.53 2.8 2.4640B G1110.650.650.690.9750a G1B1210.850.90.90.9850b B B c788.498.458.48.19960G G c77 6.72 6.5 6.97.11070F E222222(b)Scrap rates(%).No.Code[5][6][11][13][14]Proposed approach 11031.7 1.24 1.24 1.24 1.24 1.24220a 4.55 4.553 6.43 1.98 4.56320b 4.55 4.553 2.08 4.77 3.58430a00000.160.27530b 1.240.180.20.11600.276400.090.0930.56 1.980.27750a00.090.230.080.240.27850b 2.140.6940.230.6530.510.64960 2.140.214 2.5 1.47 1.98 1.781070 2.140.214 1.24 1.24 1.24 1.24 Overall scrap41.9313.9513.7813.1813.3813.35rate(%)Table4.Effects of the relative cost weighting factor on process tolerance.No Code Case1Case2Case3Case4Case5w i C pi t i(␮m)w i C pi t i(␮m)w i C pi t i(␮m)w i C pi t i(␮m)w i C pi t i(␮m)11010.833510.833510.833540.8335100.8335 220a10.667 2.010.700 2.110.667 2.040.700 2.1100.667 2.0 320b10.700 2.110.667 2.010.700 2.140.667 2.0100.700 2.1 430a1 1.000 2.41 1.000 2.41 1.000 2.44 1.000 2.410 1.000 2.4 530b 1.5 1.000 2.41 1.000 2.42 1.000 2.41 1.000 2.41 1.000 2.4 6402 1.0000.91 1.0000.94 1.0000.91 1.0000.91 1.0000.9 750a2 1.0000.91 1.0000.94 1.0000.91 1.0000.91 1.0000.9 850b30.9118.1910.9118.1950.9118.1910.9118.1910.9118.19 96030.7897.1010.7787.0050.7897.1010.7787.0010.7897.10 107010.833 2.010.833 2.010.833 2.040.833 2.0100.833 2.0 C44272596789648219263is able to provide improvements in acceptable process prob-ability,as the scrap rates are reduced,and therefore the method is advantageous in comparison with the existing methods. 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