U OPTIMUM TOLERANCE DESIGN SENSITIVITY RATIO ALGORITHM

Optimum Tolerance Design635tolerance design, some solutions are considered better than the others.Assignment of different values to the same variables may satisfy the constraints but they acquire different costs. The constraint satisfaction optimization problems (CSOP) are an extension of CSP where the optimal solutionis desired. CSOP is defined as follows:Definition 1A constraint satisfaction optimization problem is a quadruple ðV; D; C; FÞ,where V is a set of variablesD is a function mapping every variable in V to a set of possible values,D: V ! a set of possible valuesC is a set of constraints on an arbitrary subset of variables in VF is an optimization function mapping every solution to anumerical valueF: S ! a numerical value, S is the set of solutionsTOLERANCE DESIGN USING CONSTRAINT NETWORKSA careful and critical analysis of the literature has revealed that thetolerance allocation problems are not yet completely solved satisfactorilyusing conventional methods. The methods and models suggested by variousauthors can only handle certain types of simple problems. Solution ofcomplex tolerance design problems with many types of cost functions,assembly constraints, and large number of entities requires further researchand development of better methods. The current effort in this work is todevelop an algorithm, which can solve the tolerance design of any complexity. Recent studies have shown that constraint networks can be used tosimplify the complex tolerance design problems.Hierarchical interval constraint networks can be used to represent therelationship between the highest (assembly) level functional requirements andthe lowest level entities of any assembly (Yang et al. 1997). The functionalrequirements are decided by considering various factors, including customerfeedback and market demand. Each functional requirement of the assemblycan be represented as a mathematical expression (function) of a number ofattributes. For example, volume of a cylindrical vessel is a function of theinner radius and the effective inner height of the vessel. In this case, volume isthe functional requirement, whereas inner radius and height are the attributes. Attributes can also be described as functions of the mechanicaldimensions of the associated parts entities. For example, the inner radius of acylinder can be expressed as a function of the outer radius and the radialthickness. Similarly, the total height, base wall thickness, and the cylinderleg height are entities of the attribute inner height. These relationships can

636C. C. Yang and V. N. NaikanFIGURE 1. Representation of an assembly by an interval constraint network. (Direction of arrowsindicate forward propagation. Opposite direction is backward propagation.)be represented in a constraint network for tolerance analysis and synthesis.A contraint network is a diagram in which functional requirements, attributes, and entities are represented in circles called nodes of the diagram. Therelationships between entities and attributes and between attributes andfunctional requirements are represented by rectangles. Arcs are used toconnect nodes through the rectangles (constraints). Figure 1 illustrates aninterval constraint network (ICN) designed for tolerance design.The constraint functions represent the relationships between functionalrequirements and attributes and between attributes and entities. An intervalconstraint function is derived based on interval arithmetic. The intervalarithmetic is an extension of the real arithmetic and it deals with closeintervals X ¼ [xlow,xup], representing {xj xlow x xup}. Given an arithmeticoperation, , Z ¼ X Y is defined as:Z ¼ ½zlow ; zup ¼ X Y ¼ ½xlow ; xup ½ylow ; yup ¼ fx yjxlow x xup ; ylow y yup gConsistency of ICN for Tolerance DesignThe satisfaction of a constraint networks can always be described in termsof consistency of variables or constraints. The definition of consistencydepends on the applications of the constraint network. Hyvonen (1992)defined the satisfaction of interval constraint network in terms of the consistency of variables. The purpose of such definition is to refine the intervals ofthe variables in the interval constraint network as far as possible without losing

Optimum Tolerance Design637possible exact solutions of the constraints. For tolerance design, it is desirableto have the computed interval using the input intervals and the interval constraint function to be a subset of the assigned output interval. For example, thetolerances propagated from the attributes to a functional requirement throughthe corresponding interval constraint function must be a subset of the tolerance of the functional requirement. (fk ðAi ; . . . Aj; ; . . . ; An Þ Fk Þ. Therefore,the satisfaction of the interval constraint network for tolerance design isdefined in terms of the constraint consistency.The interval constraint function in ICN for tolerance design consists ofmultiple inputs and a single output function and is represented as a tripleCi ðU; k; fi ðÞÞ. U is the set of indexes for the input variables and k is the indexof the output variable for the constraint CI ; fi ðÞ is the constraint function.The definition of satisfaction of interval constraint network for tolerancedesign is as follows:Definition 2TA constraint Ci ðU; k; fi ðÞÞ is consistent if and only if j2U ð8vj 2 Vj jVj ¼ vj Þ;ð9vk jVk ¼ vk Þ such that Ci ðU; k; fi ðÞÞ is satisfied.Definition 3The ICN for tolerance design is satisfied if and only if all the constraints areconsistent.These definitions for consistency of constraints satisfaction of the ICNensure that the tolerances assigned to the entities satisfy the requirements ofthe functional tolerance of the assembly. In other words, the computed tolerance of functional requirements based on the input tolerances and consistentconstraints will be a subset of the actual functional tolerance requirements.Forward and Backward Propagation of ToleranceTolerance propagation is carried out to update the intervals (tolerances)in the network to make the interval constraints consistent. There are twotypes of tolerance propagations: forward propagation (FP) and backwardpropagation (BP). In FP, tolerances of input variables are propagatedthrough the constraints to obtain the tolerances of the output variables. FPwill detect the consistency of various constraints in the network. On the otherhand, in BP, tolerance of an output variable is propagated through theconstraint expressions to distribute it among the multiple input variables(such that the constraints are made consistent); the minimization of themanufacturing costs is also considered in the propagation.In this paper, FP is carried out to verify whether any of the functionaltolerances of the assembly is are already satisfied by the natural tolerances

638C. C. Yang and V. N. Naikan(readily available and most economical tolerance under the present processconditions) of the entities, i.e., tolerance analysis. This will provide a choice foreither eliminating the already satisfied functional tolerance requirement andthe corresponding entities (which do not affect other functional requirements)from further backward propagation or retaining them for BP. The first choicewill reduce the dimension of the model and therefore will be easier to solve.It is also logically correct to accept the first option because natural tolerance isthe most economical tolerance for any entity. (It is recommended that exercising the first choice be done carefully and only if it is appropriate to theprocess setup.) The FP is carried out with numerical values of the nominal andthe natural tolerances of the entities. In BP, the tolerances of functionalrequirements are algebraically propagated through the constraint functions toderive the algebraic functions of tolerances. These functions will relate theincrement of tolerances of assembly functions with that of the entities.Tolerance Analysis by Forward PropagationForward propagation (FP), as described in Appendix A, propagates thenumerical tolerances of entities through attributes to functions to determineif any of the constraints are consistent (satisfied). Entities, attributes, andfunctions for the satisfied constraint can be removed from further modeling.Those variables affecting the satisfied assembly function and not affecting theother functions may only be removed.Algorithm for tolerance analysis by forward propagation is below:SetF ¼ {Fi, i ¼ 1.n}Call FP, Get FFP ¼ {Fbnom-pp, Fbup-pp,Fblow-pp, i ¼ 1.n}Read Freq ¼ {Fbnom-req, Fbup-req, Fblow-req,i ¼ 1.n}For i ¼ 1.n Do{IF (Fbnom-pp ¼ Fbnom-req) AND (Fbuppp Fbup-req) AND (Fblow-pp Fblow-req)THEN g ¼ i, SetF ¼ {Fi, i ¼ 1.n, i 6¼ g}}SetA ¼ {Aj, j ¼ 1.r}For j ¼ 1.r Do {IF Fg ¼ f(Aj) AND (Fi, i ¼ 1.n,i 6¼ g) 6¼ f(Aj)THEN h ¼ j, SetA ¼ {Aj, j ¼ 1.r, j 6¼ h}}SetE ¼ {Ek, k ¼ 1.m}For k ¼ 1.m Do {IF AL ¼ f(Ek) AND (Aj, j ¼ 1.r,j 6¼ h) 6¼ f(Ek)THEN s ¼ j, SetE ¼ {Ek, k ¼ 1.m, k 6¼ s}}Write (SetF, SetA, Set E) define SetF as a Set of all assemblyfunctions FPP is the propagated values of F Freq is the required values of F If Ci is satisfied, remove Fi from SetF toupdate it define SetA as a Set of all attributes if Aj affects only Fg, remove it from SetA Define SetE as a Set of all entities if Ek affects only Ah, remove it from SetE Remaining sets of functions, attributes, andentities

Optimum Tolerance Design639Propagate the Tolerance of Functional Requirements Backward AlgebraicallyAfter removing the entities, attributes, and functional requirements of thesatisfied constraints, the algebraic relationships between the remaining entities and the remaining functional requirements are determined by backwardpropagation (BP).Step 1. Propagate the algebraic tolerances of functional requirements toattributes for all assembly functions or only for those remainingafter elimination in tolerance analysis. We get a system of algebraicexpressions, each representing the incremental tolerance of assemblyfunction in terms of the incremental tolerances of attributes.For i ¼ 1::n; i 6¼ g Do Constraint Ci in terms of Fi and related Aj s Fi ¼ fðAj Þ Taking the first order increment Fi DFi ¼ fðDAj ÞStep 2. Carry out the second level of BP (attributes to entities) for allattributes. We get another set of algebraic functions, each representing the incremental tolerance of attributes in terms of theincremental tolerances of entities.For j ¼ 1::r; j 6¼ h Do Constraint Cj in terms of Aj ; and related Ek s Aj ¼ fðEk Þ Taking the first order increment Aj DAj ¼ fðDEk ÞStep 3. Substitute the system of equations obtained in Step 2 into those inStep 1. We get a new system of equations, each representing theincremental tolerance of assembly functions in terms of the incremental tolerances of entities. Substitute the numerical value of themaximum required upper tolerance of all assembly level functions.(Observe the rule that if any positive increment of Ek results inmonotonic decrease of Fi, then DEk should be multiplied by ( 1) tocorrectly represent in the maximum upper tolerance expression.)For i ¼ 1::n; i 6¼ g DoReadðDFiup Þ Substituting DAj obtained in Step 3 into Step 2 DF1 ¼ fðDEk Þ Substituting the numerical value of the requiredf1 ðDEk Þ ¼ DFiupmaximum positive tolerance of Fi Step 4. Identify the redundant equation in the system of equations obtainedin Step 3 and eliminate them. The remaining equations are theconstraints of the optimization problem. If dth expression is redundant, update the constraint equations in Step 3 by eliminating it from

640C. C. Yang and V. N. Naikanthe system of equations. Repeat this for other redundancies. Theremaining equations are:F1 ðDEk Þ ¼ DFiupi ¼ 1::n; j 6¼ d; i 6¼ g; for all ‘d’ s and ‘g’ sThe actual number of constraints to the optimizing problem equals thenumber of original assembly-level nodes (functions) minus the number of assembly functions satisfied by FP and the number of constraints eliminated dueto redundancy.Cost of Manufacturing and TolerancesAs discussed earlier, the cost of manufacturing increases when thetolerance bands are reduced. Several models for relating cost of manufacturing with required tolerance, expressed as functions of the tolerances,have been suggested in the literature. These include a linear model, areciprocal model (cost of manufacturing is proportional to reciprocal oftolerance) (Chase and Greenwood 1988); a reciprocal-squared (Spotts1973); a reciprocal-powered (Sutherland and Roth 1975); an exponential(Speckhart 1972); and a combined exponential-reciprocal powered(Michael and Siddali 1981; 1982), among others. While all these models areempirical and based on experiences, it has been observed that the reciprocalsquared and the exponential models are more frequently used than othermodels. Chen (2001) has proposed an algorithm using neural networks forderiving the cost-tolerance relationship. The proposed algorithm in this papercan handle any form of cost model. In the illustrative example, anexponential cost model has been used for the cost function. The generalform of exponential cost model is as given below:Exponential cost tolerance relation is:CMðDEk Þ ¼ a þ b expð cDEk Þwhere DEk corresponds to the tolerance of kth part (or entity), a, b, c areconstants for a part (estimates from cost-tolerance data), and CM ðDEk Þcorresponds to the cost of manufacturing of kth entity with tolerance DEk .The objective function of the optimization problem is the total manufacturing cost function of the assembly. This is expressed as the sum of thecost functions of the individual entities. If any entity is eliminated in tolerance analysis, the cost function for that entity will not be included in theobjective function. Moreover, if any particular entity is used more than oncein an assembly, its cost function will be multiplied by the number of times it isused in the assembly. For example, suppose there are 4 similar drill holes inan assembly. The cost function of the drill holes should be multiplied by 4

Optimum Tolerance Design641while evaluating the total cost function of the assembly. All these costfunctions will be in terms of the tolerances of individual entities. Mathematically, the objective function can be expressed as follows:mXNk ðCMEk ÞCMASS ¼k¼1; k6¼0where CMASS equals the cost of manufacturing the assembly for requiredtolerances, CMEk equals the cost of manufacturing the kth entity, expressedas a function of its tolerance, and Nk equals the number of components withkth entity used in the assembly.THE RELATIVE SENSITIVITY RATIO ALGORITHMAn efficient algorithm, which has been developed based on the principlesof hierarchical interval constraint networks, is proposed in this paper foroptimum tolerance design. The constraints and the objective functions of theassembly are derived by the procedure explained in the previous subsections.This will provide the mathematical models to represent the tolerance designproblem. The procedure for solution of the rexponential problem is iterative,based on relative sensitivity ratio (RSR). RSR is defined as follows:relative sensitivity ratio ¼sensitivity of cost of manufacturingwith respect to entity tolerancesensitivity of functional tolerancewith respect to entity toleranceMathematically, relative sensitivity ratio for the ith assembly constraintfunction with respect to the kth entity, RSR (i,k), is derived as:RSRði; kÞ ¼@ðCMASS Þ@Ek@ðFt Þ@Ekwhere CMASS equals the cost of manufacturing of the assembly andmX¼CMðDEk Þ:k¼1In this algorithm, all the entities are initially assumed to have tolerancesequal to a very small number (say 0.0001 units). Then the RSR value isevaluated for each entity and the minimum value is found. The tolerance ofthat particular entity, having the minimum RSR value, is now increased byone small step (say 0.0001 units) and the functional tolerance requirementsare tested. If the constraints are not satisfied, the procedure is repeated untilall constraints are satisfied (within required accuracy). At any stage, if anyone constraint is satisfied, it will be removed from the list of constraints and

642C. C. Yang and V. N. Naikanthe present tolerance of entities of this constraint are stored as their optimumvalues. These entities may appear in other remaining constraints, but theirtolerances will not change from these values. In each iteration, the tolerancesare propagated through all unsatisfied constraint functions and entities.Selection of an entity for increasing its tolerance during the iteration is basedon the RSR value. This is based on the logic that increasing tolerance of thisparticular entity should result in the highest decrease of overall cost ofmanufacturing and lowest increase of the tolerance of assembly-level functions. Details of the algorithm are discussed in the following sections.The RSR AlgorithmStep 1. Define the accuracy required in solution.(For example, the accuracy required for each entity tolerance is0.0001 units.) accuracy ¼ 0.0001Step 2. Set the initial tolerance of each entity equal to the accuracy required.For (k ¼ 1.m), substitute DEk ¼ accuracyStep 3. Evaluate RSR for all constraint functions with respect to all entities.For (i ¼ 1.n)For (k ¼ 1.m)RSRði; kÞ ¼@ðCMASS Þ@Ek@ðFi Þ@EkStep 4. Evaluate the constraint functions for the present values of entitytolerances and evaluate the error.Forði ¼ 1 . . . nÞ CFðiÞ is the propagated value of DFi CFðiÞ ¼ fðDEk ÞErrorðiÞ ¼ CFðiÞ fðiÞ fðiÞ is the required tolerance of Fi Step 5. (A) If any of the constraints are satisfied (or the error is withinacceptable limits), remove this constraint from further consideration. The present values of the tolerances of those entities included inthis constraint are stored as their optimum values. These entitiesmay appear in other constraints, but their tolerances are now fixed.For i ¼ 1::nIf ððjErrorðiÞj ¼ eÞandðEk is included in CFðiÞÞÞ e is the maximumEoptðkÞ ¼ DEkallowable error EoptðkÞ is the optimalDEk pðiÞ ¼ pðiÞ þ i; pðiÞ is the list storingall the satisfied Fi qðkÞ ¼ qðkÞ þ k qðkÞ is the list storingall the satisfied Ek

Optimum Tolerance Design643(B) If all the constraints are satisfied or if all the entity tolerancesare stored as optimum, stop the iteration. The present values ofentity tolerances are the optimum.If (p(i) ¼ i) for all i, stop.(C) For those unsatisfied constraints do the following:(i) Evaluate RSR for unsatisfied constraintsFor (i ¼ 1.n)For (k ¼ 1.m)If ((i 6¼ pðiÞÞ and (k 6¼ qðkÞÞÞRSRði; kÞ ¼@ðCMASS Þ@Ek@ðFi Þ@Ek(ii) Find out the particular constraint and the particular entity(after eliminating the satisfied constraints and entities) forwhich the RSR is minimum.For (i ¼ 1.n)For (k ¼ 1.m)If

Literature on Tolerance Design The relationship between the functional requirements and entities of the mechanical part can be derived and expressed as F 1 ¼ fðE 1;E 2;.;E nÞ. Tolerance design consists of tolerance analysis and tolerance synthesis. In tolerance analysis, the goal is to ensure the tolerance of functional require-