Constraints. Sirisha. Sep. 6-8, 2006.

Transcription

1 Towards a Better Understanding of Equality in Robust Design Optimization Constraints Sirisha Rangavajhala Achille Messac Corresponding Author Achille Messac, PhD Distinguished Professor and Department Chair Mechanical and Aerospace Engineering Syracuse University, 263 Link Hall Syracuse, New York 13244, USA messac@syredu Tel: (315) Fax: (315) Bibliographical Informationn Rangavajhala, S, and Messac, A, "Towards a Better Understanding of Equality Constraints in Robust Design Optimization," 11th Multidisciplinary Analysis and Optimization Conference, Portsmouth, Virginia, Paper No AIAA , Sep 6-8, 2006

2 11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference 6-8 September 2006, Portsmouth, Virginia AIAA th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 6 8 Sep 2006, Portsmouth, Virginia Towards a Better Understanding of Equality Constraints in Robust Design Optimization Sirisha Rangavajhala and Achille Messac Rensselaer Polytechnic Institute, Troy, NY, 12180, USA Abstract Equality constraints in deterministic problems pose strict limitations on design feasibility because of the exactitude associated with such constraints Equality constraints in robust design optimization (RDO) problems can be classified into two types: (1) those that must be satisfied regardless of uncertainty, examples include physics-based constraints, such as F = ma, and (2) those that cannot be satisfied because of uncertainty, which are typically designer-imposed, such as dimensional constraints Our goal is to maintain design feasibility under uncertain conditions to exactly satisfy physics based equality constraints, and to satisfy designer-imposed constraints exactly or as closely as possible Whether or not a particular equality constraint can be exactly satisfied depends on the nature of the design variables that exist in the constraint In this context, the contribution of this paper is two-fold First, we present a rank-based matrix approach to interactively classify equality constraints into the above two types Second, we present an approach to incorporate designer s intra-constraint and inter-constraint preferences for designer-imposed constraints into the RDO formulation Intra-constraint preference expresses how closely a designer wishes to satisfy a particular constraint, in terms of its mean and standard deviation A designer may express inter-constraint preference if satisfaction of a particular designer-imposed constraint is more important than that of another In other words, a designer might desire higher constraint satisfaction for some equality constraints, even if it is at the expense of lower constraint satisfaction for other equality constraints The above discussed constraint satisfaction preferences give the designer the means to explore design space possibilities; and entail interesting implications in terms of decision making An example is provided to illustrate the proposed approach I Introduction Uncertainty is present in all real life engineering design problems Robust design optimization (RDO) approaches 1 9 attempt to minimize the effects of uncertainties on the design reliability and performance Uncertainty in design can exist in several forms, which include: (1) manufacturing tolerances, (2) material properties, (3) modeling uncertainty, and (4) fluctuating operating conditions Most RDO problems are formulated by appropriately incorporating uncertainty into the corresponding deterministic optimization problems The resulting solution in RDO highly depends on how the objective function and the constraints of the corresponding deterministic problem are modified to account for uncertainties In particular, equality constraints that exist in the deterministic problem impose strict limitations on the solution feasibility, and must be carefully formulated into the RDO problem In our past research, we explored the challenges and implications associated with a careful formulation of equality constraints under PhD Candidate, Department of Mechanical, Aerospace, and Nuclear Engineering, AIAA Student Member Professor, Department of Mechanical, Aerospace, and Nuclear Engineering, AIAA Associate Fellow, Corresponding Author messac@rpiedu Copyright c 2006 by Achille Messac Published by the, Inc with permission 1 of 18 Copyright 2006 by Achille Messac Published by the, Inc, with permission

3 uncertainty 10,11 In this paper, we further examine equality constraint formulation in RDO problems, with the added perspective of flexibility to the designer in terms of constraint classification and formulation We begin by providing a brief introduction and pertinent literature survey In most RDO problems, the design variables are typically modeled as random variables to account for uncertainty The objective function and the constraint functions then become functions of random variables, and are hence random variables themselves Strict satisfaction of the equality constraint is a contentious issue because of this probabilistic nature On the other hand, equality constraints representing physical laws of nature, such as static or dynamic equilibrium conditions, must be identically satisfied in order to maintain design feasibility, regardless of uncertainty Satisfying equality constraints at their mean values, relaxing the equality constraint, and elimination of the equality constraint through substitution 19 are approaches that are typically followed in RDO problems In our previous work, 10,11 we present a detailed study of equality constraint formulation for single and multiobjective RDO problems We present equality constraints in RDO problems as classified into two types: 10,11,20 (1) Type S, those that must be satisfied, regardless of the uncertainty present in the problem, and (2) Type S, those that cannot be satisfied, because of the uncertainty present in the problem In deterministic optimization problems, feasibility is possible only when the equality constraints are exactly satisfied When the constraints are transformed into the robust domain, the goal of maintaining the design feasibility translates to satisfying as many equality constraints as possible, and as closely as possible Whether or not an equality constraint can be satisfied depends on the nature of design variables (eg, independent or dependent) that exist in that constraint In our previous work, 10 we outline propositions based on which one can classify constraints into Type S and Type S For large-scale problems with several equality constraints, the classification of constraints can be challenging; a linearization based classification is also discussed in Ref 10 In this approach, a given set of equality constraints is linearized about a point of interest, and arranged in a matrix form as a linear system of equations An equation ordering scheme is implemented to classify equality constraints Once the equality constraints are classified, the next task is to appropriately formulate the equality constraints into the RDO problem Type S constraints must be formulated so that they are exactly satisfied under uncertain conditions Such equality constraints can be potentially eliminated by substituting for a dependent variable present in the constraint 10,19 For Type S constraints, we use a so-called approximate moment matching approach, 10,11 where we attempt to satisfy the equality constraint as closely as possible We restrict the mean of Type S constraints to be as close to zero as possible, and their standard deviations to be as small as possible 10,11 Figure 1 illustrates the approximate moment matching approach discussed above The optimization formulations for the approximate moment matching approach for single and multiobjective problems are given in Refs, 10,11 and are later discussed in detail The approximate moment matching approach results in a multiobjective formulation to formulate equality constraints, which entails interesting implications 11 in terms of decision making under uncertainty A Observations from the Literature Survey Motivation We observe from the literature that, while the previous work 10,11 provides an important understanding of the challenges and implications associated with equality constraints in RDO problems, associated critical issues need further examination The classification approach presented in Ref 10 uses a linearization-based technique to classify equality constraints into the above discussed two types This approach works well if the designer has prior knowledge regarding the nature of the design variables If, for example, the designer wishes to explore what-if scenarios in terms of variable types (ie, independent and dependent), the linearization based approach presented in Ref 10 might not be suitable Also, in some cases, the designer might wish to exactly satisfy a particular equality constraint, and define the type of the associated design variables In such cases, the classification of the constraints governs the classification of the variables The linearization based approach presented in Ref 10 does not provide a systematic way to do so There is a need to make the constraint classification process more interactive, where the designer has more control over the process than what is offered in Ref 10 2 of 18

4 Satisfying Constraint at the Mean µ h = 0 possibly high σ h 0 h(x) Approximate Moment Matching method mean close to zero low standard deviation 0 Mean tolerance h(x) Figure 1 Approximate Moment Matching Formulation for Type S Constraints Another issue that requires further consideration is the need to incorporate intra-constraint and interconstraint preferences in the RDO formulation Intra-constraint preference expresses how closely a designer wishes to satisfy a particular constraint, in terms of its mean and standard deviation A designer may express inter-constraint preference if satisfaction of a particular designer-imposed constraint is more important than that of another In other words, a designer might desire higher constraint satisfaction for some equality constraints, even if it is at the expense of lower constraint satisfaction for other equality constraints Intraconstraint preference can be expressed using the approximate moment matching formulation presented in Ref, 10 which can be further enhanced to incorporate inter-constraint preferences In this paper, our objectives are two fold First, we propose an interactive equality constraint classification approach, where the designer can exercise appropriate control over the constraint classification process For example, the designer can choose to make a particular constraint Type S; in which case the proposed approach ensures that the required mathematical definitions of the underlying variables are appropriately made Second, we present an RDO formulation that expresses intra-constraint and inter-constraint preferences explicitly The paper is organized as follows In Section II, we present the mathematical background needed for the interactive classification approach, which is presented in Section III Intra and inter constraint preferences are discussed in Section IV In Section V, we present a numerical example to illustrate the proposed approaches, and conclude the paper with a summary in Section VI II Equality Constraints and the Nature of Design Variables In this section, we present the necessary mathematical background needed to explain the interactive classification proposed in this paper We present the assumptions made in the study, and discuss the propositions used to classify equality constraints into the two types discussed earlier As discussed earlier, the uncertainty in the design variables in RDO problems is accounted for by modeling them as random variables We observe that two types of random variables are possible in RDO problems: (1) First, those that are independently distributed, with a statistical definition that is independent of that of other variables Such variables are independently prescribable by the designer, typically in terms of their means and standard deviations We refer to such variables as prescribable variables (2) Second, those that are jointly distributed with other variables Such variables are not independently prescribable by the 3 of 18

5 designer, and hence we refer to them as non-prescribable variables We assume that the statistical dependence of such variables on the other independent variables is given to us in the form of the equality constraint In other words, the equality constraint gives the functional form of the statistical dependence between the independent (prescribable) and dependent (non-prescribable) variables The following two important propositions are made based on which equality constraints can be classified: 10 Proposition 1: An equality constraint belongs to the Type S class if there exist at least one available non-prescribable variable and at least one prescribable variable in that constraint function Let us explain why the above proposition holds The existence of at least one non-prescribable and one prescribable variable in the equality constraint implies the following The dependent (non-prescribable) variable in the constraint, which does not have its own statistical definition, can be defined as a function of the independent (prescribable) variable(s) In other words, the statistical dependence of the non-prescribable variable on the prescribable variable(s) is defined by the equality constraint We now explain the term availability in the above proposition Consider the hypothetical case where several equality constraints contain the same and only non-prescribable variable Since the non-prescribable variable can have only one mathematical definition, it can be only used to satisfy one equality constraint Once a non-prescribable variable has been defined using a particular constraint, it is considered unavailable for the other constraints Once we establish that Proposition 1 holds for a given equality constraint, we could potentially eliminate the constraint by substituting for the available non-prescribable variable To illustrate the above discussion, we consider a simple example Two equality constraints for a cylinder of radius, R, and height, H are given as follows: volume, πr 2 H = 30 m 3, and total surface area, 2πRH+2πR 2 = 50m 2 Assume that H is a prescribable variable and R is a non-prescribable variable The variable R can have only one independent mathematical definition in terms of H, either using the volume constraint, or using the area constraint Say we choose to exactly satisfy the volume constraint Then, the variable R is unavailable to exactly satisfy the area constraint The variable R can be eliminated from the formulation using the volume constraint: R = 30 πh We now discuss the second proposition used to classify equality constraints Proposition 2: An equality constraint belongs to the Type S class if it consists of only prescribable variables Prescribable variables have a-priori definitions of their own, and cannot be again defined otherwise, eg, using an equation A function of prescribable random variables, which is a random variable, cannot exactly be equal to a constant Since such constraints cannot be exactly satisfied, our interest lies in satisfying them as closely as possible under uncertainty 10 In small-scale problems, classifying an equality constraint based on the above propositions might not be complicated However, in large-scale problems with nonlinear equality constraints, it might be difficult to understand the coupling between the prescribable and the non-prescribable variables In such cases, judging the satisfiability status of an equality constraint can be challenging In the following section, we discuss a matrix based classification approach that can be used to understand the satisfiability relations between variables and equality constraints The approach presented in the next section is particularly suitable for large-scale problems The method relies on matrix operations, and can be easily implemented using a computer program III Interactive Classification of Equality Constraints In this section, we explain the proposed interactive classification approach We first setup the mathematical framework required, and proceed to discuss how the designer can interactively change the nature of the variables and constraints The discussion is carried out with the help of an example, to promote ease of understanding of the proposed approach For some problems, the designer might have a good understanding of which variables are independently 4 of 18

6 prescribable and which are not As seen from the discussion in the previous section, defining a variable as prescribable or non-prescribable has important implications in terms of the satisfiability of equality constraints The classification of variables (into prescribable or non-prescribable) and classification of equality constraints (into Type S and Type S), therefore, generally depend on each other In this context, a designer has two options for variable and constraint classification: (1) choose the definition of the variables (prescribable or non-prescribable) first, and obtain a corresponding classification of the constraints (Type S and Type S) as per the propositions, or (2) choose the classification of constraints first, and obtain a corresponding classification of the variables that satisfies the propositions We note, however, that whichever option is taken, some iteration might be required to explore the choices possible Our previous work 10 provides the mathematical framework to implement the first option assume a definition for the variables, and classify the constraints accordingly The second option discussed above is useful when the designer has definitive preferences in terms of which constraints should be exactly satisfied, and which need not be In such cases, the nature of the variables must be appropriately defined such that the propositions are not violated In the approach presented next, we combine options 1 and 2 into one unified approach Using the proposed approach, the designer can exercise choice 1, or choice 2, or an interplay between the two We present a mathematical framework under which the designer can interactively explore the implications of making each design variable prescribable or non-prescribable We outline a step-wise procedure that can be used to study a set of what-if scenarios for classifying variables and, thereby, constraints We begin by providing the nomenclature used in the proposed approach Let X = {X 1,X 2,,X n } T be the vector of random design variables, where n is the number of design variables Let m be the number of equality constraints in the deterministic optimization problem that must be appropriately classified as Type S or Type S constraints The proposed interactive classification is discussed next A Proposed Approach We propose the following procedure for the interactive variable and constraint classification In this procedure, we define a matrix to represent the existence of variables in the set of equality constraints We perform certain operations on the resulting matrix to interactively classify equality constraints and variables In order to illustrate the proposed idea, the discussion below is carried out with the help of the following example Example: Let X = {X 1,X 2,,X 12 } T be the vector of random design variables, n = 12 At this point, we do not make any assumptions regarding the variables, being prescribable or non-prescribable A set of seven equality constraints (m = 7) involving the above 12 design variables is given as Problem 1: Discussion Example h 1 2X 2 1X 2 + 9X 4 3 8X = 0 (1) h 2 X 5 + X X 7 7X = 0 (2) h 3 4X 9 + 6X 4 10 X 11 + X = 0 (3) h 4 3X 1 X X = 0 (4) h 5 3X 5 2 X 6 + 8X = 0 (5) h 6 X 4 3 6X X = 0 (6) h 7 5X 4 + 3X 2 8 X = 0 (7) 1 Define Variable Existence Matrix (VEM) for Constraints: Define an m n variable existence matrix A It is defined as follows: A ij = 1 if the jth variable exists in the ith constraint, and A ij = 0 otherwise, where i = {1,,m} and j = {1,,n} Each row of matrix A represents the existence of the variables in the left hand side (LHS) of an equality constraint 5 of 18

7 Example: For the discussion example, the matrix A is a 7 12 matrix, given as A = X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 X 9 X 10 X 11 X 12 h h h h h h h (8) The matrix A readily conveys which variables exist in a particular equality constraint; eg, the variables X 1, X 5, and X 9 exist in equality constraint h 4 2 Arrange all known Type S constraints on the top: Say the designer knows a-priori that some equality constraints must be satisfied regardless of uncertainty, such as physics-based constraints We rearrange the above matrix A such that all such constraints are on the top Define A s as [ ] A = where A s is the matrix containing those rows of equality constraints which are known to be Type S a-priori; and A u is the matrix containing those constraints that are not yet classified as Type S or S Example: In the discussion example, let h 4 be a Type S constraint Then the matrix A s is given by The matrix A u is given by A s A u X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 [ A s = h ] (10) (9) A u = X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 X 9 X 10 X 11 X 12 h h h h h h (11) 3 Define Known Prescribable and Non-Prescribable Variables: If the designer knows the classification of some variables in the problem, he/she can specify these variable definition by defining X np and X p, where X np is the current vector of non-prescribable variables and X p is the current vector of prescribable variables The vector X then can be defined as X = {X np,x u,x p } T, where X u is the set of those variables for which the classification is not yet defined Also, for the equality constraints represented in the matrix A s that are Type S, the designer must choose prescribable and non-prescribable variables such that the propositions are satisfied Example: For the discussion example, we assume that X np = {X 1,X 6,X 11 } T are non-prescribable variables, X p = {X 9,X 10 } T are prescribable variables, and X u = {X 2,X 3,X 4,X 5,X 7,X 8,X 12 } T are variables that are not yet classified In Step 1, we represent the existence of variables in the LHS of the set of equality constraints in a matrix form However, this matrix is not readily suitable for classification of equality constraints Whether or not propositions 1 and 2 are satisfied for a given equality constraint is not immediately obvious from this form of matrix A To check if propositions are satisfied, we need to establish the existence of the appropriate types of variables in each equation In the next step, we define a transformation matrix that provides the designer with a systematic framework under which he/she can classify the variables as needed 6 of 18

8 4 Define Variable Transformation Matrix: We now define a transformation, T (not yet completely known), that maps the design variable vector X into prescribable and non-prescribable variables The definition of T is given as X 1 X 2 X n [ ] = T np T u T p X np X u X p (12) where T = [T np T u T p ] is an n n variable transformation matrix In the above equation, T np corresponds to those n np variables that are known to be non-prescribable; T np is a n n np matrix T p corresponds to those n p variables that are known to be prescribable; T p is a n n p matrix T u corresponds to those n u = n (n p + n np ) variables whose types are not yet known or assigned; T u is a n n u matrix It is by using this matrix T u that the designer can interactively classify variables and constraints The matrix T is defined as follows Consider an element X i in the design variable vector X We define the ith row of the matrix T as follows: T ij = 1, if j is the index of the variable X i in the vector {X np,x u,x p } T, and T ij = 0 otherwise, where i,j = {1,2,,n} For those variables which are not yet classified, the index of the variable in the vector {X np,x u,x p } T is not yet defined Therefore, the matrix T u is initially unknown The classification of variables represented by the matrix T u is discussed shortly Example: For the discussion example, T np is a 12 3 matrix, T p is a 12 2 matrix, and T u is a 12 7 matrix The variable transformation in Eq 12 becomes X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 X 9 X 10 X 11 X 12 = T np Tp [T u ] X 1 X 6 X 11 X u X 9 X 10 (13) The next step is to provide definitions for each of the variables in X u, ie, to define the matrix T u based on the designer s choice 5 Define Variable Type for Each Variable in X u : The variables in the vector X u have not yet been classified as non-prescribable or prescribable A key feature of the proposed approach is that using the transformation defined in Step 4, the designer can classify a given variable as prescribable or non-prescribable, and observe how the constraints are classified as per the propositions This matrix based framework gives the designer the opportunity to explore a set of what-if scenarios in terms of variable classifications and their implications 7 of 18

9 In this step, we define each variable in the vector X u as either prescribable or non-prescribable as per our choice The entries of the matrix T u can be either zero or one depending on whether a variable is prescribable or non-prescribable (see discussion in Step 4) This results in a new transformation matrix, where the elements of T u are completely defined, and all the variables now are classified into prescribable and non-prescribable variables Let n np and n p be the number of prescribable and nonprescribable variables after the classification for the elements of T u is chosen The vector X u is rearranged such that all the non-prescribable variables are arranged above the prescribable variables Equation 12 then becomes X 1 X 2 X n [ = T np T p ] [ X np where T np and T p are the sub-matrices, respectively, representing the transformations for non-prescribable and prescribable variables, after the variable types for X u are defined; and X np and X p are the correspondingly updated vectors of non-prescribable and prescribable variables, respectively Example: We assume that the variables X 2, X 3, X 5, X 7, and X 12 are prescribable variables and X 4 and X 8 are non-prescribable variables For this definition, X np = {X 1,X 6,X 11,X 4,X 8 } and X p = {X 12,X 7,X 5,X 3,X 2,X 9,X 10 } The new matrix T becomes X p ] (14) X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 X 9 X 10 X 11 X 12 = T np T p X 1 X 6 X 11 X 4 X 8 X 12 X 7 X 5 X 3 X 2 X 9 X 10 (15) For this example, n np = 5 and n p = 7 6 Combining VEM with the Variable Classification: The VEM defined in Step 1 provides information about the existence of a particular variable in a constraint The variable transformation, defined in Eq 14 displays information about the type of each variable in the vector X In this step, we combine both the variable existence and variable type information into a single matrix Multiplying the VEM of the set of equality constraints, A (m n matrix), with the variable transformation matrix, T (n n matrix), yields an m n matrix that displays both the variable existence and variable type for a particular constraint This matrix is given as A s T np A s T p (16) A u T np A u T p 8 of 18

10 In the above equation, note that the block [A s T np A s T p] represents the VEM of those equality constraints whose classification has been determined in Step 2 The block [A u T np A u T p] represents the VEM of those equality constraints whose type is to be determined Note that because of the above transformation, the propositions to determine constraint classification can be applied easily The sub-matrices matrix [A s T np], [A u T np] and [A s T p], [A u T p] indicate the presence of non-prescribable and prescribable variables in an equality constraint, respectively This is exactly the information needed to classify the constraints into Type S and Type S using the propositions Example: For the discussion example, Eq 16 becomes h h h h h h h (17) 7 Matrices of Interest for the Current Variable Classification: In the above step, we observe that the elements of A u T np and A u T p represent the existence of non-prescribable and prescribable variables, respectively, for those equality constraints whose classifications are to be determined We designate the matrix A u T np as A 1 and the matrix A u T p as A 2 Equation 16 is then written as A s T np A s T p A 1 A 2 Example: For the discussion example, the sub-matrix [A 1 A 2] is (18) A 1 h h h h h h A (19) The matrices A 1 and A 2 can be used to determine the type of each equality constraint by applying the propositions Next, we discuss how certain measures of the matrices A 1 and A 2 in Eq 18 provides information about the satisfiability of equality constraints 8 Existence of Available Non-Prescribable Variables Rank: According to proposition 1, there should exist at least one unique non-prescribable variable for each equality constraint to be Type S The matrix A 1 provides the information needed to check if proposition 1 for the existence of available non-prescribable variables is satisfied We make an interesting 9 of 18

11 observation that the rank of the matrix A 1, r 1, provides the number of constraints in A 1 with available non-prescribable variables The number of rows in A s T np, r 2, provides the number of non-prescribable variables that are used a-priori to satisfy those constraints that are defined as Type S in Step 2 The above suggests that the upper limit on the number of equality constraints in A 1 that can be Type S with the defined set of non-prescribable variables is given by r 1 r 2, assuming r 1 > r 2 If r 1 < r 2, the constraints in A 1 do not have any available non-prescribable variables, because they have been used a-priori for the Type S constraints The exact number of Type S constraints in A 1 depends upon whether or not a prescribable variable exists in a particular equality constraint Example: In the discussion example, the rank of matrix A 1 is r 1 = 5, and the number of rows in matrix A s T np is r 2 = 1 This implies that out of the six equality constraints that are not yet classified, the upper limit on the number of Type S constraints is r 1 r 2 = 4 This statement applies for the current variable classification of X np = {X 1,X 6,X 11,X 4,X 8 } and X p = {X 12,X 7,X 5,X 3,X 2,X 9,X 10 } Next, we discuss the approach used to determine the number of equality constraints that can be satisfied We discuss an approach to check if a prescribable variable exists in a particular equality constraint 9 Existence of Prescribable Variables Zero Rows: If a particular equality constraint does not consist of prescribable variables, then the corresponding entries of the matrix A 2 will be zeros Therefore, the zero rows in the matrix A 2 gives us information about those equality constraints that violate proposition 1, and hence are Type S While there are several techniques with which one could find the existence of zero rows, we propose a simple technique Because of the special nature of the variable existence matrix defined in Step 1, each element of the matrix A 2 is either zero or one Then, a given row does not consist of any prescribable variables if the sum of elements along that row is zero This technique can be easily automated for a large-scale problem with a simple routine that computes the sum along each row of the matrix A 2 Example: In the discussion example, we can readily observe that all the equality constraints contain prescribable variables This implies that none of the constraints violate proposition 1 for prescribable variables Next, we combine this step with the previous one to establish the satisfiability of an equality constraint 10 Finding the Constraint Type from the Propositions: As discussed in Step 8, the quantity r 1 r 2, if positive, gives the upper limit on the number of Type S equality constraints possible with the existing variable classification The number of Type S constraints is governed by the results of Step 9 The designer now has two options: (1) Choose any r 1 r 2 constraints that have non-zero rows in A 2 as Type S, or (2) Change the variable definitions using in the variable transformation matrix T, and repeat Steps 5 through 10 Example: For the discussion example, r1 r2 = 4 Since all the constraints contain prescribable variables, we can choose any four constraints as Type S constraints, and the remaining two constraints are Type S Using the approach outlined above, the designer can systematically explore the possible options for variable and constraint classification The approach presented above is a simple, yet powerful tool to interactively classify equality constraints and variables using matrix manipulations The use of rank can prove particularly useful in large-scale problems By using rank, laborious procedures involving manipulation of individual rows and columns can be avoided Figure 2 summarizes the interactive classification procedure proposed in this Section So far, we discussed an interactive approach using which an constraint can be classified as Type S or Type S In the next Section, we put forth the second important contribution of this paper a method to incorporate intra-constraint and inter-constraint preferences We begin by defining the above two preferences, and discuss how they can be incorporated into the RDO formulation 10 of 18

12 Define Variable Existence Mapping Arrange all known Type S constraints on the top Define Initial Variable Transformation Mapping Define elements of classification for variables in T u Obtain A 1 * and A 2 * Find r 1 *, r 2 *, and zero rows of A 2 * Choose (r 1 * - r 2 *) non-zero rows of A 2 * as Type S Is classification satisfactory? YES NO STOP Figure 2 Proposed Interactive Classification of Equality Constraints IV Intra Constraint and Inter Constraint Preferences In RDO problems, ensuring feasibility of constraints under uncertainty is a challenging task This is more challenging in the presence of equality constraints because of the exactitude of such constraints Moreover, satisfaction of equality constraints in RDO problems entails a tradeoff with objective function minimization 10,11 In other words, the higher the equality constraint satisfaction, the worse the objective function value A designer can express an intra-constraint preference if he/she wishes a certain level of equality constraint satisfaction, typically in terms of its mean and standard deviation The above discussed tradeoffs are more complicated in the case of multiple equality constraints In such cases, the constraint satisfaction of a particular equality constraint potentially trades off with not only the objective function, but also with other equality constraints This implies that the designer must choose satisfying more some equality constraints over the others, which leads to inter-constraint preferences In this Section, we present an approach that can formulate both intra-constraint and inter-constraint preferences We begin by discussing intra-constraint preferences A Intra-Constraint Preferences There are several methods available in the literature 10 for formulating equality constraints that yield different constraint satisfactions By using a particular constraint formulation approach for a given constraint, we implicitly express an intra-constraint preference how closely we choose to satisfy the constraint The following are the approaches available in the literature to formulate equality constraints 1 Satisfying the equality constraint exactly at its mean value Using Approximate matching method (discussed below), of 18

13 3 Specifying acceptable ranges for means and standard deviations of the equality constraint 11 In this subsection, we present approximate moment matching method, which comparatively offers remarkable generality In our previous work, 10 we presented the approximate moment matching formulation, where the designer can explicitly specify intra-constraint preferences regarding the mean and the standard deviation of an equality constraint Using this approach, we restrict the mean of the Type S constraint, µ h to be as close to zero as possible, and the standard deviation of the Type S constraint, σ h, to be as small as possible Two parameters, δ µ and δ σ, that control the constraint satisfaction are introduced in the objective function to be minimized The smaller the δ µ and δ σ values, the closer the equality constraint satisfaction under uncertainty A simplified formulation for the approximate moment matching method 10,11 (shown in Fig 1) is given as Problem 2: Intra-Constraint Preferences Approximate Moment Matching Method min {µ J,δ µ,δ σ } (20) µ X,δ µ,δ σ such that g(x) + ασ g (X) 0 (21) δ µ µ h δ µ (22) 0 σ h δ σ (23) x min + ασ X X x max ασ X (24) where σ g is the standard deviation of the constraint function g(x) and α indicates the shift in the inequality or side constraint Figure 3 illustrates the shifting of inequality constraints described in the above formulation The mean and the variance of a random function (in the above case, h or g), can be approximated using first order Taylor series, 21 given as Side Constraints Inequality Constraints Deterministic Deterministic x min feasible x max x feasible 0 g(x) RDO RDO Shift Shift Shift x min feasible x max X feasible 0 g(x) infeasible infeasible Figure 3 Shifting Inequality Constraints - Reducing the Feasible Design Space µ g = g(µ X ) (25) [ n x 2 g n x n x Var[g] = g X i σ Xi] + g X=µX X i X j Cov(X i,x j ) X=µX (26) i=1 i=1 j=1,i j X=µX where σ X denotes the vector of the standard deviations of X, Cov(X i,x j ) denotes the covariance between the variables X i and X j, and i,j = {1,,n x } The standard deviation of g can then be computed as σ g = Var[g] By choosing different preferences for the δ µ and δ σ parameters in the above formulation, the designer can control the equality constraint satisfaction in terms of its mean and standard deviation thereby exercising 12 of 18

14 an intra-constraint preference Satisfying the equality constraint exactly at its mean value can be done by replacing Eqs 22 and 23 in the above formulation by a single equality constraint, µ h = 0, and by removing the δ µ and δ σ parameters from the formulation Specifying acceptable ranges for means and standard deviations of the equality constraint 11 can done by specifying numerical values for δ µ and/or δ σ in the approximate moment matching method presented above Note that in such a case, δ µ and/or δ σ are not a part of the objective function, and are not design variables Next, we study how inter-constraint preferences can be incorporated into the RDO problem B Inter-Constraint Preferences As mentioned earlier, several methods have been used to formulate equality constraints in the literature 10 Although the approximate moment matching method presents a systematic approach to express intra-constraint preferences, it is not necessary that the designer must use the same formulation method for all the equality constraints in an RDO problem The above statement explains the method we use to enforce inter-constraint preferences giving the designer the option that each constraint could be formulated differently In the proposed formulation, we provide the designer three choices (discussed in the previous subsection) to formulate equality constraints in RDO problems The designer can choose to satisfy each equality constraint differently, thereby specifying both an inter-constraint and intra-constraint preference Because of the multiobjective nature of the approximate moment matching approach, it can be used to express both intra and inter-constraint preferences Here, we present a generic formulation for the above discussion Assume that we now have n S Type S constraints An RDO formulation that gives the designer an option of expressing both intra-constraint and inter-constraint preferences can be given as Problem 3: Intra and Inter-Constraint Preferences Approximate Moment Matching Method min µ X,δ µ1,δ σ1,δ µr,δ σr {µ J,δ µ1,δ σ1,δ µr,δ σr } (27) such that g(x) + ασ g (X) 0 (28) δ µi µ hi δ µi (29) 0 σ hi δ σi i = {1,2,r} (30) µ hj = 0 j = {1,2,,,q} (31) M k µ hk M k (32) 0 σ hk S k k = {1,2,p} (33) x min + ασ X X x max ασ X (34) where r constraints are formulated using approximate moment matching approach, q constraints are satisfied exactly at their means, and p constraints are provided acceptable ranges for their means and/or standard deviations, M and S, respectively Note that the values of M k and S k need not be the same for all the p constraints, thereby providing the means to enforce inter-constraint preferences In the next section, we present a numerical example where we illustrate the above RDO formulation, which can be used to specify both inter-constraint and intra-constraint preferences V Numerical Example In this section, we present an assembly tolerance allocation problem that is solved using the proposed method Using this example, we illustrate how the proposed RDO formulation approach can be used to express inter-constraint and intra-constraint preferences, and discuss the associated tradeoffs A design assembly consisting of two mating parts is considered from Ref 22 (shown in Fig 4), where optimal dimensional tolerances must be assigned from a cost perspective The following constraints are imposed on the problem The lengths x 1 and x 12 must be equal to each other, within a tolerance of ± of 18

15 x 1 x 2 x 3 x 4 x 5 x 6 θ 1 θ 2 x 8 x 9 x10 x 11 x 7 x 12 Figure 4 Tolerance Design Example inch The angles θ 1 and θ 2 must be equal to each other, within a tolerance of ±tan π 180 The clearances shown in Fig 4 between the two parts must be positive (given in constraints g 1 and g 2 below) The constraint equations are given as g 1 x 6 + x 5 + x 8 x 7 0 (35) g 2 x 3 + x 4 + x 11 x 10 0 (36) h 1 tan θ 1 tan θ 2 = 0 (37) h 2 x 1 x 12 = 0 (38) We are required to find optimum tolerances for the dimensions, where the tolerance of the ith dimension, t i, is six times the variance of the ith dimension We minimize the cost function given as C(σ i ) = 12 i=1 a i (6σ i ) bi (39) where a = [02,1,0015,0015,0008,0009,0008,0006,1,001,0015,02], b = [2,2,,2], and σ i is the standard deviation of the ith dimension The dimensions are assumed to be independent normal variables The nominal dimensions of the part dimensions are given in Ref 22 as [500, ,2005,99985,99985,300, 100,300,1005,300,400,500] In this paper, we take lower and upper bounds for the mean dimensions to be ±6σ xi inch of the nominal dimensions given above In this example, the equality constraints h 1 and h 2 are Type S constraints, since all the dimensions are assumed to be independently distributed as per proposition 2 We use the approximate moment matching formulation for the equality constraints given above The RDO formulation is given as 14 of 18

16 Problem 4: Tolerance Allocation Example min µ X,δ µ1,δ σ1,δ µ2,δ σ2 {C,δ µ1,δ σ1,δ µ2,δ σ2 } (40) such that g 1 + 6σ g1 0 (41) g 2 + 6σ g2 0 (42) δ µ1 µ h1 δ µ1 (43) 0 σ h1 δ σ1 (44) δ µ2 µ h2 δ µ2 (45) 0 σ h2 δ σ2 (46) δ µ1 tan π 180 (47) δ µ2 001 (48) x lb x x ub (49) In the above formulation, the intra-constraint preference (given by the design specifications) is given by the constraints in Eqs 47 and 48 The inter-constraint preference can be exercised using the multiobjective formulation given in Eq 40 We use a weighted sum formulation to solve the above problem A Results and Discussion In this subsection, we discuss the results of the multiobjective optimization problem presented above The problem consists of three broadly classified objectives: (1) C is the cost that must be minimized, (2) δ µ1 and δ σ1, which must be minimized; these quantities control the constraint satisfaction for h 1, which requires θ 1 = θ 2, and (3) δ µ2 and δ σ2, which must be minimized; these quantities control the constraint satisfaction for h 2, which requires x 1 = x 12 A large value of δ µ1 or δ µ2 indicates that there is a large mean shift in the equality constraint violation A large value of δ σ1 and δ σ2 indicates that there is a large variation in the constraint violation implying that the design is not robust It is therefore important to minimize both δ µ and δ σ for each constraint to obtain tight tolerances This however, potentially results in an increased cost because of the associated tradeoffs The multiobjective problem presented in the previous section can be solved using two approaches: 23,24 (1) Construct an aggregate objective function (AOF) that adequately reflects the designer s preferences for the objectives, and optimize the AOF to obtain a single optimum design This approach is known as the Integrated Generating and Choosing (IGC) approach, or (2) Generate several Pareto optimal designs first, and choose the most desirable solution later This approach is known as Generate First Choose Later (GFCL) approach In this paper, we use the IGC approach We perform two sets of tradeoff studies, which are discussed next The first tradeoff study minimizes each of the above three objectives, namely, (1) cost, (2) δ µ1 and δ σ1, and (3) δ µ2 and δ σ2, individually In the second tradeoff study, we minimize all the three objectives simultaneously 1 Tradeoff Study 1 For this tradeoff study, we solve the above problem for three different sets of preferences: (1) Minimizing the cost objective, C only, (2) Minimizing δ µ1 and δ σ1 only, and (3) Minimizing δ µ2 and δ σ2 only In each of the above cases, we observe the tradeoff between the cost and the constraint satisfaction of h 1 and h 2 We note that in the above formulation, the smaller the values of δ µ1, δ σ1, δ µ2, and δ σ2, the higher the equality constraint satisfaction To quantify the term equality constraint satisfaction, we use a metric called probability of constraint satisfaction, 10 PCS, which is defined using a Monte Carlo simulation, as PCS = N s N (50) 15 of 18

17 where N s is the number of simulation cycles for which the equality constraint, h 1 or h 2, lies within ±00001 inch, ie, h 1,h , and N = 10 6 is the total number of simulation cycles The PCS value gives an estimate of how closely the constraint is being satisfied Table 1 Comparison of Equality Constraint Satisfaction C only δ µ1 and δ σ1 only δ µ2 and δ σ2 only Cost ($) e e+006 PCS e e-004 PCS e δ µ1 (inch) δ σ1 (inch) δ µ2 (inch) δ σ2 (inch) Table 1 shows the equality constraint satisfaction and objective function values for the three cases minimizing cost only, minimizing δ µ1 and δ σ1 only, and minimizing δ µ2 and δ σ2 only In column 2 of Table 1, we minimize cost only The optimum value obtained for the cost objective is the most desirable in this case when compared to other cases However, the equality constraint satisfaction values are not satisfactory For example, the values δ µ1 = and δ µ2 = 001 values are equal to the tolerance requirements for the constraints, and indicate large mean shifts in the constraint violations The δ σ1 and δ σ2 values are large, indicating that there is a large variation in constraint violation, thereby leading to poor robustness This leads to a low probability of the tolerance requirements being satisfied, which is indicated by the low PCS value This design can be deemed not acceptable because of poor constraint satisfaction In columns 3 and 4, where the δ µ and δ σ for each constraint are minimized, we note that the cost objective is very large However, the constraint satisfaction for h 1 in column 3 and h 2 is column 4 are good, which are indicated by the respective PCS values However, this design can also be deemed not acceptable because of the high cost The three objectives, C, constraint satisfaction for h 1, and constraint satisfaction for h 2 tradeoff with each other, and a compromise must be made among the three objectives 2 Tradeoff Study 2 In this subsection, we minimize all the objectives: C, δ µ1, δ σ1, δ µ2, and δ σ2 We give different preferences to the constraint satisfactions of h 1 and h 2 by choosing appropriate weights for δ µ1, δ σ1, δ µ2, and δ σ2, thereby exercising inter-constraint preferences The advantage of the proposed formulation is that the designer can explicitly specify preferences in terms of the means and standard deviations of each equality constraint Although several Pareto solutions can be generated for this problem, we use the IGC approach as discussed earlier to solve the multiobjective problem We present five sample solutions, where different preferences are assigned to each of the objectives Table 2 summarizes the results of this tradeoff study Table 2 Comparison of Equality Constraint Satisfaction Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Cost 36110e e e e e+006 PCS e PCS e In sample 2, higher preference is given to minimizing cost In samples 3, 4, and 5, higher preference is given for constraint satisfaction As we observe the samples of data presented in Table 2, we note that as PCS2 improves, at least one of the other two objectives worsens For example, between samples 1 and 2, 16 of 18