Proceedings of DETC 99: 1999 ASME Design Engineering Technical Conferences September 115, 1999 Las Vegas, Nevada DETC99/DAC8589 A Comprehensive Robust Design Approach for Decision TradeOffs in Complex Systems Design Monu Kalsi ASME Student Member, Graduate Research Assistant, mkalsi@eng.buffalo.edu, Tel: (716) 645593 x61 Fax: (716) 6453668 Kurt Hacker ASME Student Member, NASA Graduate Research Fellow, khacker@eng.buffalo.edu Tel: (716) 645593 x61 Fax: (716) 6453668 Kemper Lewis ASME Member, Assistant Professor, Corresponding Author, kelewis@eng.buffalo.edu, Tel: (716) 645593 x3 Fax: (716) 6453668 ABSTRACT In this paper we introduce a technique to reduce the effects of uncertainty and incorporate flexibility in the design of complex engineering systems involving multiple decisionmakers. We focus on the uncertainty that is created when a disciplinary designer or design team must try to predict or model the behavior of other disciplinary subsystems. The design of a complex system is performed by many different designers and design teams, each of which may only have control over a portion of the total set of system design variables. Modeling the interaction among these decisionmakers and reducing the effect caused by lack of global control by any one designer is the focus of this paper. We use concepts from robust design to reduce the effects of decisions made during the design of one subsystem on the performance of the rest of the system. Thus, in a situation where the cost of uncertainty is high, these tools can be used to increase the robustness, or independence, of the subsystems, enabling designers to make more effective decisions. This approach includes uncertainty caused by control factor variation (Type II robust design) and uncertainty caused by unknown nonlocal design information (Type I robust design). To demonstrate the usefulness of this approach, we consider a case study involving the design of a passenger aircraft. KEYWORDS Robust Design, Multidisciplinary Design, Optimization, Game Theory NOMENCLATURE X i Design variables for each discipline i Y ij Linking variables that are evaluated by discipline i and required by discipline j as the input f i Objective functions of discipline i g i Constraints of discipline i µ f Mean of the objective function f σ f Standard deviation of the objective function f x Deviation range of design solution b Wing span, ft. l Fuselage length, ft. S Wing area, ft Wto Takeoff weight, lbs. Ti Installed Thrust, lbs. V br Best_range velocity, ft/s R fr Fuel ratio required Ld c Lifttodragratio on climb Ld l Lifttodragratio on landing Ld t Lifttodragratio on takeoff 1. INTRODUCTION In the design of a large, complex engineering system it is not uncommon for more than one design team to be involved. Often these teams are formed along disciplinary lines, each responsible for the design of a single part (subsystem) of the overall system. Quite possibly, each subsystem has its own goals and constraints that must be satisfied along with the systemlevel goals and constraints. In addition, the goals of the individual subsystems might be contradictory, i.e. a satisfactory design from the view of one subsystem is not necessarily satisfactory from the view of the others. Finally, the subsystems are often coupled, i.e. there are design variables in the disciplinary subproblems controlled by other disciplinary designers. Concurrent Engineering (CE) methods were developed to improve upon the sequential, over the wall design commonly practiced. Multidisciplinary teams are formed 1
involving engineers from every aspect of the product life cycle so that design goals and constraints from all of the subsystems can be considered and the appropriate tradeoffs made during the initial stages of the design process. Although the use of CE concepts can and has lead to increases in the efficiency of the design process, in some cases existing organizational structure within a design organization can hamper multidisciplinary interaction (Womack, Jones et al., 1990). In addition, even when a concurrent approach is taken, some iteration may still take place due to communication or geographical barriers between design teams that are not collocated or which may not even be part of the same company (Lewis and Mistree, 1996). Clearly, it would be advantageous for a designer to be able to make a decision regarding the design of a subsystem independent of the other designers decisions. This problem provides the impetus for the method developed in this paper. We explore a technique that allows for the design of a subsystem in the face of uncertainty stemming from incomplete information concerning the remainder of the system. We accomplish this by approximating the unknown design decisions as noise variables and use robust design techniques to mitigate their effect. The proposed formulation is an integration of robust design principles with models of the design process that together aid in describing the decisionmaking process and prescribing appropriate product decisions. Although pursuing concurrency is the ideal in many situations, other complex engineering systems are better suited to sequential design models. Modeling the subsytems sequentially has the advantage of reduced complexity over the concurrent approach as well as more realistically representing many actual design organizations in which the various design and manufacturing departments are separated. Due to the sequential nature of many design processes, decisions made by the designers at the beginning of the process have a strong influence on the design decisions made downstream of them. Consider a system consisting of multiple subsystems executed sequentially. Design decisions made by the first designer may make it difficult for the remaining designers downstream to find satisfactory or even feasible designs. To help mitigate this problem research is ongoing to develop mathematical models of the design process and techniques that allow designers to solve their disciplinary subproblems independent of the rest of the system (Bloebaum, Hajela et al., 199; Balling and Sobieski, 1994). One such technique developed by Chen and Lewis (Chen and Lewis, 1999) integrates game theoretic models of the design process with robust design techniques in order to reduce the effect of the coupling between the subsystems and improve the performance of the overall system. In this technique the initial designer chooses a range in which the performance is satisfactory and stable. The proceeding designers then have the freedom to select from any values within this range, yielding better performance than a point solution. Although this technique is quite effective, in cases where there is coupling in both directions, (the subsystems later in the design process need information from the designers upstream, and the upstream designers need design information for the subsystems downstream) it becomes necessary to estimate or approximate the latter. These estimates are often of the worst case variety and while work by Lewis (Lewis, 1996; Lewis and Mistree, 1997) has shown that approximating the unknown information using a game theoretic construct known as the Rational Reaction Set (Vincent and Grantham, 1981) is effective, it may require considerable computational effort and some coordination among the designers. Our approach addresses these same issues but in a different manner. We also seek to find a range where the performance is robust to changes in the control factors, but instead of guessing at the unknown design information or approximating it, we choose to treat it as a noise variable and use robust design techniques to mitigate its effect. We seek a solution that is not only robust with respect to changes in the initial designer s control variables but also stable with respect to the unknown design variables controlled by the downstream designers. The remainder of the paper is divided into three parts. In Section, we introduce the technical foundation for this paper, focusing on robust design and the mathematical models of the design process we are utilizing. In Section 3, we present our approach for mitigating the effect of uncertainty when designing for flexibility. In Section 4, we apply our approach to the design of a passenger aircraft and present the results. Section 5 is the closure of the paper.. TECHNICAL BACKGROUND.1 Mathematical Models of Multidisciplinary Design A typical model of a multidisciplinary optimization model is given below in Figure 1 (Balling and Sobieski, 1994; Chen and Lewis, 1999). The coupling in the system is represented by the linking variables, y ij, which are the design or behavior variables that are needed by the discipline j and determined during the analysis of discipline i. X 1 f1 g 1 X f g X 3 f 3 g 3 y 1 y 3 Discipline 1 y 1 Discipline y 3 Discipline 3 y 13 y 31 Figure 1. Multidisciplinary Coupling Model
Since these linking variables are essential in the solution to the disciplinary subproblems, the mechanism by which their values are communicated between the disciplinary designers is of great importance. There are a number of ways in which a designer might approach the problem of having uncertain linking variables. 1. Ignore Uncertainty A guess is made for the uncertain linking variables.. Seek out perfect information. The disciplinary designers determine the linking variables by collaboration with the other designers.. Represent and Manipulate the Uncertainty Techniques such as robust design can be used to make the solution robust to the uncertainty in the linking variables. Intuition tells us that collaboration between the designers should yield the best results, but in many cases there may be barriers which make full cooperation difficult or impossible (Vincent, 1983). Communication between the interacting disciplinary subsystems might be hampered by geographical separation (i.e. teams may not be collocated) or by the fact that the design teams are part of different departments within the same company or even different companies. Thus, although designers should seek out design information and collaborate in order to eliminate uncertainty in many cases this may not be possible. Guessing at the unknown design information is always an option, especially when based on experience, but if the guess is far off it can lead to degradation in performance and expensive and time consuming iteration. Modeling the interaction between the designers and making the solution robust to the uncertainty may be the preferred strategy in many cases. Game theory has been recognized as one way to model problems with multiple interacting disciplines that may or may not share the same objectives (Vincent, 1983). Three game theoretic protocols are developed in (Lewis, K. and Mistree, F., 1998) to describe the interactions that occur among players (designers) in a design situation. They are cooperative, noncooperative, and sequential. The sequential or leader/follower protocol is descriptive of many actual design processes and this protocol is our area of concentration. A simple model for a sequential twodesigner problem is given below in Equation 1 and Equation with designer 1 as the leader and designer as the follower. Designer Model (Follower) minimize f = (f 1 ( x, y 1 ), f ( x, y 1 ),.,f 3 ( x, y 1 )) s.t. g j ( x, y 1 ) 0 j = 1 o h k ( x, y 1 ) = 0 k = 1 p () Bounds on x Note that each subsystem has its own set of objectives and constraints, although it should be noted that there could be some which are common to both subsystems. The follower creates a RRS for his design variable in terms of the leader s design variable and this is passed to the leader. The leader uses this information to approximate the unknown linking variable and uses it to solve his model. Once he has solved his model, he passes over his solution to the follower. Though this approach could be used to model the unknown coupling variables, the construction of the RRS could be computationally expensive and time consuming and this approach does not seek to mitigate the effects of these variables on the performance of the system. In the next section, robust design is introduced as a way to mitigate uncertainty in these types of problems.. Robust Design Fundamentally, robust design is concerned with minimizing the effect of uncertainty or variation in design parameters on a design without eliminating the source of the uncertainty or variation (Phadke, 1989). In other words a robust design is less sensitive to variation in uncontrollable design parameters than the traditional optimal design point. There are two general categories of robust design (Chen, 1996). In Type I robust design, the goal is to minimize the variation caused by uncontrollable noise factors. Examples might include changes in ambient temperature, operating environment, or other natural phenomena that are impossible or prohibitively costly to control. Figure below illustrates this. In this figure the variation in performance f(x) for a traditional 'optimal' design and a robust design are compared when the design variable x varies a quantity x about its mean value µ. Designer 1 Model (Leader) minimize f 1 = (f 1 1 ( x 1, y 1 ), f 1 ( x 1, y 1 ),.,f 1 3 ( x 1, y 1 )) s.t. g 1 j ( x 1, y 1 ) 0 j = 1 n h 1 k ( x 1, y 1 ) = 0 k = 1 m (1) Bounds on x1 3
Y Performance Optimization Solution Robust Solution Equation 3 below. The goal is to bring the mean on target and minimize the variance about the mean by proper selection of x and x. : x,?x Minimize: [µ f,σ f ] M µ ± x µ ± x opt robust Design Variable x n g subject to g j (x) + k j j x i=1 x i 0, j = 1,,..., J (3) i x L + x x x U x, Figure. Illustration of Type II Robust Design In Type II robust design, the goal is to minimize variations caused by deviation in the control factors. This could result from manufacturing tolerance limitations, material quality variations, or even evolving design preferences (Sundaresan, Ishii et al., 1993). Although robust design has been traditionally applied in manufacturing there has been research recently into applying these techniques to make the design conceptually robust (Chang, Ward et al., 1994; Chen, 1996). One goal of this research is to make design decisions robust to uncertainty caused by evolving design objectives and constraints in a multidisciplinary design environment. The approach we are presenting is an integration of Type I and Type II robust design. Type I will be used to make the leader's solution robust to unknown design decisions made by the follower. Recall from Equation 1, that designer 1 must solve his disciplinary subproblem under uncertainty, not knowing how the follower will act. The fundamental difference between what we are proposing and traditional Type I robust design lies in the definition of the noise factors. As opposed to external noise factors (ambient temperature, humidity) which by their very nature are uncontrollable (or prohibitively expensive to control), we are concerned with internal noise variables which are deterministic decisions made by the other designers, but not controllable or even known by everyone. The end goal is the same, however, namely, to minimize the influence of the noise on the subsystem under consideration. We are also incorporating Type II robust design to incorporate more flexibility into the design. This approach was developed in Chen and Lewis (Chen and Lewis, 1999). In this work, the idea of a robust solution range is introduced. The first designer chooses a range instead of a traditional point solution within which subsystem performance is stable and at a satisfactory, although possibly suboptimal level. By choosing a range, designer 1 is allowing the proceeding designers more freedom to find a satisfactory solution to their disciplinary subproblems. Chen's formulation is presented in This technique is effective for problems where there is oneway coupling between the subsystems and specifically problems where the follower needs design information from the leader, but the leader needs no design information from the follower. In many systems, however, there is twoway coupling in which both the leader and the follower need design information (design or behavior variables) from the other. In such cases the designers at the beginning of the design process need design information from designers downstream in order to solve their disciplinary subproblem. Chen and Lewis (Chen and Lewis, 1998) overcome this problem by assuming that the designers downstream react rationally to the design decisions made upstream of them. This reaction is embodied in a construct from game theory called the Rational Reaction Set (Lewis, 1996). With this function the leader in a sequential design process can predict what the reaction of the follower will be to a design decision, allowing the leader s disciplinary subproblem to be solved. The difficulty in this approach lies in the actual construction of the RRS. In practice this function may be computationally expensive to obtain and its use necessitates some interaction among the designers. As this may not always be possible in real design situations, we propose to instead model this unknown information as an additional noise parameter and use Type I robust design techniques to mitigate its effect. This will be accomplished by finding values of the local design variables where the performance variation lies within a tolerable range subject to variation in the downstream subsystem linking parameters, y 1. The only a priori information needed are ranges for the downstream subsystem design variables, which we assume to be known. Fundamentally, our approach, to be presented in detail in Section 3, locates robust ranges in the design space with respect to control factor deviation within that range and with respect to the unknown downstream control factors (modeled as noise). 4
3. AN APPROACH FOR CONCEPTUAL ROBUST DESIGN Consider the system in Figure 3. It is composed of two disciplinary subsystems: designer 1 controls design vector x 1 and designer controls design vector x. We use the compromise Decision Support Problem (DSP) to model the system (Mistree, Hughes et al., 1993). The compromise DSP includes concepts from mathematical and goal programming. The objective is to minimize the deviation from the target goal values established by the designer while satisfying system constraints. We have chosen this formulation because it allows the modeling of the design goals separately and at different priority levels. This allows additional insight into the tradeoff between performance and robustness to be obtained. Consider the case in which the design process is sequential in nature and starts with designer 1 and continues to designer. The value of the linking variables, y 1, need to be determined before a solution to the Subsystem 1 can be determined. The first option in such a case is for designer 1 to guess a value for the linking variables, y 1. A strategy might be adopted to minimize the maximum value of the objective function. In other words, designer 1 would assume that designer is going to choose the worst possible values of y 1. This is known as a minmax strategy and invariably leads to less than optimal results for both subsystems, assuming a feasible solution can be found at all. Another possibility is to assume that designer will behave rationally. As described in Section 3., designer constructs an analytical function of the form y 1 =f(x 1 ), which allows a solution to Subsystem 1 to be found. This is the standard Stackelberg/LeaderFollower approach (Lewis and Mistree, 1997). Designer 1 Model The approach that is presented in this paper provides an additional option to handling the uncertainty without the need to construct the Rational Reaction Set, which can be problematic in many real design situations. The unknown linking variables, y 1, needed by designer 1 are modeled as noise variables with uniform probability distributions varying within the modified bounds: y 1Lower * y 1 y 1Upper * that lie within the true bounds of the linking variables which are known quantities. The modified bounds are found by a pattern search that is run in the nonlocal linking variables design space and are maximum ranges of the linking variables within their total bounds that guarantee a robust and feasible local design variable solution within those bounds. This idea is illustrated in Figure 4 below which shows the true and modified bounds. y1l Figure 4. Modeling y 1 with a Uniform Distribution A uniform distribution is used to model the linking variables since using this distribution it can be assumed that the other designer has an equal probability of choosing any value of the unknown variables within those bounds. A uniform distribution can be described by two parameters, mean and variance, given in Equations 4 and 5. Designer Model y1l * y1 y 1 y1u * y1u Given Bounds on y 1 Given Bounds on y 1 Control Variables x i Control Variables x i Deviation Variables d i +, d i Deviation Variables d i +, d i Constraints g j ( x 1,y 1 ) 0 Constraints g j ( x,y 1 ) 0 Goals Goals A i (x 1,y 1 ) d i + + d i = G i A i (x, y 1 ) d i + d i = G i Bounds on x i Minimize Deviation Function Z = f 1 ( d i, d i )... f n ( d i, d i ) Bounds on x i Minimize Deviation Function Z = f ( d i, d i )... f n ( d i, d i ) Figure 3. Compromise DSP formulations for leader/follower design protocol 5
µ * y 1 = y * * ( 1U + y 1L ) σ y * 1 = y * * ( 1U y 1 L ) 1 We introduce two new design parameters into the model for Subsystem 1, µ * y1 and y 1. These parameters describe the location of the modified bounds y * 1L and y * 1U within the true bounds y 1L and y 1U. The goals for Subsystem 1 can be divided into two categories; goals related to the quality of the solution or goals related to its robustness to uncertainty. Expressions for the mean and variance of the performance are given in equations 6 and 7 below. Mean of the Performance µ f1 = f (x 1,µ * y1 ) (6) Variance of the Performance σ f1 = f 1 σ (7) y 1 1 In the modified problem, the objective is to bring the mean of the performance on target and minimize the variance of the performance about the mean. These goals can be placed on different priority levels based on the relative importance of performance or robustness. The compromise DSPs for showing the flow of information between the two subsystems is shown in Figure 5, where G i is the target value of mean performance to be achieved and the variance is desired to be close to zero. To account for the variation of constraints caused by the noise factors, y 1, (or the control factors in the case of Type II robust design) we assume that the noise or control factors of the system vary simultaneously in the worst possible combinations (Sundaresan, 1993). This way we can ensure the feasibility of the design in all situations. The worst case scenario constraints and performance variance goals are constructed using Taylor series approximations. From the perspective of Subsystem, this formulation additionally constrains the feasible design space to be within (4) (5) the modified bounds chosen by designer 1, y * * 1L and y 1U which are more restrictive than the true bounds. In addition, designer 1 also dictates x 1, which may further constrain Subsystem. To help mitigate the effect these additional constraints have on the performance of Subsystem and introduce flexibility in to the design process, we include Chen's (Chen and Lewis, 1998) approach wherein the point solution x 1 is replaced by a solution range described by µ x1 and x 1. Designer then chooses the most suitable value from within this range and uses it in the solution to Subsystem. With the added flexibility of being able to select from a range of possible values for the linking variables as opposed to a point solution, designer is much more likely to be able to find a design that satisfies their constraints and goals. Within this range, the objective of Subsystem 1 is stable with respect to changes in x 1. However, the performance of Subsystem 1 will likely be worse than at the point solution because the robust optimal is often different from the standard optimal. Despite this drawback, effectively reducing the downstream coupling in a preliminary design environment has important implications. Changing design objectives and constraints in an evolving design may make necessary large amounts of redesign if the performance of one subsystem is tightly coupled and sensitive to changes in the other subsystems to which it is coupled. If the uncertainty present is not addressed, there exists the potential for a significant amount of wasted time and effort in the preliminary stages of the design process. This can take the form of design iteration among the subsystems. The approach we have presented in this section has the potential to lessen the impact of these difficulties and provide the designer with a good initial design point for later, more detailed design. In the next section we illustrate the usefulness of our approach by considering the preliminary design of a passenger aircraft. Subsystem 1 Given: Leader Leader s Solution, y 1 Bounds on y 1 : [ x 1, y * 1 Lower, y * 1 Upper ] : Constraints: g j ( x 1, y 1 ) + Goals : n g j y i = 1 y 1 0 1 A i ( x 1, y 1 ) d i + d i = G i A i σ + y 1 1 d i + d i = 0 Bounds : x 1 Lower 1 x x 1Upper Minimize: Z = f 1 ( d i, d i )... f n ( d i, d i ) Subsystem Given : Leader s Solution Follower [ y 1, x * Lower, x * Upper ] : x : Constraints: g l ( x, y 1 ) 0 Goals : A i ( x 1, y 1 ) d i + d i = G i Bounds : x * Lower x x * Upper Minimize: Z = f 1 ( d 1, d 1 )... f n ( d i, d i ) True bounds on y 1 Figure 5. Compromise DSP for Aerodynamics as Leader/Weights as Follower with Type I Robust Design Considerations 6
4. CASE STUDY: DESIGN OF A PASSENGER AIRCRAFT 4.1 Compromise DSP for Aerodynamics as leader / Weights as follower To demonstrate our approach we consider the design of a passenger aircraft previously considered in (Lewis, 1996; Chen and Lewis, 1998). The system is partitioned into two subsystems, an aerodynamics subsystem and a weight subsystem. The Compromise Decision Support Problem for each is shown in Figure 6 below. The formulations shown for each player are without any robust design considerations. The aerodynamics subsystem has three control variables; wing span (b), fuselage length (l) and wing area (s). The Weights subsystem has two control variables, takeoff weight (Wto) and installed thrust (Ti). There are three linking variables, y 1 Wto, Ti, and Rfr required by the Aerodynamics subsystem from the Weights subsystem, and five linking variables, y 1 S, Vbr, Ldt, Ldc, Ldl required by the Weights subsystem from the Aerodynamics subsystem to solve their problem. We focus on the design variables Wto, Ti, and S in our approach. The reader is directed to (Lewis, 1996) for a more detailed problem description. In Section 4., we investigate the case in which the Aerodynamics subsystem is the leader and the Weights subsystem is the follower in a sequential design process. We consider three scenarios. In Scenario I, a baseline solution is found without any robust design considerations using the Rational Reaction Set to determine the value of the unknown linking variables, Wto and Ti. In Scenario II, Wto and Ti, are modeled as random variables with a uniform distribution. In Scenario III, we combine Scenario II with Chen's (Chen and Lewis, 1999) robust design approach to increase decision flexibility for the Weights subsystem. 4. Robust Design Approach to Minimize Control and Noise Factor Deviation Scenario I: Without any robust design considerations, the design problem can be modeled with the leader/follower design protocol as described in Section.1. The compromise DSPs for this scenario are shown in Figure 6. In this case we use the RRS to determine Wto and Ti (y 1 ) for any value of S (x 1 ) (Chen and Lewis, 1999). The values of the design variables and the performance deviation function for this scenario are presented in Table 1. Design Variables AERO (LEADER) WEIGHT (FOLLOWER) b(ft) 131.138 Ti (lbs) 4178.1 S(ft ) 185.71 Wto (lbs) 09441 l(ft) 146.378 Deviation 1.4439 1.3898 Table 1. Solutions of Aero as Leader/Weights as Follower Model for Scenario I AERODYNAMICS The values of the system variables Units Wing area, S [ft ] Fuselage length, l [ft] Wing span, b [ft] The values of the deviation variables associated with the goals The system constraints The aspect ratio must be less than 10.5 The achievable climb gradient on landing must be.4 The achievable climb gradient on landing must be.7 The landing field length must be 4,500 ft. The takeoff field length must be 6,500 ft. The drag coefficient in takeoff and landing must be 0.0 The drag coefficient in cruise must be 0.0 The system goals Missed Approach Climb Gradient, landing Missed Approach Climb Gradient, takeoff Landing Field Length Takeoff Field Length Aspect Ratio The bounds on the system variables Minimize The sum of the deviation variables Z = ΣW i (d i + d i+ ) Variables needed by aero designer: Takeoff Weight, Wto Installed Thrust, Ti WEIGHTS The values of the system variables Units Takeoff Weight, Wto [lbs] Installed Thrust, Ti [lbs] The values of the deviation variables associated with the goals The system constraints The useful load must be less than 0.3 The fuel available must be the fuel required The achievable climb gradient on landing must be.4 The achievable climb gradient on landing must be.7 The landing field length must be 4,500 ft. The takeoff field length must be 6,500 ft. The system goals Productivity Index Useful Load Fraction Fuel Balance Missed Approach Climb Gradient, landing Missed Approach Climb Gradient, takeoff Landing Field Length Takeoff Field Length The bounds on the system variables Minimize The sum of the deviation variables Z = ΣW j (d j + d j + ))} Variables needed by weights designer: Wing Area, S Figure 6. Compromise Decision Support Models for the Players 7
Scenario II: In this scenario, the aerodynamics compromise DSP is modified based on robust design principles as described in Section 3. In addition to finding the value of S in the Aerodynamics subsystem model, we also find a maximum range of the nonlocal coupling variables Wto and Ti such that the solution is robust and feasible within that range. We run a preliminary pattern search to find the most promising area in the Aerodynamic subsystem's design space and then refine our analysis to maximize these bounds of the noise or coupling variables where the solution of Aerodynamics subsystem is insensitive to the perturbation of these noise variables. The design as a result may not be optimal, but we are interested in robust regions that are reasonably close to optimal. The compromise DSPs for the Aerodynamics subsystem is shown in Figure 7 below. Given Relevant constants Important Aerodynamic Relationships and Equations Upper and Lower Bounds of coupling variables of Weight player Aerodynamics Design Variables l, b, S Best: m(wto,ti), Max: D(Wto,Ti) Deviation variables associated with the goals The constraints: 7 modified aerodynamic constraints for worstcase scenario Ex: The aspect ratio (AR) must be less than 10.5 E(AR) +?AR/?Wto Wto +?AR/?Ti Ti = 10.5 Where E represents the expected value of a function The goals: 5 goals on aerodynamics mean performance A i + d i d i + = G i Where A i is the mean performance value (i=1, 5) 5 goals on aerodynamics performance variation (?A i /?Wto) σwto + (?A i /?Ti) σti + d i d i + = 0 Variable bounds (upper and lower bounds) Objective Minimize the deviation function Z = [ f 1 (d i, d i + ),..., f 10 (d i, d i + ) ] Figure 7. Robust Design Model for Aerodynamics as leader for scenario II The compromise DSP for Weights subsystem as follower for this scenario is given in Figure 8. The Weights subsystem is now constrained to find a solution within the modified bounds passed to him by the leader. Given Relevant constants Important Fuel/Weight Relationships Solution of Aerodynamics S, Ldc, Ldt, Ldl, Vbr Weights Design Variables Wto, Ti Deviation variables associated with the goals The constraints: 6 weights constraints The goals: 7 goals on weights performance A i + d i d i + = G i Where A i is the mean performance value (i=1, 7) Variable (upper and lower) bounds set by the leader Objective Minimize the deviation function Z = [ f 1 (d i, d i + ),..., f 7 (d i, d i + ) ] Figure 8: Model for Weights as follower for Scenario II Three different design cases are considered when solving Scenario II and III. In the first case we use the Archimedean formulation in which both the goals for robustness and performance are placed at same level with equal weights. In the next two cases we make use of the preemptive formulation in which the goals are placed at different priority levels (Mistree, Hughes et al., 1993). The values of the modified ranges for the coupling variables Wto and Ti are found to be approximately 16% and 1% respectively of the total variable ranges and are shown in Table along with the original bounds in the shaded rows. These values allow the Aerodynamics discipline to find acceptable solutions. Beyond this, the Taylor series approximations may become invalid. The results consisting of the design variables, linking variables, and deviation functions for the Aero subsystem are given in Table 3. The maximum standard deviation allowed in the performance is limited to 15% about the mean and forms the threshold limit for the performance variance allowed. The solution of the Weights subsystem as follower is shown in Table 4. Wto L 140000.0 Wto U 50000.0 Wto L* 174800.0 Wto U* 19300.0 Ti L 750.0 Ti U 55000.0 Ti L* 3787.5 Ti U* 4001.5 Table. Original Bounds and Ranges found for Ti and Wto by Pattern Search 8
Case A Robustness & Performance at one level Case B Level 1: Performance Level : Robustness Case C Level 1: Robustness Level : Performance Design Variables b (ft) 108.455 109.187 137.593 S (ft ) 1677.93 1510.78 498.0 l (ft) 111.671 131.675 105.946 Performance Variance Performance Variance Performance Variance Deviation Functions 1.58331 3.06E0 1.5939 3.14E0.13873.87E0 Table 3. Robust Solution of Aerodynamics as Leader for Scenario II A B C Design Variables Ti (lbs) 39061.5 38650 38897.6 Wto (lbs) 193067 19300 19300 Linking S (ft ) 1677.93 1510.78 498.0 Deviation Function 1.45984 1.69375 1.8655 Table 4. Solution of Weights as Follower for Scenario II As observed from the results obtained in the three cases that the deviation function values for both the Aerodynamics and Weights subsystem increase as compared to that obtained in Scenario I where there were no robust design considerations, but the effect of this increase in the deviation function depends on the tradeoff that exists between achieving an optimum solution and eliminating downstream coupling as explained in Section 3. We explore this tradeoff further in Scenario III. Scenario III: In addition to the Type I robust design introduced in the previous scenario, we want to provide the follower with increased design flexibility in their decisions. Since the follower is constrained to choose designs within the modified ranges passed to him by the leader, the leader providing him a range of acceptable linking variable values ( S) instead of a single design point (S), increases the chances of finding a acceptable solution (Chen and Lewis 1999). The compromise DSP for the Aerodynamics as leader for Scenario III is given in Figure 9. The formulations are the same to the previous scenario, except for the range of wing area S that is considered as an additional design variable in the Aerodynamics robust design model. The formulations for the worst case scenario constraints and the variance of the performance from Scenario II are augmented to include the effects of deviation of the control factor, S. The compromise DSP for Weights subsystem as follower for this scenario is given in Figure 10. In addition to his own design variables, the Weights subsystem now has to choose the best value of S in the range S passed to him by the Aerodynamics subsystem. Given Relevant constants Important Aerodynamic Relationships and Equations Upper and Lower Bounds of coupling variables of Weight subsystem Aerodynamics Design Variables l, b, S Best: µ(wto,ti), Max: (Wto,Ti) Range of Wing Area, DS Deviation variables associated with the goals The constraints: 7 modified aerodynamic constraints for worstcase scenario Ex: The aspect ratio (AR) must be less than 10.5 E(AR) +?AR/?S S +?AR/?Wto Wto +?AR/?Ti Ti = 10.5 Where E represents the expected value of a function The goals: 5 goals on aerodynamics mean performance: A i + d i d i + = G i Where A i is the mean performance value (i=1, 5) 5 goals on aerodynamics performance variation: (?A i /?S) σs + (?A i /?Wto) σwto + (?A i /?Ti) σti + d i d i + = 0 Variable bounds (upper and lower bounds) Objective Minimize the deviation function Z = [ f 1 (d i, d i + ),..., f 10 (d i, d i + ) ] Figure 9. Robust Model of Aerodynamics player as leader for Scenario III 9
The ranges of modified bounds of the linking variables, Wto and Ti, are the same as given in Table. The results of design variables, linking variables, and deviation functions for the Aero subsystem are given in Table 5. Once the leader has solved his model, these solutions are passed over to the follower along with the range of the linking variable S. The solution of the Weights subsystem as the follower are shown in Table 6. It can be noted that in all three cases the performance of the follower has improved as compared to scenario II with a further increase in the deviation of the leader. We have set the minimum value of S to 150 which is approximately 10% of the total range of variable S. Depending on the leader's willingness to sacrifice his performance and the need of achieving flexibility the minimum range value could be increased and additional constraints and objectives included (Chen and Lewis 1998). Given Relevant constants Important Fuel/Weight Relationships and Equations Solution of Aerodynamics S, S, Ldc, Ldt, Ldl, Vbr Weights Design Variables Wto, Ti Most favorable value of coupling variable S within DS Deviation variables associated with the goals The constraints: 6 weights constraints The goals: 7 goals on weights performance A i + d i d i + = G i Where A i is the mean performance value (i=1, 7) Variable (upper and lower) bounds set by the leader Objective Minimize the deviation function Z = [ f 1 (d i, d i + ),..., f 7 (d i, d i + ) ] Figure 10. Model of Weights player as follower for Scenario III Case A Robustness & Performance at one level Case B Level 1: Performance Level : Robustness Case C Level 1: Robustness Level : Performance Design Variables b(ft) 108.705 109.078 111.386 S (ft ) 1619.15 169.06 3.04? S 15.1 169.091 171.606 L (ft) 135.15 13.609 106.144 Performance Variance Performance Variance Performance Variance Deviation Functions 1.60866 4.3E0 1.6647 4.58E0.655 3.9E0 Table 5. Robust Solution of Aerodynamics as Leader for Scenario III A B C Design Variables Ti (lbs) 39078.7 38650 38897.6 Wto (lbs) 193095. 19300 19300 Linking S (ft ) 1771.7 1798.15 150.43 Deviation Function 1.4400 1.63067 1.6561 Table 6. Solution of Weights as Follower for Scenario III 10
As a summary, plots of results for three cases of different priority levels are shown in Figure 11. For the Aerodynamics model, we plot the performance variance against the performance mean deviation for no robust design (Scenario I), Type II (Chen and Lewis, 1999), Type I (Scenario II) and Type I & II (Scenario III) robust design to give a comprehensive overview at the tradeoffs involved in each scenario. Performance Mean Deviation 1.8 1.6 TypeI&IIRD 1.4 TypeIRD 1. w/o RD Type IIRD 1 0.8 0.6 0.4 0. 0 0 0.01 0.0 0.03 0.04 0.05 Performance Variance Performance Deviation 1.5 1.45 1.4 1.35 1.3 1.5 1. 1.15 w/o RD TypeIIRD TypeIRD TypeI&IIRD a. Aerodynamics: Equal Priority b. Weights: Equal Priority Performance Mean Deviation.5 1.5 1 0.5 0 w/o RD Type II RD Type I&IIRD Type II RD 0 0.005 0.01 0.015 0.0 0.05 0.03 0.035 Performance Variance Performance Deviation 1.8 1.6 1.4 1. 1 0.8 0.6 0.4 0. 0 w/o RD TypeIIRD TypeIRD TypeI&IIRD c. Aerodynamics: RobustnessLevel I d. Weights: RobustnessLevel I PerformanceLevel II PerformanceLevel II Performance Mean Deviation 1.8 1.6 1.4 Type IRD Type I&IIRD 1. 1 w/o RD TypeII RD 0.8 0.6 0.4 0. 0 0 0.01 0.0 0.03 0.04 0.05 Performance Variance Performance Deviation 1.8 1.6 1.4 1. 1 0.8 0.6 0.4 0. 0 w/o RD TypeIIRD TypeIRD TypeI&IIRD e. Aerodynamics: PerformanceLevel I f. Weights: PerformanceLevel I RobustnessLevel II RobustnessLevel II Figure 11: Comparison of Aero and Weights Subsystem Deviation Functions 11
The Weights subsystem performance deviation function is also plotted to observe the response to the Aerodynamics subsystem behavior. As explained in previous three scenarios, it can be seen in plots a, c and e that the deviation function for Aerodynamics is the least in scenario I with a zero variance. Including Type II robust design only improves Weight subsystem's solution and the leader has to sacrifice his performance to some extent as seen in plots a, c and e. With Type I robust design considerations, both the Aerodynamics and Weights subsystem's performances are sacrificed, but this sacrifice could be offset by a reduction in the coupling effects between them to make their design insensitive to the changes in the objectives and constraints that might take place as the design proceeds through the later stages. Combining Type I and II gives us the benefits of both scenarios where the leader can find a solution insensitive to the coupling variables and the follower is provided with a decision flexibility. The choice of the scenario depends on the objectives of the designers and their willingness to sacrifice performance and gain robustness. 5. CLOSURE In this paper, we develop a comprehensive design methodology to incorporate robust design concepts into game theoretic protocols for multidisciplinary design. We develop Type I robust design for internal noise reduction and combine that with Type II robust design to achieve a combined robust and flexible design. The concept is developed to reduce downstream coupling in a preliminary design environment making the leader's design less likely to be affected due to the changes in the noise variables as the design process evolves. By minimizing the effects of the decisions made by one discipline upon the other, we feel that iteration time can be saved, and the ability to make decisions concurrently can be improved. We feel that the integration of robust design concepts with game theoretic concepts to minimize the effect of nonlocal design information could prove extremely beneficial to complex systems design. 6. ACKNOWLEDGEMENTS We would like to acknowledge the support from NSF Grants DMI9800435, DMI970994 and NASA Fellowship Grant NGT15185. We are also grateful to Dr. Wei Chen, University of Illinois at Chicago for her support and advice. 7. REFERENCES Balling, R. J. and J. Sobieski (1994, September 79). An Algorithm for Solving the SystemLevel Problem in Multilevel Optimization. 5th AIAA/USAF/NASA/ISSMO Symposium on Recent Advances in Multidisciplinary Analysis and Optimization, Panama City, FL., pp. 794809 Bloebaum, C. L., P. Hajela, et al. (199). NonHierarchic System Decomposition in Structural Optimization. Engineering Optimization, 19, pp. 171186. Chang, T. S., A. C. Ward, et al. (1994). Distributed Design with Conceptual Robustness: A Procedure Based on Taguchi's Parameter Design. ASME Concurrent Product Design Conference, Chicago, IL.,74, pp. 199 Chen, W., J.K. Allen, D., Mavris, and F. Mistree (1996). A Concept Exploration Method for Determining Robust Top Level Specifications. Engineering Optimization, pp. 137 158. Chen, W. and K. Lewis (1999). A Robust Design Approach for Achieving Flexibility in Multidisciplinary Design. AIAA Journal of Aircraft, To appear Lewis, K. and Mistree, F. (1998). "Collaborative, Sequential and Isolated Decisions in Design." ASME Journal of Mechanical Design, Vol. 10, pp. 64365. Lewis, K. (1996). An Algorithm for Integrated Subsystem Embodiment and System Synthesis, Ph.D. Dissertation, G.W. Woodruff School of Mechanical Engineering. Atlanta, GA, Georgia Institute of Technology. Lewis, K. and F. Mistree (1997). Modeling Interactions in Multidisciplinary Design: A Game Theoretic Approach. AIAA Journal,35(8), pp. 1387139. Mistree, F., O. F. Hughes, et al. (1993). The Compromise Decision Support Problem and the Adaptive Linear Programming Algorithm. Structural Optimization: Status and Promise. M. P. Kamat. Washington, D.C., AIAA, pp. 4786. Phadke, M. S. (1989). Quality Engineering using Robust Design. Englewood Cliffs, New Jersey, Prentice Hall. Sundaresan, S., K. Ishii, et al. (1993). A Robust Optimization Procedure with variations on Design Variables and Constraints. Advances in Design Automation, 1, pp. 379386. Vincent, T. L. (1983). Game Theory as a Design Tool. ASME Journal of Mechanisms, Transmissions, and Automation in Design,105, pp. 165170. Vincent, T. L. and W. J. Grantham (1981). Optimality in Parametric Systems. New York, NY, John Wiley. Womack, J., D. Jones, et al. (1990). The Machine that Changed the World. New York, New York, Rawsan Associates. 1