Collaborative, Sequential, and Isolated Decisions in Design

Proceedings of DETC’97 1997 ASME Design Engineering Technical Conferences September 14-17, 1997, Sacramento, California

DETC97/DTM-3883

COLLABORATIVE, SEQUENTIAL, AND ISOLATED DECISIONS IN DESIGN Kemper Lewis Department of Mechanical and Aerospace Engineering State University of New York and Buffalo Buffalo, New York 14260 USA (716) 645-2593 (o) (716) 645-3668 (fax) [email protected]

Farrokh Mistree George W. Woodruff School of Mechanical Engineering Georgia Institute of Technology Atlanta, Georgia 30332-0405, USA (404) 784-8412 (o) (404) 894-9342 (fax) [email protected]

ABSTRACT The Massachusetts Institute of Technology (MIT) Commission on Industrial Productivity, in their report Made in America, found that six recurring weaknesses were hampering American manufacturing industries. The two weaknesses most relevant to product development were 1) technological weakness in development and production, and 2) failures in cooperation. The remedies to these weaknesses are considered the essential twin pillars of CE: 1) improved development process, and 2) closer cooperation. In the MIT report, it is recognized that total cooperation among teams in a CE environment is rare in American industry, while the majority of the design research in mathematically modeling CE has assumed total cooperation. In this paper, we present mathematical constructs, based on game theoretic principles, to model degrees of collaboration characterized by approximate cooperation, sequential decision making and isolation. The design of a pressure vessel and a passenger aircraft are included as illustrative examples.

1 STRATEGIC DECISION MAKING IN DESIGN: A DECISION-BASED PERSPECTIVE We approach design from a decision-based perspective wherein the principal (but not only) role of a designer is to make decisions. This seemingly limited role is useful in providing a starting point for developing design methods based on paradigms that spring from the perspective of decisions made by designers (who may use computers) as opposed to design that is assisted by the use of computers, optimization methods (computer-aided design optimization) or methods that evolve from specific analysis tools such as finite element analysis. Decisions help bridge the gap between an idea and reality. In general, decisions are characterized by information from many sources (and disciplines) and may have wide ranging repercussions. In Decision-Based Design, decisions serve as markers to identify the progression of a design from initiation to implementation to termination. In Decision-Based Design they represent a unit of communication; one that has both domain-dependent and domain-independent features. Some principal observations and beliefs from a Decision-Based Design perspective are as follows (Mistree, et al., 1990):

Key Words: Design Management, Concurrent Engineering, Game Theory, Optimization, Design Teams, Distributed Decision Making

• •

NOMENCLATURE player: A designer and their associated computer-based analysis and synthesis tools RRS: Rational Reaction Set DSP: Decision Support Problem GSE: Global Sensitivity Equations Ld(t,c,l): Lift-to-drag ratio on take-off, cruise, and landing U: Useful Load PRI: Productivity Index AR: Aspect Ratio

• •

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The principal role of an engineer or designer is to make decisions. Design involves a series of decisions some of which may be made sequentially and others that must be made concurrently. Design involves hierarchical decision making and the interaction between these decisions must be taken into account. Design productivity can be increased through the use of analysis, visualization and synthesis in complimentary roles, and by augmenting the recognized capability of computers in processing numerical information to include the processing of symbols (for example, graphs, pictures, drawings,

Copyright © 1997 by ASME

•

words) and reasoning (for example, list processing in artificial intelligence). Life-cycle considerations that affect design can be modeled in upstream design decisions.

Decomposition of complex systems into smaller more manageable problems is a common approach but also creates integration problems (Eppinger, et al., 1990, Kusiak and Larson, 1995). To illustrate, let us assume that a complex system such as an aircraft has been decomposed into disciplinary subsystems such as propulsion and structures. Assume that the propulsion designer controls x , and the parameters, p, are variables controlled by the structures designer which the propulsion designer has no control, and f i are respective propulsion objective functions. We assert that the following standard multiobjective formulation, representing the propulsion subsystem

The characteristics of decisions are governed by the characteristics associated with the design of real-life engineering systems. These characteristics are summarized by the following descriptive sentences: • • • • •

•

•

Decisions in design are invariably multileveled and multidimensional in nature. Decisions involve information that comes from different sources and disciplines. Decisions are governed by multiple measures of merit and performance. All the information required to arrive at a decision may not be available. Some of the information used in arriving at a decision may be hard, that is, based on scientific principles and some information may be soft, that is, based in the designer's judgment and experience. The problem for which a decision is being made is invariably loosely defined and open. Virtually none of the decisions are characterized by a singular, unique solution. The decisions are less than optimal and are called satisficing solutions. Design is the process of converting information that characterizes the needs and requirements of a system into knowledge about the system itself. In Decision-Based Design it is the making of decisions that brings about the transformation of information into knowledge.

minimize f(x,p) = {f1(x,p),...,fr(x,p)}

(1)

x∈ f(p) ⊂ ℜn

is the typical starting point for much of the current research and practice in systems modeling and applied optimization. And yet in specific design instances, this assertion should be boldly challenged. For example, since the propulsion designer only controls x and the structures designer controls p, how is p chosen in the propulsion design? Can the propulsion designer assume that the structural designer will always select the vector that is most advantageous to the propulsion design? If not, how should the propulsion designer respond to this conflict? This scenario describes a two-player strategic game where one player controls x and the other player controls p and where p represents all decisions which are outside the scope of the designer controlling x (Aubin, 1979, Dresher, 1981, Von Neumann and Morgenstern, 1944). In this paper, we model these types of strategic relationships using game theoretical principles. A "game" consists of multiple decision-makers who each control a specified subset of design variables and who each seek to minimize their own cost functions subject to their individual constraints (Myerson, 1991). In a game, these multiple decision makers are required to select single decision strategies to optimize their set of rewards. However, each player's reward depends on the other player's strategies, i.e., it depends on design variables that are controlled by other players (Chen, et al., 1994). The fact that players lack control over all decision variables affecting their rewards is what makes a game a game and what distinguishes it from an optimization problem.

By focusing upon decisions, we have a description of the processes written in a common “language” for teams from the various disciplines -- a language that can be used in the process of designing. In complex systems such as aircraft, the decisions are typically made by design groups organized by discipline. Ideally, a seamless Concurrent Engineering philosophy could be applied to a company's design process among disciplines. In reality, however, the simultaneous nature of information flow and cooperation, inherent in CE, among design teams makes concurrency difficult, if not impossible. The Massachusetts Institute of Technology (MIT) Commission on Industrial Productivity, in their report Made in America (M.L. Dertouzos, 1989), found that six recurring weaknesses were hampering American manufacturing industries. The two weaknesses most relevant to product development were 1) technological weakness in development and production, and 2) failures in cooperation. The remedies to these weaknesses are considered the essential twin pillars of CE: 1) improved development process, and 2) closer cooperation (Schrage and Gordon, 1992). In the MIT report, it is recognized that total cooperation among teams in a CE environment is rare in American industry, while the majority of the design research in mathematically modeling CE has assumed total cooperation.

The theoretical and mathematical foundations of games are used to abstract the processes required to design a complex system as a game. The players in this game are defined as the disciplinary design teams and their associated analysis/synthesis tools. There are various game protocols depending on the level of cooperation and behavior of the players. Certain protocols lend themselves nicely to modeling interactions in design, namely the cooperative or Pareto formulation when the players work together and communicate, the Nash or noncooperative formulation when the players act in their own self-interest or are

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isolated, and the Stackelberg or leader/follower formulation when one player dominates another in a sequential relationship. In the next section we provide the mathematical background of each protocol. The fundamental decision construct which is used as a framework to formulate, model, and solve the decisions made by each disciplinary team is the compromise Decision Support Problem (DSP) (Mistree, et al., 1993). The relevant background of the compromise DSP is provided in the next section as well.

Given An alternative to be improved, domain dependent assumptions The system parameters: n number of system variables, q inequality constraints, p+q number of system constraints, m number of system goals, gi(X) system constraint functions , fk(di ) function of deviation variables to be minimized at priority level k for the preemptive case. Find System Design Variables, Xi i = 1, …, n - + Deviation Variables, di , d i i = 1, …, m Satisfy System constraints (linear, nonlinear) g i(X) = 0 ; i = 1, .., p g i(X) ≥ 0 ; i = p+1, .., p+q System goals (linear, nonlinear) Ai(X) + d-i - d+i = Gi ; i = 1, …, m Bounds X imin ≤ Xi ≤ Ximax ; i = 1, …, n

2 FRAME OF REFERENCE: DECISION MODELS In this section, the mathematics supporting the fundamental theoretical constructs used in this paper are presented. We begin with our domain-independent decision model, the compromise DSP. The three fundamental game theoretic protocols used in this work are introduced in this section. The theory and mathematics behind the protocols are integrated with the compromise DSP and implemented and applied in a design context in Section 3. In Section 5, the resulting implications in modern design processes are explored.

di- , d+i ≥ 0, d-i . d +i = 0 ; i = 1, …, m Minimize: deviation function f = [ f1( d-i , d+i ), ..., fk ( d-i , d+i ) ] Figure 1. Mathematical Form of a Compromise DSP

2 . 1 Our Fundamental Decision Model We characterize the decisions made by each player using a compromise Decision Support Problem. The compromise DSP is a multiobjective decision model which is a hybrid formulation based on Mathematical Programming and Goal Programming (Mistree, et al., 1993). It is used to determine the values of design variables which satisfy a set of constraints while achieving a set of conflicting goals as well as possible.

2 . 2 Theoretical Development of Game Constructs For Design For this discussion, assume there are two players, P1 and P2, who each control x1 and x2 and who try to minimize their own deviation functions, f1 and f2 respectively. When 'min f' is used in the discussion, it succinctly represents the solution of a compromise DSP. The term ƒ(x1,x2) represents the certain deviation function for values of the design variables, x 1 and x 2 , of two players. In the following, we have summarized material from (Aubin, 1979, Vincent and Grantham, 1981).

The system descriptors, namely, design and deviation variables, system constraints, system goals, bounds and the deviation function are described in detail in (Mistree, et al., 1993) and are therefore not repeated here. The mathematical form of the compromise DSP is summarized in Figure 1. The strategy of a player, embodied by the compromise DSP is to minimize the deviation function. The deviation function is a measure of the difference between what a designers wants and what he/she can achieve and incorporates multiple goals. The deviation function can either be a traditional Archimedean weighting scheme, or can be Preemptively ordered if only preferences are known. The compromise DSP is solved using the Adaptive Linear Programming (ALP) algorithm (Mistree, et al., 1993) if only continuous variables are used. If discrete and continuous variables are used, the compromise DSP is solved using the Foraging-directed Adaptive Linear Programming (FALP) algorithm (Lewis and Mistree, 1996). Both algorithms are part of the decision support software DSIDES (Decision Support in Designing Engineering Systems) (Reddy, 1992).

Cooperative or Pareto solution In our cooperative model, both players have knowledge of the other player's information (they know the "hands" of the other players) and they work together to find a Pareto solution. In this model, the players interact simultaneously. The representative issues in simultaneous design are addressed in (Ishii, et al., 1989). If the players cooperate, they can be expected to obtain better solutions than when they do not. A pair (x1p, x2p) is Pareto optimal if no other pair (x1,x2) exists such that

Integration of game theoretic principles with the compromise DSP to model realistic interactions in multidisciplinary design is illustrated in the next section.

ƒ1(x1,x2) < ƒ1(x1p, x2p) & ƒ2(x1,x2) < ƒ2(x1p, x2p)

3

(2)

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Nash or noncooperative solution The Nash or noncooperative formulation occurs when coalition among players is not possible due to organizational, information, or process barriers. This model is one of isolation where the players must make decisions by assuming the other decision makers try to adversely affect their outcome (they do not know the "hands" of the other players). A strategy pair (x1N, x2N) is a Nash solution if ƒ1(x1N,x2N) = min ƒ1(x1,x2N)

Stackelberg or leader/follower solutions Consider the case when one player dominates another, i.e., the two players have a leader-follower relationship. This is a common occurrence in a design process when one discipline dominates the design (which may be the case when one discipline plays a larger role in a design), or in a design process that involves a sequential execution of interrelated disciplinary processes. This is the traditional over-the-wall realization of a design process. P1 is a leader if he/she declares his/her strategy first (solves his/her model) by assuming that the follower P2 behaves rationally. Thus, the model with P1 as leader is minimize ƒ1(x1,x2)

x 1 ∈ X1

& 1N 2N ƒ2(x ,x ) = min ƒ2(x1N,x2)

(3)

x 2 ∈ X2

This solution is difficult to compute since it is the fixed point of a nonlinear map, namely, (x1N,x2N) ∈ X1N(x2N) × X2N(x1N) (4) where

(x1,x2) ∈ X1 × X 2

satisfying x2 ∈ X2N(x1) and the model with P2 as leader is minimize ƒ2(x1,x2)

(7)

(5)

satisfying x1 ∈ X1N(x2)

(8)

(6)

where X 1N (x2) and X 2N (x1) are defined in Eqn. 5 and Eqn. 6. For two players, these Stackelberg games are special cases of bilevel models.

X1N(x2) := {x1N ∈ X1 : ƒ1(x1N,x2) = minƒ1(x1,x2)} x 1 ∈ X1

(x1,x2) ∈ X1 × X 2

and X2N(x1) := {x2N ∈ X2 : ƒ2(x1,x2N) = minƒ2(x1,x2)} x 2 ∈ X2

These protocols are implemented within and between multiple compromise DSPs to model multiple decisions being which may or may not be coupled. Practical implementation of our strategic decision models are discussed in the next section.

are called the rational reaction sets of the two players. The rational reaction set (RRS) of a player is a fundamental construct in game theory and has strong implications in design. The RRS is a set of solutions that an isolated decision maker constructs based on unknown information from other decision makers. In a noncooperative model, all the players construct their own RRS and a solution exists if the intersection of these sets produce something other than the null set (eqn. 4).

2.3 Practical Implementation Although the theoretical fundamentals presented in the previous section are straightforward, their applicability to design warrants some deeper investigation. Although implementations of game theoretical principles in a practical setting has been demonstrated in other fields where strategic decision making is paramount, the use of game theory in design where strategic decision making is also important is relatively infantile. The descriptive motivations of this work parallel reverse engineering methods which analyze existing design products in order to make design improvements in the next product evolutions (Otto and Wood, 1996). Game theoretical notions can be used to model existing processes and make improvements on process integration to meet product requirements. The practical implementation of game theoretical principles in design have far-reaching implications, some of which include: • Design teams could capture the motivations of other decision makers, they can better determine what decisions to make and why to make them. • Managers could strategically oversee the disciplinary design teams and ensure that a system level perspective is taken when necessary and a balance exists between disciplinary and system-level objectives.

The rational reaction set embodies the decision making strategy of a player as a function of the control variables of another player. However, functions of this type, x2 = f(x1), are difficult to compute. In simple problems the rational reaction set can be calculated exactly. However, in complex problems, finding exact rational reaction sets are difficult if not impossible since it involved finding functions relating the independent variables of one player as a function of the independent variables of another player. In this case, the rational reaction sets must be approximated. We approximate the rational reaction sets using response surface methodology coupled with design of experiments techniques (Box and Draper, 1987, Cartuyvels and Dupas, 1993, Lewis, 1996, Montgomery, 1991). In the pressure vessel example in Section 3.1, both the exact and approximate RRS's are shown, while in the more complex aircraft example in Section 3.2 only the approximate RRS's are used.

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• Companies could strategically schedule their design and production schedules based on necessary team interactions and reduced iteration (Eppinger, 1991). • Certain disciplinary design teams could only focus on their specialized expertise, while others could be coupled with other disciplines where the potential for interaction and iteration is high. • Certain disciplines play a larger role in the design of complex systems than others. Those disciplines should be accounted for earlier than others to avoid unnecessary rework and iteration. If some sense of rigorous mathematical investigations and developments can predict these type of scenarios and their influence on the final design, companies can design systems more effectively by assuring that the proper information is available at the right time, and more efficiently by avoiding unnecessary iteration and coupling. In effect, we are addressing design process parameters in order to best control system integration, much like design product parameter control addressed in (Otto and Antonsson, 1993). In addition, an inherent assumption in game theory is the existence of multiple objectives for multiple decision makers (Rao and Freiheit, 1991), which is also common in design (Herling, et al., 1995, Stadler, 1988, Thurston, et al., 1994). We explore the practical implementation of the four design scenarios.

This scenario is ideal and will lead to a Pareto solution as discussed in Section 2.2 if resolved properly. This is the ideal case and is difficult to achieve in a practical environment. A more typical and possibly more appropriate scenario is approximate cooperation. Approximate cooperation In this scenario, decision makers or their tools have an approximate representation of the information they need from other decision makers. This is the more practical case when designers say things like, "Well, it will be about 3-3.5 feet in diameter and approximately 11 feet in length." Many times this type of information is valuable and is easier to determine than the exact values which may require more time and resources. Therefore, in the approximate cooperation model, the necessary information is constructed using approximations of nonlocal variable information. In this paper, first order approximations are used, but higher order approximations can be constructed if warranted. The generic model for the approximate cooperation scenario is shown in Figure 3.

Cooperation Senge asserts that systems thinking should pervade modern organizations where a shared vision is common and subscribed to by all members of an organization (Senge, 1990). Establishing this shared vision and systems level thinking are the cornerstones for ensuring full cooperation in a product realization process. Although all members of a product realization team may have shared visions and cooperate personally, this does not mean that the technological backbones of the process are cooperating. Full cooperation in design requires both personal cooperation and what we call "mathematical cooperation" where the disciplinary analysis and synthesis models are communicating and cooperating in a integrated computer environment. These are the assumptions under which our cooperation model is developed. In the cooperation model, shown in Figure 2, it is assumed that all decision makers (or their associated computer tools) have all the necessary information they need in the appropriate form.

Player 1

Given Find x1 d1i+,d1iSatisfy constraints goals Minimize Z1

approximate representations

Given Find x2 d 2i+,d2i Satisfy constraints goals Minimize Z2

Figure 3. Approximate Cooperation Decision Support Model In this work, the nonlocal functions are approximated using a first-order Taylor's series expansion, of the form

s(x) ≈ s o + ∇s(x o )(x − x o ) where s represents the nonlocal variable needed locally. The appropriate first-order derivatives are constructed using the Global Sensitivity Equation (GSE) approach. Using the GSE method, the total derivatives of the dependent variables can be found with respect to the independent variables from every player, through the chain rule. Therefore, a player has an approximation of a nonlocal variable based on the influence of every player. This is the essence of approximate cooperation in design. In the context of the compromise DSP, the compromise DSPs of each player are solved simultaneously and the nonlocal approximations needed by each player are continuously updated at each solution iteration until convergence of all players has been achieved. Although it has not been investigated, the use of fuzzy set theory to model approximate cooperation among multiple decision makers could be beneficial and worth exploring.

Player 2

full representations

Player 2

Player 1

Given Find x1 d1i +,d1i Satisfy constraints goals Minimize Z1

Given Find x2 d2i+,d2iSatisfy constraints goals Minimize Z2

Similar models of approximate cooperation has been assumed in many multidisciplinary design optimization studies (Bloebaum, et al., 1992, Renaud and Gabriele, 1991, Renaud

Integrated Computer Environment

Figure 2. Full Cooperation Decision Support Model

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and Gabriele, 1994, Sobieszczanski-Sobieski, 1988). They have been shown to produce effective results under the appropriate assumptions, such as "concurrent decision making is available." Although concurrency is effective and ideal in many scenarios, it is difficult to achieve in a complex product design process. If concurrency does not exist, the assumptions for approximate cooperation are invalid. A sequential decision making construct may better model actual design decisions.

available to make a decision and assumptions must be made, usually worst-case ones Moreover, organizational barriers may exist (e.g., time zone differences, building layouts) that cause designers to be isolated from the necessary information they need. Therefore, we use the noncooperative principles from game theory to model these scenarios when isolated decisions take place. Although the noncooperative principles were developed with an economics mindset where noncooperation frequently occurs, in design, designers should always be striving for cooperation. Thus, when the term "noncooperation" is used here, it should not imply designers voluntarily turning their backs on each other (although this may still occur). Rather it should imply involuntary noncooperation through organizational or informational barriers among decision makers. The generic model for isolation in design is shown in Figure 5. There is, in effect, a wall between two decision makers. Therefore, each player determines a set of rational solutions (a player's rational reaction set) and if these sets intersect, then there is a feasible solution.

Sequential Decision Making Although many companies are breaking down the historical walls between disciplines and the iteration that occurs, true concurrency is difficult to achieve. Many times, decision makers still make their decisions sequentially. Furthermore, it may even be advantageous for certain disciplines to interact sequentially, as the influence of one discipline on another may be strongly uni-directional. Therefore, the game theoretic leader/follower principle can be used to model a design scenario where one or more disciplines make their decisions first, based on an assumption that the later disciplines will behave rationally. This assumption may not hold weight in a competitive environment in the business or political sectors, but in design, decision makers should behave rationally and according to the laws of nature. Once the leaders have made their decisions, the follower then makes their decisions. The leader has the advantage of making their decision first, but has to make the assumption that the followers will behave rationally. The follower is constrained by the leader's decision, but does not have to make any assumptions. This model is shown in Figure 4.

In the next section, we explore two design problems, each with two distinct decision makers with their own decision model and motivations. We investigate the effects of the decision making interaction (or lack of) on the form and performance of the final design.

Player 1-Leader Given Rationality of followers Find x1 d1i+,d1iSatisfy constraints goals Minimize Z1

Assumption of Rationality

Player 2

Player 1 Given Find x1 d1i+,d1iSatisfy constraints goals Minimize Z1

Given Find x2 d2i+,d2iSatisfy constraints goals Minimize Z2

Leader’s Solution Player 2-Follower Given Leader’s solution Find x2 d2i+,d2iSatisfy constraints goals Minimize Z2

Rational set of solutions for unknown actions of Player 2

Rational set of solutions for unknown actions of Player 1

Intersection represents acceptable solutions

Figure 5. Isolated Decision Support Problem Model

Figure 4. Sequential Decision Support Problem Model Isolation The final design decision model is when decision makers are acting in isolation. With time deadlines both at the system and subsystem design levels, designers are continually under pressure to make decisions expediently and effectively. However, many times the appropriate information is simply not

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EXAMPLES

3.1

Design of a Pressure Vessel As a simple study, the design of a thin-walled pressure vessel which has hemispherical ends as shown in Figure 6 is

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used. The nomenclature for this example is presented in Table 1. This case study is derived from the example studied in (Karandikar and Mistree, 1992, Rao, et al., 1996). The design variables are the radius R, the length L, and the thickness T. The vessel is to withstand a specified internal pressure P and the material is also specified. There are two objectives: to minimize the weight and to maximize the volume of the cylinder, both subject to stress and geometry constraints. It is recognized that this example is not naturally a multi-player problem, and that a single-player multiobjective formulation is normally used. A multi-player formulation is used in this paper to demonstrate the benefits of using game theoretic principles to model decision making in design. Two players are used: 1) player VOL who wishes to maximize the volume and thus controls R and L, and 2) player WGT who wishes to minimize the weight of the vessel and controls T.

Satisfy

Table 1. Nomenclature for the Pressure Vessel Example

Find Design Variables: R and L Underachievement Deviation Variable associated with the volume goal, dVSatisfy

W V R T L P St ρ σ circ TV

L

Geometric constraints:

g 2: 5T - R ≤ 0 (9) g3: R + T - 40 ≤ 0 g4: L + 2R + 2T - 150 ≤ 0 T l ≤ T ≤ Tu W - dW+ = WTV

PLAYER VOL Given

4 V(R, L) =  πR3 + πR2 L  3 

Volume

Weight of the pressure vessel, lbs. Volume, in.3 Radius, in. Thickness, in. Length, in. Pressure inside the cylinder, Klb. Allowable tensile strength of the cylinder material, Klb. Density of the cylinder material lbs./in.3 Circumferential stress lbs./in.2 Target Value for a goal

R

PR ≤ St T

g1:

Bounds: Weight Goal: Minimize dW+

σ circ =

PR ≤ St (10) T

Stress constraint:

g1:

Geometric constraints:

g 2: 5T - R ≤ 0 g3: R + T - 40 ≤ 0 g4: L + 2R + 2T - 150 ≤ 0 R l ≤ R ≤ Ru L l ≤ L ≤ Lu V + dV- = VTV

Bounds: T

R

σ circ =

Stress constraint:

Volume Goal: Minimize dV-

R R

The specific data (problem constants) for this problem is given in Table 2.

L

Table 2. Pressure Vessel Parameters R

P St ρ Ll Lu Rl Ru Tl Tu WTV VTV

Figure 6. Thin-Walled Pressure Vessel The compromise DSPs of the two players are shown below: PLAYER WGT Given Weight:

4 4 W(R,T, L) = ρ  π (R + T )3 + π (R + T )2 L − ( πR3 + πR2 L) 3 3  Find Design Variable: T Overachievement Deviation Variable associated with the weight goal, dW+

3.89 klb 35.0 klb 0.283 lbs/in3 0.1 in. 140.0 in. 0.1 in. 36.0 in. 0.5 in. 6.0 in. 0 lbs. 775,000 in3

This example is studied as a multi-player formulation in (Lewis, 1996, Rao, et al., 1996) The feasible design space can be mapped and is shown in Figure 7 (Rao, et al., 1996) In

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Figure 7, the feasible set is a vertical solid box in the positive orthant. One end of this vertical box is the polygon AJIH in the L=0 plane and the other end is formed by the inclined plane passing through the points marked as EFG. In is our focus in this section to investigate what solutions are found within this feasible design space depending upon the interaction among the two designers. We investigate the pressure vessel solutions that result if a) a cooperative relationship exists, b) a sequential relationship exists, and c) the decision makers are in isolation.

The Noncooperative Formulation As discussed in Section 2 the noncooperative solution occurs at the intersection of the players' Rational Reaction Sets. The Rational Reaction Sets of the two players from (Rao, et al., 1996) are:



TS 

t DVOLUME: R(T) = min 40 − T, P   

(12)

L(T) = 150 − 2T − 2R(T)

L 150

Player WGT as Leader

DWEIGHT:

Noncooperative

E

 PR if R ≤ 40 St and L + 2R(1 + P ) ≤ 150  P+St St . (13) T(R, L) =  St () otherwise

D g4 = 0 Player VOL as Leader; Pareto solution, emphasis on Volume

Using response surface methodology to approximate the RRS of each player, the RRS of each player are:

C

set of feasible strategies in this vertical box

B

Pareto Solution, emphasis on Weight

0.5

0

75

A

T

H

DWEIGHT: T'(R,L) = 2 + 1.75*R - 2.267*10-5*L + 3.15*10-5*R*L + 0.2445*R2 + 8.667*10-7*L2

g3 = 0

(15)

I

The exact noncooperative solution occurs at the intersection of Eqns. 12 and 13. In (Rao, et al., 1996), the intersection lies along the line represented by

g2 = 0

75 R

(14)

F

J

40

DVOLUME: R(T) = 29.29 + 14.75*T - 10.01*T2 L(T) = 85.45 - 34.45*T + 20.10*T2

G

St (150 − Lu ) 40St ≤ RN ≤ 2(P + St ) P + St . LN := 150 − 2R N SPt + 1 PR N T N := St

g1 = 0

[

Figure 7. The Feasible Strategy in the R-L-T Space The Cooperative Formulations In (Rao, et al., 1996), the set of cooperative or Pareto solutions are found. In (Rao, et al., 1996), the extreme points of the Pareto solution set are given as  40St 40P  (R,T, L) =  , , 70 , (5Tl ,Tl , Ll ) .  P + St P + St 

]

(16)

R N = 28.4 in. LN = 86.9 in. TN = 3.16 in. Weight = 24746 lbs. Volume = 316110 in3.

(17)

The solution to Eqns. 14 and 15 yields:

(11)

These two solutions using the parameters from Table 2 are (R,T, L) = (36, 4, 70 ), (2.5, 0.5, 0.1)

This solution corresponds to point C in Figure 7.

(Weight,Volume) = (39475 lbs, 480385 in3 ),(13.73 lbs,67.41 in3 ) .

The Leader/Follower Formulation Player WGT as the Leader In this protocol, player WGT dictates his strategy first by assuming or dictating that the player VOL behaves in a predetermined or rational way (i.e., player VOL must minimize his deviation function for a given thickness, T).

These solutions correspond to points B and A, respectively, in Figure 7. Since there are an infinite number of widely different solutions in the cooperative case, there must be a way to handle the compromise of each player and what cooperative solution will satisfy the most people the fullest. It is suggested that in cases like this, another player in the game must be introduced, a higher-level systems manager or someone with holistic knowledge of the system.

In other words, the follower constructs his Rational Reaction Set which the leader can use to solve his compromise DSP. The RRS of Player VOL is given by Eqns. 12 and 14,

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pareto pareto

Volume as leader noncooperative

noncooperative Weight as leader

Weight as leader

Volume as leader

Figure 8. Pressure Vessel Solutions (1:40 scale) exact and approximate, respectively. In (Rao, et al., 1996) the solution of the leader/follower problem with WGT as the leader is reported as

TS S (R,T, L) =  l t ,Tl ,150 − 2Tl (1 + l ) .  P P 

to satisfy the geometric and stress constraints while being thin and maximizing volume. The noncooperative solution is smaller than the largest Pareto solution and a little thinner, but is a significantly different configuration. Therefore, it is demonstrated what kind of effect the interactions among designers has on the design of a very simple pressure vessel. A wide range of pressure vessel configurations are generated depending upon what information a designer has. Application to a complex, large scale system is presented in the next section using the design of a passenger aircraft.

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Using the constant and parameters values (see Table 2), this corresponds to a solution of (R, T, L) = (4.5 in., 0.5 in., 140 in.). (Weight, Volume) = (635.6 lbs, 9413.9 in3) This solution is shown as point D in Figure 7.

3.2

Design of a Passenger Aircraft The next case study in this paper is the design of a 727-200 aircraft. This study is derived from (Lewis, et al., 1994, Mistree, et al., 1988). In this study, two distinct players are identified, each with their own analysis and synthesis routines: the aerodynamics player responsible for the wing and fuselage lift characteristics, and the weights player responsible for setting the thrust and take-off weight through a fuel balance. In an earlier single-level version of the 727-200 template (Lewis, et al., 1994, Mistree, et al., 1988), the existing 727-200 design was reproduced. The model presented here is an updated version. The primary difference is the inclusion of two coupled disciplinary problems, as opposed to the previously studied single level problem. Simplified forms of each players' compromise DSP are given is follows. The full form of the players' compromise DSPs are given in (Lewis, 1996).

Player VOL as the Leader In this protocol, player VOL dictates his strategy first by assuming or dictating that the player WGT behave in a predetermined or rational way (i.e., player WGT must minimize his deviation function for a given radius and length, R and L). The RRS of Player WGT is given by Eqns. 13 and 15, exact and approximate, respectively. In (Rao, et al., 1996) the solution of the leader/follower problem with VOL as the leader is reported as  40St 40P  (19) (R,T, L) =  , , 70 .  P + St P + St  Using the constant and parameters values, this corresponds to a numerical solution of (R, T, L) = (36.0 in., 4.0 in., 70.0 in.). (Weight, Volume) = (484863.0 lbs., 39772.4 in3) This solution is shown as point B in Figure 7, and also corresponds to the Pareto solution when the volume is emphasized.

PLAYER Aerodynamics Given Relevant constants Important Aerodynamic Relationships and Equations Find Aerodynamic Design variables Deviation Variables associated with Aero goals, di-,di+ Satisfy aerodynamic constraints (7 nonlinear constraints) aerodynamic goals (5 nonlinear goals) Variable Bounds (upper and lower bounds) Minimize: deviation function ZA = ∑ (di- + di+)

In Figure 8, all the solutions from each protocol are shown. The solution when Volume is the leader is the same as the largest Pareto solution. The smallest Pareto solution is basically a sphere, as the length has been decreased to its lower bound. The solution with Weight as leader is a very thin shell. The Volume player does what it can to increase the volume once the thickness has been set by the Weights player. Therefore, the pressure vessel becomes a very long and thin one

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(a) BN (b) AC (c) FC (d) AL (e) WL (f) 727-200 BN: Best Noncooperative Solution AC: Approximate Cooperative FC: Full Cooperative AL: Aerodynamics as Leader WL: Weights as Leader Figure 9. Aircraft Configurations (approximately 1:1500 scale) Therefore, only the best noncooperative solution is used in this section.

PLAYER Weight Given Relevant constants Important Fuel/Weight Relationships and Equations Find Weight design variables Deviation Variables associated with Weight goals, di-,di+ (21) Satisfy Weight constraints (6 nonlinear constraints) Weight goals (7 nonlinear goals) Variable Bounds (upper and lower bounds) Minimize: deviation function ZW = ∑ (di- + di+)

In Figure 10 the deviation functions corresponding to the protocols are shown. Some interesting observations can be made from the results. • The best "overall" results occur, as expected, when cooperation exists among the players. The term "overall" is meant to imply that both players cumulatively do well. It is interesting to note that whether full or approximate cooperation is exercised does not affect the result significantly. Using approximate cooperation provides nice results with less computational demands transfer than full cooperation, as the disciplinary analysis and synthesis are kept isolated.

Results In this section we compare the results of applying the protocol developments of the previous section to the aircraft study and to gain some insight into the role of each protocol in design. The configurations corresponding to the solutions of each protocol, along with the configuration of an existing 727-200 aircraft, are shown in Figure 9. Depending upon the simplifying assumptions made in the noncooperative formulation, different solutions are found.

Deviation Function

In the analytical models of the aerodynamics and weights designer, there are three variables required by the aerodynamics designer from the weights designer and five control variables required by the weights designer from the 0.35 727 aerodynamics designer. What are the resulting designs when each discipline has 0.3 the information it needs? When they do not BN AL WL BN AC FC AL have the information? When they are solved 727 0.25 AC FC sequentially? It is the resolution of these WL coupling and coordination challenges that are 0.2 of interest in this paper. The different game protocols studied which are used to model 0.15 these scenarios are discussed.

BN AC FC AL WL

0.1

727-200

0.05 0 Aero

Weights Player

BN: Best Noncooperative Solution AC: Approximate Cooperative FC: Full Cooperative AL: Aerodynamics as Leader WL: Weights as Leader Figure 10. Protocol Results as Compared to an Existing Design

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• Player aerodynamics does very well (same as in the cooperative formulations) when he/she is leader in the leader/follower formulation, but at the expense of the weight player. Player weight as the leader in the leader/follower formulation actually does better than he/she does in the cooperative formulations, but at the expense of the aerodynamics player.

From the system requirements in this problem, the desired values of the goals for each player are shown in Table 4. Table 4. Sample Goals and Targets of Each Player Player Weight Player

• In the existing 727-200, the aerodynamics player fares worse than every other scenario. Player weight only fares worse when he/she is the follower in the leader/follower formulation (AL) and in the best noncooperative formulation (BN). This result is not supposed to be used in any means to suggest that the 727-200 aircraft is inferior in any sense. It only shows that using this model of an aircraft (aerodynamics and weights player), the 727-200 is inferior. Certainly, aircraft design involves other disciplines as well, such as structures and controls. These disciplines were not accounted for in this work.

Aerodynamics Player

Goal Useful Load, U = 0.5 Maximize Productivity Index, PRI Maximize Lift-to-drag ratios, Ldl, Ldt, Ldc Aspect Ratio, AR = 10.5

Each of these goals and its achievement (Table 3) is investigated for the various solution cases. Useful Load, U The useful load fraction for each player in the FC, AC, WL, and 727-200 cases are close to 0.50, but in the other aircraft they are less than 0.50. This is intuitive because in the AL case, the aerodynamics player does not leave the weight player enough freedom to improve U. In the noncooperative protocol, the two players do not reach a suitable compromise, and therefore U is sacrificed.

• The existing 727-200 values do not match with any one protocol exercised in this work. This is not unexpected, as the model used in this work is a simplified model of a complete aircraft model. The simplified model is used to illustrate the rich insights and benefits that could be generated when the behavior of the disciplines is modeled as a strategic interaction using game theory. It is interesting to note that if one aircraft had to chosen as being closest to the 727-200 aircraft, it would be the aircraft from the leader/follower formulation with weights as the leader (Figure 9 (e) and (f)). The significant differences are the values of the wing span, fuselage length, which are larger in the weight as leader aircraft and thrust, which is larger in the 727-200.

Productivity Index, PRI In the FC, AC, WL, and 727-200 cases, the productivity index is the maximum, while in the others it is significantly less. When weights is the leader (WL), the PRI is high because PRI is a state variable of the weight player, and he/she strives to maximize it. In both cooperative formulations, the players cooperate and achieve the highest PRI of the scenarios. Lift-to-Drag ratios, Ld's The lift-to-drag ratios are maximum when aerodynamics is leader (AL). This is interesting, as when the players cooperate (FC and AC), the aerodynamics player sacrifices some of the lift-to-drag to benefit the weight player in U, R f, and PRI. When aerodynamics is only concerned with his own requirements, the lift-to-drag ratios are maximum, but this adversely affects the weight player, and in turn the "goodness" of the overall aircraft.

Further insight into the different aircraft can be gained by exploring the values of the goals in the aircraft problem. In Table 3, the significant goals for the aircraft are given. Table 3. State Variables of Various Solutions BN: Best Noncooperative Solution AC: Approximate Cooperative FC: Full Cooperative AL: Aerodynamics as Leader WL: Weights as Leader

Goal

BN

FC

AC

AL

WL

727-200

U PRI

0.48 158

0.49 177

0.49 177

0.46 155

0.48 174

0.49 174

Ldl

13.1

15.1

15.0

16.0

12.9

11.7

Ldt

10.0

11.8

11.8

12.6

9.9

8.8

Ldc

18.0

19.9

19.9

20.7

18.3

17.2

AR

7.2

9.66

9.65

9.89

7.91

6.86

Aspect Ratio, AR Similar to the lift-to-drag ratios, AR is closest to 10.5 in the AL case. However, when cooperation is exercised, player aerodynamics realizes that he/she can sacrifice the AR to benefit both players. 4 OBSERVATIONS AND IMPLICATIONS IN DESIGN The developments and results in Sections 3 and 4 have computational and theoretical implications in modern design processes. • The sequential leader/follower protocol embodies a philosophy that is not consistent with principles such as concurrent engineering (CE), and integrated product and process development (IPPD). However, in complex

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systems where design teams are located throughout the world and are governed by different management with different objectives and priorities, true concurrency is very difficult. Therefore, tools and methods that accept and engage in some form of sequential processes have important roles in complex systems design. • The noncooperative case is significantly inferior to the solutions in the other protocols. Therefore, noncooperation should be avoided at all costs. This, of course, is common sense. However, it has been supported in this paper using formal and rigorous decision constructs. Even largely sequential processes, as modeled in the leader/follower protocol are shown to be more advantageous to the final design than the noncooperative case. • The computational requirements of constructing a player's rational reaction set is a direct function of the number of variables needed from another player. There is a large body of work aimed at reducing the amount of coupling among decision makers. However, completely decoupling a problem in complex systems design such as aircraft is virtually impossible. Given that some coupling exists, construction of the rational reaction sets is performed using response surface methodology and design of experiments. Depending upon the amount of coupling different design of experiments strategies can be used along with varying orders or response surfaces. This determination requires a tradeoff between accuracy of the resulting approximations and the efficiency of constructing them.

model multi-level decision making processes where cooperation may not exist among decision makers in engineering design is of relatively recent origin; its usefulness in many other decision-making sectors such as economics, politics, and strategic warfare is well-established. In this paper we strive to establish the usefulness of game theory in design by abstracting complex design processes as a series of games and analyzing the resulting insights into design problem and process structure. Using two examples, we have illustrated the significant effects the interaction of decision makers has on the form and performance of the final design. Much, much more needs to be done. ACKNOWLEDGMENTS We thank Rudi Cartuyvels ([email protected]) for providing us with a copy of NORMAN® to automate the process of creating response surface models. We gratefully acknowledge the following grants: NASA NGT 51102 and NSF DMI-9420405. The cost of computer time was underwritten by the Systems Realization Laboratory of the Georgia Institute of Technology and by the State University of New York at Buffalo. REFERENCES Aubin, J. P., 1979, Mathematical Methods of Game and Economic Theory, North-Holland Publishing Company, Amsterdam. Bloebaum, C. L., Hajela, P. and Sobieski, J., 1992, "NonHierarchic System Decomposition in Structural Optimization," Engineering Optimization, Vol. 19, pp. 171-186. Box, G. E. P. and Draper, N. R., 1987, Empirical Modelbuilding and Response Surfaces, John Wiley & Sons, New York, NY. Cartuyvels, R. and Dupas, L. H., 1993, "NORMAN/DEBORA: A Powerful CAD-Integrating Automatic Sequencing System Including Advanced DOE/RSM Techniques for Various Engineering Optimization Problems," JSPE-IFIP WG 5.3 DIISM'93 Workshop (Design of Information Infrastructure Systems for Manufacturing), Tokyo, Japan. Chen, W., Rosen, D., Allen, J. K. and Mistree, F., 1994, "The Modularity and Independence of Functional Requirements in Designing Complex Systems," ASME Winter Annual Meeting, Chicago, IL, pp. 31-38. Dertouzos, M.L. et. al., and MIT Commission on Industrial Productivity, 1989, Made in America: Regaining the Productive Edge, MIT Press, Cambridge, MA. Dresher, M., 1981, Games of Strategy, Dover Publication, New York. Eppinger, S. D., 1991, "Model-Based Approaches to Managing Concurrent Engineering," Journal of Engineering Design, Vol. 2, No. 4, pp. 283-290. Eppinger, S. D., Whitney, D. E., Smith, R. P. and Gebala, D. A., 1990, "Organizing the Tasks in Complex Design Projects," Second International Conference on Design

The results and observations documented in this paper have largely been driven by descriptive motivations, as opposed to prescriptive motivations. In other words, this work in general describes the resulting designs when various design process structures are used, or when different strategies are used by different design teams. In this work we do not intended to prescribe managerial remedies to bridge noncooperative or leader/follower relationships, but describe the results if these relationships exists. And since relationships such as these certainly exist and will continue to exist in modern design of complex systems, the descriptive power of this work is beneficial to explore certain scenarios and the inherent tradeoffs between them.

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CLOSURE The design of multidisciplinary systems requires decisions which are made by multiple decision makers, design teams, or organizations. Concurrent Engineering principles have been used to facilitate this decision making process at a personal interaction level. Get everyone in the same room to bring the issues usually reserved for later in a design process up front in the process; this is the CE creed. However, this does not ensure that the decision makers have understood the physical implications underlying their personal cooperation and collaboration, or lack thereof. The use of game theory to

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Theory and Methodology, Rinderle, J., ed., Chicago, American Society of Mechanical Engineers. Herling, D., Ullman, D.G., D'Ambrosio, B., 1995, "Engineering Decision Support System (EDSS)," ASME Design Technical Conferences, Boston, MA, Vol. 2, pp. 619626. Ishii, K., Goel, A. and Adler, R. E., 1989, "A Model of Simultaneous Engineering Design," Proceedings of the Fourth International Conference on the Applications of Artificial Intelligence in Engineering, Gero, J. S., ed., Cambridge, United Kingdom, Springer Verlag, pp. 483-501. Karandikar, H. M. and Mistree, F., 1992, "Designing a Composite Material Pressure Vessel for Manufacture: A Case Study for Concurrent Engineering," Engineering Optimization, Vol. 18, No. 4, pp. 235-262. Kusiak, A. and Larson, N., 1995, "Decomposition and Representation Methods in Design," Journal of Mechanical Design, Transactions of the ASME, Vol. 117, pp. 17-24. Lewis, K., 1996, "An Algorithm for Integrated Subsystem Embodiment and System Synthesis," Ph.D. Dissertation, Georgia Institute of Technology. Lewis, K., Lucas, T. and Mistree, F., 1994, "A Decision Based Approach to Developing Ranged Top-Level Aircraft Specifications: A Conceptual Exposition," 5th AIAA/USAF/NASA/ISSMO Symposium on Recent Advances in Multidisciplinary Analysis and Optimization, Panama City, FL, pp. 465-481. Lewis, K. and Mistree, F., 1996, "Foraging-directed Adaptive Linear Programming: An Algorithm for Solving Nonlinear Mixed Discrete/Continuous Design Problems," ASME Design Automation Conference, Irvine, CA, 96DETC/DAC-1601. Mistree, F., Hughes, O. F. and Bras, B. A., 1993, "The Compromise Decision Support Problem and the Adaptive Linear Programming Algorithm," Structural Optimization: Status and Promise, Kamat, M. P., ed., AIAA, Washington, D.C., pp. 247-286. Mistree, F., Marinopoulos, S., Jackson, D. and Shupe, J. A., 1988, "The Design of Aircraft using the Decision Support Problem Technique," NASA Contractor Report, CR 88-4134. Mistree, F., Smith, W. F., Bras, B., Allen, J. K. and Muster, D., 1990, "Decision-Based Design: A Contemporary Paradigm for Ship Design," Transactions, Society of Naval Architects and Marine Engineers, Jersey City, New Jersey, pp. 565-597. Montgomery, D., 1991, Design and Analysis of Experiments, John Wiley & Sons, New York, NY. Myerson, R. B., 1991, Game Theory: Analysis of Conflict, Harvard University Press, Cambridge, Massachusetts.

Otto, K. and Antonsson, E., 1993, "Tuning Parameters in Engineering Design," Journal of Mechanical Design, Vol. 115, No. 1, pp. 14-19. Otto, K. N. and Wood, K. L., 1996, "A Reverse Engineering and Redesign Methodology for Product Evolution," ASME Design Theory and Methodology Conference, Irvine, CA, DETC/DTM-1523. Rao, J. R. J., Badrinath, K., Pakala, R. and Mistree, F., 1997, "A Study of Optimal Design Under Conflict Using Models of Multi-Player Games," Engineering Optimization, in press. Rao, S. S. and Freiheit, T. I., 1991, "A Modified Game Theory Approach to Multiobjective Optimization," Journal of Mechanical Design, Vol. 113, pp. 286-291. Reddy, R., Smith, W.F., Mistree, F., Bras, B.A., Chen, W., Malhotra, A., Badrinath, K., Lautenschlager, U., Pakala, R., Vadde, S., Patel, P., and Lewis, K., 1996, "DSIDES User Manual," Systems Realization Laboratory, Woodruff School of of Mechanical Engineering, Georgia Institute of Technology. Renaud, J. E. and Gabriele, G. A., 1991, "Sequential Global Approximation in Non-Hierarchic System Decomposition and Optimization," Advances in Design Automation, ASME, Vol. 1, pp. 191-200. Renaud, J. E. and Gabriele, G. A., 1994, "Approximation in Non-Hierarchic System Optimization," AIAA Journal, Vol. 32, No. 1, pp. 198-205. Schrage, D. P. and Gordon, M., 1992, "Management Issues and Techniques in Concurrent Engineering," AIAA Aircraft Design Systems, Hilton Head, SC. Senge, P. M., 1990, The Fifth Discipline, Doubleday/Currency, New York, NY. Sobieszczanski-Sobieski, J., 1988, "Optimization by Decomposition: A Step from Hierarchic to Non-hierarchic Systems," Second NASA/Air Force Symposium on Recent Advances in Multidisciplinary Analysis and Optimization, NASA CP 3031. Stadler, W., 1988, Multicriteria Optimization in Engineering & in the Sciences, Plenum Press, New York. Thurston, D. L., Carnahan, J. V. and Liu, T., 1994, "Optimization of Design Utility," Journal of Mechanical Design, Vol. 116, No. 3. Vincent, T. L. and Grantham, W. J., 1981, Optimality in Parametric Systems, John Wiley, New York, NY. Von Neumann, J. and Morgenstern, O., 1944, Theory of Games and Economic Behavior, Princeton University Press, Princeton, NJ.

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