8th ASCE Specialty Conference on Probabilistic Mechanics and Structural Reliability
PMC2000-217
DECISION BASED COLLABORATIVE OPTIMIZATION X. Gu and J.E. Renaud University of Notre Dame, Indiana, IN 46556
[email protected] Abstract In this research a Collaborative Optimization (CO) approach for multidisciplinary systems design is used to develop a decision based design framework for non-deterministic optimization. To date CO strategies have been developed for use in application to deterministic systems design problems. In this research the decision based design framework proposed by Hazelrigg (1996, 1998) is modified for use in a collaborative optimization framework. The Hazelrigg framework as originally proposed provides a single level optimization strategy that combines engineering decisions with business decisions in a single level optimization. By transforming the Hazelrigg framework for use in collaborative optimization one can decompose the business and engineering decision making processes. In the new multilevel framework of Decision Based Collaborative Optimization (DBCO) the business decisions are made at the system level. These business decisions result in a set of engineering performance targets that disciplinary engineering design teams seek to satisfy as part of subspace optimizations. The Decision Based Collaborative Optimization framework more accurately models the existing relationship between business and engineering in multidisciplinary systems design.
Introduction Increasing attention has been paid to the notion that engineering design is a decision making process. This notion is consistent with the definition of decision as a choice from among a set of options and as an irrevocable allocation of resources. The approach of decision based design (DBD) is built upon this notion. Rooted from more than two hundred years of research in the field of decision science, economics, operations research and other disciplines, decision-based design (DBD) provides a rigorous foundation for design, which enables engineers to identify the best trade-off and focus on where the payoffs are greatest. Decision Based Design (DBD) Framework Application of decision based design within an optimization domain, requires practitioners to formulate valid objective functions for proper decision making. An optimization solution obtained using any search method is no better than the objective function chosen for the optimization. If a mathematically defective objective function is used (Hazelrigg 1996) then there are no guarantees of any sort of solution. Therefore a primary concern in DBD is the development of a mathematically sound objective function. Recognizing that design is a decision-making process, the decision based design framework of Hazelrigg (1996, 1998) implements the concept of rational decisions. Rational decisions follow the rule that the preferred decision is the option whose expectation has the highest value. Due to the nature of engineering design, expectations on design alternatives can never be determined with certainty. It is imperative that the objective function (or utility function in the context of economy) be valid under conditions of uncertainty and risk. The von NeumannMorgenstern (vN-M) utility (von Neumann, 1953) is such a value measure. The DBD framework of Hazelrigg acknowledges the limitations imposed by Arrow’s Impossibility Theorem (Arrow, 1963) and views the objective of systems design as one of maximizing profit. Therefore a valid objective function for optimization or decision making under Gu and Renaud
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uncertainty and risk is established: the optimizer should seek to maximize the expected vN-M utility of the profit. Profit is also referred to as net revenue (NR). The relationship between profit (net revenue NR), demand q, total cost CT (cost of manufacture and all other life cycle costs), and the price P can be summarized in Equation (1). NR = ( P – C T ) ⋅ q
(1)
Multidisciplinary Enterprise Model Design is inherently a multidisciplinary process. The DBD framework of Hazelrigg, 1996, 1998 combines both engineering and business performance simulations in a single level all-at-once optimization approach. In this research the Hazelrigg framework has been decomposed into the multidisciplinary enterprise model shown in Figure 1. The decomposed system consists of two major organizations: the engineering disciplines and the business discipline. The work in the engineering disciplines focuses on predicting the performance of the product for different design configurations, as well as satisfying performance targets set in the business discipline (i.e., management). The role of the business discipline, centers on providing targets for performance improvements in order to yield higher profit. These two organizations are coupled through attributes a, total cost CT and demand q. In this research it is assumed that the demand for the product, the number of the product manufactured and the amount of the product sold are equal. Attributes a refer to the features of a product that customers tend to be interested in. Examples of attributes include speed, acceleration, quality, reliability or safety, etc. Note that the price of the product not only directly affects the amount of profit or net revenue (see Eqn. 1), it is also an important factor driving the demand q. Since price is free to be chosen by the decision maker, demand q can be modeled as a function of attributes a and the price, P Engineering Discipline Performance ya
CA1 Tool A Prediction Error
a CA2
Variabilty ∆x Engineering Design Variables x
yb
Tool B Preditction Error yc
➪
CA3 Tool C
Prediction Error
CT
Manufacturing cost and other life cydle costs CT Prediction Error
a q CT u
attributes demand total cost utility
q
Business Discipline Demand q
q
Prediction Error
Utility of Profit
y states Price
➪
u Prediction Error
Figure 1. Multidisciplinary Enterprise Model
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(see Equation (2)). q = q ( a, P )
(2)
Collaborative Optimization (CO) The Collaborative Optimization (CO) strategy was first proposed by Kroo and Sobieski (1994) and has been successfully applied to a number of different design problems. Tappeta and Renaud (1997) extended this approach and developed three different formulations to provide for multiobjective optimization of multidisciplinary systems. Collaborative Optimization (CO) is a two level optimization method specifically created for large-scale distributed-analysis applications. The system level optimizer attempts to minimize a system level objective function F while satisfying all the compatibility constraints. System level design variables consist of not only the shared variables but also auxiliary variables. These variables are specified by the system level optimization and are sent down to subspaces as targets to be matched. Each subspace, as a local optimizer, operates on its own set of design variables with the goal of matching target values posed by the system level as well as satisfying local constraints. The matching can be attained by minimizing the discrepancy between some of the local design variables and/or local states and their corresponding target values, in other words, the objective functions at subspace level are identical to the system level (compatibility) constraints. This formulation allows the use of post-optimal sensitivities at the subspace optimum as the gradients of the system level constraints. This important feature improves the overall efficiency of CO by eliminating the need to execute subspace analyses for the sole purpose of calculating system constraint gradients by finite differencing. Decision Based Collaborative Optimization (DBCO) Framework In Decision Based Collaborative Optimization (DBCO), the method of collaborative optimization (CO) is used to determine the optimal design of the multidisciplinary enterprise model (Fig. 1). The resulting decision based collaborative optimization (DBCO) framework (shown in Fig. 2) rigorously simulates the existing relationship between business and engineering in multidisciplinary systems design. In this framework, the business decisions are made at the system level. The system level optimizer attempts to increase expected utility of net revenue E(u(NR)) while satisfying compatibility constraints d. According to the analyses in the business discipline and the subspace optimization results, the system level optimizer determines price P and establishes a set of performance targets including demand q and total cost CT. These targets are then sent down to appropriate subspaces. The subspace optimizer, based on his/her expertise in the discipline analysis, tries to match these targets as close as possible and reports the discrepancy back to the system level. The subspace optimizers are subject to local design constraints. In the field of engineering design, the design constraints normally guard against failure or other unacceptable behavior. The use of constraints to prevent undesirable behavior requires designers to quantify what is undesirable. In DBD the market place is used to determine undesirability through demand models and therefore constraints related to undesirability are eliminated. Therefore the local constraints in the decision based collaborative optimization framework (DBCO) tend to be those constraints that guard against system failure. Other
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traditional engineering constraints related to consumer preference are eliminated and instead incorporated in the demand model and/or cost model. Business Discipline System Level Optimizer Maximize: f=u Subject to: d∗i = 0 0 0 0 D.V. : x = [ x shared, x aux , CT, price ]
SubSpace 2 Optimer
d∗1
x0shared , x0aux
u,q 0 0
0
x aux , CT, price
Utility of Profit
u
SubSpace 3 Optimer
CA2
d ∗c
0
0
0
0
xshared , x aux , CT, q
CA3
SubSpace Cost Optimizer
SubSpace 1 Optimizer
Min: d∗c D.V.: xssc= [ (x sh)c , (xaux) jc, x c ]
Min: d∗1 D.V.: xss1= [ (xsh)1 , (xaux)j1, x1 ]
y1j
xss1
y1j
xss1
q
Demand q
Manufacturing cost and other life cydle costs C T
CA1
Figure 2. Decision-Based Collaborative Optimization
The system level optimization problem and the subspace optimization problem for discipline 1 in Fig 2, in its standard form, are given in Equation (3). Suspace 1 Optimization Problem
System Level Optimization Problem Minimize Subject to:
F = – E ( u ( NR ) ) *
i = 1, 2, …, n ss
di = 0 o ( x sys ) min o
Minimize
≤
o x sys
o
o
≤
o
d 1 = ( ( x sh ) 1 – ( x sh ) 1 )
2
n ss
+
o ( x sys ) max
∑ ( ( xaux ) j1 – ( xaux ) j1 ) o
j=2 n ss
x sys = ( x sh, x aux ) +
P>0
∑ ( y1 j – ( xaux )1k ) o
2
2
(3)
k=2
Subject to
g1 ≥ 0 ( x ss1 ) min ≤ x ss1 ≤ ( x ss1 ) max x ss1 = ( ( x sh ) 1, ( x aux ) j1, x 1 )
Test Problem: Aircraft Concept Sizing (ACS) Problem A preliminary application of the decision based collaborative optimization framework has been tested on the Aircraft Concept Sizing (ACS) problem. This problem was originally developed by the MDO research group at the University of Notre Dame (Wujek and Renaud, 1996, Tappeta, 1996). It involves the preliminary sizing of a general aviation aircraft subject to certain performance constraints. The design variables in this problem are comprised of variables relating to the geometry of the aircraft, propulsion and aerodynamic characteristics, and flight regime. Appropriate bounds are placed on all design variables. The problem also includes a number of parameters which are fixed during the
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design process to represent constraints on mission requirements, available technologies, and aircraft class regulations. The objective in the ACS problem is to determine the least gross take-off weight within the bounded design space subject to two performance constraints. The first constraint is that the aircraft range must be no less than a prescribed requirement, and the second constraint is that the stall speed must no greater than a specified maximum stall speed. The original Aircraft Concept Sizing (ACS) problem had three disciplines: aerodynamics, weight and performance. Two disciplines have been added to fit in the DBD approach: cost and business. A demand model and a cost model have been developed for use in the business discipline and cost discipline, respectively. Optimization Results & Discussion This application study is in its preliminary stage, and focuses on the collaborative optimization features of the DBCO framework. The issues of propagated uncertainty are neglected. The utility of profit is assumed to be the profit itself. Hence the objective of the resulting deterministic optimization problem is to maximize profit (or net revenue). During the optimization, the demand q is treated as a continuous variable, other than an integer. At the end of the system optimization, q is rounded to the nearest integer. A Sequential Quadratic Programming (SQP) method was used for optimization at both the system level and the subspace level. The SQP solver, fmincon, was obtained from the Matlab Optimization Toolbox. The system level optimizer tries to maximize the negative profit and minimize the constraint violation simultaneously. At the beginning of the optimization, aiming at achieving large profit, the system level optimizer set targets on high price, high level of performance (to ensure high demand) and low cost according to the business analyses. However these targets conflict with one another and lead to a large discrepancy at the subspace level. Thus the system level optimizer, while trying to keep profit as high as possible, was forced to lower price, downgrade performance and tolerate higher cost so that the subspace discrepancy could be reduced. Gradually the system level optimizer found the best trade-off among the targets and reached a consistent optimal design. The optimization history observed in the ACS problem resembles the existing relationship between business and engineering in multidisciplinary systems design. The optimal solution is listed in Table 1. The demand model and cost model play an important role in the decision based design approach. In order to illustrate the influence of demand and cost models, a conventional all-at-once optimization was performed using the problem formulation in Wujek and Renaud (1996). The conventional optimum obtained is also listed in Table 1. It can be observed that the conventional optimum outperforms DBCO optimal design on lower weight (y3, y4). However it possess poor characteristics in many aspects such as smaller aircraft range (y5), higher stall speed (y6) and smaller fuselage volume (y7). Such an outcome is no surprise since the main concern of the conventional ACS problem is to minimize take-off weight, while the DBD approach takes into account other performance attributes, because of the DBD objective of maximizing profit. Conclusions In this research a Decision Based Collaborative Optimization (DBCO) framework which
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incorporates the concepts of normative decision based design (DBD) and the strategy of Collaborative Optimization is developed. This bi-level non-deterministic optimization framework more accurately captures the existing relationship between business and engineering in multidisciplinary systems design. The business decisions are made at the system level, which result in a set of engineering performance targets that disciplinary engineering design teams seek to satisfy as part of subspace optimizations. A preliminary application of this approach (deterministic case) has been conducted on the Aircraft Concept Sizing (ACS) test problem. The corresponding optimization results are discussed. Table 1: Optimal Solutions for ACS Problem Name (Unit)
DV Bounds
DBCO Conven. Optimum Optimum
Name (Unit)
DBCO Optimum
Conven. Optimum
x1
aspect ratio of the wing
5~9
7.968
5
y1
total aircraft wetted area (ft2)
887.21
710.3
x2
wing area (ft2)
100~300
230.3
176.53
y2
max lift to drag ratio
14.273
10.971
x3
fuselage length (ft)
20~30
21.927
20
y3
empty weight (lbs)
1556.6
1207.6
x4
fuselage diameter (ft)
4~5
4.1871
4
y4
gross take-off weight (lbs)
2185.9
1748.4
x5
density of air at cruise speed (slug/ft)
.0017~ .002378
.0023
.0017
y5
aircraft range (miles)
953.67
560
x6
cruise speed (ft/sec)
200~300
219.65
200
y6
stall speed (ft/sec)
68.525
70
x7
fuel weight (lbs)
100~2000
231.22
142.86
y7
fuselage volume (ft3)
301.92
251.33
Acknowledgements This multidisciplinary research effort was supported in part by the following grants: NSF grant DMI9812857 and NASA grant NAG1-2240. References Arrow, K.J., 1963, Social Choice and Individual Value, 2nd Edition, John Wiley and Sons, NY. Hazelrigg, G.A., 1996, Systems Engineering: An Approach to Information-Based Design, Prentice Hall, Upper Saddle River, NJ. Hazelrigg, G.A., 1998, “A Framework for Decision-Based Engineering Design,” Journal of Mechanical Design. Kroo, I., Altus, S., Braun, R., Gage, P., Sobieski, I., 1994, “Multidisciplinary Optimization Methods for Aircraft Preliminary Design,” AIAA-94-4325-CP, Proceedings of the 5th AIAA/NASA/USAF/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Panama City, Florida, September 1994. Tappeta, R.V., 1996, “An Investigation of Alternative Problem Formulations for Multidisciplinary Optimization,” M.S. thesis, University of Notre Dame, December 1996. Tappeta, R.V. and Renaud, J.E., 1997, “Multiobjective Collaborative Optimization,” ASME Journal of Mechanical Design, Vol 119, No. 3, September 1997, pp. 403-411. von Neumann, J. and Morgenstern, O., 1953, The Theory of Games and Economic Behavior, 3rd Edition, Princeton University, Princeton, NJ. Wujek, B.A. and Renaud, J.E., 1996, “Design Flow Management and Multidisciplinary Design Optimization in Application to Aircraft Concept Sizing,” AIAA-96-0713, 34th Aerospace Sciences Meeting & Exhibit, Reno, NY, January 1996.
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