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Compromise: An effective approach for the hierarchical design of structural systems
COMPROMISE: AN EFFECTIVE APPROACH HIERARCHICAL DESIGN OF STRUCTURAL J. A. ~~~~rnen~
SHuPe,t F. MIWRFZE~ and J.
FOR THE SYSTEMS
SOBEXANSK~-SOBEBKI$
of Mechanical ~ngin~~ng, University of Houston, Houston, TX 77004, U.S.A. #NASA Langley Research Center, Hampton, VA 23665, U.S.A. (Received 15 December 1986)
this paper the use of the compromise decision support problem in hierarchical design of structural systems is described. In the past we had postulated the hierarchical decision support problem
Ah&rack-In
design.In this paper, for the first time, the mathematicaltemplatethat supports the underlyingpreceptsof hierarchical design in the context of the Jkision Support problem Technique
for use in hierarchical
is presented. A structural example that demonstrates the efficacy of the approach is included.
I. ~RODU~ON AND BACRGROUND TO ~~~1~ DBCIBION MAKFNG IN ~GINR~ING DBSXGN
-present an example that establishes the efficacy of using a hierarchical DSP for solving this class of structural problems.
The
analysis and synthesis of engineering systems, such as structures, are generally too complex to be handled as a single problem. This necessitates design of the overall system by first decomposing the system into subsystems. If the system is then designed in parts (sequentially), there is no guarantee that an overall superior design will be reached. Thus, it becomes necessary to develop a methodology that will facilitate the superior design of a hierarchical system. A ~erarchi~l system is a system that contains multiple levels of interaction between a parent system and the associated subsystems. The design of a hierarchical system is based on the formulation and solution of a series of problems involving decisions to be made by the designer. This type of design has been termed “hierarchical decision making” and the difficulties inherent in accomplishing system design can be dealt with using this approach. Decision Support Problems (DSP) provide a basis upon which a designer can make the decisions encountered in hierarchical decision making. Solution of the DSP is expected to result in superior (or optimal) designs. The DSP are capable of handling multiple objectives that contain both hard and soft info~ation. In this paper, we describe the background of hierarchical design and DSP and the relationship between them, including the utilization of goal constraints to model system interactions. It is postulated that the two can be used in concert to solve problems of greater scope and difficulty than has been possible with other design techniques. To demonstrate the validity of the postulate we: -describe hierarchical design in the context of DSP; -define a DSP for the hierarchical design of a structural system;
An investigation into hierarchical design has been made by Sobieski [l]. Solution of structural hierarchical design problems by means of the Decision Support Problem Technique was first proposed by Kuppuraju et 01.[2]. An application of the DSP Technique to structures was demonstrated in [3) and subsequently Shupe and Mistree have shown how to include reserve strength, a feature of damage tolerance, in the design of structures in 141. These four papers provide material upon which this paper is based. The concepts underlying hierarchical design are described in Section 2. Section 3 contains an example in which the application of the general method of hierarchical decision making applied to a specific structural problem is shown. Finally, in Section 4, work that needs to be done in the future to fully exploit hierarchical design as a tool for designers is discussed.
DECEION SUPFDRT PROBLXM FOR HIERARCHICAL DESIGN
2. THE REPROS
An engineering system can be represented by a hierarchy of a parent system and subsystems as in Fig. 1. The hierarchical nature of system design facilitates the identification of complexity and the interactions among various levels and subsystems. The relations and bonds of a hierarchical system arc essential for the visualization of the system. It is important to note that all of the subsystems arc integral parts of the parent system. This means that changes in one subsystem aff& the parent system and other subsystems. Thus no subsystem can be considered in isolation from the rest.
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-
5 i u & % 1
PI al.
requires computers capable of parallel processing; -some decisions are inseparable and the input of decision 9” is the output of decision “i - I” and thus the decisions can be made sequentially, see Fig. 2(b); -some decisions are inseparable and are coupled and the output of one is the input for the other, see Figs 2(c) and (d). In this case the decisions could be made either ~quentially (start with a reasonable guess for the first output of “i”) or concurrently (using optimization). Design involves hierarchical decision making:
LATERAL INTERACTIONSPS - Psr*nt Syrtam SS - SubsSstm
-interaction between the various levels of subsystems exists. This interaction may go one way or both ways; -interaction between the subsystems at the same level of the same parent also exists. This interaction can also go one way or both ways.
Fig. 1. A general four-level hierarchical system [I]. At each level of the hierarchy, for each subsystem, the design process involves decisions which are qualified by the follo~ng assertions. Design is a seriesof decisions:
This leads us to postulate two classes of hierarchical design, see Fig. 3. The first class is multilevel hierarchical decision making, wherein the hierarchy of the system is modeled as separate decisions, which are
-some decisions are separable and therefore can be made concurrently, see Fig. 2(a). This of course
a - kparsble
deoslons
mode concurrently
b - lnsepsrabledecisions
c - lneeparsblrandcoupled Mclslons made srqurntlslly and requirtng ItsratIon
mode S8qUootiottY
d - lnsepareble and coupled decisions made concurrently
Fig. 2. Series of decisions.
Hierarchical design of structural systems
1
SEL;KTplDN
TYPES
1
OF DSPS
1 CDIlP650pSE
USED
]
1
IN THE DECISION
CDD”Ll;ED
1
PROCESS
Fig. 3. Hierarchical design: types of decisions.
made at the different levels at which they occur. The decisions are separable and can therefore be made concurrently [Fig. 2(a)] or sequentially [Figs 2(b) and (c)l. It is difficult in this approach to achieve a high level of interaction between the different decisions. The second class of hierarchical design is coupled hierarchical decision making. In this class the hierarchy is modeled as a single decision, Fig. 2(d). The interactions between the decisions are strongly linked, creating a tight bond between subsystems solved by the problem and hence it is relatively easy to achieve a high degree of interaction between decisions. It is this class that is described in this paper. 2.1 Types of Decision Support Problems [5] In design, there are basically three ways to derive the answers to problems encountered. These ways are analysis, synthesis and heuristic thinking. These may seem to be widely divergent but it is possible to integrate all three in the form of structured DSP. Since there are different types of decisions to be made, different categories of DSP exist. These categories are: selection [a]; compromise [3,4]; hierarchical [2]; conditional. A selection DSP is used to choose an alternative from several. The choice is made based on ratings given to multiple attributes and their relative importance. A compromise DSP involves improving an alternative, by changing design variables optimally, to find a superior solution. A hierarchical (coupled) DSP involves the combined solution of multiple selection and compromise DSP simultaneously. Decisions in which risk and uncertainty of the outcome are taken into account are facilitated through the solution of conditional DSP. Both selection and compromise DSP may be formulated and solved in the synthesis of a hierarchical system composed of parent systems and the related subsystems as shown in Fig. 1. This is an application of multilevel hierarchical decision making. The
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coupled DSP, however, is the basis of coupled hierarchical decision making. The coupled DSP may involve any combination of selection and/or compromise DSP. The interaction can consist of coupling between subsystems at the same level or at different levels. If the coupling occurs between the parent system and a subsystem it is called Vertical Interaction (VI) [l]. There are two types of VI, namely, Forward Vertical Interaction (FVI) and Reverse Vertical Interaction (RVI). If interaction occurs between two subsystems it is called Lateral Interaction (LI). These interactions are illustrated in Fig. I. The mathematical model for a coupled DSP involving both selection and compromise is presented in [7, case 61 and the mathematical model used in this paper is described in [3,4]. The method of modeling the interactions is extremely important. These are modeled as goal constraints of a compromise DSP. The approach is described in Section 3. Multilevel hierarchical decisions require the separate but possibly concurrent solution of all types of DSP. Coupled hierarchical decisions require simultaneous solution of two or more DSP as a single coupled problem. In this paper, we discuss coupled hierarchical decision making only. The multilevel hierarchical problem requires further research to be done as well as some new hardware that allows multitasking or parallel processing.
3. HIERARCHICAL DESIGN OF A PORTAL FRAME
The efficacy of the coupled selection-compromise DSP in structural design has been demonstrated for the classical three-bar truss problem in [8,9]. Hierarchical structural design also involves the coupling of two (or more) compromise DSPs. In this section, therefore, the various interactions in hierarchical design are demonstrated by the use of a coupled compromise-compromise DSP. The portal frame, used as an example, was originally used by Sobieski [l] to demonstrate top-down decomposition without Reverse and/or Lateral Interaction. Sobieski used the portal frame example to show the efficacy of optimization by decomposition with linking through sensitivity. Our present ap preach is different and this distinction is amplified in Section 3.5. The portal frame, in this paper, is used to demonstrate the effect of Vertical (both Forward and Reverse) and Lateral Interactions by using the coupled DSP. All interactions are formulated as goal constraints [3,7,8] of a compromise DSP. It is emphasized that we are not trying to demonstrate the savings associated with the use of decomposition techniques per se; rather we intend to demonstrate the efficacy of using the compromise DSP to model the interactions between parent and subsystems (VI) as well as between one subsystem and another 0-I).
Fig.
4.
Hierarchical portal frame probiem.
3.1Res~r~~t~#~of the portal frame
t, = top flange height I~= web width h = web height.
Given a three-member portal frame as presented in Fig. 4, the pertinent system ~~n~~~~~s are:
These are the principal dimensions of an I-beam. 3.2 Organization of the portal frame design
The objjective is to blaze the overall maas of the frame, white being subjected to static toads, f and M. AIsoo, the system is subject to certain constraints covering normal stress, bending stress, shear stress and buckling in each member. There are two types of design variables; one type for the parent system and another type for the subsystems. The parent system design varibles [_4 and I) are each member’s crosssectional area and moment ofinertia. Each subsystem has six design variables, reprexnting: 6, = bottom flange width 8, = bottom flange height 4 = top flange width
The approach used to solve a multilevel hierarchical DSP is detaihxi in this section. As indicated earEel, the approach we take in adding structural hierarchy is different from that proposed by Sobieski \I]. Some of these differences are highlighted. The coupted DSPs for the four cases were solved by using the SLiP2 program [7,lO]. The SLIP2 aigor&&n is different from those that are currently favored in optimum structural design. A schematic of the integration of a finite element analysis program with SLIP2 is shown in Fig. 5. The step-by-step procedure is as fohows: t* 2. 3. 4.
Defme geometry, mateliai properties and loads. Define parent system stress constraints. Initiaiize detailed cubists dimensions. From detailed sub~yst~ ~mension~, calculate A and I for each beam. 5. Using a frame analysis subroutine, analyze the
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Hierarchical design of structural systems
portal frame (parent system) to find the member responses (N, M, r). 6. Begin the synthesis cycle. Iiolding member responses constant, formulate the nonlinear DSP. This requires the quantification of the following constraints:
FORHULATE
“,“,““”
I NONLINEAR
-parent system stress constraints -subsystem combined stress constraints -subsystem shear stress constraints -subsystem buckling constraints -parent system goal (minimum mass) -VI goal constraints (these goals maintain the equality of the parent system variables (A and I) to the subsystems’ areas and moments of inertia as functions of the subsystem variables (detail dimensions)) -bounds. Formulate the linearized DSP [7, IO] and solve using the Revised Dual Simplex algorithm. Repeat synthesis cycle, if necessary. 7. Repeat frame analysis and synthesis cycle, if necessary.
-*St70
DESIGN-MALYGiS
DSP
-i
Fig. 5. Implementation
of the SLIP2 algorithm.
variables (A and 1) and the subsystem variables (b, r, Observe that the finite element analysis is performed only in the analysis/synthesis cycle (see Fig. 5). For the cases presented in this paper no finite element analysis was performed within the synthesis cycle, The response of the structural system, obtained from finite element analysis, is used to formulate the nonlinear DSP. This nonlinear DSP is linearized and solved using a Revised Dual Simplex algo~t~. Note that the four cases include different types of structural hierarchy and each of the DSPs is different. Each DSP, however, is solved using the same scheme outlined in Fig. 5 and unlike the approach presented in [i] no decomposition of the structural system is involved. 3.3 Strategies for exploring the hierarchical frame problem In this section, we deal with two types of VI, namely Reversed VI and Forward VI. RVI has been demonstrated in the portal frame by Sobieski [ I,1 l]. Although FVI had been demonstrated fill, LI remained to be demonstrated. In this section we discuss strategies for demonstrating all of these interactions using the DSP technique. The basis of the strategy for studying the hierarchical problem is that the solution of the problem is approached one case at a time. This is done so that the interactions between different levels can be understood in isolation first, before their effects are combined into a comprehensive formulation. The interactions, VI, that occur between the parent system and each of its subsystems, as well as the interactions that occur between each subsystem, LI, are shown in Fig. 4. The Vi necessitates the inclusion of constraints that match the parent system design
h). The LI necessitates the inclusion of constraints
that match the subsystem variables to their counterparts in the other subsystems. (Note that there are no interactions shown between subsystem 1 and subsystem 3 since there is no physical connection between the two.) The “matching” that the interactions perform is modeled mathematically by goal constraints, An interaction goal ~nstraint constrains one subsystem variable to be equal to its counterpart in the other subsystem. For example, in Fig. 4, the individual dimensions of the center beam (subsystem 2) of the portal frame should match those of the beams (subsystems 1 and 3) on either side. Thus, the LI equality constraints are created to handle this, The deviation from this equality is measured by the goal constraint’s deviation variables (underachievement, d- and overachievement, d+). Thus the interaction goal constraint for the parent system and subsystem 1 is written as R,+d-
-d”
=A(b,t,h),
where A, is the cross-sectional area of member 1 (parent system) and A(b, t, h), is the cross-sectional area of member 1 as a function of the subsystem variables. The interaction goal constraint provides an effective and efficient approach for mainlining the interactions between the parent system and its subsystems and between individual subsystems. The exploration of the hierarchical problem follows the following path. -First, a compromise DSP is solved for the problem as a whole system. This is called the Regular System (RS) case. There is no coupling and thus
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there arc no interaction goal constraints. All constraints are in terms of the design variables at the subsystem Ievel. In terms of Fig. 4, this example involves only the variables b, t and h for each beam, and does not make use of the parent system variables, A and I. This example establishes the baseline for the rest of the investigation. The coupled DSP for this example includes 18 system variables, two deviation variabies, 33 constraints and one goal. -The next example involves a compromise DSP that involves the coupling of a parent system and its attendant subsystems. This DSP is referred to as the Vertical Interaction case. This DSP includes VI goaf constraints to maintain the interface between the parent system design variables and the subsystem design variables (Fig. 4). This example is used to demonstrate the feasibility of establishing interactions between a patent system and its subsystems using goal constraints and deviation variables to control the interaction. This example uses all the variables depicted in Fig. 4, but the only interaction is between the parent system variables and their related subsystem variables. The coupled DSP in this example involves 24 system variables, 14 deviation variables, 33 constraints and seven goals. -In the third example the portal frame is modeled as three separate subsystems and this is again formulated as a coupled DSP. This example is referred to as the Lateral Interaction case, This DSP includes LI goal constraints to regulate the interface between the three subsystems. In this example, the parept system variables depicted in Fig. 4 are not used at all, so there is no VI; the interactions are
Given:
i
l
el
al.
bctwecn the subsystems only. Notice however, that there is no interaction between the first and third subsystems since there is no physical connection. This example is an importanl demonstration of the need foi interaction between seemingly independent subsystems. The coupled DSP for this example includes 18 system variables, 26 deviation variables, 33 constraints and 13 goals. -The final exampte in the process is the ~mbination of all the VI constraints and the LI constraints. Thus, this problem is designated the Comprehensive (C) case. The example employs all the variables and interactions depicted in Fig. 4. This problem is a comprehensive formufation of the coupled hierarchical DSP. Size of problem: 24 system variables, 38 deviation variables, 33 constraints, 18 goals. It is important to note that the number of goals and the number of deviation variables are directly proportional to the number of interactions being modeled. A generic mathematical formulation for the hierarchical design problem is given in Section 3.4, followed by the results in Section 3.5. 3.4 Coupled compromise DSP for the portal jiiame The mathemati~l fo~uIation of a coupfed compromise DSP is detailed for the Comprehensive case in this section. This formulation is for the portal frame shown in Fig. 4 and is prepared with reference to the mathematical form of the Compromise DSP presented in [3,7’j. Since the following coupled DSP is for the Comprehensive case the DSP for the other cases can be derived by deleting the appropriate constraints and/or goals.
index for element number
S(A+ i) - combined normal and bending stresses
for
member I MA,) = total mass of frame as a function of the cross sectlonal areas HE = expected value for mass. A( b, t, h Ii = c~~-~ct~~l area of the I-th element as a functfon of the subsystem dimension varlabies I( b, t, h ), moment of inertia for the i-th element as a l
function of’the subsystem dimension variables UA,Uf - maximum ~~~~~ble ValUeSforAt and II LA, Ll = mlnlmum permissible Values for Al and I I
b, t, h = dlmenslon variables determined at subsystem level Ubl Ut, uh - maxims permissible values for b, t and h Lb, Lt,Lh - minims perfnfsstble values for b, t and h f= I ,2,S: member number
Hierarchical design of structural systems
j= 1 - Indicates left end of member t
2 - indicates rtght end of member t
index for devfatlon vartables; corresponds to goal constrarnt m = I ,2: First Lateral lnteractlon lndex n = 2,3,..., 13: Second Lateral Interaction lndex
k =
FM
Ai - cross sectlonal area of the I-th element Ii - moment of inertia of t-th element bll,b2i
-
flange widths
t 1i,tzi - flange thfcknesses t3j - web thickness hi - height of web Satisfy System Constraints: Parent system - combined stress constraints SfAi,li) i Smax System Constraints Subsystems - combined stress in the top flange oa / db,t+hl,j 1 t i- 1,2,3; f= I,2 - cubit stress in the bottom f Mqe cr, / o( b,t,h Ii4 i I i= 1,2,3; j= 132 - shear stress constraint t,/d b,t,h Irj 1 1 i- 1,2,3; jm I,2 - f iange buckling constraint aa,(b, ,,t, ,) ! 1a( b,t,h 1111 1 I* 1,2,3 ta(b\\,tli)
/ Id bat&h Ii IL 1 Is 12J
~*{~~,t2i
I/ I a( b,t,h Ji I 2 1 i- 1,2,3
tatb2i8t2j
1 I id b,t,h )t i ;I 1 irn 1,2,3
Goal Constants - mass goal: minimize mass of system M(A$‘+d-l -d+$ =O - vertlcar ~ntera~ti~ Goats Ai + d jtl - d+i+, l Af b,t,h II
I= 1,283
11l d-1+4 - d+,tJ = I( b,t,h ), I= I ,2,3 - Lateral interaction Goals (n = Im- II * 6) bj,m + d”2+n* d’p+n’bl,m+l m-1,2; b2,m + d-3tn - d43+n * b2mt 1 mrlSk tt,m+d-q+n’d’~tn’tl,mtI
m-182;
t2,m + d-S+n _ d’S+n t t2,m* 1 mR13; tj,m * d_6+n _ d’6+n mt3,mt f mw12
~td-,+n-d’~+n=~+~ 8ounds on oesrgn Variabtes LA,AI
cUA
L’,I(
mmt,2;
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et
al,
Bounds on Deviatlon Variables 0.0 (d-k,d*k
5 1.0 k- I,2 ,...,7
tlfnimize Z(~-,d+l= P,(d-,
+ d+, I+ P2(d-2 + dt2 + dt3 + ... + d-7 + dt7)
P t B P2 where B lndlcates preference.
Mass is an indirect measure of economic efficiency whereas the interactions are secondary (strength, buckling, etc. being primary) measures of technical efficiency. Hence, in the preceding a preference is shown towards minimizing mass at the expense of the interactions. 3.5 Results The different cases are identified in Table 1. In making computer runs to get results for this investigation, several questions are asked, in order to direct the course of the investigation. These questions are given here to begin the discussion of the results: What effect do different initial designs have on the solution? Will the problem as formulated handle infeasible initial points. How infeasible can the initial points be? What are the differences between the Regular System, VI, LI and Comprehensive solutions? These questions, when answered, will illustrate how the interaction constraints can be manipulated to handle hierar~hi~l problems. The answers to these questions are presented in Sections 3X1-3.5.3. 3.5.1 The effect of different initial designs. The initial question to be answered concerns whether or not the choice of an initial starting design is important. Ideally, the choice of an initial design should not matter, but it does make a difference in some cases. As shown in Table 2, there were three initial designs used, an initial infeasible design with a volume of 26,469 cm’, a high initial feasible design with a volume of 275,OOOcm’ and a low initial feasible design with a volume of 104,910cm3. The &al designs derived from the initiai designs vary, but the total volume values of the final solutions differ by as little as 0.05% (the difference between final designs in cases C 2 and C 3 in Table 6) to as much as 6.5% (the difference between final designs in problems VI I and
Table 1. Titles for cases
CatCgOliCS
104,910 cm’ (feasible)
Regular system VI Ll Comprehensive
Prb, Prb, Prb. Prb.
t/RS I 4/Vf i l/L1 1 10/C 1
275,000 cm’ (feasible)
26,469 em’ (infeasible)
Prb. Prb. Prb. Prb.
Prb. Prb. Prb. Prb.
2/RS 2 s/VI 2 8jLT 2 I r/C 2
Table 2. Initial designs
VOfIBIle (cm’)
3iRS 3 6/W 3 9/LI 3 lZ/C 3
Low feasible initial design 104,910
High feasible initial design 275,000
Infeasrble initial design 26,469
Parent sysfem 4
(cm’)
1, (cm’) A, (cm21 1, (cm’) A, (em*) 1, (em’)
52.14 37,993.o 58.3 ~,497.0 20.5 6135.0
110.0 49,436.0 110.0 49,436.0 110.0 49,436.0
10.59 78.1 10.59 780.1 10.59 780.1
Subsplem I (member 1-dimensions w cm) b, 11.0 30.0 tt 0.5 1.0 b1 9.0 30.0 82 0.53 f.6 *3 0.53 5;:: fr 79.0
11.0 0.275 5.5 0.275 0.275 22.0
Subsystem 2 (member 2-dimehms b, 11.0 4 bz 8.;
in cm) 30.0 I.0 30.0
11.0 0.275 5.5
12
0:s
I3
0.5 95.7
1.0 I.0 50.0
0.275 0.275 22.0
h
Initial designs Problem
VI 3 in Table 4). There is, however, a great deal of difference in the individual member values. The only final designs that really match up are the Comprehensive designs. These Comprehensive designs combine LI and VI goal constraints. Since the results are so close, the Comprehensive fo~ulation is the only one to achieve global convergence. Thus, different initial designs do effect the result, unless the formulation is complete. This is typical of problems solved using SLIPZ. 3.5.2 infetz.viblei&&f designs. The next question to be dealt with concerns the ability of the program to handle infeasible initial designs. The program not only handled an infeasible initial design, but a highly infeasible one as well. The low initial volume design (26,469 cm’) was a very infeasible initial design. Of 18 stress and buckling constraints, this design violated eight ~~str~nts initially, yet yielded a design within 6.5% of the low initial feasible design problem’s final designs and within 0.3% of the high initial feasible design problem’s final designs.
Subsysrem 3 (member 3-dimensions in cm) b, 11.0 30.0
4 b, ‘2 f3
It
0.4 6.0 0.3 0.3 47.8
1.0 30.0 1.0 1.0 50.0
Il.0 0.275 5.5 0.275 0.275 22.0
Hierarchical design of structural systems 3.5.3 The dl#ierences between the four solutions.
Finally, the differences and similarities of the Regular System, VI, LI and Comprehensive solutions are discussed. The solutions are given in Tables 3-6. The histories of the design cycles are presented in graphical form in Figs 6-9. As can be seen from the figures, the Regular System and VI final solutions are in general agreement for all three initial designs. ,The instances where there is a little agreement mostly occur in the third member, as seen in Tables 3 and 4. This member is subjected to lower end loads and moments and, as a result, was the member most easily reduced in size. Since an important goal is the minimization of total volume, the third member was the member most affected by the solution process. Thus, there are some large differences in design variables here that do not occur in the other members. However, the final volumes for both the Regular and VI solutions are very close. The LI final designs do not closely agree with either of the other two types of solutions, which can be seen by comparing Fig. 7 with the other figures. This is a reasonable result, however. The LI problem has been given the stipulation that each I-beam dimension (subsystem design variable) in a member must match its counterpart in the contiguous member. Given this requirement, it makes more sense to examine the LI problem internally. In other words, do the design variables match their counterparts? By examining Table 5a, it is seen that the answer is “yes”. The design variables do match to a very close tolerance. However, in this category, there is still disagreement between the solutions found for the different initial designs. This indicates that LI constraints work between subsystems, but they should not be used alone. From the overall design point of view, it is interesting to note how the VI and LI cases compare to Table 3. Final Regular daigns Volume (cm’)
RS 1 79,219
Subsysrem 1 (member b, II b, 1s 13 h
l-dimensions in cm) 18.34 12.38 0.846 0.531 18.76 12.33 0.858 0.641 0.402 0.450 43.82 62.81
Subsystem 2 (member 2-dimensions
b, ‘I bz
‘1 ‘I h
16.05 0.833 18.92 0.851 0.338 55.51
Subsystem 3 (member 3-dimen.rions
b, r, bz ‘2 ‘3 h
RS 2 19,531
10.0 0.323 11.8 0.278 0.275 10.45
RS 3 19,286 12.97 0.558 12.98 0.635 0.439
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Table 4. Final VI designs Volume (cm’) Parent system A, (cm2) I, (cm’) i2(cm2j 12
b-f)
A, (cm’) 1, (cm’) Subsysfem b,
1, b, ‘2
I,
i
VI I 76,130
VI 2 80,883
VI 3 81,085
46.19 26.923.0
42.15 19.583.0 22.96 8643.7 35.91 22,071.o
39.10 28.061.O ’ 27.40 12,911.O 30.98 16,749.O
38,0% 7.0 182.0
1 (member l-dimensions 11.57 0.619 13.53 0.619 0.411 64.13
in cm) 10.97 0.545 5.86 0941 0.464 69.82
Subsystem 2 (member 2-dbnension.v in cm) 12.71 10.0 b, 0.563 0.418 11
b2 ‘I ‘I h
14.81 0.656 0.387 15.42
10.72 0.415 0.275 49.16
Subsystem 3 (member 3-dimensions in cm) 10.0 12.76 b, 0.219 0.565 1, 1.18 12.29 bz ‘2
‘3 h
0.275 0.275 14.13
0.544 0.344 64.09
14.95 0.609 12.64 0.587 0.451 65.14 10.0 0.443
12.22 0.541 0.292 56.04 11.86 0.525 9.29 0.574 0.318 60.35
the Regular System case in terms of total volume, since volume can be considered the economic indicator for this example. The VI final volumes are very close to the final volumes in the Regular System case. In fact, the final volume for the low initial feasible design (VI 1 in Table 4) is less than its counterpart in the regular case (RS 1) in Table 3. In the LI case, however, all the final volumes in Table 5 are nearly 25% greater than their counterparts in the Regular System case. Neither result is surprising. The VI just matches the parent system, where we want the volume reduced, to the subsystems, where we are trying to maintain sufficient levels of strength to withstand the applied loads. The LI case causes an increase in volume due to the need for more material to make all the subsystem dimensions match (i.e. the top flange should be the same width and thickness for two
60.08
in cm) 10.97 0.492 13.11 0.579 0.313 59.50
11.23 0.498 13.28 0.685 0.301 56.29
in cm) 11.86 0.526 11.08 0.491 0.282 52.84
11.84 0.524 11.1 0.491 0.296 55.73
0 + + +
Fig. 6. Results of Regular System problem.
Prc4lmml PfdhmZ Prcam3
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a
2
4 o&n
%-iUPE .fJ cd
6 cyci.
s
10
.Fig. 9. Results of VI problem.
adham members, etc.). The modei thus behaves p#&t&ly. The best sohnions are the Comprehensive designs. The VI work well, and the LI work better than when used alone in the previous case. The designs are stable and feasible. Convergence is achieved rapidly for eases C 1, C 2 and C 3 (see Fig. 9). Most im~rta~tIy, alf three problems achieve similar final designs PabIe 6). The onfy negative side to these designs are that
~ Subsystem
1 (member
:
l-dimensions 13.13 13.93 0.717
in cm) 14.43 IS.?2 0.725
r, (3 k
Q73f 0.496 50.73
0.735 0.4@9 M.24
b,
Subsystem
4
2 (member
tl h $2 f) h Subsysrem b, iz
3 {member
2~irne~~Q~
in cm)
13.12 0.724 13.96 0.735 0.405 45.16
14.42 0.728 15.72 8.737 0.409 50.21
14.55 0.823 0.387 45.94
3-dimensions 13.12 13.98 0.731
in cm) 14.42 15.71 0.730
12.16 0,928
12.13 0.933
14,56 8,
0.739
O.739
4 R
0.404
0.347
1
46.12
36.10
they use morel material than the previous designs. This is of little consequence, however, if the desired design must have members of similar sizes so that it can be fabricated. The Camprehensive category is aptly named; it is a comprehensive f~~~~~~on in a mathematical programm~n~ sense because it atives at one soI~tion from any starting point and does so ~~~~~y. It is also com~~hensive in the hierarchi~f dads sense because it controls the inte~~~t~ons between all levels of the system. Thus, it is an accurate model of hierarchicaf decision making. There is one f&I comment that can be made about the results for alf problems: the interaction constraints, which have been formulated as goal constraints here, are very effective. They control the differences between the parent system and each subsystem. There are a few exceptions, resulting in the diI%rens.es in member 3 as noted above. ~tkwl!se, the ~ntera~ti~~s are modeled accurately by the goal constraints, and this seems to be a promising formulation of the coupled DSP. 4. SJMMARY AND FUTURE WORH papes, we have postufated that the conceprs of hierarchical decision making and the compromise Decision Support Problem can be used together to solve problems of greater scope and difficulty than was previously possible. In support of this postulate the following has been achieved. h-i ahis
0.825
0.324 42.73
-Hierarchicaf design has been described in the GOBtext of decision making.
1037
Hierarchical desrgn of structural systems
+
Plobkmll
01
0
4
2
Dwlgn
6
6
10
cycle
Fig. 9. Results of Comprehensive problem. Table 6. Final Comprehensive designs Volume (cm’) Parent system A, (cm’)
I,’ &tfj
A, (cm2) 12 (cm’) A, (cm’) 1, (cm’)
Cl 100,590 40.26 20,675.O 40.2 20,650.O 40.21 20,628.O
c2 100,480 40.21 20445.0 40.18 20.419.0 40.16 20,426.O
&&system I (member l-dimensions
6, ‘I
6, ‘2 ‘3
h
12.34 0.546 13.0 0.575 0.436 59.77
Subsystem 2 (member 24imensions
6, II 4 ‘2
‘1
h
12.33 0.546 13.0 0.575 0.436 59.77
‘2
‘I
h
0.546 12.97 0.575
0.436 59.75
40.17 20,208.O 40.13 20,190.o 40.12 20,161.O
in cm)
12.51 0.555 13.1 0.583 0.434 59.1
12.72 0.563 13.36 0.591 0.431 58.34
in cm) 12.5 0.554 13.1 0.583 0.433 59.09
12.72 0.563 13.35 0.591 0.431 58.33
Subsystem 3 (member 3-dimensions in cm) 12.32 12.49 6,
‘I 6,
c3 100.430
0.554 13.12 0.583
0.433 59.08
12.71 0.563 13.34 0.591
0.43 1 58.32
-DSP have been defined in terms of the requirements of hierarchical design. This includes the generation of a template for the DSP for coupled hierarchical decision making. -A coupled DSP has been formulated, the results obtained and discussed. The example provides a basis from which a designer could formulate other structural design problems that involve hierarchy as coupled DSP. The appropriateness of the method for multi-level hierarchical decisions (as described in Section 1) still needs to be demonstrated. As a precursor to this, SLIP2 must be modified so that it will be usable for solving problems associated with the interactions between the different levels. Once the modification is accomplished, the method for multi-level hierarchical
decisions can he implemented. This method will he most useful for large systems that require decomposition for efficient solution. The development of the multi-level method and the modification of SLIP2 will have to be performed concurrently, so that the method can be correctly implemented. In this paper, the efficacy of solving hierarchical design problems via the compromise DSP has been shown. At this time, the efficacy of the approach has been demonstrated for coupled hierarchical decisions as defined in Section 1 of this paper. The demonstration has included the features described as VI and LI as defined in Section 2. Each of these interactions have been demonstrated independently of each other. REFERENCE.5
1. 1. S. Sobiaki, A linear decomposition method for large optimization problems-blueprint for development. NASA Technical Memorandum 83248 (1982). _ 2. N. KUDDUraiU. S. Ganesan. F. Mistree and 1. S. Sobieski, I%era&al decision’ making in system design. Engng Optimizat.8, 223-252 (1985). 3. N. Kuppuraju and F. Mistree, Compromiscan effective approach for solving multiobjective structural design problems. Comput. Struct. 22, 857-865 (1986). 4. J. A. Shupe and F. Mistree, Compromise: an effective approactt for the design of damage tolerant structural systems. Comput. Struct. (in press). 5. F. Mistree and D. Muster, The Decision Support Problem Technique for design. Inr. /. appl. Engng Education (in press). 6. N. Kuppuraju, P. lttimakin and F. Mistree, Design through selection: a method that works. Design Studies 6,91-106 (1985). 7. F. Mistree and 0. F. Hughes, Adaptive linear programming: an algorithm for solving multi-objective decision support problems. Eng. Optimizat.(submitted). 8. J. A. Shupe, Application of the Decision Support Problem Technique to the design of structural systems. M.S. thesis, Department of Mechanical Engineering, University of Houston (Februarv 1985). 9. J. A. Shupe, F. Mistree and R. HI Allen: Models for tire design of damage tolerant structural systems. Proc. 1984 Ship Structures Symp., Arlington, Virginia, pp. 62-78. SNAME publication SY-20 (1984). 10. F. Mistree, 0. F. Hughes and H. B. Phuoc, An optimization method for the design of large, highly constrained complex systems. Engng Optimizat. 5, 179-197 (1981). 11. J. Sobiesxczanski-Sobieski,B. B. James and A. R. Dovi, Structural optimization by multi-level optimization. AIAA J. 23, 1775-1782 (1985).
SHuPe,t F. MIWRFZE~ and J.
FOR THE SYSTEMS
SOBEXANSK~-SOBEBKI$
of Mechanical ~ngin~~ng, University of Houston, Houston, TX 77004, U.S.A. #NASA Langley Research Center, Hampton, VA 23665, U.S.A. (Received 15 December 1986)
this paper the use of the compromise decision support problem in hierarchical design of structural systems is described. In the past we had postulated the hierarchical decision support problem
Ah&rack-In
design.In this paper, for the first time, the mathematicaltemplatethat supports the underlyingpreceptsof hierarchical design in the context of the Jkision Support problem Technique
for use in hierarchical
is presented. A structural example that demonstrates the efficacy of the approach is included.
I. ~RODU~ON AND BACRGROUND TO ~~~1~ DBCIBION MAKFNG IN ~GINR~ING DBSXGN
-present an example that establishes the efficacy of using a hierarchical DSP for solving this class of structural problems.
The
analysis and synthesis of engineering systems, such as structures, are generally too complex to be handled as a single problem. This necessitates design of the overall system by first decomposing the system into subsystems. If the system is then designed in parts (sequentially), there is no guarantee that an overall superior design will be reached. Thus, it becomes necessary to develop a methodology that will facilitate the superior design of a hierarchical system. A ~erarchi~l system is a system that contains multiple levels of interaction between a parent system and the associated subsystems. The design of a hierarchical system is based on the formulation and solution of a series of problems involving decisions to be made by the designer. This type of design has been termed “hierarchical decision making” and the difficulties inherent in accomplishing system design can be dealt with using this approach. Decision Support Problems (DSP) provide a basis upon which a designer can make the decisions encountered in hierarchical decision making. Solution of the DSP is expected to result in superior (or optimal) designs. The DSP are capable of handling multiple objectives that contain both hard and soft info~ation. In this paper, we describe the background of hierarchical design and DSP and the relationship between them, including the utilization of goal constraints to model system interactions. It is postulated that the two can be used in concert to solve problems of greater scope and difficulty than has been possible with other design techniques. To demonstrate the validity of the postulate we: -describe hierarchical design in the context of DSP; -define a DSP for the hierarchical design of a structural system;
An investigation into hierarchical design has been made by Sobieski [l]. Solution of structural hierarchical design problems by means of the Decision Support Problem Technique was first proposed by Kuppuraju et 01.[2]. An application of the DSP Technique to structures was demonstrated in [3) and subsequently Shupe and Mistree have shown how to include reserve strength, a feature of damage tolerance, in the design of structures in 141. These four papers provide material upon which this paper is based. The concepts underlying hierarchical design are described in Section 2. Section 3 contains an example in which the application of the general method of hierarchical decision making applied to a specific structural problem is shown. Finally, in Section 4, work that needs to be done in the future to fully exploit hierarchical design as a tool for designers is discussed.
DECEION SUPFDRT PROBLXM FOR HIERARCHICAL DESIGN
2. THE REPROS
An engineering system can be represented by a hierarchy of a parent system and subsystems as in Fig. 1. The hierarchical nature of system design facilitates the identification of complexity and the interactions among various levels and subsystems. The relations and bonds of a hierarchical system arc essential for the visualization of the system. It is important to note that all of the subsystems arc integral parts of the parent system. This means that changes in one subsystem aff& the parent system and other subsystems. Thus no subsystem can be considered in isolation from the rest.
1027
1028
J. A.
SHUPE
T :: 0 h a
-
5 i u & % 1
PI al.
requires computers capable of parallel processing; -some decisions are inseparable and the input of decision 9” is the output of decision “i - I” and thus the decisions can be made sequentially, see Fig. 2(b); -some decisions are inseparable and are coupled and the output of one is the input for the other, see Figs 2(c) and (d). In this case the decisions could be made either ~quentially (start with a reasonable guess for the first output of “i”) or concurrently (using optimization). Design involves hierarchical decision making:
LATERAL INTERACTIONSPS - Psr*nt Syrtam SS - SubsSstm
-interaction between the various levels of subsystems exists. This interaction may go one way or both ways; -interaction between the subsystems at the same level of the same parent also exists. This interaction can also go one way or both ways.
Fig. 1. A general four-level hierarchical system [I]. At each level of the hierarchy, for each subsystem, the design process involves decisions which are qualified by the follo~ng assertions. Design is a seriesof decisions:
This leads us to postulate two classes of hierarchical design, see Fig. 3. The first class is multilevel hierarchical decision making, wherein the hierarchy of the system is modeled as separate decisions, which are
-some decisions are separable and therefore can be made concurrently, see Fig. 2(a). This of course
a - kparsble
deoslons
mode concurrently
b - lnsepsrabledecisions
c - lneeparsblrandcoupled Mclslons made srqurntlslly and requirtng ItsratIon
mode S8qUootiottY
d - lnsepareble and coupled decisions made concurrently
Fig. 2. Series of decisions.
Hierarchical design of structural systems
1
SEL;KTplDN
TYPES
1
OF DSPS
1 CDIlP650pSE
USED
]
1
IN THE DECISION
CDD”Ll;ED
1
PROCESS
Fig. 3. Hierarchical design: types of decisions.
made at the different levels at which they occur. The decisions are separable and can therefore be made concurrently [Fig. 2(a)] or sequentially [Figs 2(b) and (c)l. It is difficult in this approach to achieve a high level of interaction between the different decisions. The second class of hierarchical design is coupled hierarchical decision making. In this class the hierarchy is modeled as a single decision, Fig. 2(d). The interactions between the decisions are strongly linked, creating a tight bond between subsystems solved by the problem and hence it is relatively easy to achieve a high degree of interaction between decisions. It is this class that is described in this paper. 2.1 Types of Decision Support Problems [5] In design, there are basically three ways to derive the answers to problems encountered. These ways are analysis, synthesis and heuristic thinking. These may seem to be widely divergent but it is possible to integrate all three in the form of structured DSP. Since there are different types of decisions to be made, different categories of DSP exist. These categories are: selection [a]; compromise [3,4]; hierarchical [2]; conditional. A selection DSP is used to choose an alternative from several. The choice is made based on ratings given to multiple attributes and their relative importance. A compromise DSP involves improving an alternative, by changing design variables optimally, to find a superior solution. A hierarchical (coupled) DSP involves the combined solution of multiple selection and compromise DSP simultaneously. Decisions in which risk and uncertainty of the outcome are taken into account are facilitated through the solution of conditional DSP. Both selection and compromise DSP may be formulated and solved in the synthesis of a hierarchical system composed of parent systems and the related subsystems as shown in Fig. 1. This is an application of multilevel hierarchical decision making. The
1029
coupled DSP, however, is the basis of coupled hierarchical decision making. The coupled DSP may involve any combination of selection and/or compromise DSP. The interaction can consist of coupling between subsystems at the same level or at different levels. If the coupling occurs between the parent system and a subsystem it is called Vertical Interaction (VI) [l]. There are two types of VI, namely, Forward Vertical Interaction (FVI) and Reverse Vertical Interaction (RVI). If interaction occurs between two subsystems it is called Lateral Interaction (LI). These interactions are illustrated in Fig. I. The mathematical model for a coupled DSP involving both selection and compromise is presented in [7, case 61 and the mathematical model used in this paper is described in [3,4]. The method of modeling the interactions is extremely important. These are modeled as goal constraints of a compromise DSP. The approach is described in Section 3. Multilevel hierarchical decisions require the separate but possibly concurrent solution of all types of DSP. Coupled hierarchical decisions require simultaneous solution of two or more DSP as a single coupled problem. In this paper, we discuss coupled hierarchical decision making only. The multilevel hierarchical problem requires further research to be done as well as some new hardware that allows multitasking or parallel processing.
3. HIERARCHICAL DESIGN OF A PORTAL FRAME
The efficacy of the coupled selection-compromise DSP in structural design has been demonstrated for the classical three-bar truss problem in [8,9]. Hierarchical structural design also involves the coupling of two (or more) compromise DSPs. In this section, therefore, the various interactions in hierarchical design are demonstrated by the use of a coupled compromise-compromise DSP. The portal frame, used as an example, was originally used by Sobieski [l] to demonstrate top-down decomposition without Reverse and/or Lateral Interaction. Sobieski used the portal frame example to show the efficacy of optimization by decomposition with linking through sensitivity. Our present ap preach is different and this distinction is amplified in Section 3.5. The portal frame, in this paper, is used to demonstrate the effect of Vertical (both Forward and Reverse) and Lateral Interactions by using the coupled DSP. All interactions are formulated as goal constraints [3,7,8] of a compromise DSP. It is emphasized that we are not trying to demonstrate the savings associated with the use of decomposition techniques per se; rather we intend to demonstrate the efficacy of using the compromise DSP to model the interactions between parent and subsystems (VI) as well as between one subsystem and another 0-I).
Fig.
4.
Hierarchical portal frame probiem.
3.1Res~r~~t~#~of the portal frame
t, = top flange height I~= web width h = web height.
Given a three-member portal frame as presented in Fig. 4, the pertinent system ~~n~~~~~s are:
These are the principal dimensions of an I-beam. 3.2 Organization of the portal frame design
The objjective is to blaze the overall maas of the frame, white being subjected to static toads, f and M. AIsoo, the system is subject to certain constraints covering normal stress, bending stress, shear stress and buckling in each member. There are two types of design variables; one type for the parent system and another type for the subsystems. The parent system design varibles [_4 and I) are each member’s crosssectional area and moment ofinertia. Each subsystem has six design variables, reprexnting: 6, = bottom flange width 8, = bottom flange height 4 = top flange width
The approach used to solve a multilevel hierarchical DSP is detaihxi in this section. As indicated earEel, the approach we take in adding structural hierarchy is different from that proposed by Sobieski \I]. Some of these differences are highlighted. The coupted DSPs for the four cases were solved by using the SLiP2 program [7,lO]. The SLIP2 aigor&&n is different from those that are currently favored in optimum structural design. A schematic of the integration of a finite element analysis program with SLIP2 is shown in Fig. 5. The step-by-step procedure is as fohows: t* 2. 3. 4.
Defme geometry, mateliai properties and loads. Define parent system stress constraints. Initiaiize detailed cubists dimensions. From detailed sub~yst~ ~mension~, calculate A and I for each beam. 5. Using a frame analysis subroutine, analyze the
1031
Hierarchical design of structural systems
portal frame (parent system) to find the member responses (N, M, r). 6. Begin the synthesis cycle. Iiolding member responses constant, formulate the nonlinear DSP. This requires the quantification of the following constraints:
FORHULATE
“,“,““”
I NONLINEAR
-parent system stress constraints -subsystem combined stress constraints -subsystem shear stress constraints -subsystem buckling constraints -parent system goal (minimum mass) -VI goal constraints (these goals maintain the equality of the parent system variables (A and I) to the subsystems’ areas and moments of inertia as functions of the subsystem variables (detail dimensions)) -bounds. Formulate the linearized DSP [7, IO] and solve using the Revised Dual Simplex algorithm. Repeat synthesis cycle, if necessary. 7. Repeat frame analysis and synthesis cycle, if necessary.
-*St70
DESIGN-MALYGiS
DSP
-i
Fig. 5. Implementation
of the SLIP2 algorithm.
variables (A and 1) and the subsystem variables (b, r, Observe that the finite element analysis is performed only in the analysis/synthesis cycle (see Fig. 5). For the cases presented in this paper no finite element analysis was performed within the synthesis cycle, The response of the structural system, obtained from finite element analysis, is used to formulate the nonlinear DSP. This nonlinear DSP is linearized and solved using a Revised Dual Simplex algo~t~. Note that the four cases include different types of structural hierarchy and each of the DSPs is different. Each DSP, however, is solved using the same scheme outlined in Fig. 5 and unlike the approach presented in [i] no decomposition of the structural system is involved. 3.3 Strategies for exploring the hierarchical frame problem In this section, we deal with two types of VI, namely Reversed VI and Forward VI. RVI has been demonstrated in the portal frame by Sobieski [ I,1 l]. Although FVI had been demonstrated fill, LI remained to be demonstrated. In this section we discuss strategies for demonstrating all of these interactions using the DSP technique. The basis of the strategy for studying the hierarchical problem is that the solution of the problem is approached one case at a time. This is done so that the interactions between different levels can be understood in isolation first, before their effects are combined into a comprehensive formulation. The interactions, VI, that occur between the parent system and each of its subsystems, as well as the interactions that occur between each subsystem, LI, are shown in Fig. 4. The Vi necessitates the inclusion of constraints that match the parent system design
h). The LI necessitates the inclusion of constraints
that match the subsystem variables to their counterparts in the other subsystems. (Note that there are no interactions shown between subsystem 1 and subsystem 3 since there is no physical connection between the two.) The “matching” that the interactions perform is modeled mathematically by goal constraints, An interaction goal ~nstraint constrains one subsystem variable to be equal to its counterpart in the other subsystem. For example, in Fig. 4, the individual dimensions of the center beam (subsystem 2) of the portal frame should match those of the beams (subsystems 1 and 3) on either side. Thus, the LI equality constraints are created to handle this, The deviation from this equality is measured by the goal constraint’s deviation variables (underachievement, d- and overachievement, d+). Thus the interaction goal constraint for the parent system and subsystem 1 is written as R,+d-
-d”
=A(b,t,h),
where A, is the cross-sectional area of member 1 (parent system) and A(b, t, h), is the cross-sectional area of member 1 as a function of the subsystem variables. The interaction goal constraint provides an effective and efficient approach for mainlining the interactions between the parent system and its subsystems and between individual subsystems. The exploration of the hierarchical problem follows the following path. -First, a compromise DSP is solved for the problem as a whole system. This is called the Regular System (RS) case. There is no coupling and thus
1032
J. A.
SHIJPE
there arc no interaction goal constraints. All constraints are in terms of the design variables at the subsystem Ievel. In terms of Fig. 4, this example involves only the variables b, t and h for each beam, and does not make use of the parent system variables, A and I. This example establishes the baseline for the rest of the investigation. The coupled DSP for this example includes 18 system variables, two deviation variabies, 33 constraints and one goal. -The next example involves a compromise DSP that involves the coupling of a parent system and its attendant subsystems. This DSP is referred to as the Vertical Interaction case. This DSP includes VI goaf constraints to maintain the interface between the parent system design variables and the subsystem design variables (Fig. 4). This example is used to demonstrate the feasibility of establishing interactions between a patent system and its subsystems using goal constraints and deviation variables to control the interaction. This example uses all the variables depicted in Fig. 4, but the only interaction is between the parent system variables and their related subsystem variables. The coupled DSP in this example involves 24 system variables, 14 deviation variables, 33 constraints and seven goals. -In the third example the portal frame is modeled as three separate subsystems and this is again formulated as a coupled DSP. This example is referred to as the Lateral Interaction case, This DSP includes LI goal constraints to regulate the interface between the three subsystems. In this example, the parept system variables depicted in Fig. 4 are not used at all, so there is no VI; the interactions are
Given:
i
l
el
al.
bctwecn the subsystems only. Notice however, that there is no interaction between the first and third subsystems since there is no physical connection. This example is an importanl demonstration of the need foi interaction between seemingly independent subsystems. The coupled DSP for this example includes 18 system variables, 26 deviation variables, 33 constraints and 13 goals. -The final exampte in the process is the ~mbination of all the VI constraints and the LI constraints. Thus, this problem is designated the Comprehensive (C) case. The example employs all the variables and interactions depicted in Fig. 4. This problem is a comprehensive formufation of the coupled hierarchical DSP. Size of problem: 24 system variables, 38 deviation variables, 33 constraints, 18 goals. It is important to note that the number of goals and the number of deviation variables are directly proportional to the number of interactions being modeled. A generic mathematical formulation for the hierarchical design problem is given in Section 3.4, followed by the results in Section 3.5. 3.4 Coupled compromise DSP for the portal jiiame The mathemati~l fo~uIation of a coupfed compromise DSP is detailed for the Comprehensive case in this section. This formulation is for the portal frame shown in Fig. 4 and is prepared with reference to the mathematical form of the Compromise DSP presented in [3,7’j. Since the following coupled DSP is for the Comprehensive case the DSP for the other cases can be derived by deleting the appropriate constraints and/or goals.
index for element number
S(A+ i) - combined normal and bending stresses
for
member I MA,) = total mass of frame as a function of the cross sectlonal areas HE = expected value for mass. A( b, t, h Ii = c~~-~ct~~l area of the I-th element as a functfon of the subsystem dimension varlabies I( b, t, h ), moment of inertia for the i-th element as a l
function of’the subsystem dimension variables UA,Uf - maximum ~~~~~ble ValUeSforAt and II LA, Ll = mlnlmum permissible Values for Al and I I
b, t, h = dlmenslon variables determined at subsystem level Ubl Ut, uh - maxims permissible values for b, t and h Lb, Lt,Lh - minims perfnfsstble values for b, t and h f= I ,2,S: member number
Hierarchical design of structural systems
j= 1 - Indicates left end of member t
2 - indicates rtght end of member t
index for devfatlon vartables; corresponds to goal constrarnt m = I ,2: First Lateral lnteractlon lndex n = 2,3,..., 13: Second Lateral Interaction lndex
k =
FM
Ai - cross sectlonal area of the I-th element Ii - moment of inertia of t-th element bll,b2i
-
flange widths
t 1i,tzi - flange thfcknesses t3j - web thickness hi - height of web Satisfy System Constraints: Parent system - combined stress constraints SfAi,li) i Smax System Constraints Subsystems - combined stress in the top flange oa / db,t+hl,j 1 t i- 1,2,3; f= I,2 - cubit stress in the bottom f Mqe cr, / o( b,t,h Ii4 i I i= 1,2,3; j= 132 - shear stress constraint t,/d b,t,h Irj 1 1 i- 1,2,3; jm I,2 - f iange buckling constraint aa,(b, ,,t, ,) ! 1a( b,t,h 1111 1 I* 1,2,3 ta(b\\,tli)
/ Id bat&h Ii IL 1 Is 12J
~*{~~,t2i
I/ I a( b,t,h Ji I 2 1 i- 1,2,3
tatb2i8t2j
1 I id b,t,h )t i ;I 1 irn 1,2,3
Goal Constants - mass goal: minimize mass of system M(A$‘+d-l -d+$ =O - vertlcar ~ntera~ti~ Goats Ai + d jtl - d+i+, l Af b,t,h II
I= 1,283
11l d-1+4 - d+,tJ = I( b,t,h ), I= I ,2,3 - Lateral interaction Goals (n = Im- II * 6) bj,m + d”2+n* d’p+n’bl,m+l m-1,2; b2,m + d-3tn - d43+n * b2mt 1 mrlSk tt,m+d-q+n’d’~tn’tl,mtI
m-182;
t2,m + d-S+n _ d’S+n t t2,m* 1 mR13; tj,m * d_6+n _ d’6+n mt3,mt f mw12
~td-,+n-d’~+n=~+~ 8ounds on oesrgn Variabtes LA,AI
cUA
L’,I(
mmt,2;
1033
J. A.
1034
SHUPE
et
al,
Bounds on Deviatlon Variables 0.0 (d-k,d*k
5 1.0 k- I,2 ,...,7
tlfnimize Z(~-,d+l= P,(d-,
+ d+, I+ P2(d-2 + dt2 + dt3 + ... + d-7 + dt7)
P t B P2 where B lndlcates preference.
Mass is an indirect measure of economic efficiency whereas the interactions are secondary (strength, buckling, etc. being primary) measures of technical efficiency. Hence, in the preceding a preference is shown towards minimizing mass at the expense of the interactions. 3.5 Results The different cases are identified in Table 1. In making computer runs to get results for this investigation, several questions are asked, in order to direct the course of the investigation. These questions are given here to begin the discussion of the results: What effect do different initial designs have on the solution? Will the problem as formulated handle infeasible initial points. How infeasible can the initial points be? What are the differences between the Regular System, VI, LI and Comprehensive solutions? These questions, when answered, will illustrate how the interaction constraints can be manipulated to handle hierar~hi~l problems. The answers to these questions are presented in Sections 3X1-3.5.3. 3.5.1 The effect of different initial designs. The initial question to be answered concerns whether or not the choice of an initial starting design is important. Ideally, the choice of an initial design should not matter, but it does make a difference in some cases. As shown in Table 2, there were three initial designs used, an initial infeasible design with a volume of 26,469 cm’, a high initial feasible design with a volume of 275,OOOcm’ and a low initial feasible design with a volume of 104,910cm3. The &al designs derived from the initiai designs vary, but the total volume values of the final solutions differ by as little as 0.05% (the difference between final designs in cases C 2 and C 3 in Table 6) to as much as 6.5% (the difference between final designs in problems VI I and
Table 1. Titles for cases
CatCgOliCS
104,910 cm’ (feasible)
Regular system VI Ll Comprehensive
Prb, Prb, Prb. Prb.
t/RS I 4/Vf i l/L1 1 10/C 1
275,000 cm’ (feasible)
26,469 em’ (infeasible)
Prb. Prb. Prb. Prb.
Prb. Prb. Prb. Prb.
2/RS 2 s/VI 2 8jLT 2 I r/C 2
Table 2. Initial designs
VOfIBIle (cm’)
3iRS 3 6/W 3 9/LI 3 lZ/C 3
Low feasible initial design 104,910
High feasible initial design 275,000
Infeasrble initial design 26,469
Parent sysfem 4
(cm’)
1, (cm’) A, (cm21 1, (cm’) A, (em*) 1, (em’)
52.14 37,993.o 58.3 ~,497.0 20.5 6135.0
110.0 49,436.0 110.0 49,436.0 110.0 49,436.0
10.59 78.1 10.59 780.1 10.59 780.1
Subsplem I (member 1-dimensions w cm) b, 11.0 30.0 tt 0.5 1.0 b1 9.0 30.0 82 0.53 f.6 *3 0.53 5;:: fr 79.0
11.0 0.275 5.5 0.275 0.275 22.0
Subsystem 2 (member 2-dimehms b, 11.0 4 bz 8.;
in cm) 30.0 I.0 30.0
11.0 0.275 5.5
12
0:s
I3
0.5 95.7
1.0 I.0 50.0
0.275 0.275 22.0
h
Initial designs Problem
VI 3 in Table 4). There is, however, a great deal of difference in the individual member values. The only final designs that really match up are the Comprehensive designs. These Comprehensive designs combine LI and VI goal constraints. Since the results are so close, the Comprehensive fo~ulation is the only one to achieve global convergence. Thus, different initial designs do effect the result, unless the formulation is complete. This is typical of problems solved using SLIPZ. 3.5.2 infetz.viblei&&f designs. The next question to be dealt with concerns the ability of the program to handle infeasible initial designs. The program not only handled an infeasible initial design, but a highly infeasible one as well. The low initial volume design (26,469 cm’) was a very infeasible initial design. Of 18 stress and buckling constraints, this design violated eight ~~str~nts initially, yet yielded a design within 6.5% of the low initial feasible design problem’s final designs and within 0.3% of the high initial feasible design problem’s final designs.
Subsysrem 3 (member 3-dimensions in cm) b, 11.0 30.0
4 b, ‘2 f3
It
0.4 6.0 0.3 0.3 47.8
1.0 30.0 1.0 1.0 50.0
Il.0 0.275 5.5 0.275 0.275 22.0
Hierarchical design of structural systems 3.5.3 The dl#ierences between the four solutions.
Finally, the differences and similarities of the Regular System, VI, LI and Comprehensive solutions are discussed. The solutions are given in Tables 3-6. The histories of the design cycles are presented in graphical form in Figs 6-9. As can be seen from the figures, the Regular System and VI final solutions are in general agreement for all three initial designs. ,The instances where there is a little agreement mostly occur in the third member, as seen in Tables 3 and 4. This member is subjected to lower end loads and moments and, as a result, was the member most easily reduced in size. Since an important goal is the minimization of total volume, the third member was the member most affected by the solution process. Thus, there are some large differences in design variables here that do not occur in the other members. However, the final volumes for both the Regular and VI solutions are very close. The LI final designs do not closely agree with either of the other two types of solutions, which can be seen by comparing Fig. 7 with the other figures. This is a reasonable result, however. The LI problem has been given the stipulation that each I-beam dimension (subsystem design variable) in a member must match its counterpart in the contiguous member. Given this requirement, it makes more sense to examine the LI problem internally. In other words, do the design variables match their counterparts? By examining Table 5a, it is seen that the answer is “yes”. The design variables do match to a very close tolerance. However, in this category, there is still disagreement between the solutions found for the different initial designs. This indicates that LI constraints work between subsystems, but they should not be used alone. From the overall design point of view, it is interesting to note how the VI and LI cases compare to Table 3. Final Regular daigns Volume (cm’)
RS 1 79,219
Subsysrem 1 (member b, II b, 1s 13 h
l-dimensions in cm) 18.34 12.38 0.846 0.531 18.76 12.33 0.858 0.641 0.402 0.450 43.82 62.81
Subsystem 2 (member 2-dimensions
b, ‘I bz
‘1 ‘I h
16.05 0.833 18.92 0.851 0.338 55.51
Subsystem 3 (member 3-dimen.rions
b, r, bz ‘2 ‘3 h
RS 2 19,531
10.0 0.323 11.8 0.278 0.275 10.45
RS 3 19,286 12.97 0.558 12.98 0.635 0.439
1035
Table 4. Final VI designs Volume (cm’) Parent system A, (cm2) I, (cm’) i2(cm2j 12
b-f)
A, (cm’) 1, (cm’) Subsysfem b,
1, b, ‘2
I,
i
VI I 76,130
VI 2 80,883
VI 3 81,085
46.19 26.923.0
42.15 19.583.0 22.96 8643.7 35.91 22,071.o
39.10 28.061.O ’ 27.40 12,911.O 30.98 16,749.O
38,0% 7.0 182.0
1 (member l-dimensions 11.57 0.619 13.53 0.619 0.411 64.13
in cm) 10.97 0.545 5.86 0941 0.464 69.82
Subsystem 2 (member 2-dbnension.v in cm) 12.71 10.0 b, 0.563 0.418 11
b2 ‘I ‘I h
14.81 0.656 0.387 15.42
10.72 0.415 0.275 49.16
Subsystem 3 (member 3-dimensions in cm) 10.0 12.76 b, 0.219 0.565 1, 1.18 12.29 bz ‘2
‘3 h
0.275 0.275 14.13
0.544 0.344 64.09
14.95 0.609 12.64 0.587 0.451 65.14 10.0 0.443
12.22 0.541 0.292 56.04 11.86 0.525 9.29 0.574 0.318 60.35
the Regular System case in terms of total volume, since volume can be considered the economic indicator for this example. The VI final volumes are very close to the final volumes in the Regular System case. In fact, the final volume for the low initial feasible design (VI 1 in Table 4) is less than its counterpart in the regular case (RS 1) in Table 3. In the LI case, however, all the final volumes in Table 5 are nearly 25% greater than their counterparts in the Regular System case. Neither result is surprising. The VI just matches the parent system, where we want the volume reduced, to the subsystems, where we are trying to maintain sufficient levels of strength to withstand the applied loads. The LI case causes an increase in volume due to the need for more material to make all the subsystem dimensions match (i.e. the top flange should be the same width and thickness for two
60.08
in cm) 10.97 0.492 13.11 0.579 0.313 59.50
11.23 0.498 13.28 0.685 0.301 56.29
in cm) 11.86 0.526 11.08 0.491 0.282 52.84
11.84 0.524 11.1 0.491 0.296 55.73
0 + + +
Fig. 6. Results of Regular System problem.
Prc4lmml PfdhmZ Prcam3
1036
J. A
a
2
4 o&n
%-iUPE .fJ cd
6 cyci.
s
10
.Fig. 9. Results of VI problem.
adham members, etc.). The modei thus behaves p#&t&ly. The best sohnions are the Comprehensive designs. The VI work well, and the LI work better than when used alone in the previous case. The designs are stable and feasible. Convergence is achieved rapidly for eases C 1, C 2 and C 3 (see Fig. 9). Most im~rta~tIy, alf three problems achieve similar final designs PabIe 6). The onfy negative side to these designs are that
~ Subsystem
1 (member
:
l-dimensions 13.13 13.93 0.717
in cm) 14.43 IS.?2 0.725
r, (3 k
Q73f 0.496 50.73
0.735 0.4@9 M.24
b,
Subsystem
4
2 (member
tl h $2 f) h Subsysrem b, iz
3 {member
2~irne~~Q~
in cm)
13.12 0.724 13.96 0.735 0.405 45.16
14.42 0.728 15.72 8.737 0.409 50.21
14.55 0.823 0.387 45.94
3-dimensions 13.12 13.98 0.731
in cm) 14.42 15.71 0.730
12.16 0,928
12.13 0.933
14,56 8,
0.739
O.739
4 R
0.404
0.347
1
46.12
36.10
they use morel material than the previous designs. This is of little consequence, however, if the desired design must have members of similar sizes so that it can be fabricated. The Camprehensive category is aptly named; it is a comprehensive f~~~~~~on in a mathematical programm~n~ sense because it atives at one soI~tion from any starting point and does so ~~~~~y. It is also com~~hensive in the hierarchi~f dads sense because it controls the inte~~~t~ons between all levels of the system. Thus, it is an accurate model of hierarchicaf decision making. There is one f&I comment that can be made about the results for alf problems: the interaction constraints, which have been formulated as goal constraints here, are very effective. They control the differences between the parent system and each subsystem. There are a few exceptions, resulting in the diI%rens.es in member 3 as noted above. ~tkwl!se, the ~ntera~ti~~s are modeled accurately by the goal constraints, and this seems to be a promising formulation of the coupled DSP. 4. SJMMARY AND FUTURE WORH papes, we have postufated that the conceprs of hierarchical decision making and the compromise Decision Support Problem can be used together to solve problems of greater scope and difficulty than was previously possible. In support of this postulate the following has been achieved. h-i ahis
0.825
0.324 42.73
-Hierarchicaf design has been described in the GOBtext of decision making.
1037
Hierarchical desrgn of structural systems
+
Plobkmll
01
0
4
2
Dwlgn
6
6
10
cycle
Fig. 9. Results of Comprehensive problem. Table 6. Final Comprehensive designs Volume (cm’) Parent system A, (cm’)
I,’ &tfj
A, (cm2) 12 (cm’) A, (cm’) 1, (cm’)
Cl 100,590 40.26 20,675.O 40.2 20,650.O 40.21 20,628.O
c2 100,480 40.21 20445.0 40.18 20.419.0 40.16 20,426.O
&&system I (member l-dimensions
6, ‘I
6, ‘2 ‘3
h
12.34 0.546 13.0 0.575 0.436 59.77
Subsystem 2 (member 24imensions
6, II 4 ‘2
‘1
h
12.33 0.546 13.0 0.575 0.436 59.77
‘2
‘I
h
0.546 12.97 0.575
0.436 59.75
40.17 20,208.O 40.13 20,190.o 40.12 20,161.O
in cm)
12.51 0.555 13.1 0.583 0.434 59.1
12.72 0.563 13.36 0.591 0.431 58.34
in cm) 12.5 0.554 13.1 0.583 0.433 59.09
12.72 0.563 13.35 0.591 0.431 58.33
Subsystem 3 (member 3-dimensions in cm) 12.32 12.49 6,
‘I 6,
c3 100.430
0.554 13.12 0.583
0.433 59.08
12.71 0.563 13.34 0.591
0.43 1 58.32
-DSP have been defined in terms of the requirements of hierarchical design. This includes the generation of a template for the DSP for coupled hierarchical decision making. -A coupled DSP has been formulated, the results obtained and discussed. The example provides a basis from which a designer could formulate other structural design problems that involve hierarchy as coupled DSP. The appropriateness of the method for multi-level hierarchical decisions (as described in Section 1) still needs to be demonstrated. As a precursor to this, SLIP2 must be modified so that it will be usable for solving problems associated with the interactions between the different levels. Once the modification is accomplished, the method for multi-level hierarchical
decisions can he implemented. This method will he most useful for large systems that require decomposition for efficient solution. The development of the multi-level method and the modification of SLIP2 will have to be performed concurrently, so that the method can be correctly implemented. In this paper, the efficacy of solving hierarchical design problems via the compromise DSP has been shown. At this time, the efficacy of the approach has been demonstrated for coupled hierarchical decisions as defined in Section 1 of this paper. The demonstration has included the features described as VI and LI as defined in Section 2. Each of these interactions have been demonstrated independently of each other. REFERENCE.5
1. 1. S. Sobiaki, A linear decomposition method for large optimization problems-blueprint for development. NASA Technical Memorandum 83248 (1982). _ 2. N. KUDDUraiU. S. Ganesan. F. Mistree and 1. S. Sobieski, I%era&al decision’ making in system design. Engng Optimizat.8, 223-252 (1985). 3. N. Kuppuraju and F. Mistree, Compromiscan effective approach for solving multiobjective structural design problems. Comput. Struct. 22, 857-865 (1986). 4. J. A. Shupe and F. Mistree, Compromise: an effective approactt for the design of damage tolerant structural systems. Comput. Struct. (in press). 5. F. Mistree and D. Muster, The Decision Support Problem Technique for design. Inr. /. appl. Engng Education (in press). 6. N. Kuppuraju, P. lttimakin and F. Mistree, Design through selection: a method that works. Design Studies 6,91-106 (1985). 7. F. Mistree and 0. F. Hughes, Adaptive linear programming: an algorithm for solving multi-objective decision support problems. Eng. Optimizat.(submitted). 8. J. A. Shupe, Application of the Decision Support Problem Technique to the design of structural systems. M.S. thesis, Department of Mechanical Engineering, University of Houston (Februarv 1985). 9. J. A. Shupe, F. Mistree and R. HI Allen: Models for tire design of damage tolerant structural systems. Proc. 1984 Ship Structures Symp., Arlington, Virginia, pp. 62-78. SNAME publication SY-20 (1984). 10. F. Mistree, 0. F. Hughes and H. B. Phuoc, An optimization method for the design of large, highly constrained complex systems. Engng Optimizat. 5, 179-197 (1981). 11. J. Sobiesxczanski-Sobieski,B. B. James and A. R. Dovi, Structural optimization by multi-level optimization. AIAA J. 23, 1775-1782 (1985).