Computation of Preference in DVS and PVS
4.4 The Computation of the Overall Preference
One thing that should be noticed is that this new extension of the LIA can be used to induce any preference function from the DVS to the PVS. The new extension of the LIA improves the accuracy at the expense of the increase in the computational cost. If the metamodel of the mapping function is used, the increased computational cost is reasonable as demonstrated in Section 5.3.1.
Then∀αk ≤µ∗o, Dαd
k ⊇Dµo∗o, becauseµd(d~∗)≥µ∗o ≥αk,∀d~∗ ∈Doµ∗o. Because the larger theαk, the smaller the size ofDdαk, the search space ofDoµ∗
o is now restricted toDdα∗
kwhereα∗kis the largest αkamong allαk≤µ∗o.
Although the backward path succeeded for the engine design problem, there are some difficulties for a general problem without the three special conditions. First, if there is compensation in the aggregation function ofµdandµp, the search space ofDµo∗
o is the whole DVS. Second, the brute- force search is not feasible if the DVS is not a finite set of design alternatives. When there is compensation in the aggregation function ofµdandµp, theα-cut ofµomay not be limited to theα- cut ofµd. There are two ways to compute theα-cut ofµofrom theα-cut ofµd: either by intersection ofPαdk andPαPk in which way some region ofPαok is lost, or reconstructµd(~p)fromPαdk’s which is not trivial. Finally, iff~(d)~ is nonmonotonic, it is possible thatf~−1(Pµo∗o)⊃Doµ∗o, wheref~−1 is not a single-valued function and defined as
f~−1(P) ={d~∈ X |f(~ d) =~ p, ~~ p∈P} ∀P ⊆ Y (4.17) whereYis the set of all vectors of performance variables with valid values.
Becausef~(d)~ is nonmonotonic, ∃d1, d2 ∈DV S such that f~(d1) =f~(d2) =~p∗. Without loss of generality, assume that µd(d2)> α2 > µd(d1)> α1>0. According to the extension princi- ple,µd(~p∗)≤max(µd(d~1), µd(d~2)) =µd(d~2)> α2. Assumeα2 > µp(~p∗)> α1, then there exists P∗ 6= maxsuch thatµo(~p∗) =P∗(µd(~p∗), µp(~p∗))> α2, i.e.:
~
p∗ ∈Pαo2 and ~d1, ~d2∈f~−1(Pαo2) (4.18) But because µd(d~1) < α2 and µp(~p∗) < α2, from the idempotency and monotonicity ofP, µo(d~1) =P(µd(d~1), µp(f~(d~1)) =P(µd(d~1), µp(~p∗))<P(α2, α2) =α2, which means that
d~1 ∈/Doα2 (4.19)
From Equation 4.18 and Equation 4.19, it can be concluded thatf~−1(Pµo∗o) ⊃Dµo∗o for aggregation functionP∗andµ∗o > α2.
All above difficulties and errors stem from the usage of f~−1. They can be avoided if f~−1 is no longer needed. The following method can compute Dαok and Pαok without f~−1, and so it will produce more accurate results with less computational cost.
• The new method to computeDαo
k andPαo
kwithoutf~−1:
1. Specifyd,~ ~p, X, Y,µdi(di)’s, andµpi(pi)’s. Decide the trade-off strategy, i.e., theP used to compute the overall preferenceµo =P(µd, µp).
2. Compute µo(d)~ the overall preference in DVS, and use the method in Section 4.3 to generateDoα2k.
3. ComputePαo2
k withDoα2
k by the extension of the LIA in Section 4.3.
The biggest difference between the new method and the old method is that the new method does not needf~−1. In the old method computations are carried out in both the DVS and the PVS, and finallyPαo
k is generated beforeDoα
k, which is why it needsf~−1. But in the new method, although µp(f~(d))~ can be considered computations in the PVS, the final aggregation for the overall preference is in DVS. Computingµp(f~(d))~ ford~∈ X is the basis of the new method. The LIA is a discrete approximate implementation of the extension method. When apply LIA toDαo2k,µo(d)~ is needed to be computed only at a finite number of points in DVS. Even iff~is not invertible over the PVS but it can be considered invertible at any single point in the PVS because thef~−1 in Equation 4.17 is single-valued forP ={f~(d)~} ⊆ Y. The new method can be understood as applying the two-step old method at a finite number of design points where f~−1 is well defined. The usage of f~−1 is avoided by transferring the information of the functional requirements in the the PVS to the DVS by using the equivalent inverse off~at individual points in the PVS.
To demonstrate the new method, it will be applied to the example in Section 4.3 with a two- dimensional DVS and a two-dimensional PVS. The design preferences for both design variables are defined in Figure 4.1. The performance function is shown in Equation 4.9. The design preferences and functional requirements are shown in Figure 4.10. The aggregation functions for the combined design preference, combined functional requirement, and the overall preference in the DVS are
µd(d)~ = min (µd1(d1), µd2(d2) ) (4.20) µp(d)~ =
h
µp1(f1(d~)) +µp2(f20(d~)) i
/2 µo(d)~ =
q
µd(d~)·µp(d~)
The overall preference in DVS is shown in Figure 4.11. Hereαk = 0.5is chosen, because the α-cut will be more complicated than those forαk=ε or 1.0.
−1 −0.5 0 0.5 1 0
0.5 1
d1 µ d1
−1 −0.5 0 0.5 1
0 0.5 1
d2
µ d2
−4 −2 0 2 4
0 0.5 1
p1 µ p1
−4 −2 0 2 4
0 0.5 1
p2
µ p2
Figure 4.10:µd1(d1),µd2(d2),µp1(p1)andµp2(p2).
−0.8 −1
−0.4 −0.6 0 −0.2
0.4 0.2 0.8 0.6
1
−1
−0.5
0
0.5
1 0
0.2 0.4 0.6 0.8 1
d2
d1
µ o
Figure 4.11: The shape ofµo(d).~
−1 −0.5 0 0.5 1
−1
−0.5 0 0.5 1
S = 4
−1 −0.5 0 0.5 1
−1
−0.5 0 0.5 1
S = 8
−1 −0.5 0 0.5 1
−1
−0.5 0 0.5 1
S = 16
−1 −0.5 0 0.5 1
−1
−0.5 0 0.5 1
S = 32
Figure 4.12:D0o.25(S)for different values ofS.
−2 −1 0 1 2
−3
−2
−1 0 1 2 3
p1
p 2
S = 4, U = 8
−2 −1 0 1 2
−3
−2
−1 0 1 2 3
p1
p 2
S = 8, U = 16
−2 −1 0 1 2
−3
−2
−1 0 1 2 3
p1
p 2
S = 16, U = 32
−2 −1 0 1 2
−3
−2
−1 0 1 2 3
p1
p 2
S = 32, U = 64
Figure 4.13:P0o.25(S,U)for different values ofSandU.
Now the new method is used to compute D0o.25 andP0o.25. Following procedure is listed in Sec- tion 4.4,D0o2.5should be computed first. Here differentT’s are chosen as4,8,16and32to compare the effect ofS. TheseD0o2.5’s are shown in Figure 4.12. Among the fourDo02.5’s,Do02.5(32)is of course the most accurate, and D0o2.5(16)also catches many details in D0o.25(32), evenDo02.5(8)has enough details for the use in preliminary stages. The P0o.25’s are also induced with U = 2S as shown in Figure 4.13. Not surprisingly,P0o.25(32,64)is the best one. And theP0o.25(16,32)is almost the same asP0o.25(32,64). P0o.25(8,16) is the most cost-effective result among these four P0o.25’s. Although Do02.5(4)is rough,P0o.25(4,8)is still on the right track if compared with others.
Now the DVS is continuous, and f~is not invertible over the whole DVS. The backward path can not be used to compute D0o.5. So the result of the new method and the result of the forward calculation can be compared. (S,U) = (8,16)is chosen because it is the cost-effective setting.
P˜0d.25(8,16)andP˜0o.25(8,16)are computed by the forward calculation and are shown in Figure 4.14.
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
−3
−2
−1 0 1 2 3 4 5
p1
p 2
. . . .
. . . .
. . . .
. . . . . . ..
. . . . . . ..
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.. . . . . . . .
.. . . . . . . .
.. . . . . . . .
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. . . . . .
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. . . . .
. . . . . α−cut of µo α−cut of µd
Figure 4.14:P0d.25(8,16)andP0o.25(8,16)by the forward calculation.
In order to simplify the computation, only the subregions within P˜0d.25(8,16) are used to compute P˜0o.25(8,16), although there is compensation inµo =√µd·µp.
Figure 4.15 shows theP0o.25(8,16)and theP˜0o.25(8,16), along with theα-cut generated by the for- ward calculation with the original LIA, which can be considered as P˜0o.25(1,1). Several test points are also drawn in Figure 4.15, where ~p× =f~0(d~×) =f~0(−0.2,0.2), ~p∗ =f~0(d~∗) =f~0(0.6,0.4), and p~+=f~0(d~+) =f~0(0.48,0.36). µo(~p×)≥µo(d~×) = 0.79201> α= 0.5 indicates the error ofP˜0o.25(8,16) in the region around ~p× which may be introduced by not including points outside P˜0.5d2(8,16). µo(d~∗) = 0.40825< α= 0.5 indicates possible error of P˜0o.25(1,1) in the region around~p∗. Andµo(d~+) = 0.40825< α= 0.5indicates possible error ofP0o.25(8,16)at its upper- right corner around~p+.
Among these three α-cuts of the overall preference atα-level 0.5, the result of the forward calculation with the original LIA,P˜0o.25(1,1), is the least accurate one. If the LIA was replaced by the extension with8subregions in each DV and16subregions in each PV, theα-cut was refined.
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
−3
−2
−1 0 1 2 3 4 5
p1
p 2
. . . .
. . . .
. . . .
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.. . . . . . . .
.. . . . . . . .
.. . . . . . . .
. . . . . .
. . . . . .
. . . . . . µo=0.79201
µo=0.40825 µo=0.4886
α−cut by the original LIA . . . . .
. . . . .
. . . . . α−cut by the Forward Calculation α−cut by the new method
Figure 4.15: P0o.25(8,16)by the new method andP0o.25(1,1)andP0o.25(8,16)by the forward calcula- tion.
But becauseµo(~p)was not calculated for~p /∈P˜0d.25(1,1),P˜0o.25(8,16)will be smaller than the actual P0o.5. For the resultingα-cut by the new methodP0o.25(8,16), there are errors around its boundary becauseS = 8andU = 16. IncreasingS andU can increase the accuracy ofP0o.25.