The Revised LIA

Computation of Preference in DVS and PVS

4.3 The Revised LIA



 p₁ p₂



=



 f₁(d₁, d₂) =d₁+d₂ f₂(d₁, d₂) =d₁+ 2·d₂



 (4.7)

Within the region [−1 .. 1]×[−1 .. 1], the mapping functions, f₁ and f₂, are single-valued and bounded, have no singularities, and are also monotonic. So the results of [p_1,min .. p_1,max]and [p₂,min .. p₂,max]using the LIA are accurate. The result is thatP^0d_αk is a Cartesian product of the intervals onp₁andp₂.

Becausef₁andf₂are both linear functions ofd₁andd₂, an alternate way to computeP_α^d_kis to map the corner points ofD^d_αk to the PVS and connect them in the same order as in the DVS. The two sets ofP_α^d_k’s forα = {ε,0.5,1.0}are shown in Figure 4.6. There are significant differences between theα-cuts generated by the different methods because of the strong dependence between f₁ and f₂. The stronger the dependence between mapping functions, the bigger the difference between the results from the two methods. The situation will become more complicated if the mapping functions are not linear, or there are other errors caused by the anomalies of the LIA or the aggregation function in DVS.

the extrema in the reduce search space of only those nonlinearf_j’s.

The linear approximation of all fj’s can also be used to relax the assumption that P_α^d_+k is a hypercube. The linear approximation is in the form as [31]:

f~⁰(d)~ = A·(d~−d~ctr) +∆~ (4.8)

=









a₁₁ · · · a_1n ... . .. ... a_q1 · · · a_qn







·









d₁−d^ctr₁ ... d_n−d^ctr_n







+









∆₁ ...

∆_n









A polyhedron can be generated to approximate the actual P_αk^d . Consider the column vectors in matrixAas the principle directions, and the extrema points found through linear approximation or optimization as the corner points of the polyhedron. The result of the extension of the LIA by principle directions off~_linear⁰ (d)~ is denoted asP_α^d]

k. For the example in section 4.2.3, the upper-right corner points are the maxima forp₁andp₂, the lower-left corner points are the maxima forp₁andp₂ because both performance functions are linear. Therefor, the actualα-cuts will be found. However, if any extremum is generated by optimization, a special adjustment is needed. The resultingα-cut by the extension of the LIA with linear approximations and optimizations on nonlinear performance functions is denoted asP_α^d\_k.

If the f₂ in Equation 4.7 is changed to make a newf~^?(d₁, d₂) as in Equation 4.9, then the linear approximation off₂^? is thef₂ in Equation 4.7, and theP_ε^d] is the same as that in Figure 4.6.

If optimization is applied on nonlinear performance functions, the extrema ofp₂ will be found as p₂,min = f₂^?(−1.0,−0.75) = −3.5andp₂,max = f₂^?(1.0,0.5) = 3.5. The extrema ofp₁ are the upper-right and the lower-left corner points in Figure 4.6. TwoP_ε^d’s from two different extensions are shown in Figure 4.7. The optimizations add two more regions toP_ε^d], and some adjustments are needed if convexity is desired. From the location of(f₁(0.8,0.8), f₂^?(0.8,0.8)), it can be seen that theP_ε^d\is more accurate thanP_ε^d], because(0.8,0.8)∈D_ε^d.



 p₁ p₂



=











f₁(d₁, d₂) =d₁+d₂

f₂^?(d₁, d₂) =













d₁−d₂+ 3.0 ifd₂ ≥0.5

d₁+ 4d₂+ 0.5 if−0.75≥d₂ >0.5 d₁−2d₂−4.0 if−0.75> d₂











(4.9)

−3 −2 −1 0 1 2 3

−4

−3

−2

−1 0 1 2 3 4

p1,min

p1,max

p2,min

p2,max

p1

p 2

Extrema (f1,f

2

*) at (0.8,0.8) Principle Directions Adjustment Line α−cut using Linear Approximation

Figure 4.7:P_ε^d]andP_ε^d\in a 2-D PVS from a 2-D DVS.

For a mapping between a two-dimensional DVS and a two-dimensional PVS, the result gen- erated by the extrema and principle directions with adjustment lines is highly accurate. However for a mapping between an n-dimensional DVS and aq-dimensional PVS, ifn > q, then P_α^d

k has q degrees of freedom and its boundary has q−1degrees of freedom. Forq ≥ 3, it is difficult to generate the boundary ofP_α^d_k from the principle directions which have only 2 degrees of freedom, and it is also difficult to add the adjustment patches for the resulting extrema from the optimizations.

Although the extension of the LIA by a linear approximation of f~(d)~ can improve the per- formance of the LIA, the resulting α-cut is not presented in a desired form. The operations on q-dimensional polyhedron are not trivial even if the difficulty of constructing the polyhedra is ig- nored, and patches are added, when some extrema are the results of optimizations. On the other hand, the results from Equation 4.4 and Equation 4.6 are less accurate but expressed in a much sim- pler form,n-cubes. Operations onn-cubes are much easier than those onn-dimensional polyhedra.

Furthermore, the errors caused by invalid assumptions about the aggregation function or the inde- pendence betweenf_j’s and the chance thatf_j’s have local extrema within DVS are all proportional to the size or volume of the DVS or PVS. Based on above the observations, an alternative way to improve the performance of the LIA by dividing the DVS and PVS into smaller regions is proposed below.

First, divide the relevant range of design variabled_iintos_isubregions by{d_i,0,· · ·, d_i,s

i}, The subregion fordi, the sub-hypercube in the DVS and its center points, are denoted by

Xi,r_i = [d_i,ri−1, d_i,ri] (4.10)

X_~r^∗ = X₁,r₁ × · · · × Xn,r_n

~c_~r = c_~r,1· · · c_~r,n

= (d_1,r1 −d_1,r1−1)/2,· · ·,(d_n,rn −d_n,rn−1)/2 where ~r= (r₁,· · ·, r_n), 1≤r_i ≤s_i, 1≤i≤n

Now each sub-hypercube will have its local design preference, which includes the effect of the aggregation function:

µ_di,~r(di) = P µd₁(c_~r,1),· · ·, µd_i(di),· · ·, µd_n(c_~r,n) (4.11)

where ~r= (r₁,· · ·, r_n), 1≤r_i ≤s_i, 1≤i≤n

Then find theα-cut of the local design preferenceµ_di,~r(d_i)in eachX~ras[d_i,r^min

i , d_i,r^max

i ], and use Equation 4.4 computeD_~r,α^d

k. The wholeα-cut in the DVS, denoted as D_α^d2_k(~s), is the union of all D_~^d_r,α

k over all sub-hypercubes in the DVS. Ifs_i =S, ∀1≤i≤n, the wholeα-cut is denoted as D^d_α²_k(S).

D_~r,α^d

k = [d^α_1,r^k

1,min, d^α_1,r^k_1,max]× · · · ×[d^α_n,r^k

n,min, d^α_n,r^k_n,max] (4.12) D_α^d2

k(~s) = ^[

∀~r

D_~^d_r,α

k

where ~r= (r₁,· · ·, r_n), ~s= (s₁,· · ·, s_n) and 1≤r_i ≤s_i, 1≤i≤n

The P_~r,α^d

k can be computed in a similar way. First, compute D_α^d2

k as described above. Then divide the relevant range of each design variable pi into ui subregions by{p_i,0,· · ·, p_i,ui}. The subregion forp_j and the sub-hypercube formed by these subregions are denoted by

Yj,t_j = [p_j,tj−1, p_j,tj] (4.13)

Y~t^∗ = Y1,t₁ × · · · × Yq,t_q

where ~t= (t₁,· · ·, tq), 1≤tj ≤uj, 1≤j≤q

Then apply the original LIA to eachD_~r,α^d

k to get theα-cut on eachpj in eachY_~t^∗as [p_j,t^αk

j,~r,min, p_j,t^αk

j,~r,max]. The endpoints of theα-cut ofµ_d(p_j)inY_~t^∗ are the union of theα-cuts in allX_~r^∗’s:

p_j,t^αk

j,min = min

∀~r p_j,t^αk

j,~r,min (4.14)

p_j,t^αk

j,max = max

∀~r p_j,t^αk

j,~r,max

where ~r = (r₁,· · ·, r_n), 1≤r_i≤s_i, 1≤i≤n and 1≤t_j ≤u_j, 1≤j ≤q

Then compute theα-cut in each sub-hypercube in the PVSP_~t,α^d

k using Equation 4.6. Theα-cut in the whole the PVS,P_α^d_k²(~s, ~u), is the union of allP_~r,α^d

k over all sub-hypercubes in PVS. And if s_i = S, ∀ 1 ≤ i ≤ nand u_j = U, ∀ 1 ≤ j ≤ q, theα-cut for the whole PVS is denoted as P_α^d2

k(S,U):

P_~t,α^d

k = [p^α_1,t^k

1,min..p^α_1,t^k_1,max]× · · · ×[p^α_q,t^k

q,min..p^α_q,t^k_q,max] (4.15) P_α^d2

k(~s, ~u) = ^[

∀~t

P_~t,α^d

k

where ~t= (t₁,· · ·, t_q), ~u= (u₁,· · ·, u_q) 1≤t_j ≤u_j, 1≤j≤q

and ~s= (s₁,· · ·, s_n),

This extension of the LIA will be demonstrated on the examples in Section 4.2.2 and in Sec- tion 4.2.3. For the problem of µ_d(d) =~ P(µ_d1, µ_d2), the support of each design variable is divided equally into 10 subregions, and the interval of each design variable is computed from µ_di,~r(d_i) in each sub-rectangle in the DVS. Then theα-cut is generated by Equation 4.13. The D^d_0.²₅(10)’s for4different aggregation functions are shown in Figure 4.8 with the actualα-cuts. For P = min(d₁, d₂), the result from the LIA with the hypercube assumption is correct, so is the result from the extended LIA. ForP = max(d₁, d₂), the result from the extended LIA is the same as the actual α-cut because the boundaries of the actual α-cut are parallel to the boundaries of the sub-rectangle. ForP = (d₁+d₂)/2andP =√

d₁·d₂, the results from the extended LIA are not the same as the actualα-cuts because now the boundaries of the actual α-cuts are not parallel to the boundaries of the sub-rectangle. But the the results from the extended LIA approximate the actual α-cuts with good accuracy.

For the problem with multiple performance variables, the same dividing method is applied to the modifiedf~^?(d₁, d₂)as the one used in the beginning of this section to demonstrate the extension of the LIA with linear approximations and optimizations. In short, the result of the new extension of the LIA withSdesign variable subregions andU performance variable subregions is denoted as P_α^d2

k(S,U). Also in order to show the effects of the changes inSandU, results from different values ofS andU are shown in Figure 4.9. Because of the anticipated complexity of thef~^?(d₁, d₂),U is chosen as2S. The mark “×” in each figure isp~_×=f~^?(−0.88,0.48)and it can be seen~p_×∈/ P_α^d\

k

−1 −0.5 0 0.5 1

−1

−0.5 0 0.5 1

d1

d2

µ = min(µ_d1,µ_d2)

−1 −0.5 0 0.5 1

−1

−0.5 0 0.5 1

d1

d2

µ = max(µ_d1,µ_d2)

−1 −0.5 0 0.5 1

−1

−0.5 0 0.5 1

d1

d2

µ = (µ_d1+µ_d2)/2

−1 −0.5 0 0.5 1

−1

−0.5 0 0.5 1

d1

d2

µ = (µ_d1 * µ_d2)^1/2

Figure 4.8:D₀^d_.5(10)’s by different aggregation functions.

−2 −1 0 1 2

−3

−2

−1 0 1 2 3

p1

p 2

S = 2, U = 4

−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

Figure 4.9:P_ε^d2(S,U)with different values ofT andU.

butp~_×∈P_α^d_k²(S,2S), S = 2,4,8,16. IfP_α^d2_k(16,32)is considered as the actualα-cut, it can be seen that evenP_α^d_k²(2,4)is much better thatP_α^d\_k, andP_α^d_k²(4,8) already has acceptable accuracy, and with almost no big difference betweenP_α^d2

k(8,16)andP_α^d2

k(16,32).

P_α^d_k²(S,U) is better than P_α^d]_k and P_α^d\_k with respect to flexibility of aggregation function and dimensionality of the PVS. The anomaly thatP_α^d2

k ⊂ P_α^d2

l whereα_k < α_l may still be triggered by the nonmonotonicity off~(d)~ although the chances can be reduced by dividing the DVS and the PVS. To solve this problem, the following operation for multi-α-cuts withkincreasing from1to M−1can be added:

P_α^d_k+1² = P_α^d_k+1^{2 [}P_α^d_k², ∀ 1≤k≤M−1 (4.16) where α_k < α_l, ∀ 1≤k < l≤M

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.