multiple criteria optimization: some introductory...
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Multiple Criteria Optimization:Some Introductory Topics
Ralph E. SteuerDepartment of Banking & Finance
University of GeorgiaAthens, Georgia 30602-6253 USA
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Tchebycheff contour
probing direction
feasible region in criterion space
1 1max{ (x) }
max{ (x) }x
k k
f = z
f = zs .t . S∈
M
Z
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Production planningmin { cost }min { fuel consumption }min { production in a given geographical area }
River basin management
achieve { BOD standards }min { nitrate standards }min { pollution removal costs } achieve { municipal water demands }min { groundwater pumping }
Oil refiningmin { cost }min { imported crude }min { environmental pollution } min { deviations from demand slate }
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Sausage blending
min { cost }max { protein }min { fat } min { deviations from moisture target }
Portfolio selection in finance
min { variance }max { expected return }max { dividends }max { liquidity }max { social responsibility }
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Discrete Alternative Methods
Multiple Criteria Optimization
1 1max{ (x) }
max{ (x) }x
k k
f = z
f = zs .t . S∈
M
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1. Decision Space vs. Criterion Space2. Contenders for Optimality3. Criterion and Semi-Positive Polar Cones4. Graphical Detection of the Efficient Set5. Graphical Detection of the Nondominated Set6. Nondominated Set Detection with Min and Max
Objectives7. Image/Inverse Image Relationship and Collapsing8. Unsupported Nondominated Criterion Vectors
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1 1max{ (x) }
max{ (x) }x
k k
f = z
f = zs .t . S∈
M
11max{ c x }
max{ c x }x
kk
= z
= zs .t . S∈
M
But if all objectives and constraints are linear, we write
in which case we have a multiple objective linear program (MOLP).
In the general case, we write
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1 2 1
1 2 2
max{ }max{ 2 }. . x
x x zx x z
s t S
− =− + =
∈
z1
z2
Z
z1
z2z3
z4 = (3, -2)
= (0, 3)= (-1, 3)
1. Decision Space vs. Criterion Space
S
x1
x2
c1
c2
x1
x2
x3
x4 = (4, 1)
= (3, 3)
= (1, 2)
criterion objective outcome attribute evaluation image
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S
x1
x2
Morphing of S into Z as we change coordinate system
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z1
z2
Z
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2. Contenders for Optimality
Points (criterion vectors) in criterion space are either nondominated or dominated.
Their points in decision space are either efficient or inefficient.
We are interested in nondominated criterion vectors and their efficient points because only they are contenders for optimality.
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3. Criterion and Nonnegative Polar Cones
Criterion cone -- convex cone generated by the gradients of the objective functions.
S
x1
x2
c1
c2
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S
x1
x2
c1
c2
The larger the criterion cone (i.e., the more conflict there is in the problem), the bigger the efficient set.
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S
x1
x2
c1
c2
Nonnegative polar of the criterion cone -- set of vectors that make 90o or less angle with all objective function gradients. In the case of an MOLP, given by
{y : c y 0, 1, , }n iR i = k∈ ≥ K
Contains all points that dominate its vertex.
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4. Graphical Detection of the Efficient Set
Example 1
observe the criterion cone.
Sx1
x2
c1
c2
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form nonnegative polar cone
Sx1
x2
c1
c2
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Sx1
x2
c1
c2
x1
x2
x3
move it around
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Sx1
x2
c1
c2
x1
x2
x3
1 2 2 3
1 2 3x
efficient set E = [x x ] [x x ]set of efficient extreme points E = {x , x , x }
, ,≡ γ ∪ γ
≡
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Sx1
x2
c1
c2
x1
x2
x3
1 2 2 3
1 2 3x
efficient set E = [x x ] [x x ]set of efficient extreme points E = {x , x , x }
, ,≡ γ ∪ γ
≡
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Sx1
x2
c1
c2
x1
x2
x3
1 2 2 3
1 2 3x
efficient set E = [x x ] [x x ]set of efficient extreme points E = {x , x , x }
, ,≡ γ ∪ γ
≡
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Sx1
x2
c1
c2
x1
x2
x3
Only when there is no intersection at other than the vertex of the cone
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S
x1
x2
c1
c2
Graphical Detection of the Efficient Set
Example 2
observe criterion cone
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S
x1
x2
c1
c2
form nonnegative polar cone
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1 2 3 4 5E = [x x ) {x } (x x ], ,∂ ∪ ∪ ∂
S
x1
x2
c1
c2
x1
x2
x3
x4
x5
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1 2 3 4 5E = [x x ) {x } (x x ], ,∂ ∪ ∪ ∂
S
x1
x2
c1
c2
x1
x2
x3
x4
x5
(observe that x2 and x4 are not efficient)
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Graphical Detection of the Efficient Set
Example 3
x1
x2
x1
x2
x3
x4
x5
x6
c1
c2
Note small size of criterion cone and that S consists of only 6 points.
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x1
x2
x1
x2
x3
x4
x5
x6
c1
c2
Small criterion cone results in a large nonnegative polar cone.
(this makes it harder for points to be efficient).
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6E = {x }Moving nonnegative polar cone around.
x1
x2
x1
x2
x3
x4
x5
x6
c1
c2
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z1
z2
Z
5. Graphical Detection of the Nondominated Set
To determine if a criterion vector in Z is nondominated, translate nonnegative orthant in Rk to the point.
move nonnegative orthant around
max max
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try to identify the entire nondominated set
z1
z2
Z
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z1
z2
Z
z1
z2
z3
1 2 2 3nondominated set N = [z z ] [z z ], ,γ≡ ∂ ∪
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z1
z2
Z
Now, move nonnegative orthant around.
max max
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z1
z2
Z
z1
z2 z3
z4
z5
z6
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z1
z2
Z
z1
z2 z3
z4
z5
z6
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z1
z2
Z
z1
z2 z3
z4
z5
z6
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z1
z2
Z
z1
z2 z3
z4
z5
z6
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z1
z2
Z
z1
z2 z3
z4
z5
z6
1 2 3 4 5 6N = [z z ) [z z ] (z z ], , ,∂ ∪ ∂ ∪ ∂
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z1
z2
6. Nondominated Set Detection with Min and Max Objectives
max max
Z
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z1
z2
Z
max max
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z1
z2
Z
max max
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z1
z2
Z
max max
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z1
z2
Z
max max
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z1
z2 min max
Z
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z1
z2
Z
min max
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z1
z2
Z
min max
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z1
z2
Z
min max
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z1
z2min min
Z
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z1
z2
Z
min min
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z1
z2 max min
Z
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z1
z2
Z
max min
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Let z . Then z is if and only if there doesnot exist another z Z such that for all and for
at least one Otherwise z is .
Let x . Then x is if and
i i j j
Z nondominatedz z i z z
j . , dominated
S efficient
∈∈ ≥ >
∈ only its criterion vector zis nondominated. Otherwise x is ., inefficient
In other words,
image of an efficient point is a nondominated criterion vector inverse image of a nondominated criterion vector is an eff point
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1 2 3 1
1 2 3 2
max{3 2 }max{ }. . x unit cube
x x x zx x x z
s t S
+ − =− + + =∈ =
7. Image/Inverse Image Relationship and Collapsing
dimensionality of S is n, but dimensionality of Z is k.
x1x5
x2 x6
x4 x8
x3 x7
x1
x2
x3
c1
c2
= (1, 1, 0)
S
z4
z1
z2
z1
z3
z6
z2
z5
z7
z8
2
Z = (2, 1)
= (4, 0)
= (-2, 1)
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8. Unsupported Nondominated Criterion Vectors
A nondominated criterion vector is supported or unsupported.
Unsupported if dominated by a convex combination of other feasible criterion vectors.
Unsupported nondominated criterion vectors are typically hard to compute.
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1 2 1
1 2 2
max{ 9 }max{ 3 8 }. . x
x x zx x z
s t S
− + =− =∈
z1
z2x2
x115
c2
c1
2
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1 2 1
1 2 2
max{ 9 }max{ 3 8 }. . x
x x zx x z
s t S
− + =− =∈
z1
z2x2
x115
c2
c1
2
-15
x1 x2 x3
z3
z2
z1
x4
x5
z4
z5
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1 2 1
1 2 2
max{ 9 }max{ 3 8 }. . x
x x zx x z
s t S
− + =− =∈
supp 1 2 3 4 5
unsupp supp
N = N {z , z , z , z , z }
N N
Z
Z=
= −
z1
z2x2
x115
c2
c1
2
-15
x1 x2 x3
z3
z2
z1
x4
x5
z4
z5
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z1
z2
Z z1
z2
supp 1
unsupp 2
N {z }N {z }
=
=
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Multiple Criteria Optimization:An Introduction (Continued)
Ralph E. SteuerDepartment of Banking & Finance
University of GeorgiaAthens, Georgia 30602-6253 USA
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Recall9. Ideal way?10. Contours, Upper Level Sets and Quasiconcavity11. More-Is-Always-Better-Than-Less vs. Quasiconcavity12. ADBASE13. Size of the Nondominated Set14. Criterion Value Ranges over Nondominated Set15. Nadir Criterion Values16. Payoff Tables17. Filtering18. Stamp/Coin Example19. Weighted-Sums Method20. e-Constraint Method
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1 2 1
1 2 2
max{ }max{ 2 }. . x
x x zx x z
s t S
− =− + =
∈
z1
z2
Z
z1
z2z3
z4 = (3, -2)
= (0, 3)= (-1, 3)
S
x1
x2
c1
c2
x1
x2
x3
x4 = (4, 1)
= (3, 3)
= (1, 2)
Recall
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Z
zo
z1
z2
1max{ ( , , )}. . (x) 1, ,
x
k
i i
U z zs t f z i k
S= =∈
K
K
Assess a decision maker’s utility function UUUUUUaand solve: kU R R→
9. Ideal Way?
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Maybe not good for four reasons.
1. Difficulty in assessing U
2. U is almost certainly nonlinear
3. Generates only one solution
4. Does not allow for learning
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A U is quasiconcave if all upper level sets are convex.
10. Contours, Upper Level Sets, and Quasiconcavity
1 2 3 4 5 6 7 8
U
z1
z2
z1
6
4
2
8
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1 2 3 4 5 6 7 8
U
z1
z2
z1 z2
z2z1
6
4
2
8
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1 2 3 4 5 6 7 8
U
z1
z2
z1 z2
z2z1
6
4
2
8
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1 2 3 4 5 6 7 8
U
z1
z2
z1 z2
z2z1
6
4
2
8
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1 2 3 4 5 6 7 8
U
z1
z2
z1 z2
z2z1
6
4
2
8
Quasiconcave functions have at most one top.
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11. More-Is-Always-Better-Than-Less (i.e, Coordinate-wise increasing) vs. Quasiconcavity
1 3 5 7 9
U
z1
z2
6
4
2
8
Z
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More-is-always-better-than-less does not imply that all local optima are global optima
1 3 5 7 9
z1
z2
Zz1
z2
z1 is a local optimum, but z2 is the global optimum.
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1 3 5 7 9
U
z1
z2
6
4
2
8
Z
z3
z4
z4z3
More-is-always-better-than-less does not imply quasiconcavity
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1 3 5 7 9
U
z1
z2
6
4
2
8
Z
z3
z4
z4z3
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1 3 5 7 9
U
z1
z2
6
4
2
8
Z
z3
z4
z4z3
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• Nondominated set N -- set of all potentially optimal criterion vectors.
• Efficient set E -- set of all potentially optimal solutions.
Assuming that U is coordinate-wise increasing:
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12. ADBASE
In an MOLP, of course, efficient set is a portion of the surface of S, and nondominated set is a portion of the surface of Z.
ADBASE is for MOLPs. It computes all of the extreme points of S that efficient, and hence all of the vertices of Z that are nondominated in an MOLP.
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13. Size of the Efficient and Nondominated Sets
MOLP ave efficient aveproblem size extreme pts CPU time3 x 100 x 150 13,415 363 x 250 x 375 285,693 5,5734 x 50 x 75 19,921 245 x 35 x 45 15,484 145 x 60 x 90 414,418 1,223
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14. Criterion Value Ranges over the Nondominated Set
Obj1 Obj2 Obj3 Obj4 Obj5
100
0
50
The lower bounds on the ranges are called nadir criterion values.
If know nondominated set ahead of time, can “warm up” decision maker with following information.
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Obj1 Obj2 Obj3 Obj4 Obj5
100
0
50
max
nad
(110, 90,100, 50, 40)estimated ( 30, 10, 60, 20, 20)
=
= − −
zz
15. Nadir Criterion Values
If don’t know nondominated set ahead of time, true nadir criterion vector can be difficult to obtain.
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Obj1 Obj2 Obj3
15
-15
0
16. Payoff TableObtained by individually maximizing each objective over S. But minimum column values often over-estimate nadir values.
x1
x2
c2
c3
c1
S
= (3, 0)
= (-1, 3)
= (-1, -3)
x1
x2
x3
x4
15 1 -11
12 8 -16
12 -4 -4
z1
z2
z3
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Obj1 Obj2 Obj3
10
-15
0
In this problem E = S. True nadir value for Obj1 is 0 not 12.
x1
x2
c2
c3
c1
S
= (3, 0)
= (-1, 3)
= (-1, -3)
x1
x2
x3
x4
15 1 -11
12 8 -16
12 -4 -4
z1
z2
z3
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The larger the problem, the greater the likelihood that the payoff table column minimum values will be wrong.
After about 5 x 20 x 30, most will be wrong.
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17. Filtering
Reducing 8 vectors down to a dispersed subset of size 5
z1
z2
z3
z4
z5
z6
z7
z8
z1
z2
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First point z1
always retained by filter.
z1
z2
z3
z4
z5
z6
z7
z8
z1
z2
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z2
retained by filter, but z3 and z5 discarded.
z1
z2
z3
z4
z5
z6
z7
z8
z1
z2
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z4
retained by filter, but z8
discarded.
z1
z2
z3
z4
z5
z6
z7
z8
z1
z2
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z6
retained by filter, but z7
discarded.
z1
z2
z3
z4
z5
z6
z7
z8
z1
z2
Wanted 5 but got 4. Reduce neighborhood, then do again. After a number of iterations, will converge to desired size.
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Z
z1
z2 (Coins)
(Stamps)
z1
z2
z3
18. Stamp/Coin Example
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Z
z1
z2 (Coins)
(Stamps)
z1
z2
z3
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19. Weighted-Sums Method
1. decision-maker’s preferences.
2. scale in which the objectives are measured (e.g., cubic feet versus board feet of lumber).
3. shape of the feasible region
But how to pick the weights because they are a function of
max{ Cx}. . x
T
s t Sλ∈
11max{ c x }
max{ c x }. . x
kk
z
zs t S
=
=∈
M
May also get flip-flopping behavior.
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Purpose of weighted-sums approach is to obtain information from the DM to create a λ-vector that causes “composite gradient” λTC in the weighted-sums program
to point in the same direction as the utility function gradient.
max{ Cx}. . x
T
s t Sλ∈
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Boss says to go with 50/50 weights.
2.
x1
x2
S
c1
c2
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Boss likes resulting solution and is proud his 50/50 weights. Then asks that second objective be changed from cubic feet to board feet of timber production.
x1
x2
S
c1
c2
x1
λTC
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With 50/50 weights, this causes composite gradient to point in a different direction. Get a completely different solution.
x1
x2
S
c1
c2
x1
x2
λTC
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3.
Boss says use 60/40 weights.
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Then a constraint needs to be changed slightly. Get a completely different solution.
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z1
z2
Z
z1
z2
z3
Utility function is quasiconcave. Assuming we get perfect information from DM, weighted-sums method will iterate forever!
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11max{ c x }
max{ c x }. . x
kk
z
zs t S
=
=∈
M
20. e-Constraint Method
max{ c x }
. . c xx
jj
ii
z
s t e i jS
=
≥ ≠∈
Basically trial-and-error
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max{ c x }
. . c x
c xx
pp
mm
ee
z
s t e
eS
=
≥
≥∈
x1
cmce
cp
x2eeem
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22. Overall Interactive Algorithmic Structure23. Vector-Maximum/Filtering24. Goal Programming25. Lp-Metrics26. Weighted Lp-Metrics27. Reference Criterion Vector28. Wierzbicki’s Aspiration Criterion Vector Method29. Lexicographic Tchebycheff Sampling Program30. Tchebycheff Procedure (overview)31. Tchebycheff Procedure (in more detail)32. Tchebycheff Vertex λ-Vector33. How to Compute Dispersed Probing Rays34. Projected Line Search Method35. List of Interactive Procedures
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set controlling parameters for the 1st iteration
solve optimizationproblem(s)
examine criterion vector results
done
reset controlling parametersfor the next iteration
stopy
start
Controlling Parameters:
weighting vector ei RHS values aspiration vector others
22. Overall Interactive Algorithmic Structure
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23. Vector-Maximum/Filtering
Let number of solutions shown be 8, convergence rate be 1/6.
Solve an MOLP for, say 66,000, nondominated extreme points.
Filter to obtain the 8 most different among the 66,000.
• Decision maker selects z(1), the most preferred of the 8.
Filter to obtain the 8 most different among the 11,000 closest to z(1).
• Decision maker selects z(2) , the most preferred of the new 8.
Filter to obtain the 8 most different among the 1,833 closets to z(2).
• Decision maker selects z(3) , the most preferred of the new 8.
And so forth.
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24. Goal Programming
11
22
33
max{ c x }
achieve{ c x }
min{ c x }. . x
z
z
zs t S
=
=
=∈
1 1 2 2 2 2 3 31
1 12
2 2 23
3 3
min{ }
. . c x
c x
c xx
all , 0i i
w d w d w d w d
s t d t
d d t
d tS
d d
− − − − + + + +
−
− +
+
− +
+ + +
+ ≥
+ − =
+ ≤∈
≥
Must choose a target vector and then select deviational variable weights.
Goal programming uses weighted L1 -metric.
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z1
z2
Z
tt2
t1
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z1
z2
Z
tt2
t1
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z1
z2
Z
tt2
t1
z1
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z1
z2
Z
tt2
t1
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z1
z2
Z
tt2
t1
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z1
z2
Z
tt2
t1
z1
z2
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25. Lp-Metrics
**z
{ }
1
**1**
**
1
1, 2,
max
ppki ii
p
i ii k
z z pz z
z z p
=
≤ ≤
⎧ ⎡ ⎤− =⎪⎪ ⎢ ⎥⎣ ⎦− = ⎨⎪ − = ∞⎪⎩
∑ K
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{ }
1
**1**
**
1
1, 2,
max
ppki ii
p
i ii k
z z pz z
z z p
=
≤ ≤
⎧ ⎡ ⎤− =⎪⎪ ⎢ ⎥⎣ ⎦− = ⎨⎪ − = ∞⎪⎩
∑ K
**z
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{ }
1
**1**
**
1
1, 2,
max
ppki ii
p
i ii k
z z pz z
z z p
=
≤ ≤
⎧ ⎡ ⎤− =⎪⎪ ⎢ ⎥⎣ ⎦− = ⎨⎪ − = ∞⎪⎩
∑ K
**z
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26. Weighted Lp-Metrics
( ){ }
1
**1**
**
1
1, 2,
max
ppki i ii
p
i i ii k
z z pz z
z z p
λ λ
λ
=
≤ ≤
⎧ ⎡ ⎤− =⎪ ⎢ ⎥⎪ ⎣ ⎦− = ⎨⎪ − = ∞⎪⎩
∑ K
**z
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( ){ }
1
**1**
**
1
1, 2,
max
ppki i ii
p
i i ii k
z z pz z
z z p
λ λ
λ
=
≤ ≤
⎧ ⎡ ⎤− =⎪ ⎢ ⎥⎪ ⎣ ⎦− = ⎨⎪ − = ∞⎪⎩
∑ K
**z
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( ){ }
1
**1**
**
1
1, 2,
max
ppki i ii
p
i i ii k
z z pz z
z z p
λ λ
λ
=
≤ ≤
⎧ ⎡ ⎤− =⎪ ⎢ ⎥⎪ ⎣ ⎦− = ⎨⎪ − = ∞⎪⎩
∑ K
**z
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( ){ }
1
**1**
**
1
1, 2,
max
ppki i ii
p
i i ii k
z z pz z
z z p
λ λ
λ
=
≤ ≤
⎧ ⎡ ⎤− =⎪ ⎢ ⎥⎪ ⎣ ⎦− = ⎨⎪ − = ∞⎪⎩
∑ K
**z
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z1
z2
Z
3
27. Reference Criterion Vector
Constructed so as to dominate every point in the nondominated set
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z1
z2
zref
Z
3
= (5, 4)
usually good enough to round to next largest integer
ref maxi i iz z ε= +
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Z
z1
z2
28. Wierzbicki’s Reference Point Procedure
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Z
z1
z2 zref
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Z
z1
z2 zref
q(1)
First iteration
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Z
z1
z2 zref
q(1)
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Z
z1
z2 zref
q(1)
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Z
z1
z2 zref
q(1)
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Z
z1
z2 zref
q(1)
z(1)
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Z
z1
z2 zref
q(2)
Second iteration
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Z
z1
z2 zref
q(2)
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Z
z1
z2 zref
q(2)
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Z
z1
z2 zref
q(2)
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Z
z1
z2 zref
q(2)
z(2)
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Third iteration
Z
z1
z2 zref
q(3)
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Z
z1
z2 zref
q(3)
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Z
z1
z2 zref
q(3)
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Z
z1
z2 zref
q(3)
z(3)
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1lex min{ , }
. . ( ) 1, ,(x) 1, ,
x
kii
refi i i
i i
z
s t z z i kf z i k
S
α
α λ=
−
≥ − == =∈
∑K
K
Geometry carried out by lexicographic Tchebycheff sampling program
Minimizing α
causes non-negative orthant contour to slide up the probing ray until it last touches the feasible region Z.
1 1, ,1 1i kλ λ⎛ ⎞
−⎜ ⎟⎝ ⎠
K
Direction of the probing ray emanating from zref is given by
Perturbation term 1
kii
z=
−∑ is there to break ties.
29. Lexicographic Tchebycheff Sampling Program
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z1
z2zref
Z
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z1
z2zref
Z
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z1
z2zref
Z
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z1
z2zref
Z
z1
z2
Two lexicographic minimum solutions, but both nondominated
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30. Tchebycheff Method (Overview)
zref
z1
z2
Z
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First iteration
zref
z1
z2
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zref
z1
z2
z(1)
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Second iteration
zref
z1
z2
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zref
z1
z2
z(2)
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Third iteration
zref
z1
z2
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zref
z1
z2
z(3)
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set controlling parameters for the 1st iteration
solve optimizationproblem(s)
examine criterion vector results
done
reset controlling parametersfor the next iteration
stopy
start
Controlling Parameters:
target vector, weights q(i) aspiration vectors λi multipliers
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31. Tchebycheff Method (in more detail)
Let P = number of solutions to be presented to the DM at each iteration = 4
Let r = reduction factor = 0.5
Let t = number of iterations = 4
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Now, form reference criterion vector zref.
z1
z2
Z
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z1
z2
Z
zref
Now, form Λ(1) and obtain 4 dispersed λ-vectors from it.
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Now, solve four lexicographic Tchebycheff sampling programs (one for each probing ray).
z1
z2
Z
zref
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Now, select most preferred, designating it z(1).
z1
z2
Z
zref
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Now, form Λ(2) and obtain 4 dispersed λ-vectors from it.
z1
z2
Z
zref
z(1)
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32. Tchebycheff Vertex λ-Vector
zref
z(1)
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33. How to Compute Dispersed Probing Rays
zref
z(1)
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1
(1)(1) (1)
1
1 1k
i ref refji i j jz z z z
λ−
=
⎡ ⎤= ⎢ ⎥
− −⎢ ⎥⎣ ⎦∑
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Now, solve four lexicographic Tchebycheff sampling programs.
z1
z2
Z
zref
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Now, select most preferred, designating it z(2).
z1
z2
Z
zref
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Now, form Λ(3) and obtain 4 dispersed λ-vectors from it.
z1
z2
Z
zref
z(2)
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Now, solve four lexicographic Tchebycheff sampling programs.
z1
z2
Z
zref
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Now, select most preferred, designating it z(3)
z1
z2
Z
zref
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Now, form Λ(4) and obtain 4 dispersed λ-vectors from it.
z1
z2
Z
zref
z(3)
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And so forth.
z1
z2
Z
zref
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34. Projected Line Search Method
z1
z2
z(1)
Like driving across surface of moon.
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z1
z2
z(1)
q(2)
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z1
z2
z(1)
q(2)
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z1
z2
z(1)
z(2)
q(2)
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z1
z2
z(1)
z(2)
q(3)
q(3)
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z1
z2
z(1)
z(2)
q(3)
z(3)
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z1
z2
z(1)
z(2)
z(3)
Drive straight awhile, turn, drive straight awhile, turn, drive straight awhile, and so forth.
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1. Weighted-sums (traditional)
2. e-constraint method (traditional)
3. Goal programming (mostly US, 1960s)
4. STEM (France & Russia, 1971)
5. Geoffrion, Dyer, Feinberg procedure (US, 1972)
6. Vector-maximum/filtering (US, 1976)
7. Zionts-Wallenius Procedure (US & Finland, 1976)
8. Wierzbicki’s reference point method (Poland, 1980)
9. Tchebycheff method (US & Canada, 1983)
10. Satisficing tradeoff method (Japan, 1984)
11. Pareto Race (Finland, 1986)
12. AIM (US & South Africa, 1995)
13. NIMBUS (Finland, 1998)
35. List of Interactive Procedures
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The End