Dagstuhl Seminar. Practical Approaches to Multi-Objective Optimization. Dec 10-15, 2006.
Book Project on Multicriteria Optimization
8) MO Procedures with Progressive Elicitation of Preferences
A. P. Wierzbicki, K. Miettinen, F. Ruiz
Trade-off based Methods
Dagstuhl Seminar. Practical Approaches to Multi-Objective Optimization. Dec 10-15, 2006.
Trade-offs based methods
The concept of trade-off
•Objective: The change in one objective in relation to the change
in another one, when moving from one feasible (efficient)
solution to another one.
•Subjective: How much of one objective does the DM consider
desirable to sacrifice in order to improve another objective to a
certain quantity.
Dagstuhl Seminar. Practical Approaches to Multi-Objective Optimization. Dec 10-15, 2006.
Trade-offs based methods
Definitions of trade-off
Objective
Subjective
Ratio of
Change (i / j)
(total/ partial)
tradeoff
Indifference
trade-off
Limit
(t/p) trade-off
rate
tij(x*)
Indifference
trade-off rate
(MRS), mij(x*)
Dagstuhl Seminar. Practical Approaches to Multi-Objective Optimization. Dec 10-15, 2006.
Trade-offs based methods
Graphical interpretation of the trade-off rates: the slope
of the efficient frontier at the current iteration x*, f*
f(S)
f*
N
N  ( t i1 ( x *),  , t ii 1 ( x *),  1, t ii 1 ( x *),  , t ik ( x *))
Dagstuhl Seminar. Practical Approaches to Multi-Objective Optimization. Dec 10-15, 2006.
Trade-offs based methods
Graphical interpretation of the MRS: the slope of the
indifference curve at the current iteration x*, f*
(f) = *
f(S)
f*
N’
N
N '  (  m i1 ( x *),  ,  m ii 1 ( x *),  1,  m ii 1 ( x *),  ,  m ik ( x *))
Dagstuhl Seminar. Practical Approaches to Multi-Objective Optimization. Dec 10-15, 2006.
Trade-offs based methods
How are the trade-offs used in interactive methods?
Almost all the trade-off based interactive methods assume the
existence of a value function of the DM, .
•Type 1: The method calculates objective (efficient) trade-offs at
the current iteration and the DM is asked to analyze them
somehow. The next iteration is built on the base of this
information. The stopping criterion is achieved if all trade-offs at
the current solution are indifferent (or less preferred) for the DM.
•Type 2: The DM is asked to give the MRS at the current
iteration. This information is used to search increasing directions
for . The stopping criterion is achieved when the MRS coincide
with the objective trade-offs at the current solution.
Dagstuhl Seminar. Practical Approaches to Multi-Objective Optimization. Dec 10-15, 2006.
Trade-offs based methods
Graphical interpretation of the stopping criterion: the
efficient frontier and the indifference curve are tangent.
f*
f(S)
N = N’
t ij ( x *)   m ik ( x *)
ji
Dagstuhl Seminar. Practical Approaches to Multi-Objective Optimization. Dec 10-15, 2006.
Trade-offs based methods
How to determine objective trade-offs over the
efficient frontier?
•Linear case: The trade-offs are given by the shadow prices of the
simplex tableau. Zionts and Wallenius (1976) propose a method
to determine the efficient adjacent vertices.
Dagstuhl Seminar. Practical Approaches to Multi-Objective Optimization. Dec 10-15, 2006.
Trade-offs based methods
How to determine objective trade-offs over the
efficient frontier?
•Nonlinear case (2nd order conditions hold): Given a reference
function fi, and a vector of bounds , we solve
Pi ( )
 min

 s.t.


fi (x)
f j (x)   j
ji
xS
If ij* are the optimal Kuhn-Tucker multipliers with respect to ,
then – ij* is a total or partial tradeoff rate (depending on whether
there exist 0 multipliers or not) between fi and fj.
N  (   ,  , 
*
i1
*
ii 1
,  1,  
*
ii  1
,  ,  )
*
ik
Dagstuhl Seminar. Practical Approaches to Multi-Objective Optimization. Dec 10-15, 2006.
Trade-offs based methods
How to determine objective trade-offs over the
efficient frontier?
•Nonlinear case (no 2nd order conditions): We solve
 min

 s.t.



 i ( fi (x)  zi )  
*
i  1,  , k
xS
where i = 1/(fih – zi*), z* is the ideal vector, and fh is the current
iteration. Let i* be the optimal multiplier corresponding to .
Then the normal vector is:
N  (   1 ,  ,   k  )
*
1
*
k
Dagstuhl Seminar. Practical Approaches to Multi-Objective Optimization. Dec 10-15, 2006.
Type 1 methods
The Zionts-Wallenius method (Zionts and Wallenius, 1976)
The method is based on linear approximations. We consider
 min

 s.t.


T
T
(c1 x ,  , c k x )
Ax  b
x0
and the value function  ( f 1 ,  f k )  
k

i
fi
i 1
Given a guess h, the corresponding value function is
maximized over the feasible set.
Dagstuhl Seminar. Practical Approaches to Multi-Objective Optimization. Dec 10-15, 2006.
Type 1 methods
The Zionts-Wallenius method (Zionts and Wallenius, 1976)
STEPS
1.
Give an initial guess 1. Make h = 1.
2.
Maximize the value function over the feasible set. Let xh be the optimal
solution.
3.
Determine all the adjacent efficient vertices not visited yet.
4.
For each one, determine the corresponding trade-off.
5.
Ask the DM about the desirability of all the trade-offs (“yes”, “no” or
“indifferent”).
6.
If all the answers are “no” or “indifferent”, stop. Else, find a new vector of
weights h+1, such that  is increasing in the direction of the preferred tradeoffs, decreasing in the direction of the non preferred ones, and constant in the
direction of the indifferent ones. Make h = h+1. Go to step 2.
Dagstuhl Seminar. Practical Approaches to Multi-Objective Optimization. Dec 10-15, 2006.
Type 1 methods
The Interactive Surrogate Worth Trade-off (ISWT)
method (Chancong and Haimes, 1978)
ASSUMPTIONS
•
The implicit value function is assumed to be continuously differentiable and
monotonically decreasing.
•
The objective and the constraint functions are twice continuously
differentiable.
•
The feasible region is compact.
•
Regularity and second order conditions hold at the efficient solutions.
Dagstuhl Seminar. Practical Approaches to Multi-Objective Optimization. Dec 10-15, 2006.
Type 1 methods
The ISWT method (Chancong and Haimes, 1978)
STEPS
1.
Choose a reference function fi, and give a initial guess for 1 . Make h = 1.
2.
Solve Pi(h), and denote by xh its optimal solution. Determine the trade-offs
making use of the optimal Kuhn-Tucker multipliers.
3.
For each j  i, the DM is asked to rank the corresponding trade-off in a scale
from – 10 to 10 (surrogate worth).
4.
If a stopping criterion is satisfied, stop.
5.
Update h+1. Make h = h+1. Go to step 2.
Dagstuhl Seminar. Practical Approaches to Multi-Objective Optimization. Dec 10-15, 2006.
Type 2 methods
The Geoffrion, Dyer and Feinberg (GDF) method
(Geoffrion, Dyer and Feinberg, 1972)
ASSUMPTIONS
•
The implicit value function is assumed to be continuously differentiable,
monotonically decreasing and concave.
•
The objective functions are continuously differentiable and convex.
•
The feasible region is compact and convex.
Dagstuhl Seminar. Practical Approaches to Multi-Objective Optimization. Dec 10-15, 2006.
Type 2 methods
The GDF method (Geoffrion, Dyer and Feinberg, 1972)
STEPS
1.
Choose an initial feasible point x1. Make h = 1.
2.
Ask the DM to give the MRS at xh.
3.
Solve a search direction problem for the value function. Denote by yh the
corresponding optimal solution.
4.
If yh = xh, stop. Else, make dh = yh – xh.
5.
Calculate the optimal step-length th.
6.
Make xh+1 = xh + th dh. Make h = h+1. Go to step 2.
Dagstuhl Seminar. Practical Approaches to Multi-Objective Optimization. Dec 10-15, 2006.
Type 2 methods
The Sequential Proxy Optimization Technique
(SPOT) (Sakawa, 1982)
ASSUMPTIONS
•
The implicit value function is assumed to be continuously differentiable,
monotonically decreasing and concave.
•
The objective functions are twice continuously differentiable and convex.
•
The feasible region is compact and convex.
•
Regularity conditions hold at the efficient solutions.
Dagstuhl Seminar. Practical Approaches to Multi-Objective Optimization. Dec 10-15, 2006.
Type 2 methods
The SPOT method (Sakawa, 1982)
STEPS
1.
Choose a reference function fi, and give a initial guess for 1 . Make h = 1.
2.
Solve Pi(h), and denote by xh its optimal solution. Determine the trade-offs
making use of the optimal Kuhn-Tucker multipliers, ijh.
3.
The DM gives the MRS at xh, mijh.
4.
Check the consistency of the MRS. If they are not consistent, go to step 3.
5.
If a stopping criterion is achieved, stop. Else, let sjh = –(mijh – ijh), j  i.
6.
Select a proxy function and determine consistent parameters.
7.
Determine the optimal step-length *. Let h+1 = h + *sh. Make h = h+1. Go
to step 2.
Dagstuhl Seminar. Practical Approaches to Multi-Objective Optimization. Dec 10-15, 2006.
Type 2 methods
The PROJECT method (Luque, Yang and Wong, 2006)
ASSUMPTIONS
•
The implicit value function is assumed to be continuously differentiable,
monotonically decreasing.
•
The objective functions are continuously differentiable.
•
The feasible region is compact.
•
Regularity conditions hold at the efficient solutions.
Dagstuhl Seminar. Practical Approaches to Multi-Objective Optimization. Dec 10-15, 2006.
Type 2 methods
The PROJECT method (Luque, Yang and Wong, 2006)
STEPS
1.
Choose an initial efficient solution x1. Make h = 1.
2.
Determine the normal vector, Nh, to the efficient frontier at xh.
3.
The DM gives the MRS at xh, mh.
4.
Calculate the projection of mh onto a tangent plane to the efficient frontier at
xh, dfh.
5.
If a stopping criterion is achieved, stop. Else, determine the optimal steplength, and set qih = fih + *dfih.
6.
Project qh onto the efficient frontier, using a reference point scheme. Let x*, f*
be the projection.
7.
If the DM prefers f* to fh, then set xh+1 = x*, h = h+1 and go to step 2. Else,
actualize the step-length * and go to step 6.
Dagstuhl Seminar. Practical Approaches to Multi-Objective Optimization. Dec 10-15, 2006.
Thank you!