Progressively Interactive Evolutionary Multi-Objective Optimization ...

DM one or more pairs of alternative points found by an EMO
algorithm and expected the DM to provide some preference
information about the points. Some of the work in this
direction has been done by Phelps and Köksalan [10], Fowler
et al. [15], Jaszkiewicz [11], Branke et al. [6] and Korhonen
et al. [7], [8].
In a couple of recent studies [1], [2], the authors have
proposed a progressively interactive approach where the
information from the decision maker is elicited and used to
direct the search of the EMO in a preferred region. The first
study fits a quasi-concave value function to the preferences
provided by the decision maker and then uses it to drive
the EMO. The second study uses the preference information
from the decision maker to construct a polyhedral cone which
is again used to drive the EMO procedure towards the region
of interest.
A. Approximating Decision Maker’s Preference Information
with a Value Function
Information from the decision maker is usually elicited
in the the form of his/her preferences. The DM (decision
maker) is required to compare certain number of points in
the objective space. The points are presented to the DM and
pairwise comparisons of the given points result in either a
solution being more preferred over the other or the solutions
being incomparable. Based on such preference statements, a
partial ordering of the points is done. If Pk, k ∈ {1, . . . , η}
represents a set of η points in the decision space then
for a given pair (i, j) the i-th point is either preferred
over the j-th point (Pi ≻ Pj), or they are incomparable
(Pi ≡ Pj). This information is used to fit a value function
which matches the DM’s preferences. A number of available
value functions can be chosen from the literature and the
preference information can be fitted. Here, we describe the
preference fitting task with three different value functions.
The first value function is the CES [16] value function and
the second one is the Cobb-Douglas [16] value function.
These two value functions are commonly used in economics
literature. The Cobb-Douglas value function is a special form
of the CES value function. As the two value functions have a
limited number of parameters, they can be used to fit only a
certain class of convex preferences. In this study we propose
a generalized polynomial value function which can be used
to fit any kind of convex preference information. A special
form of this value function has been suggested in an earlier
study by Deb, Sinha, Korhonen and Wallenius [1]. They used
this special form of the polynomial value function in the PI-
EMO-VF procedure. In this section we discuss the process of
fitting preference information to any value function and then
incorporate the generalized value function in the PI-EMO-VF
procedure in the later part of the paper.
1) CES Value Function:
V (f1, f2, . . . , fM) = ( M ρ 1
i=1 αifi ) ρ ,
such that
αi ≥ 0, i = 1, . . . , m
M i=1 αi = 1
where fi are the objective functions
and ρ, αi are the value function parameters
(1)
2) Cobb-Douglas Value Function:
V (f1, f2, . . . , fM) = M αi
i=1 fi ,
such that
αi ≥ 0, i = 1, . . . , m
M i=1 αi = 1
where fi are the objective functions
and αi are the value function parameters
(2)
3) Polynomial Value Function: A generalized polynomial
value function has been suggested which can be utilized to fit
any number of preference information by choosing a higher
degree polynomial.
V (f1, f2, . . . , fM) = p M j=1 i=1 (αijfi + βj)
such that
0 = 1 − M i=0 αij, j = 1, . . . , p
Sj = M i=1 (αijfi + βj) > 0, j = 1, . . . , p
0 ≤ αij ≤ 1, j = 1, . . . , p
where fi are the objective functions
αij, βi, p are the value function parameters
and Sj are the linear product terms in
the value function
(3)
A special form of this value function suggested by Deb,
Sinha, Korhonen and Wallenius [1] used p = M, where M is
the number of objectives. Choosing a value of p = M makes
the shape of the value function easily deductible with each
product term, Sj, j = 1, . . . , M, representing an asymptote
(a hyper-plane). However, any positive integer value of p
can be chosen. More the number of parameters in the value
function, more is the flexibility and any type of quasi-concave
indifference curve can be fitted by increasing the value of
p. Once the preference information is given, the task is to
figure out the parameters of the value function which capture
the preference information optimally. Next, we frame the
optimization problem which needs to be solved to figure out
the value function parameters.
4) Value Function Optimization: Following is a generic
approach which could be used to fit any value function to
the preference information provided by the decision maker.
In the equations, V represents the value function being
used and P is a vector of objectives. V (P ) represents a
scalar assigned to the objective vector P such that the
scalar represents the utility/value of the objective vector. The
optimization problem attempts to find such parameters for
the value function for which the minimum difference in the
value function values between the ordered pairs of points is
49