Progressively Interactive Evolutionary Multi-Objective Optimization ...

4. Check for termination. The termination check (described in Section 4.2) is based on the distance
of the current best solution from the previous best solutions and requires a parameter
ǫu. If the algorithm terminates, the current best point is chosen as the final solution.
5. An offspring population at the upper level is produced from the parent population at the
upper level using a modified domination principle (discussed in Section 4.3) and HBLEMO
algorithm’s search operators.
6. The parent and the offspring populations are used to create a new parent population for
the next generation using the modified domination based on the current value function and
other HBLEMO algorithm’s operators. The iteration counter is incremented as t ← t + 1
and the algorithm proceeds to Step 2.
The parameters used in the PI-HBLEMO algorithm are τ, η and ǫu.
4.1 Step 3: Elicitation of Preference Information and Construction of a Polynomial
Value Function
Whenever a DM call is made, a set of η points are presented to the decision maker (DM). The
preference information from the decision maker is accepted in the form of pairwise comparisons
for each pair in the set of η points. A pairwise comparison of a give pair could lead to three
possibilities, the first being that one solution is preferred over the other, the second being that
the decision maker is indifferent to both the solutions and the third being that the two solutions
are incomparable. Based on such preference information from a decision maker, for a given
pair (i, j), if i-th point is preferred over j-th point, then Pi ≻ Pj, if the decision maker is
indifferent to the two solutions then it establishes that Pi ≡ Pj. There can be situations such
that the decision maker finds a given pair of points as incomparable and in such a case the
incomparable points are dropped from the list of η points. If the decision maker is not able to
provide preference information for any of the given solution points then algorithm moves back
to the previous population where the decision maker was able to take a decisive action, and uses
the usual domination instead of modified domination principle to proceed the search process.
But such a scenario where no preference is established by a decision maker is rare, and it is
likely to have at least one point which is better than another point. Once preference information
is available, the task is to construct a polynomial value function which satisfies the preference
statements of the decision maker.
Polynomial Value Function for Two Objectives
A polynomial value function is constructed based on the preference information provided by the
decision maker. The parameters of the polynomial value function are optimally adjusted such
that the preference statements of the decision maker are satisfied. We describe the procedure for
two objectives as all the test problems considered in this paper have two objectives. The value
function procedure described below is valid for a maximization problem therefore we convert
the test problems used in this paper into a maximization problem while implementing the value
function procedure. However, while reporting the results for the test problems they are converted
back to minimization problems.
V (F1, F2) = (F1 + k1F2 + l1)(F2 + k2F1 + l2),
where F1, F2 are the objective values
and k1, k2, l1, l2 are the value function parameters
The description of the two objective value function has been taken from [10]. In the above
equations it can been seen that the value function V , for two objectives, is represented as a
product of two linear functions S1 : R 2 → R and S2 : R 2 → R. 1 The parameters in this
1 A generalized version of the polynomial value function can be found in [20].
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