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Answering the question: Explain in more details the joint use of "utopia plane" and "\epsilon-constraint".
Answering the question: Explain in more details the joint use of "utopia plane" and "\epsilon-constraint".
First, for ease of understanding, we show the example of a bi-objective optimization problem (continuous). As in the following Fig 2.b in [32], the utopia line (the utopia plane for 2 objectives) is the line between \mu1^star and \mu2^star. And the NC approach uses the evenly-distributed reference points on the line between \mu1^star and \mu2^star. And they project the RPs back onto the Pareto Front (the lower curve between \mu1^star and \mu2^star) to get the non-dominated solutions.
First, for ease of understanding, we show the example of a bi-objective optimization problem (continuous). As in the following Fig 2.b in [32], the utopia line (the utopia plane for 2 objectives) is the line between \mu1^star and \mu2^star. And the NC approach uses the evenly-distributed reference points on the line between \mu1^star and \mu2^star. And they project the RPs back onto the Pareto Front (the lower curve between \mu1^star and \mu2^star) to get the non-dominated solutions.
However, the original NC method [32] was developed on continuous solution space. In the case of IP, it can generate dominated solutions (see the following Fig 3.a in our paper) where point A is the solution generated using NC, but there exists a point B outside the shaded feasible region, dominating A.
However, the original NC method [32] was developed on continuous solution space. In the case of IP, it can generate dominated solutions (see the following Fig 3.a in our paper) where point A is the solution generated using NC, but there exists a point B outside the shaded feasible region, dominating A.
Epsilon-constraint combined with CWMOIP can avoid generating the dominated solutions in integer cases. Therefore, we propose to first generate reference points evenly distributed on the utopia plan (Fig 3(a)) and then apply Epsilon-constraint + CWMOIP that uses of the coordinates of these points as the constraints (Fig 3(b)) to convert the MOO into a SOO problem, solved by CPLEX. In such way, we could obtain the solutions that are 1) non-dominated and 2) evenly distributed (as shown in Fig 3(c)).
Epsilon-constraint combined with CWMOIP can avoid generating the dominated solutions in integer cases. Therefore, we propose to first generate reference points evenly distributed on the utopia plan (Fig 3(a)) and then apply Epsilon-constraint + CWMOIP that uses of the coordinates of these points as the constraints (Fig 3(b)) to convert the MOO into a SOO problem, solved by CPLEX. In such way, we could obtain the solutions that are 1) non-dominated and 2) evenly distributed (as shown in Fig 3(c)).
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