In the multi-objective optimization problems we often say that the objective functions are conflicting in nature. In what sense the objective functions are said to be conflicting with each other? Also, how it could be proved numerically that the objective functions in our multi-objective problem are conflicting in nature?
Dear Syed Aqib Jalil
see this link to find the explanation for this equation
Article Solving multi-objective optimization problems in conservatio...
That an objective f_1 is in conflict with another objective f_2 means precisely that they are not identical (or, to be more precise, that f_2 is not equal to k*f_1 for some positive scalar k. This may not mean that the two objective functions CANNOT have the same optimal solutions, but that will typically not be the case. One heuristic way to test it is to solve the problem for a convex combination of f_1 and f_2, that is, to use p*f_1 + (1-p)*f_2, for many values of p in the interval [0.1]. If the optimum changes while you vary the parameter p then you do have two conflicting objectives.
Dear Michael
These details that are specified apply only to two functions but what if there are more than two functions of how the subject should be handled?
Dear Rana,
I think then it would be done by the function sum{p_i*f_i}. Where sum of all p_i would be equal to one.
Am I right Prof. Michael.
If you are, for example, designing automobile you might have many objectives. E.g. you want for automobile to be big (spacious) from inside, but also to have smaller dimensions from the outside so it is easier to park the automobile. Obviously inside size and outside size are very dependent in a way that you might can not considerably increase inside space without increasing outer size. So there you have conflicting objectives.
So if improvement in one objective lead to worsening of the other, then this two objectives are conflicted.
Conflicting objective functions = the optimization of one objective function comes at the expense of another. Such as the hight H and width W of a rectangle of a set volume V=H*W. Maximization of H demends minimization of W and vice versa. This is a trivial example in which the relationship of the 2 optimization objectives W and H is straight forward. Other multi-objective optimization problems consist of a much more complex relationship, which can be seen numerically.
Parks uses a nondominated search which illustrates the conflicting nature of some objectives in nuclear core design:
Article Multiobjective Pressurized Water Reactor Reload Core Design ...
It means that when one objective value is either increased or decreased it comes at the expense of the other in the opposite way. For example, consider the problem of minimizing both truss weight and maximum nodal displacement in multi-objective problems. Obviously, when the weight of the truss is reduced, i.e., less material, (a good thing), the structure deflects more (a bad thing) and vice versa. In this case, we say that the two objectives are conflicting. The best bet moving forward is to produce a number of tradeoff solutions, called the Pareto solutions, where designers can pick and choose from based on their perception of how important each of the objectives are to their project.
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