Objective function

The third task in the formulation procedure is to find the objective function in terms of design varibles and other problem parameters. the common engineering objectives involve minimization of overall cost of manufacturing, or minimization of net profit earned, or maximization total life of a product; or others.Althoigh most of the above objective function can be quantified (expressed in a mathematical form), there are some objectives that must be qantified easil.

Moreover, in any real-world optimization problem, there could be more than one objective that the designer may want to optimize simultaniously. even though a few multiobjective optimization algorithms exist in the litures they are complex and computationally expensive. Thus,in most optimal design problem,multiple design problem, multiple objactives are avoided.Instead, the designer chooses the most important objective as the objective function of the optimization problem,and the other objective are included as constraints by restricting their values within a certain range.

The objective function can be of two types:
Either

  • the objective function is to be maximized or
  • it has to be minimized.

    • Unfortunitely, the optimization algorithms are usually written either for minimization problems or for maximization problems. Although in some algorithms,some minor structural changes would enable to perform either minimization or maximization problems this extensive knowledge og the algorithm. Moreover, if all optimization software is used for the simulation,the modified software needs to be compiled before it can be used for simulation. fortunately, the the duality principle helps by allowing the same algorithm to be used for minimization or maximization with minor change in the objective function instead to change in the objective function instead of change in the entire algorithm. If the algorithm is developed for solving a minimization problem by simply multiplying the objective function by -1 and vice varsa.

    For Example:
    Consider the maximization of single-varible function

    f(x)=x*x(1-x)

    The maximum point happens to be at x*=0.066.
    The duality principle suggest that the above problem equivalent to minimization of function
    F(x)=-(x*x)(1-x).
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