Objective function conflicting

Consider the multi-criteria optimization problem defined in Eq. (11). Because of the fact that these objective functions usually conflict with each other in practice, the optimization of one objective implies the sacrifice of other targets it is thus impossible to attain their own optima, Js, s e <5 = [1,..., 5], simultaneously. Therefore, the decision maker (DM) must make some compromise among these goals. In contrast to the optimality used in single objective optimization problems, Pareto optimality characterizes the solutions in a multi-objective optimization problem [13]. 

From these results, it is found that the proposed multi-criteria synthesis strategy can attain a definite and compromised solution for a problem with assorted conflict objectives. The preference intervals of various objectives have significant effects on final HEN structures. Such acceptable and/or preference intervals can also reflect the importance of different objective functions. Should one specific objective is emphasized, a tighter restriction or shrinking span should be placed on its acceptable ranges. 

Sometimes it is possible to find experimental conditions which satisfy all the specified criteria sometimes there are conflicts between certain criteria and a compromise solution must be found. It is evident that such problems can be difficult to solve. A special branch of mathematics, "Optimization theory" is devoted to this type of problem. In this area it is assumed that the object function (the "theoretical" response function) is perfectly known, and that the final solution can be reached by using mathematical and numerical methods. [17] From the discussions in Chapter 3, it is evident that mathematical optimization theory is difficult to apply in the area of organic synthesis, especially when new ideas are explored. Conclusions must be drawn from observations in suitably designed experiments. The response surface models thus obtained are local and approximate, and definitefy not perfectly known. Nevertheless, we shall see that we can use experimentally determined models to find solutions to the problems sketched above. 

In what follows, we describe and summarize research on multi-objective optimization in chemical engineering reported in Hakanen (2006) and Haka-nen et al. (2004, 2005, 2006, 2008, 2007). These studies have focused on offering chemical engineering an efficient and practical way of handling all the necessary aspects of the problem, that is, to be able to simultaneously consider several conflicting objective functions that affect the behaviour of the problem considered. Therefore, they have been solved using the interactive NIMBUS method. 

Generally, two or more objective functions are defined for gene expression profiling and gene network analysis. Usually, these objectives are conflicting in nature. Use of traditional single objective optimization techniques to solve these multi-objective optimization problems suffer from many drawbacks. Single objective problems either use penalty function approach or use some of the objectives as constraints. Both of these approaches have user-defined biases. Thus, multi-objective optimization techniques are definitely needed to model and solve these and similar other problems. 

A multi-criteria optimization problem generally consists of n decision variables, m constraints, and k evaluation functions. Thereby the evaluation functions can be in conflict with each other, making it difficult to find the global optimum. To find this optimum, the solution space /I C R is created by the decision variables x = (xi, , x ) of the decision space Q with the objective function vector F O A The objective function vector F x) = (/ j(x), , / (x), X e O is optimized considering the constraints / (—x) > 0 = 1,... m x Q (Muschalla 2006). The individual fitness value of each fitness function then can be processed by the evaluation function in different ways. This can be done by a scalarization method or a Pareto dominance-based approach. 

From optimization point of view it is desired that Us(x) < Uo during a mission time Tm, i-e. there is defined a maximal value of system unavailability that cannot be overstepped. System unavailabiUty function depends partly on graph structure and partly on component s unavailability functions. We will assume that the structure of AG, as well as component hazard rate is invariant system characteristics. On the other side, the other component characteristics, as Test intervals (Tls) or repair rates can be changed within a reasonable range. Just these component characteristics may he used as decision variables because they influence hoth conflicting functions, that is unavaUahility Us(x) and cost Cs (i.e. objective function/(x)). 

The first two terms of (5) address probability of Failure in demand, while the last term of (5) depicts the relation between cost and failure rate of each component. The figure 1 demonstrates the conflict between the first two terms and last term of objective function. 

As mentioned in Sect. 1, two important and conflicting objective functions are considered in the formulation of the supply chain network design problem (1) maximizing supply chain profit and (2) maximizing supply density. 

In this section, we solve the bi-criteria model using fuzzy goal programming approach. The results confirm that the two objective functions, maximization of the supply chain profit and maximization of the supply chain density, are conflicting objectives. The increase in supply chain density leads to a decrease in supply chain profit and vice versa. The graphical representation of an efficient frontier is in Fig. 5. 

The main purpose in single-objective optimization problems is to find the values of design parameters at which the value of one objective function is optimum. While, in multi-objective optimization (which is also called the vector optimization), the problem is to find the optimum value of more than one objective function, which are usually in conflict with each other in engineering optimization problems, such that improvement of one leads to the worsening of the others. Therefore, multi objective optimization offers the optimal set of solutions, rather than an optimal response. In this set we cannot find any answer which dominants the others. The optimal solutions are called Pareto points or Pareto Front (Atashkari et al., 2005). 

It is obvious from Figure 8 that the point with the lowest value of Jj has high value of objective fimction (Figure 9), this issue is tme about the point with the lowest value of in comparison with its value for Jj, so there is a coirflict between Jj and Likewise, there is conflict between Jj and J3, also and J3 as a result, three objective fimctions are in conflict with each other. This subject shows the Pareto concept. Therefore, selected point with the lowest value of J 2 is a good compromise point, because ithas intermediate value of three objective functions. The maximum values of the displacement, velocity, and acceleration of each floor of the building under... 

The confliction exists between the objective functions lets the designer to choose the proper point for designing with establishment compromise between the objective functions. 

Multi-Objective Optimization In this method, the main purpose is to find the optimum values of more than one objective function, which are usually in conflict with each other in engineering optimization problems, so that the improvement of one of them leads to worsening the others. 

In the formulation of the objective function there are two conflicting elements to produce as much valuable product as possible, but using as little energy as possible. For a given feed, the cost function is defined as the amount of propane, butane and pentane from the LP column (at 0.99 mol% or more) multiplied by the relevant product prices, minus the cost of boilup ... 

The ideal solution is an extremum for all objective functions. Because a characteristic feature in multi-objective optimization is that the criteria are in conflict, then the ideal solution is outside the feasible region. That is indicated by point A in Figure 12.6, in which both objective functions Fj and have their maximal values at this point, but outside region P(Q). 

These objective functions, usually, conflict or compete with each other. In the case of no conflict between the objectives, any traditional single objective function technique can be used easily to solve the problem since optimising one objective will ensure that all the other objectives are optimised within the same direction of minimisation or maximisation. 

For most real world applications, the optimization problem is in fact a non-linear multiobjective optimization problem, i.e., it is concerned with several objective functions (often conflicting) that must be optimised simultaneously. Assuming a minimization problem, its mathematical formulation is ... 

The objective functions (Cy) may conflict with one another, and the difficulty is to maximize all the objectives simultaneously. 

The barrier model can be used to set safety performance objectives for conflict removal and collision avoidance, and thus contributes to the definition of the function and performance requirements for the systems which iirqilement the barriers -for example the necessary accmacy of surveillance systems. The barrier model can also be used to set integrity targets for the incorrect operation of the barriers. This can be illustrated by the conceptual model in Figure 2, which shows the risk reduction achievable by the correct operation of each group of barriers (left-pointing arrows) and the risk increase from incorrect operation of the barriers (rightpointing arrows). 

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