Non-dominated solutions

In contrast to single-objective problems where optimization methods explore the feasible search space to find the single best solution, in multi-objective settings, no best solution can be found that outperforms all others in every criterion (3). Instead, multiple best solutions exist representing the range of possible compromises of the objectives (11). These solutions, known as non-dominated, have no other solutions that are better than them in all of the objectives considered. The set of non-dominated solutions is also known as the Pareto-front or the trade-off surface. Figure 3.1 illustrates the concept of non-dominated solutions and the Pareto-front in a bi-objective minimization problem. 

Fig. 3.1. A MOP with two minimization objectives and a set of solutions represented as circles. The rank of each solution (number next to circle) is based on the number of solutions that dominate it (i.e. are better) in both objectives. The area defined by the dashed lines of each solution contains the solutions that dominate it. Non-dominated solutions are labelled 0 . Point (0, 0) represents the ideal solution to this problem.

Fig. 4.12 Plots of non-dominated solutions obtained with NSGA-II-JG after 240,000 -300,000 function evaluations (fn. evals.) and for NSGA-II for 320,000 - 400,000 fn. evals. for the ZDT4 problem. Note that /j and I2 extend over [0, 1] (global Pareto set) for NSGA-II-JG only after about 300,000 function evaluations, and do not show this characteristic for NSGA-II...

Since File 2 contains all the non-dominated solutions till the current iteration, there is no concept of elitism in MOSA. Also, Step A in Step 2 is omitted in MOSA. For SSA, Steps A, B (in Step 2) and 4 are omitted. 

Evolutionary algorithms (EAs) have been successfully applied to a range of multi-objective problems. They are particularly suitable for multiobjective problems as they result in a set of non-dominated solutions in a single run. Furthermore, EAs do not rely on functional and slope continuity and thus can be readily applied to optimization problems with mixed variables. However, EAs are essentially population based methods and require evaluation of numerous candidate solutions before converging to the desired set of solutions. Such an approach turns out to be computationally prohibitive for realistic Multidisciplinary Design Optimization problems and... 

A population size of 100 has been used to solve ZDTl problem and the population is allowed to evolve over 101 generations. The results of the algorithm with and without surrogate assistance are presented in Fig. 5.1. It is clear from Fig. 5.1 that with surrogate assistance, the algorithm could generate a better set of non-dominated solutions that are very close to the actual Pareto front and are well distributed along the Pareto front. SAEA performed 2,882 actual evaluations and 7,218 approximations. 

Inverted Generational Distances for problems ZDTl, ZDT2 and ZDT3 are given in Table 5.1 for a better understanding. The true Pareto sets for ZDT test problems are represented by 100 solutions on the Pareto front. It is seen that IGD for the non-dominated solutions obtained by SAEA is smaller than that of NSGA-II. Other metrics can also be used to determine and compare results between different optimization algorithms. 

For both optimization problems a population of size 40 is evolved over 31 generations. The results of Case A are shown in Fig. 5.7(a). SAEA performed 426 actual evaluations and NSGA-II was run with 440 evaluations. The non-dominated solutions obtained by SAEA have a much better spread as compared to the non-dominated solutions obtained by NSGA-II. 

The results of all the test problems support the fact that better non-dominated solutions can be delivered by the SAEA as compared to NSG A-II for the same number of actual function evaluations. Although the algorithm incurs additional computational cost for solution clustering and periodic training of RBF models, such cost is insignificant for problems where the evaluation of a single candidate solution requires expensive analyses like finite element methods or computational fluid d3mamics. 

For the following optimization problems find the set of non-dominated solutions using NSGA-II and SAEA. 

Fig. 11.6 Non-dominated solutions (ERPs) in case of a BLEVE phenomenon in an LPG storage tank ( capacity of the transportation network 1, warning time 1800 sec).

Some of the non-dominated solutions for different values of indirect links are shown in Table 12.5 (supplementary information on CD). Each solution represents a unique network topology. The solution closest to the origin (corresponding to the minimum value of /l+/2+/3 /3) obtained from the Pareto-optimal set is enclosed in a box in Fig. 12.6(a). The optimal input parameters used for this run are listed in Table 12.6 (supplementary information on CD). 

Figure 14.6 An example of Pareto ranking in two objectives, fi and /2 (which could be, for example, diversity and drug likeness). Pareto optimisation seeks to find the set of non-dominated solutions. A non-dominated solution is one where an improvement in one objective results in deterioration in one or more of the other objectives when compared with the other solutions in the population. In Pareto ranking, an individual s rank corresponds to the number of individuals in the current population by which it is dominated. The black coloured library has four libraries that are better than it in one or both of the design criteria, so is given a Pareto rank of 5. The white dots with black outline are non-dominated, so are given a Pareto rank of 1, and represent the set of best solutions found by the algorithm.

A solution X e O is called a Pareto-optimal solution with respect to Q if no x e O exists for that v = F(x ) =/i(x ), , fy x )) dominates u = F(x) = (fi(x), , fk(x)). The set of all Pareto-optimal solutions is called the Pareto front. Based on this Pareto front, each solution can be linked with a so-called rank. Based on the total amount of solutions, all non-dominated solutions can be determined, which represent the Pareto front. All of these individuals will receive a rank of 1 and will be removed from the solution space. For the remaining individuals of the solution space, the Pareto front is determined again. The individuals of the second front will receive the rank 2 and will also be removed from the set. In this manner, the procedure is repeated until each solution of the space is linked to a rank, which represents its quality without a scalarization. 

The MOO was carried out with a population size of 100, and the non-dominated solutions found at generations 100, 250, 500 and 1000 are presented in Figure 7.8. As can be seen, there is significant improvement in the front from generation 100 to 250, and acceptable Pareto-optimal front is obtained at generation 250 in about 60 s of computational time. As expected, MOO results in Figure 7.8 confirm that the two objectives are conflicting... 

The HEN corresponding to the non-dominated solution at the discontinuity in the front (circled in Figure 7.12) is presented in Figure 7.13. Cost of retrofitting is US 253,328, and utility cost of this network is US 12,238,781/year payback period for this retrofitting is just 0.15 years. The hot and cold utilities used are 84,055 kW and 47,085 kW, respectively. Minimum approach temperature of the network is 18.9 °C (compared with the specified constraint of 15 °C) in exchanger 8 between streams H5 and Cl these are different... 

Figure 10.9 Non-dominated solutions for maximization of utility cost savings and minimization of capital cost for retrofitting EF to an HMD system, considering reboiler duty as utility credit (plot a) optimal values of decision variables corresponding to the Pareto-optimal front are in plots b-e.

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