*4.3.3.2 Objective pair - NOX emissions and total operating cost*

The Pareto front corresponding to the NO*<sup>X</sup>* emissions and the total operating cost objective pair appears in **Figure 10** and has 24 non-dominated designs. The non-dominated designs have different geometric design variable values that best match the different discrete greener aircraft technologies to arrive at the trade-off between the NO*<sup>X</sup>* emissions and the total operating cost.

The design with minimum NO*<sup>X</sup>* emissions and maximum total operating cost (ND1) employs a three-engine configuration with one fuselage-mounted and two wing-mounted engines, along with a composite wing. The laminar flow technologies on this design include NLF technology on wings and HLFC technology on the nacelles and tail. The maximum NO*<sup>X</sup>* emitting design with minimum total operating cost (ND24) employs a two-engine configuration with wing-mounted engines, along with NLF technology on the wings and HLFC technology on the tail, and a composite nacelle. All the non-dominated designs, except the one with maximum NO*<sup>X</sup>* emissions, employ NLF technology on the wings and HLFC technology on both the nacelle and tail. As we move from left to right along the Pareto frontier in

**Figure 10.** *The non-dominated set for objective pair – NOX emissions versus total operating cost.*

*A Hybrid Approach for Solving Constrained Multi-Objective Mixed-Discrete Nonlinear… DOI: http://dx.doi.org/10.5772/intechopen.97054*

**Figure 10**, the aircraft engines change from a three-engine configuration (two wing-mounted and one fuselage-mounted) to a two-engine (wing-mounted) configuration, thereby reducing the total operating cost.

An interesting region from the airline's point of view is the near the points ND2, ND3, ND4 and ND5, where a nearly vertical portion is visible in the top left portion of the Pareto frontier (refer to **Figure 10**). Moving from left to right in this region, a substantial decrease in total operating cost is possible for a marginal increase in the NO*<sup>X</sup>* emissions of the aircraft. A plausible design from an airline's perspective– among the obtained non-dominated designs–would be the ND10 design. The reason for this observation is that a substantial increase in total operating cost will be incurred if further reduction in NO*<sup>X</sup>* emissions are desired, while any effort to further reduce the total operating cost will lead to very high NO*<sup>X</sup>* emissions, which is not desired from an environment standpoint.

Given there is some degree of randomness associated with the genetic operations in the GA, subsequent runs of the hybrid GA for the two objective pairs find a slightly different number of non-dominated designs points. However, the basic trait of the Pareto frontier, in terms of the discrete choices, did not alter; only the density of points in the Pareto frontier varied with different runs.

#### **5. Conclusions**

This chapter describes a hybrid multi-objective algorithm that makes use of an efficient gradient-based SQP algorithm for fitness evaluation inside a GA in a learning approach. The combination allows the GA to evolve a population of designs in the direction of the Pareto frontier while the SQP algorithm enforces constraints, eliminating the need for penalty multipliers or other special constraint handling methods and refines the values of the continuous design variables. The selective parent mixing and unique sets of goal point assignment to the individual lead to a distinct improvement in convergence and the quality of the Pareto frontier from a previous variation of this approach. When applied to various constrained MDNLP engineering design problems, the hybrid algorithm shows the ability to identify promising designs.

Although the ability of the hybrid approach to solve difficult constrained MDNLP problems is demonstrated in this chapter, the methodology relies heavily on the constraint enforcing ability and efficient searching of the continuous design space via the local gradient-based SQP algorithm that requires some estimates (either numerically or analytically) of the gradients of the objectives and the constraints with respect to the continuous design variables. A major advantage of a gradient-based approach besides being able to enforce the problem constraints (hence, the motivation to hybridize) is that the computational cost needed to compute the gradients is nearly independent of the number of design variables [41] when using adjoint-based methods to estimate the derivatives. This allows the gradient-based approach to efficiently solve problems with a very large number of design variables. However, if the objectives are encapsulated in a black-box function and are computationally very expensive to evaluate, then it may not be possible to directly implement a gradient-based search and may require a surrogate-based design optimization approach [40, 42, 43].

#### **Nomenclature**


