**2. Literature review**

The ability of the EAs to evolve to a Pareto-frontier as the generation progresses makes them an ideal choice for several multi-objective optimization problems. Vector Evaluated GA (VEGA), proposed by Schaffer [8] back in 1985, is one of the earlier versions of multi-objective GA. Several multi-objective EAs are developed since then including Multi-Objective Genetic Algorithm (MOGA) [9], Strength Pareto Evolutionary Algorithm [10], Non-dominated Sorted Genetic Algorithm (NSGA) [11] to mention a few popular ones.

Coello [2, 12] has conducted comprehensive literature surveys of various evolutionary multi-objective techniques. Konak et al. [13] compared various multiobjective optimization algorithms and provides a set of guidelines to follow while developing a multi-objective algorithm. Their effort primarily lies in guiding researchers with very little background in MOGA and making them familiar with the ideas and approaches of multi-objective optimization.

One such multi-objective algorithm named Non-dominated Sorting Genetic Algorithm (NSGA), developed by Srinivas and Dev [11] – arguably one of the most widely used multi-objective EAs – uses the concept of non-dominated sets originally proposed by Goldberg in his book on Genetic Algorithm and Machine Learning [14]. The NSGA approach maintains sets of non-dominated individuals, with the first set of individuals not dominated by any other individuals in the population. The second set finds the new set of non-dominated individuals after excluding the individuals from the first set. This step continues until all the individuals in the population are categorized inside the non-dominated sets.

A majority of these multi-objective algorithms, in some form, require an assignment of a scalar measure of a fitness value to the individuals in the population. As an example, MOGA [9] and NSGA [11] assign a fitness value based on a ranking scheme depending on the individual's levels of domination. The two-branch tournament selection genetic algorithm presented by Crossley et al. [15] uses a tournament selection scheme that chooses parents considering both the objectives directly in the fitness functions. The individuals are evaluated based on their fitness across both the objectives. The overall process remains the same as that of a traditional GA. However, the only difference appears in the tournament selection operator. During the tournament selection step, the algorithm selects 50% of the parents based on the fitness value associated with the first objective, that is, the individuals are evaluated solely with respect to the first objective without consideration of the other objective. These selected parents are by nature strong in objective 1, or Φ1-strong. Similarly, the tournament selects the remaining 50% of the parents based on the fitness value associated with the second objective. This second 50% are Φ2-strong parents. With this parent selection approach, randomly choosing the selected parents to pair off for crossover, ideally would result in the following distribution of matches: 25% Φ<sup>1</sup> Φ<sup>1</sup> type parents, 25% Φ<sup>2</sup> Φ<sup>2</sup> type parents, and 50% are mixed i.e., Φ<sup>1</sup> Φ<sup>2</sup> type parents.

The hybrid approach, presented in this chapter, uses this two-branch tournament selection GA as the global search optimizer and combines with a gradientbased approach to refine the search using a novel information sharing concept in the process of hybridization. The unique tournament selection strategy of the twobranch tournament GA allows to understand the underlying trait of the parents, i.e., if they are Φ<sup>1</sup> or Φ<sup>2</sup> strong, and this information is later leveraged during the crossover step to obtain children with certain desired traits.

Another challenge with multi-objective EAs is their ability to enforce constraints. Unlike gradient-based methods, which use constraint gradient information to guide the search in the feasible direction, no such constraint gradient is available for EAs. There have been several efforts to handle the constraints for EAs; however, not all of these methods strictly or directly enforce the problem constraints. The penalty function approach is arguably the most widely known of the various approaches to handle constraints in EAs. Assuming a minimization problem, this approach adds a penalty to the objective function when constraints are violated [14].

Another simple approach includes ignoring any infeasible design solution; because this does not differentiate between constraints that are close to the constraint boundaries and those that are far apart, this constraint handling method is inefficient.

Binh and Korn [16] suggested a method to assign fitness to individuals based on combining both the objective function vector as well as the degree to which the individual violates the constraint. Infeasible individuals are categorized into different classes based on how close or how far they are to the constraints boundaries.

Fonseca and Fleming [17] proposed a priority-based constraint handling strategy where search is first driven for feasibility followed by optimality by assigning high priority to constraints and low priority to objective functions. Although there are various techniques to "handle" constraints in EAs, "enforcing" them in a robust way is still an open issue. This is another motivation to pursue the hybrid approach that leverages the efficacy of gradient-based search to enforce the problem constraints.

Further, these population-based searches have issues with computational cost and rate of convergence to the Pareto frontier. After some number of generations, the candidate solutions may begin to exhibit little or no improvement. Modified versions of the algorithms work to improve the convergence rate [4, 18]; however, hybridizing EAs or GAs with an efficient gradient-based algorithm can significantly improve the convergence rate, thereby reducing the computational cost. Hybridization of an EA or GA with a gradient-based local search algorithm is not new. There are numerous references demonstrating how hybridization may improve the quality of the search for both single objective and multi-objective problem formulations; these include, but are not limited to, those appearing in [3, 19–32]. The local search can be considered as the local learning that takes place in an individual throughout its lifespan. Some of the approaches apply the local search to the final non-dominated set, while some techniques apply local search to all or many individuals of the population as the generation progresses.

The effort here extends the previous effort by Lehner and Crossley [27] to include a multi-objective formulation and combine the advantage of the hybrid approach with an novel information sharing technique between the global and the local search. The two-branch tournament selection GA algorithm globally explores the design space handling both discrete and continuous type variables, while the gradient-based approach sees only the continuous variables in a goal attainment formulation and seeks to efficiently refine the population based on the information passed on by the top-level GA while enforcing all the problem constraints.

### **3. Methodology and approach**

The hybrid approach presented in this chapter combines the two-branch tournament GA (see **Figure 1**) for the global search [15] and the goal attainment SQP

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

**Figure 1.**

*Original two branch tournament selection GA for two objective problems. Adapted from reference [15].*

algorithm provided in the function *fgoalattain* available from the MATLAB Optimization Toolbox [33] as the local search. For solution via hybrid approach, the problem statement contains two levels, as appears below.

#### **3.1 Level I: Two-branch tournament genetic algorithm**

The top level of the problem, which the GA sees as its optimization problem, is a bound constrained (i.e., only side constraints on the continuous design variables) multi-objective minimization problem that uses the two-branch tournament selection technique with some modification to include the local search. This level includes both discrete and continuous design variables of the original problem. The continuous variables in this level, *x*<sup>0</sup> *<sup>c</sup>* , are the initial values (starting point) for the local search problem. This way, the GA acts like a guide for a sequential multi-start approach as it searches the combined discrete and continuous design space. The top level formulation appears below:

$$\begin{aligned} \text{Maximize}: \left\{ \begin{aligned} &f\_1(\mathbf{x}\_d, \mathbf{x}\_c^0) \\ &f\_2(\mathbf{x}\_d, \mathbf{x}\_c^0) \end{aligned} \right\} \\ \text{Subject to :} \\ &(\mathbf{x}\_c)\_i^L \le (\mathbf{x}\_c)\_i \le (\mathbf{x}\_c)\_i^U \end{aligned} \tag{1}$$

ð Þ *xd <sup>i</sup>* ∈ A, B, C, D, … ð Þ discrete variables

In the original two-branch tournament selection GA, the tournament step selects 50% of the parents based on the fitness value associated with the first objective.

These parents are by nature strong in objective 1 or Φ1-strong. Similarly, the tournament selects the remaining 50% of the parents based on the fitness value associated with the second objective. This second 50% are Φ2-strong parents. With this parent selection approach, randomly choosing the selected parents to pair off for crossover would result, on average, in the following distribution of matches: 25% Φ1-Φ<sup>1</sup> type parents, 25% Φ2-Φ<sup>2</sup> type parents, and 50% are mixed i.e., Φ1-Φ<sup>2</sup> type parents. This has the effect of generating many compromise solutions near the middle of the Pareto frontier, potentially limiting the spread and quality of the Pareto-front. The approach described in this chapter improves the spread and quality of the Pareto front by pairing off the parents in a more prescribed manner. A flowchart depicting how the modified two-branch tournament GA interacts with the gradient-based (SQP) for local search appears in **Figure 2**.

With a given goal *f G <sup>i</sup>* , a starting point for the continuous variables *x*<sup>0</sup> *<sup>c</sup>* and a set of discrete values *xd*, the goal attainment problem formulation, for each individual in the GA-level population, seeks to find the optimal design *x*<sup>∗</sup> *<sup>c</sup>* . The goal attainment problem formulation also assigns the fitness value to the individuals, thereby waiving off the need of fitness evaluation at the GA-level (level I). Using the fitness information of these populations, a new set of goals are generated for the next iteration following the tournament selection, crossover and mutation steps of the two-branch tournament GA algorithm. The resulting design *x*<sup>∗</sup> *<sup>c</sup>* need not conform to the binary Gray coding scheme implemented to represent the chromosome of each individual in the population. The effort here employs a Lamarckian strategy [20], that updates the chromosomes of the individual to conform to the gray-coding scheme of the GA.

**Figure 3** demonstrates the parent selection process of the new two-branch tournament selection GA and the goal assignment technique with a simple example. The approach starts with a population size of 8*n*, where *n* is any positive integer (**Figure 3** assumes a population size of 8; i.e., *n* ¼ 1). After the two-branch tournament selection process, 4*n* parents are Φ1-strong and the other 4*n* parents are Φ2 strong. These parent groups remain in two separate parent pools. An additional step after the two branch selection process further categorizes these parents into

**Figure 2.** *Modified two branch tournament selection GA and SQP interaction.*

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

**Figure 3.** *Selective parent mixing strategy.*

sub-pools to ensure a prescribed mix of Φ1-strong and Φ2- strong parents for crossover. To begin, half of the parents from pool 1 (which contains Φ1-strong parents) are randomly moved to sub-pool 1. This sub-pool contains Φ1-Φ<sup>1</sup> type parents paired for crossover and leads to offspring that will likely be Φ1-strong. Similarly, half of the parents from pool 2 are randomly moved to sub-pool 2 to form Φ2-Φ<sup>2</sup> type paired parents. Sub-pool 3 pairs parents so that a Φ1-strong parent and a Φ2-strong parent form children via the crossover operation. These would create children that have features from both Φ<sup>1</sup> and Φ<sup>2</sup> strong parents. This modification to the original two-branch tournament selection approach leads to a more prescribed, yet diversified, set of parents in each pool for the crossover, somewhat analogous to the idea of breeding for plant hybridization.

#### **3.2 Level II: sequential quadratic programming**

The lower-level problem presented to the SQP algorithm refines the population of the GA by searching the continuous variable space and helps the hybrid algorithm converge to the Pareto frontier at a faster rate. The *fgoalattain* algorithm, available in MATLAB, converts the multi-objective algorithm into a single-objective optimization problem by converting all the objectives into a set of inequality constraints and minimizes a slack variable *γ* (also called the attainment factor) as the objective.

$$\begin{aligned} \text{Given}: & \boldsymbol{x}\_{\boldsymbol{c}}^{0}, \boldsymbol{x}\_{d}, \boldsymbol{f}\_{i}^{G} \\ \text{Minimize}: & \boldsymbol{\gamma} \\ \text{Subject to}: & \boldsymbol{f}\_{i}(\boldsymbol{\chi}\_{\boldsymbol{c}}) - a\_{i} \boldsymbol{\gamma} \leq \boldsymbol{f}\_{i}^{G} \\ & \boldsymbol{g}\_{j}(\boldsymbol{\chi}\_{\boldsymbol{c}}) \leq \mathbf{0} \\ & h\_{k}(\boldsymbol{\chi}\_{\boldsymbol{c}}) = \mathbf{0} \\ & (\boldsymbol{\chi}\_{\boldsymbol{c}})\_{i}^{L} \leq (\boldsymbol{\chi}\_{\boldsymbol{c}})\_{i} \leq (\boldsymbol{\chi}\_{\boldsymbol{c}})\_{i}^{U} \end{aligned} \tag{2}$$

This goal attainment formulation seeks to attain values for the objectives close to a set of predefined goal values, *f G <sup>i</sup>* , without violating any of the problem constraints *gi* ð Þ *x* ≤0 and *hk*ð Þ¼ *x* 0. The weight values, *αi*, are set as the absolute of the corresponding goal values, *f G <sup>i</sup>* , based on the guidance in [34]. This prevents scaling issues with objectives of various dimensions and magnitudes. The solution to this problem describes the set of continuous variables *x*<sup>∗</sup> *<sup>c</sup>* that minimizes *γ* and satisfies all constraints; the values of *fi x*<sup>∗</sup> *c* –the fitness value of the individual in the population– are returned to the GA-level for the use in the two-branch tournament selection.

The *fgoalattain* formulation needs a defined goal point in the objective space, and the algorithm tries to find a design as close as possible to these goal values. **Figure 4** illustrates the goal point assignment task for each newly created individual in the sub-pools following the example presented in **Figure 3**. In **Figure 4**, the points indicate the child "designs" from a set of parents; e.g., C1ð Þ <sup>1</sup>�<sup>4</sup> is the first child from the crossover of parent 1 and parent 4. The color of the symbol indicates the parent sub-pool from which the child designs were generated. Therefore, in this example, there are two children generated from parents 1 and 4 in sub-pool 1, which are indicated with the light blue color to match **Figure 3**. There are four children generated from sub-pool 2 and two children from sub-pool 3.

To assign the goal point values, the hybrid approach first identifies the local ideal point in each generation. This ideal point is the combination of the lowest *f* <sup>1</sup> and *f* <sup>2</sup> values in the current population. For this effort, the utopia point (which includes some tolerance to give the utopia smaller–or better– *f* <sup>1</sup> and *f* <sup>2</sup> values than the ideal point with the intent of encouraging under achievement in the goal attainment problem) is set as 0.95 times the local ideal point. In subsequent generations, any new objective value smaller than the corresponding value in the current utopia point replaces that current value in the utopia point. This makes the utopia point dynamic with each generation. For two-objective problems, two perpendicular lines originate from the utopia point and extend infinitely into the objective space. These straight lines appear as dashed lines in **Figure 4**.

To assign a goal point to an individual, the approach defines a vector that originates from an individual and ends to where the vector intersects with either of the dotted lines. The point of intersection becomes the goal point for that individual. Children of parents from sub-pool 1, the Φ1-strong sub-pool, receive a goal vector with slope of zero in the objective space. These are the horizontal arrows in **Figure 4**.

**Figure 4.** *Goal assignment technique.*

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

These goal points would seek the most improvement along the direction of objective 1. Similarly, children of parents from sub-pool 2 receive a goal vector with 90 degree slope in the objective space. This ensures improvement along the direction of objective 2. Lastly, the children from sub-pool 3 parents receive goal vectors relative to their spatial location in the objective space. An individual closer towards objective 1 will have a vector inclined more towards improvement in objective 1 and vice versa.

Referring back to **Figure 3**, parents 1 and 4 from sub-pool 1 create children C1ð Þ <sup>1</sup>�<sup>4</sup> and *C*2ð Þ <sup>1</sup>�<sup>4</sup> . This indicates the first child of parent 1 and 4 and the second child of parent 1 and 4 respectively. During the SQP search, these children have goal points that will minimize along the direction of *f* <sup>1</sup> without increasing their current values of *f* <sup>2</sup>. C1ð Þ <sup>5</sup>�<sup>7</sup> and C2ð Þ <sup>5</sup>�<sup>7</sup> result from sub-pool 2, and the local search will seek to improve *f* <sup>2</sup> without increasing *f* <sup>1</sup>. C1ð Þ <sup>3</sup>�<sup>6</sup> , C2ð Þ <sup>3</sup>�<sup>6</sup> , C1ð Þ <sup>2</sup>�<sup>8</sup> and C2ð Þ <sup>2</sup>�<sup>8</sup> all result from sub-pool 3, and they will have different goal points for their local searches to improve both *f* <sup>1</sup> and *f* <sup>2</sup>. This modified parent selection and goal assignment strategy, via the hybrid formulation, seeks to exploit the tournament selection process of the two-branch tournament GA and tailor the local search for children, depending on traits of their parents.

Although the approach seems robust in enforcing constraints via goal attainment formulation, there may be instances when no feasible solution exists to the goal attainment formulation for a given set of discrete variables. In such cases, the local search will not be able to return a feasible solution and the fitness function receives a severe penalty in the GA-level in an effort to discard such discrete design choices from the population. This severe penalty has some resemblance to the approach of ignoring infeasible designs that was criticised above; however, because the situation where no locally-feasible design exists results from a specific combination of discrete variables, there is no analog to having a "nearly feasible" design with a slightly violated constraint. Severely penalizing such infeasible designs for certain combinations of discrete variable choices, in this context, is appropriate.

### **4. Application to engineering design problems**

To demonstrate the efficacy of the hybrid approach in solving constrained multi-objective MDNLP problems, we solve three different engineering test problems with varying difficulties - a three-bar truss, a ten-bar truss, and greener aircraft design problem.

#### **4.1 Three-bar truss problem**

For the three-bar truss problem (see **Figure 5**), the problem formulation includes the objectives of minimizing the weight of the truss and minimizing the deflection of the free node. The deflection of a node is calculated as the resultant of the deflections in both the x and y directions. The problem consists of six design variables, of which three are continuous and three are discrete. The continuous variables describe the cross-sectional area of the three bars while the discrete variables describe the material selection properties of these bars. The details of the continuous design variables and their design bounds appear in **Table 1**. For this problem, four discrete material selection choices are available for each element and include aluminum, titanium, steel, and nickel options. The yield stress for every bar acts as a constraint for the problem (total three constraints), not allowing the stress in the bar to go beyond that upper limit. References [35, 36] provide more details about the three-bar truss problem. For the hybrid approach, the GA population is limited to 8 individuals while setting the upper limit for the number of generations

**Figure 5.** *Three bar truss problem.*


#### **Table 1.**

*Continuous variables for three-bar truss problem.*

to 50. The probability of crossover is set to 0.5 and the mutation rate is fixed at 0.005. The continuous and discrete variables uses 8 and 2 bits respectively in the Gray-coded binary scheme.

The resulting Pareto frontier for the three-bar truss problem appears in **Figure 6(a)**. The plot shows the Pareto frontier has a good spread, leading to a total of 248 nondominated points as solutions to the optimization problem. The visible trend in the non-dominated design set indicates that as the weight of the three bar truss system increases, they are accompanied by similar increases in the cross-sectional area of the bars with the material selection choice gradually shifting to steel for all the three bars. Aluminum or nickel never appeared as the material selection choice in the first two bars. The designs visible in the top left corner of the Pareto front in **Figure 6(a)** correspond to high displacement and low weight designs. The separated cluster of points (six designs) visible at the bottom right corner of the Pareto frontier corresponds to low displacement and high weight designs, with the maximum weight design having a material combination of all steel bars.

For the three-bar truss problem, only 64 possible combinations of discrete design variables exist. Hence, it is possible to perform a complete enumeration of the discrete design space and get a sense of the shape of the true Pareto front and help assess the performance of the hybrid approach. This led the authors to compare the hybrid approach (and the original two-branch tournament selection GA2 ) with a gradient-based weighted sum approach for this three-bar truss problem. The weighted sum approach converts the multi-objective problem formulation into a single objective problem by assigning weights to both the objectives and solves the single objective problem with the gradient-based approach using MATLAB's *fmincon* solver [33].

First, the objectives are normalized using the utopia point. Next, objective 1 is assigned a weight *w* that varies from 0 to 1 in a step increment of 0.05. The weight for the second objective is set to 1 *w*. For each possible combination of discrete

<sup>2</sup> The original two-branch tournament selection GA was proposed for unconstrained problems. In this example, the problem constraints in the original two-branch tournament selection GA (used for comparison) are handled using an exterior penalty approach.

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

**Figure 6.**

*Pareto front for the three-bar truss problem and its comparison with the other approaches. (a) Pareto front for the three-bar truss problem using the hybrid approach. (b) Comparison of Pareto frontier obtained using the hybrid approach, a weighted sum approach and the original two-branch tournament GA approach.*

variable choice and a given weight pair, the approach leads to a single point in the objective space. The weighted sum approach then conducts gradient-based search for all 21 different weight pairs corresponding to each of the 64 possible discrete combination choices. The resulting Pareto frontier using the weighted sum approach is compared with the hybrid approach and the original two-branch tournament GA approach in **Figure 6(b)**. The original two-branch tournament GA finds an inferior set of solutions, possibly due to the lack of local search feature, and the set of solutions also has a reduced spread across the Pareto frontier. On the other hand, the weighted sum approach with complete enumeration on the material selection choices has a slightly better spread compared to the hybrid approach but with fewer non-dominated points.

**Figure 7** compares how the Pareto frontier evolved with generations using the original two-branch tournament GA and the proposed hybrid approach. As expected, without the local search feature, the original two-branch tournament selection GA shows distinct improvement in both the quality and the spread of the Pareto front as the generation progresses. That is, the black diamonds (nondominated set after second generation) are replaced with better non-dominated designs as the generation progresses. However, in the hybrid case, we start to see the shape of the final Pareto front immediately after the second generation. As the generation progresses further, more points get added to the list of non-dominated designs. This is due to the multi-start approach where the top-level GA populates various possible combinations of the discrete material selection choices and the local gradient-based search then improves these designs by varying the continuous design variables. The hybrid approach is able to rapidly get to the final Pareto front

#### **Figure 7.**

*Evolution of the non-dominated sets as the generation progresses. (a) Original two-branch tournament selection GA. (b) Proposed hybrid approach.*

at the expense of increased number of function evaluations needed by the gradientbased local search.

### **4.2 Ten-bar truss problem**

Next, the hybrid approach solves a more difficult and challenging version of the three-bar truss problem – a ten-bar truss. Similar to the three-bar truss problem, the ten-bar truss has the competing objectives that include minimizing the weight of the ten-bar truss system and minimizing the resultant displacement of any of the free nodes. The displacement is taken as the absolute of the maximum calculated displacement among all the bar elements. This problem consists of twenty design variables – ten continuous type and ten discrete type. The continuous variables describe the cross-sectional diameters of the ten bars, ranging from 0.1 cm<sup>2</sup> to 40 cm2 , while the discrete variables specify the material selection properties of these bars. Like the three-bar problem, the four discrete material choices available for each bar include aluminum, titanium, steel, and nickel. However, this problem has over one million possible combinations of the discrete choices (410 <sup>¼</sup> 1, 048, 576) making complete enumeration of the discrete design space computationally prohibitive, unlike the three-bar truss. References [35, 36] provide more details about the ten-bar truss problem considered in this study.

**Figure 8(a)** compares the Pareto front obtained using the hybrid approach after 20 GA generations with the Pareto frontier obtained using the two-branch tournament selection GA after 100 generations. The figure shows both the approaches performed well for this problem with the two-branch tournament selection GA resulting a better spread in the low weight/high displacement region of the objective space, whereas the hybrid GA has a better spread in the low displacement/high weight region. **Figure 8(b)** shows how the non-dominated set evolved as the generation progresses using the hybrid approach. We see a similar trend as that of the three-bar truss problem. That is, there is not much significant change in the final shape of the Pareto front other than the increase in the number of non-dominated designs as the generation progresses. However, this time there is slight improvement in the quality of the Pareto front (the red non-dominated set obtained after generation 20 is slightly better than the blue or the black non-dominated designs obtained at generation 5 and 2 respectively).

For the three-bar example, a majority of the improvements across the objective space are due to the gradient-based local search's ability to obtain designs with better cross-sectional area. With only 64 possible material selection combinations,

#### **Figure 8.**

*Ten bar truss problem results. (a) Comparison between the original two-branch tournament GA (after 100 GA generation) and the hybrid approach (after 20 GA generations) for the 10 bar truss problem. (b) Evolution of non-dominated set as the generation progresses for the 10 bar truss problem using the hybrid approach.*

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

there are not many discrete material selection options to explore. On the other hand, for the ten-bar truss problem, a vast majority of the improvement is due to the ability of the GA to find a better material selection combination rather than fine-tuning the cross sectional variables. It is not possible to seek further improvement in the Pareto front just by varying the continuous variables, so the local search saturates as appear in the case of black (diamonds) and blue (squares) non-dominated designs. After few more GA iterations, the algorithm is able to find better combinations of material selection that lead to further improvement in the Pareto front (red dots).

#### **4.3 Greener aircraft design problem**

The third application problem solved using the hybrid approach is the greener aircraft design problem. Here, a "greener" aircraft design problem provides an example to demonstrate the efficacy of the hybrid algorithm and its ability to solve such MDNLP problems. The intent is to find aircraft designs that represent the best possible trade-offs among performance, economics, and environmental metrics which essentially makes this a multi-objective problem. Further, with the inclusion of discrete technologies, the problem becomes MDNLP in nature.

The aircraft design optimization problem employs the NASA sizing code FLOPS [37] to evaluate discrete design configurations and perform the sizing and performance calculations of the candidate aircraft designs. The sizing code accepts both continuous and discrete design variables as input and returns the aircraft gross weight along with environmental metrics (fuel weight, which corresponds to CO2 emissions, and NO*<sup>X</sup>* emissions) and total operating cost. Simple models simulating the potential "greener technologies" are modeled in MATLAB [33] and then integrated with FLOPS for the performance calculations. The goal of the aircraft sizing problem is to develop an aircraft with 2940 nmi design range with a seat capacity of 162 seats in two classes. A brief description of the greener aircraft design optimization problem appears below. For more details about the aircraft design problem, we encourage the readers to see Ref. [38].

#### *4.3.1 Description of the continuous variables*

The problem includes ten continuous variables that define the wing and the engine parameters of the aircraft. The details of these continuous design variables and their design bounds appear in **Table 2**.

#### *4.3.2 Simulating the discrete technologies*

This aircraft design optimization study models three types of discrete technologies. **Table 3** lists the set of discrete technologies considered in this study. To model composite material selection choice on various aircraft components, the approach here uses a binary variable for each of the aircraft components that includes wing, fuselage, tail, and nacelle. A value of one represents composites being present while a value of zero represents no composite materials in that structure. The second discrete variable includes the eight possible combinations of the location and the number of engines. Lastly, eight combinations of laminar flow technologies are included for this problem, depending on whether it is natural laminar flow (NLF) or hybrid laminar flow control (HLFC) technology and the number of components on which it is applied (as listed in **Table 3**). References [38–40] describe the various discrete technologies used in this study in further detail.
