Section 2 Societal Applications

## **Chapter 2**

## Application of Genetic Algorithms in Health Sciences

*Rohollah Fallah Madvari*

### **Abstract**

In this section, we introduce genetic algorithm (GA) and some of its applications in various health fields. Although GA and some other meta-heuristics are inspired by biology, they are more familiar to experts in other sciences, and these methods are often used to solve complex problems. The use of GAs has promising implications in various health specialities, including occupational health, environmental health, HSE, occupational medicine, industrial safety, ergonomics, toxicology, health care management, etc. This section of the book presents applications of GAs in disease screening, diagnosis, prognosis and health care management, and enables professionals to envision possible applications of this meta-heuristic method in their health professions. In the following, we discuss some applications of GAs in predicting, measuring and controlling factors that affect health.

**Keywords:** health sciences, genetic algorithms, optimisation, health, safety, environment

## **1. Introduction**

The GA is an evolutionary method for constrained and unconstrained optimisation problems that uses the principles of Darwin's natural selection to find the optimal formula to predict or match the model. In general, GA are based on iteration, with most of the parts being selected as random processes. In nature, better generations result from the combination of better chromosomes. In the meantime, there are sometimes mutations in the chromosomes that can make the next generation better. GA also use this idea to solve problems. The GA starts its entire process with an initial population of random samples. Each sample in the population represents a potential solution to the problem at hand. The samples are evolved through successive iterations, called generations, and evaluated against fitness criteria during each generation. The population of the next generation is built by genetic operators and the iteration process continues until the final state is reached.

In the following, we discuss some applications of GA in predicting, measuring and controlling factors affecting health.

## **2. Application of genetic algorithm in acoustics**

Noise is an unwanted and uncomfortable sound that has been the focus of much research as a harmful factor. Many studies have been conducted on the effects of sound, most of which point to the negative effects of sound on health [1, 2].

In general, the adverse effects of sound can be divided into three categories: psychological effects, interference with activities and physiological effects [1, 3]. Factors such as the level of sound exposure, the duration of exposure and the frequency spectrum determine the extent and nature of sound effects.

In recent years, metal foams have emerged as an attractive area of research from a scientific, industrial and audio application point of view [4]. Acoustic absorption is one of the most important functional properties of metal foams [5]. Porous metal is one of the most promising materials because porous metal has higher mechanical strength and hardness, resistance to heat, corrosion and weathering than non-metallic porous materials such as glass wool and urethane foam [6]. The sound absorption behaviour of porous metal depends on the cell structure, which is mainly divided into two types: open-cell structure and closed-cell structure [7]. Porous metals with excellent absorption properties have an open cell structure because the sound wave propagates inside the material. On the other hand, a porous metal with a closed cell, which has a wavy cell wall, does not absorb sound. Sound absorption occurs due to air adhesion friction at the boundary between the matrix and the air, and part of the sound energy is converted into thermal energy. Therefore, the absorption of sound by porous metal is related to the behaviour of air diffusion inside the cell, and therefore the characteristic of sound absorption by porous metal is strongly dependent on its cell structure, which is determined by the manufacturing methods and conditions [8, 9]. Many parameters can affect the sound absorption coefficient of aluminium foams, including porosity (Ω), pore size (*D*), pore opening (*d*), thickness (*t*), resistance to static flow, etc. (**Figure 1**) [10–12].

Various optimisation algorithms have been proposed, including the genetic algorithm (GA) [13]. The GA algorithm checks the neighbourhood by repeatedly expanding the search domain in the neighbourhood of the current solution and moving from the current solution to the increasing neighbourhood. This process is repeated until the current solution cannot be improved. This process continues until the optimal location is reached.

#### **Figure 1.**

*The morphology of the metal foam sample, where D represents the pore diameter and d represents the opening diameter of the pores [12].*

*Application of Genetic Algorithms in Health Sciences DOI: http://dx.doi.org/10.5772/intechopen.112653*

**Figure 2.** *Optimised sound absorption coefficient chart at various frequencies using GA in the thickness of 5–40 mm [13].*

GA is a problem domain-independent method and quickly searches the search space for an optimised point with a quality function. GA has a distinct advantage over other stochastic methods. It is very easy to parallelise the algorithm. This is because the calculations of each iteration are independent of each other.

In general, the (Ω), (*D*), (*d*) and (*t*) are not regulated in the manufacture of sound-absorbing foam [14]. Therefore, approximate sizes of existing foam are tested through trial and error to find the best sound-absorbing foam. However, if the shape of the metal foam can be pre-determined for constant sound absorption with an optimum SAC, it is a great step towards the intelligent production of porous foam. We discuss how to use GA to improve an optimal set of metal foaming parameters including the (Ω), (*D*) and (*d*) at any thickness and frequency to obtain an optimal foam [13]. In **Figure 2**, the value of the SAC optimised for different thicknesses is shown for each frequency.

The results in **Figure 2** show that, in order to increase the amount of sound absorption, the thickness of the panel must be increased at frequencies below 2000 Hz. However, at higher frequencies, for each thickness, it is possible to find conditions (the value of *d*, *D*, Ω), so that the amount of sound absorption reaches the maximum.

## **3. Optimisation of sound power transaction of multi-wall panels using genetic algorithm**

Tanyo et al. [15] used a GA to optimise the layout of structural layers according to the number of layers and corresponding thicknesses. The optimisation process was performed by selecting materials from a given list that included a limited

number of solid, liquid and foam materials. One of the advantages of this approach is that there is no need to create new materials for a material designed for a specific application.

The research by Shojaeifard et al. [16] began by explaining the theory behind the acoustic analysis of multilayer panels using the transfer matrix method. After comparing with the existing laboratory results in this field and ensuring the correctness of the modelling, in the next step, using this valid model, a multi-objective optimisation of the acoustic behaviour of the panel was carried out. The acoustic behaviour of the multi-wall acoustic panel is optimised by finding the optimal material and thickness for the layers, and at the same time considering the acoustic behaviour criteria, minimising the weight of the panel is considered during the optimisation process. Since in the present work, the optimisation variables include both continuous and discrete variables, therefore the GA, as one of the most powerful algorithms that can manage continuous and discrete variables together, has been selected to solve the present problem.

## **4. Using genetic algorithm in multi-objective optimisation of external louvres in office buildings**

Inadequate lighting will be associated with some degree of perceptual error such as sleepiness [17]. Improving the comfort and convenience of the interior space by optimising the level of natural light is one of the most important issues in the renovation and improvement of space, especially in office buildings [18]. With the increase in energy consumption, the need for multi-objective optimisation and efforts to reduce consumption is increasing, especially in developing countries, and this has led designers towards excellent architecture and maximum use of renewable energy, optimal use of energy and lighting. During the day, the sun not only provides a favourable environment for users but also reduces the energy used to cool and heat the environment. As a result, one of the most important factors in improving the energy efficiency of the building is to control the amount of light entering the space, and considering that the only part of the building that directly receives sunlight from the sun is the window, the use of louvres to control the amount of sunlight entering the space is essential. The use of daylight in many cities in Iran, including the city of Tehran, is remarkable due to its favourable geographical location and the availability of many sunny hours throughout the year [18]. The purpose of this research is to apply artificial intelligence and algorithmic programming to estimate the proportions and technical specifications of external louvres and to propose a model for the design of southern openings of office space in the direction of efficiency and intelligent consumption of energy and providing the required level of light in the interior space. The research method in this research is simulation and logical reasoning, therefore, using office space simulation, parametric design of louvres and optimisation of parameters [degree of rotation (θ), length (*x*), distance from the window (*a*), amount of reflection (*R*) and number of louvres (*n*)] using GA to design the south window has been analysed and studied in accordance with the conditions of solar radiation in Tehran. The results show that the use of external louvres is very efficient in controlling and improving the quality of light (**Figure 3**) [18].

*Application of Genetic Algorithms in Health Sciences DOI: http://dx.doi.org/10.5772/intechopen.112653*

**Figure 3.** *The results of the optimised brightness of the simulated sample [18].*

## **5. Multi-objective optimisation of window shape in order to simultaneously provide visual comfort and energy efficiency components through genetic algorithm**

Zhang et al. [19] used a multi-objective GA to optimise the thermal and daylighting performance of a school building in the cold climate of Tianjin, China. First, different plan designs, i.e. one-way open corridors, one-way closed corridors and two-way corridors, were considered the main examples because they represent the structures of school buildings in China. The main results show that the best case is related to the design of the two-way corridor, mainly towards the south, in order to take advantage of the absorption of radiation during winter. The Galapagos algorithm and Octopus are optimisation tools in the Grasshopper environment [20]. Galapagos algorithm as an extension for the single-purpose GA execution engine and Octopus algorithm as an extension for the multi-objective GA execution engine in Grasshopper software. As a multi-objective optimisation tool, the GA helps the production of plans and forms to move towards the optimum, and even when the local optimum occurs, it will cause a deviation from it by creating a genetic mutation [21]. For single objective optimisation, the objective function is the minimum or maximum value of each of these performance indicators. The GA is then used to explore the relationship between the design components of the window shape and the daylight and energy performance indicators and to generate new design options for better performance. Multi-objective optimisation involves finding intermediate solutions for different objectives. Firoozeh and Sayyed Majid [21] multi-objective optimisation of window shape to simultaneously provide the components of visual comfort and energy efficiency through GA provide the components of visual comfort (increasing brightness and reducing glare) and energy efficiency (reducing energy consumption). In order to achieve the optimal values of window design parameters, the optimisation process based on simulation through GA was performed automatically in Grasshopper software as multi-objective. Finally, multi-objective optimisation by visualising the boundaries of the solution space can significantly reduce the complexity of the problem and help the designer to achieve a set of variables that simultaneously consider relatively good values of all the objective functions and the possibility

of choosing options with the priority of each of the conflicting objectives to achieve a match between the project expectations and the final design [21].

## **6. Optimisation of the roof of a three-story residential building with the help of genetic algorithm**

The walls of a building are important in terms of heat exchange and control of the building's energy consumption, as they are the outermost envelope of the building in direct contact with the air and temperature changes [22]. The body of the roof is more important than the other walls of the building because its heat exchange is more exposed to sunlight and other factors than other walls due to its area and time. The aim of this section is to reduce energy consumption in a residential building in Shiraz city and to achieve thermal comfort in the building spaces by optimising the roof of the building. In this context, the following question was raised What is the most optimal design of the roof of the building (materials and passive design methods) so that the energy consumption of the building is minimised and the spaces of the building are placed within the range of thermal comfort? The amount of reduction in the energy consumption of the building has also been considered in this research. The research method is quantitative and the energy consumption of the building and the thermal comfort index were done with Energy Plus software and the optimisation process was done with GA [22]. The building roof variables were extracted and defined in three general categories: passive energy system, physical characteristics of the roof and location of the roof. The results obtained from the building simulation calculations and the objective function output of the GA showed that the best roof models provided reduced energy consumption by 50% and the average thermal comfort index was 0.9 and 0.68, respectively in the warmest and coldest months of summer and winter.

The GA is one of the methods used in many architectural optimisation projects as an optimisation method when the number of variables is large [22]. Compared to other optimisation methods, this algorithm has advantages that have made it used in different fields, including the fact that it deals with discrete values and is not limited to continuous values. The GA is inspired by the idea of natural evolution and Darwin's principle of survival. In general, GAs are population-based algorithms that can be classified as global optimisation algorithms, which include the operation of searching for the global optimal solution as well as the operation of improving local solutions. The iterative process of GAs leads to better solutions based on mating and crossing of higher-performing parents. The genetic coding of humans is called the genotype, and the coded information of individual characteristics is called the phenotype. The operations related to the genetic coding of the parents produce results for the next generation. A basic GA has three main operations that are performed at each iteration: inheritance, crossover and mutation [22].

## **7. Accurate modelling and prediction of PM2.5 concentration or use of genetic algorithm**

Many large cities face the problem of air pollution, partly from mobile sources such as vehicles and partly from stationary sources such as industry [23]. The gradual *Application of Genetic Algorithms in Health Sciences DOI: http://dx.doi.org/10.5772/intechopen.112653*

and long-term effects of air pollution have led authorities and people to pay less attention to it. It should be noted that the increasing trend in the number of deaths, cancers and heart attacks caused by air pollution indicates that air pollution has caused the gradual death of people [23].

There are limitations in measuring air pollution concentration, one of the important limitations is the number and spatial distribution of air pollution monitoring stations. Due to the high cost and lack of necessary facilities, it is not possible to cover the entire city with air pollution monitoring stations; therefore, interpolation and prediction algorithms are used to model air pollution. The accuracy of these models is of particular importance and various research works have been conducted to improve the quality of air pollution modelling.

It is used to select effective parameters for estimating the concentration of air pollution. This algorithm uses biogeography-based optimisation and GA for optimisation [23].

## **8. Optimising air distribution in ventilation network by using genetic algorithm method**

Ventilation is one of the most important support activities in the production chain of underground mining and construction activities, providing and distributing the air required by the various sectors [24]. The main purpose of a ventilation system is to economically supply the required fresh air at a speed and volume sufficient to rapidly dilute and remove contaminants from work areas. Pollutants and disturbance factors in drilling and mining operations include a wide range of flammable and noxious gases, dust, toxic gases from the explosion of explosives, heat and moisture and the acceptable levels of these pollutants are defined by a number of standards. Inadequate ventilation during the mine's operating life can cause problems with extraction and even halt production. The design of ventilation systems is usually based on technical considerations and economic analysis is rarely used as a decision criterion.

Optimum air distribution in mining operations can be described as the most important practical solution for reducing the operating and capital costs of the ventilation network, which can be achieved by selecting the correct position and resistance of the regulating doors, as well as the position and specifications of the booster fans, for real and adequate distribution of airflow in the network. It can be achieved.

In order to optimise the air distribution, a GA can be used to search for the optimum values of fan allocation, damper pressure drop and flow intensity of each branch of the ventilation network.

The GA was determined by analysing the optimal fan power and the amount of pressure drop required for the control dampers. The effect of the GA parameter values of mutation rate and grafting rate, together with the population size, on achieving the optimum response was then investigated. It was found that increasing the population to a certain extent increases the probability of achieving the optimal solution. As the linkage and mutation coefficients increase, the accuracy of the calculations decreases and the time to reach the optimal solution also increases. To complete the research, it is proposed to investigate GA coding in combination with the Hardy Cross method for the optimal design of air distribution in a mine ventilation network [24].

## **9. Scheduling employees' work shifts using a genetic algorithm approach**

The general objective of this section is to apply human factors engineering to scheduling theory in order to exploit the optimal performance of employees [25]. The problem of scheduling the work shift of employees with variable performance is investigated in this section. The objective function of the mathematical model for employee scheduling presented in this section is to minimise labour costs and attempts to assign efficient employees to work shifts in order to meet the work demand of the organisation. The important feature of the presented mathematical model is the consideration of ergonomic dimensions of employees, including learning, forgetting and fatigue caused by work. GA was used to solve the presented mathematical model in a reasonable computation time. In order to verify the efficiency and effectiveness of the GA compared to the exact problem-solving methods, the performance of the algorithm was compared with the performance of the LINGO software and the lower limit of the presented examples. The results of the study by Akbari et al. showed that the presented model has the ability to model human factors and provides favourable work shifts. This study also showed that the human parameters studied have an impact on the efficiency of the employees and consequently on the planning of the organisation's work schedules. Therefore, it is suggested that managers in organisations should use the proposed model to study the effect of human factors on employees' efficiency and provide an optimal schedule for employees' work shifts [25].

The set of analyses showed that the change in human parameters has an effect on the planning of work shifts, and therefore it is suggested that managers use the presented model to study these effects and provide suitable work schedules for employees. The presented model also has the ability to be used in a dual way to plan job rotation with the aim of improving employees' health, which can be investigated in future research. In general, the results showed that the effect of human parameters on the efficiency of employees and organisations in scheduling work shifts can be mathematically modelled using exponential and hyperbolic functions, and the presented GA has the necessary power to solve the optimisation model. Modelling other human factors such as motivation, stress, etc. in employee scheduling problems and applying other meta-heuristic methods and comparing the results are topics that can be explored in future research.

## **10. Providing the optimal model for urban waste management system using genetic algorithm based on fuzzy logic**

In recent years, all kinds of models have been studied and used to evaluate the waste management subsystems of the city of Tehran and to select the best waste management option [26]. However, the problem of final waste disposal in Tehran is one of the most important issues related to the environmental management of this metropolis. The aim of the present research is to provide a model to allocate the optimal annual amount of Tehran's waste to waste management subsystems in order to achieve maximum efficiency, reduce costs and increase the system's revenue. Firstly, the data required for the research was collected by referring to the Aradkoh Complex in Tehran and personal interviews with experts and using the information recorded in the Aradkoh Complex. Then, the proposed research model for the purpose of allocating the optimal amount of annual waste taking into account all constraints to

#### *Application of Genetic Algorithms in Health Sciences DOI: http://dx.doi.org/10.5772/intechopen.112653*

five recycling subsystems, aerobic compost, anaerobic digester, waste incinerator and sanitary landfill using the GA based on fuzzy logic with the aim of reducing the total cost of waste management system Shahri was implemented in MATLAB environment and its results were analysed. According to the results of the optimal model proposed by the research, it is necessary to allocate the optimal flow and process of the annual waste of Tehran city among recycling, aerobic composting, anaerobic digestion, waste incineration and sanitary landfill systems with more accuracy in order to increase the annual efficiency of the waste management system. The city of Tehran should be followed.

In this research, it was tried to present an optimal model for Tehran's urban waste management system with the aim of reducing costs and increasing income using the improved GA method by fuzzy logic controller, and based on this, the annual optimal values of Tehran's waste were allocated to each of five under the recycling system, aerobic compost, anaerobic digester, waste incinerator and sanitary landfill, so that the presented model can be fully implemented to increase the annual efficiency of Tehran's waste management system. The results of this research showed that the optimal model of the urban waste management system is a combination of different waste management options in order to achieve maximum productivity and reduce costs and increase revenues, and using only one waste management option is not cost-effective. It can also be said that increasing the capacity allocated to each of the subsystems does not mean reducing costs and increasing revenue generation, and the most optimal point in each of the subsystems is the point where the objective function is optimised [26].

According to the results of the implementation of the proposed research model using fuzzy GA, increasing the capacity of subsystems with lower costs and higher revenues up to a certain amount can lead to the highest profitability and the capacity increase of these subsystems after this amount is not specified. It can be affordable. Also, by examining the changes in the total cost of the system, it can be concluded that increasing the capacity of the subsystems in total up to a certain amount will optimise the urban waste management system, and after that, it will not have a positive effect on reducing the cost of the system and increasing income generation. The comparison of the results of the GA implementation alone with the GA improved by the fuzzy logic controller shows that the fuzzy system plays a positive role in the efficiency of the GA in reaching a more optimal solution with a higher speed, and therefore it can be used to make the algorithm more dynamic. Genetics makes good use of the fuzzy system in this model in the direction of greater efficiency to reach the optimal solution.

## **11. Conclusion**

The idea of a GA, like an artificial neural network, is inspired by nature. Another such innovation is evolution. By "simulating" the process of evolution in nature, GAs search the "space of candidate solutions" to find the best possible solution to a problem. In the search for the optimal solution, a set or population of initial solutions is first generated. Then, in successive "generations", a set of modified solutions is produced (in each generation of the GA, certain changes are made to the genes of the chromosomes that make up the population). The initial solutions are usually modified in such a way that, in each generation, the population of solutions "converges" towards the optimal solution.

This branch is inspired by the field of "artificial intelligence", which is based on the mechanism of evolution of living organisms and the production of more successful and graceful species in nature. In other words, the main idea of GAs is "survival of the fittest". In general, it is an algorithm based on repetition, most of its parts are selected as random processes, and these algorithms consist of the functional parts of adaptation, representation, selection and modification. GAs can help many health sciences and even medicine in the future.

## **Author details**

Rohollah Fallah Madvari Department of Occupational Health Engineering, School of Public Health, Shahid Sadoughi University of Medical Sciences, Yazd, Iran

\*Address all correspondence to: fallah134@gmail.com

© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Application of Genetic Algorithms in Health Sciences DOI: http://dx.doi.org/10.5772/intechopen.112653*

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## **Chapter 3**

## Using Group Theory to Generate Initial Population for a Genetic Algorithm for Solving Traveling Salesman

*Dharm Raj Singh*

## **Abstract**

In this chapter, we propose a novel algorithm that uses Genetic algorithm with group theory for initial population generation and also propose a novel crossover for solving Traveling Salesman Problem. In the group tour construction method, each individual/initial tour has distinct start city provided that population size is equal to total number of cities. In the initial population, each individual/tour has a distinct starting city. The distinct starting cites of each tour provide genetic material for exploration for the whole search space. Therefore, a heterogeneous starting city of a tour in initial population is generated to have rich diversity. Proposed crossover based on greedy method of sub-tour connection drives the efficient local search, followed by 2-opt mutation for improvement of tour for enhanced/optimal solution. The result of the proposed algorithm is compared with other standard algorithms followed by conclusion.

**Keywords:** genetic algorithms, traveling salesman problem, group theory for population generation, 2-opt mutation, group theory

### **1. Introduction**

#### **1.1 Genetic algorithm (GA)**

Genetic algorithm draws the idea from natural selection and natural genetics principles for searching and optimization algorithm. In this method, we use survival of the fittest rule of natural evolution. Invention of GA was done in 1960 by John Holland [1]. It consists of population of chromosomes, with each chromosome representing a solution to the particular problem. Each chromosome is evaluated to obtain its fitness value of the chromosome against some given fitness function. A set of chromosomes are selected for genetic operation(s) (selection, crossover, and mutation) in order to get new chromosomes. Chromosomes are selected according to their fitness values to reproduce/generate the next generation by genetic operations, which generate new chromosomes. To achieve this, two transformations, namely

crossover (generates new chromosomes by overlapping genes of two chromosomes) and mutation (creates a new chromosome by making changes of genes in a single chromosome), are used. After performance of crossover and mutation operation, we generate a new chromosome called child. The process continues by selecting fit chromosomes from parent and child population. Whole process of genetic algorithm is repeated until best individual is obtained or desired number of iterations completed, providing an optimal/suboptimal solution to the problem [2].

We developed a genetic algorithm for Traveling Salesman Problem (TSP) to provide balance between exploration and exploitation for the search space. For this, all the components of the genetic algorithms were carefully examined.

#### **1.2 Literature review**

Traveling Salesman Problem is a famous combinatorial optimization problem, which is still NP-complete [3, 4]. There is no clear evidence for its origin. However, credit goes to Irish mathematician Hamilton and British mathematician Thomas Krikman for its mathematical formulation, which is discussed in detail in "Graph Theory 1736-1936" book by Wilson, Biggs and Lloyd 2 [5]. The general form was firstly studied by mathematician Karl Menger [6]. Alexander Schrijver [7] pointed out the connection between the works of Whitney and Menger along with growth of TSP in his paper "On the history of combinatorial optimization (till 1960)." Merrill Flood [8] while searching for the solution of school bus routing problem used TSP mathematically for the first time. Hassler Whitney [9] of Princeton University introduced the name traveling salesman problem. The popularity of TSP increased considerably during 1950s and 1960s when prizes were offered by RAND Corporation for solving steps of the problem. As a result of it, prime contributions were made by Fulkerson, Dantzig and Johnson [10] from the RAND Corporation who used branch and bound algorithm [11] for solving integer linear program for which they developed the cutting plane method [12] for its solution. Later 532 and 2392 city TSP was solved by M. Padberg and Rinaldi [13] in 1987, and 1000 city TSP by solved M. Grotschel and O. Holland [14]. In 1991, Reinelt [15] introduced TSPLIB, which is still providing problem instances for TSP. In last few decades, many heuristic and meta-heuristic algorithms have been developed with some of the following notable algorithms: Ant colony optimization [16–25], Neural network [26–29], Self-organizing maps [30], Particle swarm optimization techniques [31, 32], Simulated annealing [33], Weed optimization [34, 35], Genetic algorithm [36–39].

The rest of chapter is divided into following sections: Details of Genetic algorithm are given in Section 1. Section 2 describes our proposed hybrid methods. In Section 3, experimental result is presented followed by conclusion in Section 4.

### **2. Proposed hybrid method**

The methods used here are hybrid because we have used a proposed group theory tour construction algorithm and proposed crossover with 2-opt mutation Croes (1958) [40]. The framework of the proposed algorithm is shown in Algorithm 1.

The main idea of the first step is to generate a population of chromosomes (tours) by using proposed group theory approach. Clearly, each chromosome of the population is same but which start city unique providing rich diversity of genetic materials for exploration.

*Using Group Theory to Generate Initial Population for a Genetic Algorithm for Solving… DOI: http://dx.doi.org/10.5772/intechopen.109049*

Fitness value of each chromosome in the population is calculated in the second step. In the third step, select two parent chromosomes (selected randomly) from the population and replace the first chromosome with minimum fitness value (tour cost). After that, apply proposed crossover operator on the selected two chromosomes with crossover probability rate. And finally, apply 2-opt mutation operator on selected parent chromosome or new pair of chromosomes generated after crossover, with mutation probability rate. Mutation operator helps in generate new population, which is then replacing new population with the previous population by the new population. Whole process is repeated until termination condition is satisfied.

#### **Algorithm 1: Proposed Algorithm**

```
1: Generate initial population of the tour with population size P using Group
  theory.
2: Gen = 1;
3: while (Gen ≤ NGen) do.
4: Calculate the fitness of each tour in P.
5: Bs = Best tour in P;
6: Randomly select two parents S1 and S2 tour in P;
7: S1new = Bs;
8: S2new = S2;
9: rnd1 = rand (0,1];
10: if (rnd1 < crossover probability rate (pc)) then.
11: Perform proposed crossover on selected two parents Bs and S2 to generate two
   new children C1 and C2;
12: end if.
13: S1new = C1;
14: S2new = C2;
15: rnd2 = rand (0,1];
16: if (rnd2 < mutation probability rate (pm)) then.
17: Perform 2-opt optimal mutation operator on S1new and S2new
18: update new population P0
                              ;
19: P P0
          ;
20: end if.
21: Gen = Gen + 1;
22: end while
```
#### **2.1 Proposed group theory for population generation**

There are various possible methods for generating the initial population [41–43]. One of the simplest ways is generating the initial population randomly using random number generator. Zhang [42, 43] proposed greedy tour construction heuristic with Karp-patching for feasible tour construction and used for solving Assignment problem [44]. We proposed the group tour construction heuristic for initial population generation. In this method, nodes of graph are label using group of integers, Zn with integer modulo n operation

$$\mathbf{a} +\_{\text{n}} \mathbf{b} = (\mathbf{a} + \mathbf{b}) \mathbf{m} \mathbf{d} \text{ n} \tag{1}$$

#### **Figure 1.** *Population generated using Group Modulo.*

where "+n" is operator that represents addition modulo of n, and (a + b) represents the normal addition of integers. This helped in generating the group table shown in **Figure 1**. In group table, no two row or column elements in the same position are identical. The function used for generating initial population of chromosomes (P) is as follows:

$$\mathbf{P(a)} = \text{mod}((\mathbf{a} + \mathbf{b}), \mathbf{n}) + \mathbf{1}.\tag{2}$$

where a represents population size whose value is from a = 1 to population size, and b = 1 to n (As mentioned in earlier, we are taking population size equal to number of cities, therefore population size = n). For n = 10, the initial population generated using Group theory is as follows: Each chromosome in the initial population being unique (group theory technique) provides a wide diversity of genetic materials for exploring of search space.

#### **2.2 Proposed crossover operator for GA**

Sharing information between a pair of chromosomes is called crossover [2]. In this process genes of parent's chromosomes are swapped to generate offspring. The selection of the parent chromosomes is with the possibility that good chromosomes may generate better offspring. Goldberg described several order-based operators, such as the Partially Matched Crossover (PMX) [45]. The order crossover (OX) was suggested by Syswerda [46]. The position-based crossover (PBX) was introduced by [39]. The cycle crossover (CX) was suggested by Oliver, Smith & Holland [47]. Freisleben & Merz introduced a distance preserving crossover (DPX) [37]. Inspired by DPX crossover, we propose a new crossover in this chapter. In the proposed crossover the cities that are identical for the same position in both parents (s1 and s2) will not change in child c1 and c2 as shown in **Figure 2**. The remaining cities will change accordingly Algorithm 2.

*Using Group Theory to Generate Initial Population for a Genetic Algorithm for Solving… DOI: http://dx.doi.org/10.5772/intechopen.109049*


1: c1 = zeros (1, n); 2: c2 = zeros (1, n); 3: for i = 1: n do 4: for j = 1: n do 5: if (s2(i)==s1(j)) then; 6: c1(i) = s2(j); 7: end if 8: if (s1(i) == s2(j)) then 9: c2(i) = s1(j); 10: end if 11: end for 12: end for

#### **2.3 Example**

Given a complete weighted graph with 5 nodes obtain weighted (cost) matrix of graph is in **Figure 3**.

**Figure 2.**

*Example of Proposed crossover.*

**Figure 3.** *Example of a complete weighted graph with 5 nodes.*


**Figure 4.**

*Generate initial population with Population size =5 using Group Theory method.*

Initial set Population size = 5, Crossover probability rate (*pc*) = 0.8 and Mutation probability rate (*pm*) = 0.2. Generate initial population with Population size = 5 using Proposed Group Theory method as shown in **Figure 4**.

Generate two random number between (1-Population size) is 4 and 2, then select two chromosome 4 and 2 from population after then replace first chromosome with minimum cost therefore selected chromosome with cost is


Generate a random number rnd1 = 0.3243. if (rnd1 < = *pc*), then apply proposed crossover, after crossover generate two new chromosomes is


again, generate a random number rnd2 = 0.0161. if (rnd2 < = *pm*), then apply 2-opt mutation, the operation 2-opt mutation on both chromosome one by one given as Apply 2-opt mutation on first chromosome = 4 5 1 2 3

> a ¼ 3, b ¼ 4, c ¼ 5, d ¼ 1, and set zmin ¼ 0, i ¼ 1, j ¼ 1 z ¼ dmat a, c ð Þþ dmat b, d ð Þ–dmat c, d ð Þ–dmat a, b ð Þ;

Where dmat(a, c) represent cost from node a to c.

$$\mathbf{z} = \mathbf{6} + \mathbf{9} \\ \mathbf{-9} - \mathbf{5} = \mathbf{1}$$

if z < zmin false, then a = 3, b = 4, c = 1, d = 2, and i = 1, j = 4.

$$\mathbf{z} = \mathbf{4} + \mathbf{7} \\ \mathbf{-8} - \mathbf{5} = -\mathbf{2}$$

if z < zmin true, then set zmin = �2, imin = 1, jmin = 4, and a = 3, b = 4, c = 2, d = 3, and i = 1, j = 5.

*Using Group Theory to Generate Initial Population for a Genetic Algorithm for Solving… DOI: http://dx.doi.org/10.5772/intechopen.109049*

$$\mathbf{z} = \mathbf{6} + \mathbf{5-6} - \mathbf{5} = \mathbf{0}$$

if z < zmin false, then a = 4, b = 5, c = 1, d = 2, and i = 2, j = 4.

z ¼ 9 þ 10–8 � 4 ¼ 7

if z < zmin false, then a = 4, b = 5, c = 2, d = 3, and i = 2, j = 5.

z ¼ 7 þ 6–6 � 4 ¼ 3

if z < zmin false, then a = 5, b = 1, c = 2, d = 3, and i = 3, j = 5.

$$\mathbf{z} = \mathbf{10} + \mathbf{4} \mathbf{-6} - \mathbf{9} = -\mathbf{1}$$

if z < zmin false.

if zmin <0 Then apply 2-opt mutation between imin to jmin �1 on first selected chromosome = 4 5 1 2 3, after then we get a new chromosome is 1 5 4 2 3.

Again apply 2-opt mutation on new chromosome 1 5 4 2 3. a = 3, b = 1, c = 5, d = 4, and set zmin = 0, i = 1, j = 3.

$$\mathbf{z} = \mathbf{6} + \mathbf{9} \\ \mathbf{-4} - \mathbf{4} = 7$$
 
$$\mathbf{z} = \mathbf{d} \\ \mathbf{mat}(\mathbf{a}, \mathbf{c}) + \mathbf{d} \\ \mathbf{mat}(\mathbf{b}, \mathbf{d}) - \mathbf{d} \\ \mathbf{mat}(\mathbf{c}, \mathbf{d}) - \mathbf{d} \\ \mathbf{mat}(\mathbf{a}, \mathbf{b});$$

if z < zmin false, then a = 3, b = 1, c = 4, d = 2, and i = 1, j = 4.

$$\mathbf{z} = \mathbf{5} + \mathbf{8} \\ \mathbf{-7} - \mathbf{4} = \mathbf{2}$$

if z < zmin false, then a = 3, b = 1, c = 2, d = 3, and i = 1, j = 5.

$$\mathbf{z} = \mathbf{6} + \mathbf{4} \mathbf{-6} - \mathbf{4} = \mathbf{0}$$

if z < zmin false, then a = 1, b = 5, c = 4, d = 2, and i = 2, j = 4.

z ¼ 9 þ 10–7 � 9 ¼ 3

if z < zmin false, then a = 1, b = 5, c = 2, d = 3and i = 2, j = 5.

$$\mathbf{z} = \mathbf{8} + \mathbf{6} \\ \mathbf{-6} - \mathbf{9} = -\mathbf{1}$$

if z < zmin true, then set zmin = �1, imin = 2, jmin = 5, and a = 5, b = 4, c = 2, d = 3, and i = 3, j = 5.

$$\mathbf{z} = \mathbf{10} + \mathbf{5-6} - \mathbf{4} = \mathbf{5}$$

if z < zmin false.

if zmin <0, then apply 2-opt mutation between imin to jmin �1 on first selected chromosome = 1 5 4 2 3, after then we get a new chromosome is 1 2 4 5 3.

Again apply 2-opt mutation 1 2 4 5 3 and a = 3, b = 1, c = 2, d = 4, and set zmin = 0, i = 1, j = 3.

$$\mathbf{z} = \mathbf{6} + \mathbf{9} \\ \mathbf{-7} - \mathbf{4} = \mathbf{4}$$

if z < zmin false, then a = 3, b = 1, c = 4, d = 5, and i = 1, j = 4.

$$\mathbf{z} = \mathbf{5} + \mathbf{9} \\ \mathbf{-4} - \mathbf{4} = \mathbf{6}$$

if z < zmin false, then a = 3, b = 1, c = 5, d = 3, and i = 1, j = 5.

z ¼ 6 þ 4–6 � 4 ¼ 0

if z < zmin false, then a = 1, b = 2, c = 4, d = 5, and i = 2, j = 4.

z ¼ 9 þ 10–4 � 8 ¼ 8

if z < zmin false, then a = 1, b = 2, c = 5, d = 3 and i = 2, j = 5.

$$\mathbf{z} = \mathbf{9} + \mathbf{6} \\ \mathbf{-6} - \mathbf{8} = \mathbf{1}$$

if z < zmin false, then a = 2, b = 4, c = 5, d = 3, and i = 3, j = 5.

$$\mathbf{z} = \mathbf{10} + \mathbf{5} \mathbf{-6} - \mathbf{7} = \mathbf{2}$$

if z < zmin false and completed loop, then we get new first child chromosome is 1 2 4 5 3.

Similarly Apply 2-opt mutation on second chromosome = 1 2 3 4 5 and completed loop, then we get new second child chromosome is 3 1 2 4 5.

After one iteration is completed, the new population is updated as


### **3. Experimental results**

#### **3.1 Experimental setup**

For evaluation purpose, results were generated on 2.20 GHz Intel Core i5 machine with 4 GB RAM.

#### **3.2 Experimental design**

The following standard benchmark data set was taken from TSPLIB: Eil51, berlin52, St70, Eil76, Pr76, Kroa100, Eil101, ch150, and ts225, were taken for performance comparison.

For the experiment, the experimental parameters crossover probability rate (*pc* = 0.8) and mutation probability rate (*pm* = 0.2) values were set to and respectively with number of iterations = 500. Population size of each instance equal to number of cities was generated by group tour construction method. For result comparison, Percentage Best Error (% Best Err.) is used. The Percentage Best Error is calculated as follows:

*Using Group Theory to Generate Initial Population for a Genetic Algorithm for Solving… DOI: http://dx.doi.org/10.5772/intechopen.109049*

$$\text{PDbest} = \frac{[(\text{Best path cost from n trial}) - (\text{Best known Solution} \text{(BKS)})]}{(\text{Best known Solution} \text{(BKS)})} \times 100$$

#### **3.3 Experimental result**

For each standard TSP data taken, we performed n = 20 trails for first set of comparison shown in **Table 1** and **Figure 5**. Best results are shown in bold. Results are taken from [48]. It can be clearly seen that proposed method is better than Hierarchic method used for comparison for every dataset taken in to consideration. Proposed method although not exact but do provide good heuristic solution.

The pictorial presentation for performance comparison in terms of Mean (average) solution for different methods is shown in **Figure 5**. **Table 2** reports the outcome of some large size of instances. If size of instances is increases then increases the population size because population size of each instances is equal to number of cities in


**Table 1.**

*Performance comparison of proposed algorithm and hierarchic algorithm.*

#### **Figure 5.**

*Performance comparison of Average distance (Mean) (over 20 trails) between Proposed algorithm and Hierarchic algorithm for the 9 TSPLIB instances.*


#### **Table 2.**

*Performance of proposed algorithm for large instances size.*

instances therefore increases the time complexity and space complexity due to increases population size.

### **4. Conclusion**

This chapter proposed a Genetic algorithm, which works by taking features of Group theory for tour construction, proposed crossover, and 2-opt mutation. Although the other reported methods can usually find a better solution for the TSP, their solution is dependent on the quality of the random initialization of the population and parameters. In order to have our proposed method uses heterogeneous population for process initialization with population size equal to number of cities, we applied group tour construction method. In group tour construction method generates all solution(tour) in the initial population has a distinct starting node that provides the initial exploration of the search space. After this, the proposed crossover and 2-opt mutation are applied. In order to maintain local optimality, crossover and mutation operators are used. By using crossover operator, new starting points were defined for a local search using information of the current population. The proposed crossover utilizes a greedy method for the duplicated paths in the parents for connecting subtours into the solution. However, 2-opt mutation can easily get stuck in a local optimum to improve the tour quality. The combination of the proposed method is required as 2-opt mutation easily gets stuck in local optimum. From the experimental results, one can easily find that our proposed algorithm gives better performance in comparison of [48].

As future work, we would like to extend the group theory for population initialization in other heuristics such as scheduling algorithm, network problems, estimation of distribution algorithm, etc.

### **Conflict of interest**

The authors declare that they have no conflict of interest.

*Using Group Theory to Generate Initial Population for a Genetic Algorithm for Solving… DOI: http://dx.doi.org/10.5772/intechopen.109049*

## **Author details**

Dharm Raj Singh Department of Computer Applications, Jagatpur P.G. College, Varanasi, Uttar Pradesh, India

\*Address all correspondence to: dharmrajsingh67@yahoo.com

© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Section 3
