genetic Algorithm particulars coding scheme

genetic Algorithm particulars

1. coding scheme

As described above the goal of the current effort is to find a cost-effective design of the radiator having a desired performance using a genetic algorithm. In this study the parameters that define the performance and cost are the number of tubes (n_t), the fin pitch (p_f), the length of the cross-section of tube (l_t) and the width of the cross-section of tube (w_t). The length of the binary string, which represents these four parameters using standard binary concatenated coding, is found by specifying the accuracy of each parameter. The minimum and maximum values for each parameter are problem specific and depend on the constraints that exist on the design. Table 1 shows pertinent information about the coding used in this study.
Table 1: Sub-string length for each parameter based on chosen accuracy and maximum and minimum values.

PARAMETER	Ap	Umin	Umax	SUBSTRING LENGTH
nt	1	29	60	5
pf (per inch)	1	8	11	2
lt (mm)	1	8	15	3
wt (mm x 10-1)	1	15	30	4

The sub-string lengths of each of the four parameters are concatenated, resulting in the total string length of fifteen. Each string represents one possible solution to the problem. The number of tubes and the fin pitch significantly affect the performance and cost of the radiator. Thus, these two parameters are placed adjacent to one another to reduce the likelihood of destroying good combinations of these two parameters by crossover. Therefore the binary string obtained is described in Table 2.
Table 2: Position of each parameter in string.

PARAMETER	nt	pf	lt	wt
Positioning in string	1 - 5	6 - 7	8 - 10	11 - 14

2. FITNESS FUNCTION

The parameters are sent to a mathematical model for evaluation of the performance and the cost effectiveness of the radiator, and in return a fitness value is assigned defining the quality of solution represented by a given binary string. The genetic algorithm then attempts to achieve the desired performance with the minimum number of tubes and fin pitch combination together with the best profile of the cross-section of the tube. Here the fitness function is defined as
Minimise {k₁*[(UA - UA_desired)² +1]^1/2 + k₂*n_t + k₃* p_f + k₄*(17 -l_t) + k₅*(5-w_t) } …(1)
where

UA_desired = desired value of UA for radiator performance k₁ = a weighting factor of 10⁷

k₂ = a weighting factor of 10⁵

k₃ = a weighting factor of 10³

k₄ = a weighting factor of 10

k₅ = a weighting factor of 1

The difference between the UA and UA_desired is weighted by maximum factor as radiator’s under-performance and over performance is highly undesired, while the higher weightage is given to number of tubes represented in the string, p_f represents the fin pitch value in the string while the higher weightage is given to the number of tubes as compared to the fin pitch the number of tubes affect the cost more severly. The profile of tube is given a low penalty for poor design.

In the solution of this problem, a genetic algorithm first generates a population of strings of given length using the user-defined constraints. Each string is decoded to yield actual parameters. The fitness of each string in the population is found by evaluating the fitness function as defined in Equation (1). Then, reproduction using tournament selection, single-point crossover, and mutation are used to generate subsequent generations and search for acceptable values for n_t, p_f, l_t and w_t which minimise the fitness function of Equation (1). Tournament selection is executed by picking 15% of the strings from the current population at random and comparing their fitness values. The string with the lowest fitness value is placed into the mating pool for the new population. Single-point crossover is accomplished by randomly picking two strings from the mating pool; then randomly picking the crossover location in the string length based on a probability of crossover of 0.9 and crossing the strings at the location. A mutation operator with a probability of 0.01 is used to introduce new genetic material into the population.

5. result

The accuracy of the genetic algorithm is tested by comparing the solution obtained using a genetic algorithm to the known practical result; one that is implemented successfully in 1998 in India.

The goal set for the of the genetic algorithm was to provide a cost-effective solution for the radiator, when the performance desired from the radiator is 817 WC (product of overall heat transfer coefficient and total heat transfer area). The genetic algorithm was run for twenty-five generations, using single-point crossover and a simple mutation operator. An initial population of 100 strings was randomly selected, where each string represented one possible solution. The fitness value of the best string in a generation is plotted against the function evaluations. The results of this case are displayed in Figure 2.

Figure 2: Best Fitness produced by the genetic algorithm vs. function evaluation

Particulars of Simple Genetic Algorithm:

Population size=100

Number of generations=25

Probability of crossover=0.9

Probability of mutation=0.01

Chromosome length=15

Tournament size =15

Desired performance of radiator =817 WC

As seen in Figure 3, the offline performance shows better convergence than the online performance. This is because of the larger pool of diverse schemata are available in larger population.

Figure 3: Comparison of online and offline performance of genetic algorithm vs. function evaluation

When the genetic algorithm was run and compared to the known practical solution the following results were obtained (Table 3):
Table 3: Comparison of genetic algorithm to the known practical solution for the 3-row radiator with straight fins and inline tubes. (population size=100, number of generations=25, probability of crossover=0.9, probability of mutation=0.01)

PARAMETER	PRACTICAL	GENETIC ALGORITHM
Number of Tubes	48	51
Fin pitch (per inch)	10	11
Length of cross-section of tube	12	11
Width of cross-section of tube	2.5	1.8
Fitness Value	1481052.5	1511163.2

The configuration resulting from the genetic algorithm is slightly less economical than the practical known solution. However, the result is near optimal and hence the genetic algorithm is successful in providing a near-optimal solution to the problem.

Based on these results, the genetic algorithm can be used to determine the configuration of a radiator, for which we have no solution, when the performance desired from the radiator is know, say 1500 WC (product of overall heat transfer coefficient and total heat transfer area). Here a genetic algorithm is run for forty generations with a population size of 50, initially selected at random. The results of this case are displayed in Figure 4.
Figure 4: Best Fitness produced by the genetic algorithm vs. function evaluation

The configuration of radiator obtained after running the genetic algorithm is given in Table 4.

Table 4: Solution provided by genetic algorithms for the 5-row radiator with straight fins and inline tubes. (population size=100, number of generations=100, probability of crossover=0.9, probability of mutation=0.1)

PARAMETER	SOLUTION PROVIDED BY GENETIC ALGORITHM
Number of Tubes	52
Fin pitch (per inch)	9
Length of cross-section of tube	10
Width of cross-section of tube	3
Fitness Value	1520972

6. CONCLUSIONS

A genetic algorithm was developed to search for the optimal design of a radiator with pre-defined performance characteristics and cost constraints. The validity of the approach was tested against a problem with a known solution. The genetic algorithm produced near-optimum result for the problem; a solution which matched the best-known practical solution. Thereafter, the genetic algorithm is used for finding the optimal design parameters for a radiator with desired performance criteria and cost constraints. Therefore, it is concluded that a genetic algorithm can be used successfully to find near-optimum solutions in the realm of radiator design.

References

1. R. A. Beard and G. J. Smith, A Method of Calculating the Heat Dissipation from Radiators to Cool Vehicle Engines, Society of Automobile Engineers – 710208

2. Charles N. Kurland, Computer Program for Engine Cooling Radiator Selection, Society of Automobile Engineers – 710209

3. Performance of Aluminium Automotive Radiators, Society of Automobile Engineers – 790400

4. Engine Cooling System Design for Heavy Duty Trucks, Society of Automobile Engineers – 770023.

5. J. P. Holman (1986). Heat Transfer.

6. P. L. Balaney. Thermal Engineering.

7. G. F. Hewitt, G. L. Shires and T. R. Bott. Process Heat Transfer.

8. M. Necati Ozisik (1985). Heat Transfer - A Basic Approach, New York, McGraw-Hill Inc.

9. Jiunn P. Chiou. Correction Factor to Unit Core Heat Transfer Capability of Heat Exchanger Core Due to Variation of Tube Length, Society of Automobile Engineers – 750884

10. K. D. Emmenthal and W. Hucho. A Rational Approach to Automotive Radiator Systems Design, Society of Automobile Engineers – 740088

11. D. E. Goldberg (1989). Genetic Algorithms in Search, Optimization, and Machine Learning.

12. C. L. Karr, J. C. Phillips. Scheduling and Resource Allocation with Genetic Algorithms, Presentation at the Society of Mechanical Engineers Annual Meeting.

13. B. K. Hodge, Robert P. Taylor, (3^rd edition) Analysis & Design of Energy Systems.

14. W. M. Kays and A. L. London, 2^nd ed. (1964), Compact Heat Exchangers.