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A new method called mutable smart bee (MSB) algorithm proposed for cooperative optimizing of the maximum power output (MPO) and minimum entropy generation (MEG) of an Atkinson cycle as a multiobjective, multi-modal mechanical problem. This method utilizes mutable smart bee instead of classical bees. The results have been checked with some of the most common optimizing algorithms like Karaboga’s original artificial bee colony, bees algorithm (BA), improved particle swarm optimization (IPSO), Lukasik firefly algorithm (LFFA), and self-adaptive penalty function genetic algorithm (SAPF-GA). According to obtained results, it can be concluded that Mutable Smart Bee (MSB) is capable to maintain its historical memory for the location and quality of food sources and also a little chance of mutation is considered for this bee. These features were found as strong elements for mining data in constraint areas and the results will prove this claim.

The Atkinson cycle was designed by James Atkinson in 1882 [

Recently, researchers focused on analyzing and optimizing Atkinson cycle using different optimization techniques and intelligent controlling systems. Leff [

Metaheuristic algorithms are population-based methods working with a set of feasible solutions and trying to improve them gradually. These algorithms can be divided into two main parts: evolutionary algorithms (EAs) which attempt to simulate the phenomenon of natural evolution and swarm intelligence base algorithms [

Other branches of population-based algorithms which are called swarm intelligence focused on collective behavior of some self-organized systems in order to develop some metaheuristics procedures which can mimic such system’s problem solution abilities. The interactive behavior between individuals locally with one another and with their environment contributes to the collective intelligence of the social colonies [

There are also some algorithms that improved the performance of swarm base algorithms by utilizing some natural concepts. In 2009, Yang and Deb [

One of the other improved algorithms which is used in this paper was produced in 2009 by Łukasik and Zak [

In this paper entropy generation and power output of air standard Atkinson cycle will be analyzed in different situations as a multiobjective problem using MSB algorithm. It will be proved that different types of constraints should be considered to derive to an acceptable engineering solution. Besides, the performance of proposed algorithm will be compared to some other well-known optimization techniques such as Karaboga’s original ABC [

Recently, many researchers focused on the interactive behavior of bees that occur through a waggle dance during the foraging process. Successful foragers share the information about the direction and the distance to patches of flower and the amount of nectar with their hive mates. Foragers can recruit other bees in their society to search in productive locations for collecting nectars with higher quality. These procedures suggest a successful data mining mechanism.

For the first time Seeley proposed a behavioral model for a colony of honey bees [

Thereafter, many researchers focused on the honey bee organism and several metaheuristics were proposed based on the peculiar intelligent behavior of honey bee swarms. Yonezawa and kikuchi proposed ecological algorithm (EA) which was focused on the description of the collective intelligence based on bees’ behavior [

Many real-world optimization problems involve inequality and equality constraints. It is hard and also takes a long time to find a feasible solution in searching space which optimizes a constraint problem with traditional strategies. Since one of the crucial problems is to gain a feasible answer in the searching spans, different concepts proposed by researchers and a variety of methods implemented for different optimization situations [

Recently, Stanarevic et al. [

Here, we will analyze some features that make this algorithm really strong for optimizing multi-modal problems.

In classical ABC proposed by Karaboga and Basturk [

In SB-ABC algorithm a different style was used to modify the solution:

One of the other advantages in this method is hiring smart bees. These artificial insects can memorize the position of the best food source and its quality which was found before and replace it to new candidate solution if the new solution is unfeasible or the new solution has a lower fitness than the best-saved solution in the SB memory.

Another important advantage of this method is the time duration for smart bee’s data processing procedure. This feature will make the algorithm more durable when high amount of these artificial organisms being hired for searching the solution space. To overcome this problem, we utilized a low amount of smart bees in constraint searching space. Besides, we add a new mutation operator to SB-ABC for overcoming subsequence fast convergence. In each of the iterations, bees that exceed from a finite number of trials will be sent to a container and participate in mutation process based on their mutation probability. The results show that the global solution can be obtained faster and by adapting a dynamic mutation probability (

The pseudocode of MSB-ABC is given in the following:

initialize the population of solutions

evaluate the population;

cycle = 1;

repeat;

produce new solutions (food source positions)

if cycle

apply selection process based on Deb’s method;

calculate the probability values

for each onlooker bee, produce a new solution

apply selection process between

determine the abandoned solutions (source), if exists, and perform mutation on each abandoned solution by following formula:

memorize the best food source position (solution) achieved so far;

cycle = cycle + 1;

until cycle = maximum cycle number.

Here the performance of an air standard Atkinson cycle with heat-transfer loss, friction, and variable specific-heats of the working fluid will be analyzed precisely. According to (P-V) diagram in Figures

P-V diagram of the theoretical air standard Atkinson cycle.

T-S diagram of theoretical air standard Atkinson cycle.

According to [

It is assumed that air is an ideal gas that consists of 78.1% nitrogen, 20.95% oxygen, 0.92% argon, and 0.03% carbon dioxide.

Heat added to the working fluid in isochoric process 2

Heat rejected by the working fluid in isobaric process 4

According to [

By combusting an amount of energy received by working fluid that is calculated by following linear equation:

Now the thermal efficiency of the Atkinson cycle engine can be expressed as following:

The amounts of

After obtaining appropriate equations and data for calculating the power output of the Atkinson cycle, the relations between obtain parameters and entropy generation will be checked. Figure

T-S diagram of real air standard Atkinson cycle.

Process 1

As it is shown in Figure

And the effectiveness of the hot-and-cold side heat exchangers can be written as following:

As it was mentioned before, we have to minimize the unexpected entropy generation and maximize the power output to obtain an efficient performance of the Atkinson cycle. In order to achieve a suitable engineering solution for optimizing the cycle under different situations, we have to face different types of constraints, and under these constraints in searching space it will be harder to find the feasible solution. In this section the efficiency of the Atkinson cycle will be checked using proposed mutable smart bee (MSB-ABC) algorithm and compared to different methods of optimizing and the results will be shown in tables at the end.

The objective functions are defined as following:

In this work,

Due to (

According to [

Once the constraints and the equations are obtained, the essentials for the optimizing with the mutable smart bee algorithm are prepared. This method will find a suitable answer that is enabling to satisfy all of the constraints. Like any other evolutionary computation methods, the answer which is found by mutable smart bee algorithm is not the definite best answer; actually there are slight differences between them. These differences are usually acceptable and in engineering applications these small differences can be disregarded, Moreover in practical works these answers provide a better performance for the systems comparing to answers which are concluding from experimental works.

The difference between the algorithm answer and the real answer can be extended by finding the local optimization instead of global optimization. For avoiding this matter a suitable probability of mutation is necessary. Indeed mutation it can developed the search space for finding the answer and avoid local optimization. Although mutation is necessary to find a global optimization and seek a wide variety of answers but in latest generations can be reduce the convergence rate. Thus, as the algorithm go ahead, the mutation probability should be decreased for a better convergence in answers. A suitable mutation probability is effective on the speed of the algorithm. All the topics that were mentioned in precede will be shown later. Note that all of algorithms and programs are implemented in Matlab software with a computer with 2.21 GHZ and with 1.00 GB RAM memory.

As an initial setting for running mutable smart bee algorithm, the following values for the basic algorithm parameters were selected: maximum cycle number = 2000, number of colony size (

For bee algorithm (BA) following parameters being set: number of scout bees in hive (

For Lukasik firefly algorithm the parameters set due to [

Initial parameters for self-adaptive penalty function genetic algorithm set as

Arithmetic experiments were repeated 30 times, starting from a random population with different seeds [

At the first step the performance of the Atkinson cycle analyzed in

Performance of tested algorithm in

Parameters | Entropy generation | Power output | CPU time | ||||

MSB-ABC | 360 | 1023 | 2015 | 755.8 | 0.0008 | 0.3912 | 5.2 |

ABC | 360 | 1030 | 2019 | 701.4 | 0.0015 | 0.2331 | 16.7 |

BA | 360 | 1029 | 2081 | 737.0 | 0.0011 | 0.2054 | 18.4 |

IPSO | 360 | 1036 | 2043 | 799.8 | 0.0012 | 0.2759 | 12.7 |

LFFA | 360 | 1039 | 2016 | 789.3 | 0.0013 | 0.2690 | 12.3 |

SAPF-GA | 360 | 1041 | 2090 | 699.2 | 0.0009 | 0.3759 | 22.2 |

Optimum performance reported in [ | 0.0012 | 0.3112 | — |

It is obvious that the proposed algorithm performs better than others and in some cases we find self-adaptive penalty function genetic algorithm (SAPF-GA) as well as proposed MSB-ABC algorithm after 30 times running but this algorithm use more time (22.2 seconds) for reaching to optimum solution comparing to other algorithms because this algorithm hire more than 60 chromosomes for performing efficient search in constraint spaces. As the table shows the MSB-ABC algorithm reached to fitter maximum power output and lower entropy generated during the performance of the Atkinson cycle and also because of hiring just 8 bees for searching in the constraint area of our problem, it takes acceptable CPU time (just 5.2 seconds) for finding the optimal condition. IPSO and LFFA show similar results and also the results show that they consume equal CPU time. Karaboga’s classical artificial bee colony find acceptable solution in this case but as it is shown it takes noticeable time for reaching to fit solution and this matter refers to hiring 30 bees in the searching space. Bee algorithm finds an acceptable amount of entropy generation but it was weak in finding optimum power output and it takes 18.4 seconds for optimizing process. At the end of the first step the power output of the Atkinson cycle and the performance of each algorithms are shown in the following plots and after that the convergence rate of each algorithm and the capability of each of them will be discussed briefly due to obtained plots.

In the first step the performance of each algorithm for finding the maximum power output will be analyzed and the maximum power out will be shown in Figure

Performance of tested algorithm in

In Figure

Comparison of performance of SPFA-GA, L-FFA, I-PSO, BA, ABC, and MSB-ABC in efficiency analyzing of the Atkinson cycle.

Compression of capability of algorithms to escape from restricted area.

The results indicate that MSB-ABC and SAPF-GA are more capable to escape from constraints and also BA and ABC have lower performance to escape from restricted area and spend more time for this process. L-FFA and I-PSO are very similar in beating the tricks both in quality and duration. According to initial setting of the parameters it seems that MSB-ABC must try harder than other algorithms to escape from local convergence because of its low amount of initial searcher agents, but when we set a fit mutation probability and limit, MSB-ABC performs really efficient for escaping from unfeasible region.

Also the performance of the Atkinson cycle will be shown in different compression ratios in Figures _{1} the algorithms found fitter power output comparing to experimental data in [

performance of tested algorithm in

Parameters | Entropy generation | Power output | CPU time | ||||

MSB-ABC | 360 | 954.9 | 2126.7 | 714.6 | 0.0013 | 0.3148 | 10.05 |

ABC | 360 | 908.9 | 2067.0 | 721.6 | 0.0012 | 0.3009 | 15.30 |

BA | 360 | 1007.6 | 2200.0 | 466.7 | 0.0019 | 0.2376 | 13.22 |

IPSO | 360 | 974.3 | 2200.0 | 782.6 | 0.0017 | 0.2444 | 14.12 |

LFFA | 360 | 991.9 | 2112.9 | 689.2 | 0.0012 | 0.3070 | 17.21 |

SAPF-GA | 360 | 992.4 | 2008.1 | 699.1 | 0.0014 | 0.2912 | 44.09 |

Optimum performance reported in [ | 0.0012 | 0.3327 | — |

performance of tested algorithm in

Parameters | Entropy generation | Power output | CPU time | ||||

MSB-ABC | 360 | 1100.1 | 2200.0 | 779.4 | 0.0009 | 0.3253 | 11.05 |

ABC | 360 | 1050.1 | 2138.2 | 726.5 | 0.0008 | 0.3848 | 17.34 |

BA | — | — | — | — | — | — | — |

IPSO | 360 | 1323.2 | 2090.1 | 703.1 | 0.0015 | 0.3155 | 15.01 |

LFFA | 360 | 1125.7 | 1823.9 | 630.0 | 0.0012 | 0.2889 | 14.32 |

SAPF-GA | 360 | 1253.7 | 1902.8 | 570.3 | 0.0010 | 0.2773 | 44.72 |

Optimum performance reported in [ | 0.0012 | 0.3202 | — |

Effect of

Effect of

In the next step, the performance of the Atkinson cycle will be analyzed under

Performance of tested algorithm in

Parameters | Entropy generation | Power output | CPU time | ||||

MSB-ABC | 360 | 953.4 | 2137.6 | 751.8 | 0.0012 | 0.3740 | 12.88 |

ABC | 360 | 881.9 | 2052.1 | 766.4 | 0.0015 | 0.3490 | 19.1 |

BA | 360 | 1615.7 | 2200.0 | 481.7 | 0.0007 | 0.2052 | 15.2 |

IPSO | 360 | 920.9 | 2052.5 | 741.1 | 0.0014 | 0.3464 | 22.3 |

LFFA | 360 | 903.8 | 2095.8 | 757.7 | 0.0012 | 0.3368 | 21.9 |

SAPF-GA | 360 | 892.5 | 2142.8 | 792.5 | 0.0016 | 0.3584 | 34.2 |

Optimum performance reported in [ | 0.0012 | 0.3165 | — |

Again the MSB-ABC shows promising results. The time duration for finding optimal solution is acceptable and also it finds better power output. This time bees algorithm (BA) finds the minimum entropy generation rate, but it was not successful in finding maximum power output. SAPF-GA finds near optimal solution but it performs weaker than other algorithms. In fact it reaches to a local optimum solution. Figure

According to Figures

Performance of tested algorithm in

Comparison of performance of SPFA-GA, L-FFA, I-PSO, ABC, and MSB-ABC in efficiency analyzing of the Atkinson cycle.

Performance of tested algorithm in

Comparison of performance of SPFA-GA, L-FFA, I-PSO, BA, ABC, and MSB-ABC in efficiency analyzing of the Atkinson cycle.

For the last case, the performance of the Atkinson cycle will be checked in

Performance of tested algorithm in

Parameters | Entropy generation | Power output | CPU time | ||||

MSB-ABC | 360 | 992.3 | 1889.3 | 831.1 | 0.0012 | 0.3572 | 10.05 |

ABC | 360 | 1987.6 | 1984.8 | 612.3 | 0.0009 | 0.2368 | 16.23 |

BA | 360 | 1002.2 | 1760.5 | 566.0 | 8.05 | ||

IPSO | 360 | 959.9 | 1730.9 | 698.9 | 0.0013 | 0.3069 | 12.41 |

LFFA | 360 | 864.8 | 2200.0 | 764.8 | 0.0014 | 0.3489 | 12.79 |

SAPF-GA | 360 | 1199.5 | 2112.1 | 618.3 | 0.0012 | 0.3262 | 35.23 |

Optimum performance reported in [ | 0.0012 | 0.3148 | — |

As it is shown, the Bee Algorithm find lower entropy generation, however, it does not find an acceptable power output. LFFA and MSB-ABC perform promising both in maximizing power output and minimizing the unexpected amount of entropy generation. MSB-ABC consumes lower CPU time to find the optimal solution and this feature leads the MSB-ABC algorithm to perform as s superior algorithm in this case.

One of the other important aspects that prove the advantage of MSB-ABC algorithm is the capability of this algorithm to escape from local optimal values. This claim will be demonstrated in the following plots which indicate the rate of convergence for algorithms during the optimizing process.

For analyzing the convergence ratio of these algorithms this parameter should be defined as:

Figure

Convergence rate of MSB-ABC in the first step.

For that we tune the mutation probability as following:

According to Figure

Convergence rate of MSB-ABC in the second step.

At the end the convergence rate of BA, L-FFA, and I-PSO are shown (Figure

Analytic comparison of convergence ratio of BA, LFFA, and IPSO.

According to the results it is obvious that when we use adaptive parameters, MSB-ABC shows better reaction for escaping from local optimum regions comparing to other algorithms. It seems that bee algorithm (BA) suffers from fast local optimum convergence during optimizing process and again L-FFA and I-PSO have similar behavior. The obtained results demonstrate that MSB-ABC algorithm is one of the most applicable algorithms for optimizing multimodal problems since it is capable to balance the intensive local search strategy and an efficient exploration of the whole search space simultaneously.

In this paper, a new method called MSB algorithm proposed for optimizing a well-known multimodal engineering problem, based on the reaction of mutable smart bees during the procedure. Thereafter, proposed algorithm has been compared with some famous optimization methods such as self-adapting penalty function genetic algorithms and improved particle swarm optimization. The results illustrate that MSB algorithm is superior or equal to these existing algorithms for optimizing multimodal problems in most cases. This issue refers to the fine tuning of the parameters that may results efficient searching in feasible space. Furthermore, our simulations indicate that because of adaptive mutation that occurs in smart bee, the algorithm has a suitable convergence rate that leads the algorithm to escape from local optimum solution. Subsequently, it seems that MSB algorithm is more generic and robust for many constraint optimization problems, comparing to other metaheuristic algorithms.

Isobaric molar specific heat (kJ/kg K)

Isochoric specific heat(kJ/kg K)

Effectiveness of hot heat exchanger

Effectiveness of cold heat exchanger

Heat transfer surface area (m^{2})

Molar mass of working fluid (kg/mol)

Heat added to working fluid (kW)

Heat leakage (kW)

Heat rejected from working fluid (kW)

Molar gas constant

Specific compression ratio

Compression ratio

Volume in state one (m^{3})

Volume in state two (m^{3})

Volume in state three (m^{3})

Volume in state four (m^{3})

Output power (kW).

Specific heat ratio

Heat transfer coefficient (kW/Km^{2})

Thermal efficiency

Entropy generation of the cycle.

The authors would like to thank S. Noudeh and P. Samadian for their precious collaboration.