Abstract: In this paper, Turbojet engine will be optimized in ideal condition by multi target genetic algorithm. Target functions are specific thrust (ST), specific fuel consumption (SFC) and thermal efficiency (η t ) that will be optimized simultaneously according to design variables and in two by two way and their Pareto points will be showed. Design variables included inlet Mach number and total compressor pressure ratio. Then according to Pareto points important relations between target functions will be introduced. It is obvious that these relations without using these methods are inaccessible

Khameneh

M Abdollahi

M R Andalibi

P Mohajeri

S H Azizi

CiteSeerX

Australian Journal of Basic and Applied Sciences, 5(12): 1594-1601, 2011 
ISSN 1991-8178 
Corresponding Author: Department of Mechanical Engineering, Bandar Lengeh Branch, Islamic Azad 
University, Bandar Lengeh, 79715-363, Iran. 
  Tel/Fax: +98-711-6322383   Mob: +98-917-7033910  
  E-mail: andalibi.mr@hotmail.com 
1594 
 
Multi Target Optimization of Turbojet Engine with Multi Target Genetic Algorithm 
 
1M.R.Andalibi, 2S.H.Azizi, 1P.Mohajeri.Khameneh, 1M.Abdollahi 
 
1Department of Mechanical Engineering, Bandar Lengeh Branch, Islamic Azad University, Bandar 
Lengeh, Iran. 
2Young Researchers Club, Department of Mechanical Engineering, Bandar Lengeh Branch, Islamic 
Azad University, Bandar Lengeh, Iran. 
 
Abstract: In this paper, Turbojet engine will be optimized in ideal condition by multi target genetic 
algorithm. Target functions are specific thrust (ST), specific fuel consumption (SFC) and thermal 
efficiency (ηt) that will be optimized simultaneously according to design variables and in two by two 
way and their Pareto points will be showed. Design variables included inlet Mach number and total 
compressor pressure ratio. Then according to Pareto points important relations between target 
functions will be introduced. It is obvious that these relations without using these methods are 
inaccessible. 
 
Key words: Genetic Algorithm, Pareto, Multi target optimization, Crossover, Mutation, Turbojet engine. 
 
INTRODUCTION 
 
 As a matter of fact optimization procedure is defined as a way to find numerical collection for design vector 
variables. These are various numerical methods included Gradient methods to find optimum   points. However 
some basic problems such as their great dependency to first assumptions can move the problem toward local 
optimization than absolute optimization. On the other hand in non continues or non derivative functions by 
using gradient methods seems improbable so other optimization methods especially genetic algorithm can solve 
this problem (Goldberg, D.E., 1989; Back, T.D.B., Z. Fogel, 1997).  
 These evolutionary algorithms are inspired by nature and their main deference with old ones is that in these 
methods, we do optimization by function not by their gradients. And they will help us to escape from local 
optimization. In multi target optimization, several targets will be optimized simultaneously. These targets may 
be in disagreement with each other, thus optimum of a target may deteriorate that of another target. Pareto was 
an Italian economist that revealed the context of multi target optimization (Pareto, V., 1896). Pareto points do 
not have any superiority toward each other but comparing to other points, they are superior in research. NSGA 
method was suggested by Deb (1994) and SPEA method was introduced by Zitzler and Thiele (Fonseca, C.M., 
1993).  
 In this paper, design variables such as inlet Mach number and total compressor pressure ratio are 
considered.  
 Selective multi target in ideal subsonic turbo jet included specific thrust, specific fuel consumption and 
thermal efficiency and with considering design variables will be optimized two by two. The results will be 
revealed by Pareto curves. In this paper, our goal is decreasing fuel consumption and increasing thrust and 
thermal efficiency. Can we find a design vector that is minimum in fuel consumption and maximum in thrust 
and thermal efficiency? 
 
2.Turbo Jet Thermodynamic Model: 
 Operating fuel in turbo jet engine is air which by changing in kinetic energy in inlet comparing with outlet 
can create thrust.  
 Ideal turbojet engine equations are shown in table A (Jack D. Mattingly, 1996). Inlet parameters in this 
cycle included flight Mach number (M0), inlet air temperature (T0, K), temperature coefficient(γ ), heating value 
(hpr, kj.kg-1), burner exit total temperature (Tt4,K), total compressor pressure ratio (πc ).  
 Outlet parameters involves specific thrust (ST, N.kg-1.S-1), fuel/air ratio (f), thrust specific fuel 
consumption(TSFC, kg.S-1.N-1 ) and thermal efficiency ( ηt).   
 In this paper hpr=48000 kj.kg
-1, γ=1.4, Tt4=1666K, T0=216.6K. Flight Mach number 0<M0≤1 and total 
compressor pressure ratio 1≤π≤40 are considered as design variables (Jack D. Mattingly, 1996). 
 
 
 
Aust. J. Basic & Appl. Sci., 5(12): 1594-1601, 2011 
1595 
 
3.  Multi Target Genetic Algorithm: 
3.1. Multi Target Optimization: 
 In multi target optimization problems, we are looking for vector design X*=[x*1, x
*
2,…, xn
*]T which is 
member of Rn that target functions are 
 
F=[f1(X), f2(X), f3(X)]T                          (1) 
 
Member of Rk according to m number condition  
 
gi(X)≤0 , i=1,2,…,m (2) 
                                  
And p number of equal condition  
 
hj(X)=0 , j=1,2,…,p (3) 
                            
Will optimize (Pareto, 1896; Oseyezka, A., 1985). 
 
3.2. Defining Predominant Pareto: 
 The vector U=[u1, u2,…, uk]ε Rk is predominant to vector V if and if 
                                                              
(4) jjii vukjvuki  :},...,2,1{},,...,2,1{
     
3.3. Defining optimum Pareto: 
 A point like X*ε Ω (Ω is an accepted design region which satisfy 2, 3 equations) is called optimum Pareto if 
and only if F(X*)<F(X) or on the other hand  
 
(5) 
)()(:},...,2,1{
)()(}{},,...,2,1{
*
1
*
1
*
XfXfkj
XfXfXXki
jj 

 
 
3.4. Defining Pareto Collection: 
 In multi target optimization problems, a Pareto collection (Θ*) included all design vectors of optimum 
Pareto. On the other hand 
 
(6) )}()(:|{ ''* XFXFXX  
 
3.5. Defining Pareto Front: 
 Vectors including target functions which are made from vectors of Pareto collections (Θ*) are called Pareto 
Front. 
 The results of multi target optimization have no superiority toward each other and are called non superior 
results. 
 In figure (1) for example can see the Pareto points, in this figure by moving from A to B (or vise versus), 
any improvement in condition of any target functions can deteriorate the condition of at least one target function 
of problem, (the goal is to minimize or maximize both target functions).  
Pareto optimum points almost are located in boundary lines of design region or are over lapped points of target 
functions. In figure (1) the bold line shows such boundary line of two target functions which its component 
points are called Pareto Front. 
 
 
Fig. 1: Pareto points in a curve form. 
Aust. J. Basic & Appl. Sci., 5(12): 1594-1601, 2011 
1596 
 
3.6. Multi Target Genetic Algorithm (NSGAІІ): 
 First stage of this method, N is the primary population which is generated randomly (Rt). Then these 
populations will be categorized and vectors which satisfy the condition of equation (4) will be categorized in 
lower levels, then among these populations, some populations will be selected randomly for crossover and 
mutation. The population which is created by crossover and mutation (Qt) will be added to primary population 
and again the total population will be classified.   
 The base of NSGAІІ is that while reproduction is continuing, the number of population must be constant, so 
the population in higher levels will be eliminated. 
 The end of a reproduction is an improved population. Diagram (a) shows a reproduction in NSGAІІ. 
 
 
 
Fig. a: Reproduction procedures. 
 
4.  Multi Target Optimization of Turbo Jet Engine With Multi Target Genetic Algorithm Two by Two Way: 
4.1. Optimization According to Thermal Efficiency And Specific Fuel Consumption Target Functions  
(ηt, SFC): 
 Figure (2) is the collection of points resulting from optimization according to two target functions SFC and 
ηt which is generated by NSGAІІ. In this figure five optional design vectors are shown.  
 By comparing design vectors 2 and 3, it can be proved that by increasing 94% in SFC, thermal efficiency 
will be increased only 13.6%. Since our target is decreasing SFC and increasing ηt, so it obvious that design 
vector 2   comparing with 3 is more important in design. Also by comparing design vectors 2 and 4, we can 
show that by increasing 16.1% in SFC, thermal efficiency will be increased 29.8%. 
 Why multi target optimization is superior to single target optimization? 
 In optimization which was according to ηt and SFC, we see that increasing thermal efficiency can lead to 
increasing fuel consumption, since the target is increasing ηt and decreasing SFC, if we do single target 
optimization, the results will be points one and five. While the point one has the minimum SFC and thermal 
efficiency and point five has the maximum SFC and thermal efficiency. It should be mentioned that point one is 
according to single target optimization of SFC and point five is according to single target optimization of ηt. 
 
4.2. Optimization According to Specific Thrust and Thermal Efficiency Target Functions (ST,ηt): 
 Figure (3) shows the Pareto points according to ST and ηt. With comparing the design vectors one and two, 
we can see that with increasing 21.6% in specific thrust, thermal efficiency will increase only 5.68%. Also by 
comparing design vectors two and three, we consider that with increasing 11.9% in specific thrust, thermal 
efficiency will decrease 36.2%. So in design viewpoint, point two is more valuable than point three. Design 
vector one amount all design vectors has the least ST and the most ηt, while design vector seven has the most ST 
and the least ηt. The design vector one is made in single target optimization ηt and design vector seven is made 
in single target optimization ST. 
 
4.3. Optimization According to Specific Thrust And Specific Fuel Consumption Target Functions (ST,SFC): 
 Our goal is finding design vector with the least SFC and the most ST. Figure (4) shows the Pareto points 
according these two target functions that obtained from multi genetic algorithm. Considering this figure, we 
understand that increasing ST will happen with increasing SFC.  
 On the other hand optimization a target function can cause to deteriorate another one. (Decreasing SFC can 
cause decreasing ST). With comparing design vectors two and three, we find that 0.1% increasing in ST, SFC 
will increase 0.93% and it means that points two and three almost have a same value in design. 
Aust. J. Basic & Appl. Sci., 5(12): 1594-1601, 2011 
1597 
 
SFC&ettaT
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.00001 0.00002 0.00003 0.00004 0.00005 0.00006 0.00007 0.00008
SFC
et
ta
T
∆SFC
∆etta T
1
2
3
4
5
 
 
Fig. 2: Pareto points of thermal efficiency and fuel consumption. 
 
ST&etteT
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
690 740 790 840 890 940 990 1040 1090
ST
et
ta
T
1
3
2
4
5
6
7
∆ettat T
∆ST
 
 
Fig. 3: Pareto points of thrust and thermal efficiency. 
 
5. Optimization According to Three Target Functions (ST, SFC, ηt): 
 Pareto points are shown two by two way. It's obvious that these design vectors which are product of 
optimization are representative of 3D curve that are showed two by two way. Similar points to one number are 
shown in figures (5), (6), (7). 
 
Aust. J. Basic & Appl. Sci., 5(12): 1594-1601, 2011 
1598 
 
ST&SFC
0.0000204
0.0000206
0.0000208
0.000021
0.0000212
0.0000214
0.0000216
1084.5 1085 1085.5 1086 1086.5 1087 1087.5 1088 1088.5
ST
S
F
C
1
2
3
4
∆SFC
∆ST
 
Fig. 4: Pareto points of thrust and fuel consumption. 
 
3SFC&ettaT
0
0.1
0.2
0.3
0.4
0.5
0 0.00002 0.00004 0.00006 0.00008
SFC
et
ta
T
12
3
4
 
 
Fig. 5:Pareto points of thermal efficiency and fuel consumption according to three target functions. 
 
3ST&ettaT
0
0.1
0.2
0.3
0.4
0.5
0 200 400 600 800 1000 1200
ST
et
ta
T
1 2
3
4
 
 
Fig. 6: Pareto points of thermal efficiency and thrust according to three target functions. 
 
 
Aust. J. Basic & Appl. Sci., 5(12): 1594-1601, 2011 
1599 
 
3ST&SFC
0
0.00001
0.00002
0.00003
0.00004
0.00005
0.00006
0.00007
0.00008
420 620 820 1020
ST
S
F
C
1
2
3
4
 
Fig. 7: Pareto points of thrust and fuel consumption according to three target functions. 
 
 Comparing two and three design vectors, it can be inferred that by 18% increasing ST, SFC will decreased 
to 8% but thermal efficiency will fall to 47%. 
 By comparing one and two design vectors, we conclude that increasing 85% in ST will be accompanied 
with loss of 5.6% in thermal efficiency and fall of 66% in SFC. So it seems that design vector two is 
considerably more important than design vector one.  
 
Conclusions:  
 Several functions may optimize simultaneously in optimization. Now if we consider optimization problem 
as a single target, we can not see the results of the other functions in optimum vector. So the optimum point of a 
target function may be the weak point of others. 
 In high pressure coefficient 30<πc≤40 and 0<M0≤0.98, specific thrust and fuel consumption will be 
improved but in this range, we can't expect that thermal efficiency be more than 42%. 
 Around M0=1 and in low pressure coefficient (1≤πc<2.1), we can reach to the highest thermal efficiency 
(60%), however in this range fuel consumption will be in its worst condition (4.55897≤SFC×105≤7.3457). 
 It obtained from Pareto points that fuel consumption is effected by pressure coefficient than Mach number 
(Table 1).             
 According to Pareto figures which are shown in this paper, we can conclude that SFC is almost proportional 
with square of ST (SFC α ST2) and thermal efficiency is proportional with ST (ηt α ST). Other important results 
are summarized two by two in table (1). 
 
Table 1: 
STt ,
1074705
44.00


ST
t  
0M                 11.0 0  M
c                     345.2  c  
SFC,ST 
10891084
2.21005.2 5


ST
SFC  
0M                        1.00 M  
c                      4030  c  
SFCt ,
34.71056.4
6.047.0
5 

SFC
t
34.21006.2
42.00
5 

SFC
t  
1.21
10


c
M

      Flight Mach number )( 0M  98.00 0  M  
Pressure ratio )( c                     40c  
 
 
 
Aust. J. Basic & Appl. Sci., 5(12): 1594-1601, 2011 
1600 
 
 Finding results included specific boundaries for target functions and design variables. For example if 
optimization is according to ηt, SFC: if 0< ηt<0.42, fuel consumption must be to [2.056e-5, 2.344e-5] and flight 
Mach number must be [0, 0.98] and total compressor pressure ratio must be 40. 
 
Index 
0a     Velocity of sound at inlet…………………..m/s           
0m     Mass flow rate…………………………..….kg/s       
4tT     Burner exit total temperature………………K                                                                                 
0T      Inlet temperature……………………………K 
prh    Heating value …………………………..….kJ/kg 
0V     Air velocity at inlet…………………..…….m/s           
cg     Newton's constants………………………...kg-m.      
R       Gas constants………………………………J/kg.K      
F        Thrust……………………………….……..N 
 
0M     Flight Mach number                                                     
r     Total static temperature ratio at inlet             
t      Burner exit/inlet total temperature ratio             
     Burner exit total enthalpy/inlet total enthalpy 
c  Compressor exit total temperature/Compressor inlet temperature 
  f         Fuel/air ratio 
  ST      Specific thrust 
SFC      Specific fuel consumption        
 t       Thermal efficiency                 
*X     Vector of optimal design variables 
c      Total compressor pressure ratio                                   
F(X)    Vector of objective function 
 
REFERENCES 
 
Arora, J.S., 1989. Introduction to Optimum Design, McGraw-Hill, New York. 
Back, T.D.B., Z. Fogel, 1997. Hand book of Evolutionary Computation, Oxford University press, New 
York/Oxford. 
Fonseca, C.M., 1993. P.J. Fleming, Genetic Algorithms for multiobje-ctive optimization: 
Formulation,discussion,san Mateo, CA,  pp: 416-423. 
Goldberg, D.E., 1989. Genetic-Algorithms in search, Optimization, and Machine Learning, Addison-
Wesley, Reading, MA. 
Jack D. Mattingly, 1996. Elements of Gas Turbine Propulsion, McG-raw-Hill series in mechanical 
engineering. 
Oseyezka, A., 1985. Multicriteria optimization for engineering design, in:J.S Gero(Ed.), Design 
optimization, Acadamic press, New York, pp: 193-227.  
Pareto, 1896. coursd's economic politique, Rouge, Lausanne, Switzerland. 
Pareto, V., 1896. Cours d'economic politique, Lausanne, Switzerland. 
Rao, S.S., 1996. Engineering Optimization: Theory and Practice, Wiley, New York. 
Srinivas, N., K. Deb, 1994. Multiobjective optimization using nondo-minated sorting in genetic algorithms, 
Evolutionary Comput., 2(3): 221-248. 
 
Table (A): 
Turbojet equations (Oseyezka, A., 1985). 
Aust. J. Basic & Appl. Sci., 5(12): 1594-1601, 2011 
1601 
 
cr
t
cr
pr
p
c
tcr
cr
c
r
t
cc
t
r
c
P
ST
f
SFC
h
TC
f
M
a
V
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a
m
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ST
a
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Multi Target Optimization of Turbojet Engine with Multi Target Genetic Algorithm

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Multi Target Optimization of Turbojet Engine with Multi Target Genetic Algorithm

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