https://doi.org/10.31449/inf.v44i1.1890 Informatica 44 (2020) 23–33 23 
 
Design Optimization Average-Based Algorithm 
João Barradas Cardoso and Artur Barreiros 
Department of Mechanical Engineering, Instituto Superior Técnico, University of Lisbon 
Av Rovisco Pais 1, 1049-001 Lisbon, Portugal 
E-mail: barreiros@tecnico.ulisboa.pt, https://tecnico.ulisboa.pt/en/ 
Keywords: heuristic, randomness, population-based, optimization 
Received: October 6, 2017 
This article introduces a metaheuristic algorithm to solve engineering design optimization problems. 
The algorithm is based on the concept of diversity and independence that is aggregated in the average 
design of a population of designs containing information dispersed through a variety of points, and on 
the concept of intensification represented by the best design. The algorithm is population-based, where 
the population individual designs are randomly generated. The population can be normally or uniformly 
generated. The algorithm may start either with points randomly generated or with a designer preferred 
trial guess. The algorithm is validated using standard classical unconstrained and constrained 
engineering optimum design test problems reported in the literature. The results presented indicate that 
the proposed algorithm is a very simple alternative to solve this kind of problems. They compare well 
with the analytical solutions and/or the best results achieved so far. Two constrained problem analytical 
solutions not found in the literature are presented in annex. 
Povzetek: Članek uvaja metahevristični algoritem za reševanje problemov optimizacije. Algoritem 
temelji na populaciji, kjer se naključno generirajo posamezni modeli posameznika in končno agregirajo 
v primerne rešitve. 
1 Introduction 
With the advent of fast, cheap and reliable computing 
power over the last decades, in addition to the application 
of classic optimization to larger and larger size problems, 
new alternative algorithms operating in a different 
fashion have been developed. The classical optimization 
algorithms have shortcomings and are not suitable for all 
optimization problems. The new alternative algorithms 
allow attacking optimization problems either too costly 
or not applicable to classical algorithms. 
The purpose of heuristic algorithms applied to 
optimization problems is to search a solution to them by 
trial-and-error in a satisfying amount of computing time. 
The optimum solution is not guaranteed but a near 
optimum solution is accepted as a good solution. 
Metaheuristics refers to higher-level algorithms 
combining lower-level techniques and tactics for 
exploration and exploitation of the design space. That is, 
these algorithms, on one hand, must be able to generate a 
range of points in the whole design space including 
potentially optimum ones; on the other hand, they 
intensify the search around the neighborhood of an 
optimum or near optimum points (Yang, 2008). 
Exploration and exploitation are the two important 
components of the metaheuristic algorithms. They are 
also called diversification and intensification (Glover, 
Laguna, 1997). A good balance of these components is 
required. Too much weight in diversification risks slow 
convergence with the solutions jumping around the 
potentially optimum ones; too much weight in 
intensification restricts the design space to a local region 
and risks the convergence to a local optimum (Blum, 
Roli, 2003). The heuristic algorithms start typically either 
with guess solutions randomly generated or with a 
designer preferred trial solution. The diversification is 
gradually reduced as the algorithm proceeds; 
simultaneously, the intensification is increased. 
One of the roles of injected randomness in stochastic 
search is to allow for movements to unexplored areas of 
the search space that may contain an unexpectedly good 
design. This is especially relevant when the search is 
stalled near a local solution. Injected randomness may 
also be used for the creation of simple random quantities 
that act like their deterministic counterparts, but which 
are much easier to obtain and more efficient to compute. 
Metaheuristics algorithms are either population-
based or trajectory-based. Examples of the former ones 
are the genetic algorithms (Holland, 1975) or the particle 
swarm optimization (Cagnina, Esquivel, Coello, 2008; 
Kayham, Ceylan, Ayvaz, Gurarslan, 2010) the last ones 
include the simulated annealing optimization 
(Kirkpatrick, Gelatt, Vecchi, 1983) or the harmonic 
search (Geem, Kim, Loganathan, 2001). 
If a sufficient numerous and diversified group of 
people is asked to decide on subjects of general interest, 
the decisions of the group are better than the decisions 
that an isolated individual would take (Surowiecki, 
2005). Well informed and sophisticated an expert is, his 
or her advice and predictions should be pooled with those 
of others to get the most out of him or her. Practical 
examples, simple and complex, are described in 
(Surowiecki, 2005), remarking the principle of group 
think, and the concept that the masses are better problem 

24 Informatica 44 (2020) 23–33 J.B. Cardoso et al. 
 
solvers, forecasters, and decision makers than any one 
individual. A classic example of group intelligence is the 
jelly-beans-in-the-jar experiment, in which invariably the 
group’s average estimate for the number of the jelly 
beans in the jar is superior to the vast majority of the 
individual guesses. 
The theory that groups are remarkably intelligent and 
often smarter than the smartest people in them, 
demonstrates the significant impact on how businesses 
operate, how knowledge is increased, how economies are 
structured, and how people live their daily lives 
(Williams, 2006). The necessary conditions for a crowd 
to be wise include diversity, independence, and a specific 
type of decentralization. These conditions are essential to 
making good decisions which are the result of 
disagreement and contest rather than consensus or 
compromise. 
Diversity means individuals have some private 
information or their own interpretation of known facts. 
Diversity helps because it actually adds perspectives that 
would otherwise be absent and because it takes away, or 
at least weakens, some of the destructive characteristics 
of group decision-making. Homogeneous groups are 
great at doing what they do well, but they become 
progressively less able to investigate alternatives. 
Independence means freedom from the influence of 
others. It keeps the individual mistakes from becoming 
correlated. Diversity is essential to preserving this 
independence. In the jelly-beans-in-the-jar experiment, 
most group members are not talking to each other or 
solving problems together. 
Decentralization means people draw on local 
knowledge. It encourages individuals to make important 
decisions, not just in one location based only on one 
specific type of information, but dispersed through a 
variety of locations from where local knowledge is 
drawn and shared. The information coming out of a 
decentralized group must be aggregated throughout the 
system, to maintain a balance between local and global 
counterparts. Aggregation needs a mechanism that turns 
individual judgments into a collective decision. For 
instance, in a free market, the aggregating mechanism is 
price; in the jelly-beans-in-the-jar experiment, individual 
guesses were aggregated and then averaged, i.e., the 
aggregating mechanism is the average guess. 
The aim of the present article is the proposal of an 
algorithm based on the concept just described. The 
algorithm considered is stochastic in the sense that it 
relies on random numbers and that different results may 
be obtained upon running the algorithm repeatedly. The 
algorithm is population-based, where the population 
individual designs are randomly generated. These 
individuals are diversified and independent since their 
design variables values are chosen stochastically without 
any correlation. 
The individual designs are also decentralized since 
the design variables are chosen all over the entire design 
space. Finally, the different values of the design variables 
are aggregated as the plain or weighted average of those 
values. However, generating a diverse set of possible 
solutions isn’t enough. The designer, as the body of 
people, also has to be able to distinguish the good 
solutions for the bad. So, at each of the iterations, the 
present algorithm selects two designs: the best design, 
the one with the best objective so far; and the averaged 
design, the one which design variable values are the 
mean of those variable values for the iteration. 
The best design represents the intensification 
component of the algorithm; the averaged design 
represents the diversity part. In the current article, the 
optimization problem is understood as a minimization 
problem, where the function of merit to be assessed is an 
extended cost function that takes in account penalization 
due to violated constraints. A reference design is 
considered as a linear combination of both the best and 
the averaged design variables. A simple recurrence 
formula, centered on the reference design, is used to 
actualize the design for the next randomly generated 
population. The population can be normally or uniformly 
generated. 
In optimization, there is traditionally a concern with 
developing a good stopping criterion. Unfortunately, the 
quest for an automatic means of stopping an algorithm 
with a guaranteed level of accuracy seems doomed to 
failure in general stochastic search problems. The 
fundamental reason is that in nontrivial problems, there 
will always be a significant region within the design 
space that will remain unexplored in any finite number of 
iterations, and there is always the possibility the optimum 
could lie in this unexplored region. 
A danger arises in making broad claims about the 
performance of an algorithm based on the results of 
numerical studies. Performance can vary tremendously 
under even small changes in the form of the functions 
involved or the coefficient settings within the algorithms 
themselves. Outstanding performance on some types of 
functions is consistent with poor performance on other 
types of functions. This is a manifestation of no free 
lunch theorems (Spall, 2003; Wolpert, Macready, 1997). 
The present algorithm is applied to several 
constrained and unconstrained test functions as well as to 
typical engineering design problems. 
2 The optimal design problem 
The optimal design problem may be formulated in a 
generalized fashion as 
 
( )
( )
0
min
..
0; 1,2,...,
; 1,2,...,
j
il i iu
st
jm
b b b i n

  =
  =
b
b
b
 (1) 
where 
0
 is the cost or objective function, 
j
 are the 
constraints in a dimensionless form, ( )
12
, ,...,
n
b b b  b is 
the design vector, 
il
b and 
iu
b are respectively the lower 
and upper bounds of the design variable 
i
b . 
In order of applying the present algorithm, the 
formulation of the Eq. (1) is replaced by the following 
one: 

Design Optimization Average-Based Algorithm Informatica 44 (2020) 23–33 25 
 
 
0 0 0
b
min
..
; 1,2,...,
il i iu
P
st
b b b i n
   + 
  =
 (2) 
In Eq. (2), P stands for the penalty factor which 
value depends on the violation of the constraints as 
 
1
0, for satisfied constraints
, for violated constraints
m
P
m




=


=

 



 (3) 
where 0   and m  is the number of violated 
constraints. In practical terms, the constraint 
j
 can be 
considered violated if 
j
  , with  a very small 
number. Note that the absolute value of the objective is 
considered in the Eq. (2) in order to accommodate 
problems with negative objective functions. 
3 The average concept algorithm 
The present algorithm may be described by the flowchart 
of Fig. 1 as well as in the following manner: 
1. Start with a preference design guess b and an 
imposed standard deviation, estimated as 
i iu il
s b b =− 
( ) 1, 2,..., in = . Compute the starting values of 
0
 , 
j
 ( ) 1, 2,..., jm = and 
0
 , and establish the 
starting reference design 
R
b as the preference guess, 
i.e. 
R
 bb . 
 Start iterates as 
2. Launch a normally or uniformly random population 
of N designs as 
 
k R k
i i i i
b b s r =+ (4) 
 where 1,2,..., in = , 1, 2,..., kN = and 
k
i
r is the k-th 
random number (with mean value 0 and standard 
deviation 1) related to the design variable 
i
b . 
 Do 
i il
bb = if 
i il
bb  and 
i iu
bb = if 
i iu
bb  . 
3. Evaluate 
0
 , 
j
 ( ) 1, 2,..., jm = and 
0
 for the 
entire population of N designs. 
4. Evaluate the best design 
B
b corresponding to the 
minimum value 
min
0
 of the extended function. 
 Evaluate the averaged design 
A
b for the distribution 
of designs as 
 
11
NN
Ak
kk
kk
pp
==
=

bb (5) 
 where 
k
b is an arbitrary design vector in the 
population and 
k
p is the weight accounted for design 
k
b on the average. The weights are selected by the 
designer. If a plain average is chosen, then 1
k
p = for 
every design. 
 Another choice used in this work is to compare the 
 
Evaluate the Best design and 
the Averaged design 
Compute starting objective, 
Constraints and Extended 
objective 
END 
Optimum design 
Optimum objective 
Constraint violation? 
Objective improvement? 
it = it + 1 
Starting design 
Starting standard deviation 
Iteration (it) = 0 
 
Launch a random population 
of designs 
Compute current objectives, 
Constraints and Extended 
objectives 
Evaluate the Reference design 
by linear combination best and 
averaged design 
N 
Y 
 
Figure 1: Flowchart of the algorithm. 
θ 
Starting Point 
(-2.5, -2.5) (-2.5, 2.5) (2.5, -2.5) (2.5, 2.5) 
1.00 
(0.08986,-0.71283) 
-1.0316283 
at iteration 38 
(0.09140,-0.71266) 
-1.0316191 
at iteration 39 
(-0.09162,0.71276) 
-1.0316163 
at iteration 36 
(0.08887,-0.71247) 
-1.0316248 
at iteration 31 
0.85 
(0.08979,-0.71267) 
-1.0316285 
at iteration 15 
(0.08997,-0.71265) 
-1.0316285 
at iteration 21 
(-0.08991,0.71265) 
-1.0316285 
at iteration 21 
(-0.08976,0.71272) 
-1.0316285 
at iteration 15 
0.25 
(0.08973,-0.71262) 
-1.0316285 
at iteration 36 
(-0.08992,-0.71259) 
-1.0316285 
at iteration 29 
(0.08974,-0.71262) 
-1.0316285 
at iteration 26 
(-0.08973,0.71262) 
-1.0316285 
at iteration 37 
0.00 
(0.09002,-0.71401) 
-1.0316136 
at iteration 989 
(-0.09020,0.71196) 
-1.0316237 
at iteration 617 
(0.09020,-0.71196) 
-1.0316237 
at iteration 617 
(-0.09002,0.71401) 
-1.0316136 
at iteration 989 
Table 1: Optimum points and function values for different θs and starting points. 

26 Informatica 44 (2020) 23–33 J.B. Cardoso et al. 
 
values of the extended function 
0
 and of the best 
extended function 
min
0
 in the previous iteration; and 
then assign a weight  2
k
p = if the first value is 
smaller than the second one, and a weight 1
k
p = if it 
is equal or larger. 
5. If there are no constraint violations and there are no 
improvements of the objective function within a 
prescribed number of iterations, go to 7. 
6. Evaluate a reference design as the linear combination 
 ( ) 1
R B A
 = + − b b b (6) 
 with 01   and assume the distribution of designs 
k
b centered at 
R
b with a standard deviation vector 
evaluated as 2
AB
=− s b b . 
 Go to 2. 
 End the iterates. 
7. Stop. 
In order of handling tabular discrete value design 
variables, the Eq. (4) is rewritten in these cases as 
 int
Rk
k i i i
i
b s r
b
 +
=



 (7) 
where  is the difference value between two consecutive 
design variables. 
4 Numerical applications 
In this Section one is going to solve unconstrained as 
well as constrained optimization problems by applying 
the formulation and the algorithm presented on the 
previous sections. With respect to the unconstrained 
problems, the applications are the minimizations of the 
benchmark following test functions: Six-Hump 
Camelback function, Rosenbrock and Michalewicz’s 
functions. 
Concerning to the constrained problems, one solves 
three well-known engineering design optimization test 
problems: the welded beam design, the pressure vessel 
design and the tension-compression spring design. For all 
the applications normally distributed populations were 
used. A total of 30 independent runs were performed per 
problem. In the Sections 4.1 to 4.6 the runs of the 
algorithm were performed with a particular initial seed 
for different population sizes and different weights on 
the calculation of the average and reference designs. 
The results are compared with analytical solutions 
and/or heuristic and nonlinear programming algorithms. 
4.1 Six-Hump Camelback function 
This function is one of the typical test functions in 
unconstrained global optimization (Dixon, Szego, 1975; 
Lee, Geem, 2005). It is mathematically expressed as 
 
2 4 6 2 4
0 1 1 1 1 2 2 2
1
4 2.1 4 4
3
x x x x x x x  = − + + − + (8) 
Within the bounded region, this function has six 
local minima. Two of them are global minima located at 
either ( ) 0.08984,0.71266 − or ( ) 0.08984, 0.71266 − , 
each with the corresponding function value equal to 
0
1.0316285

 = − . 
For the algorithm presented here, a population of 
size 20 N = and a plain average in the step 4 were used 
within the design space 
12
2.5 , 2.5 xx −   . For different 
values of the θ-factor and two starting points, the Table 1 
gives the corresponding achieved solutions and the 
number of iterations needed for convergence. 
We should note that the best and fastest solutions are 
obtained with 0.85  = , i.e., giving respectively at each 
iteration, the weights 0.85 and 0.15 to the best and 
average points of the prior iteration in the formation of 
the current reference design. 
With the starting point ( ) 2.5, 2.5 , corresponding to 
the cost value 
0
161.8489685 = , and with 0.85  = , 
the present algorithm achieves the minimum cost value 
0
1.0316285

 = − at the point ( ) 0.08976,0.71272 − after 
15 iterations. However, after 9 iterations, the algorithm 
gets the cost value 
0
1.0316285  = − at the point 
( ) 0.08799,0.71382 − , not far off the analytical solution. 
We should also note that the worst results exist for 
0  = , i.e., only considering the average design as the 
design of reference. For 1  = , respecting to the selection 
at each iteration of the best design as the reference one, 
the solutions are also not so good 
For 20 N = , 0.85  = , starting point ( ) 2.5, 2.5 and 
the weighted average (second) option on step 4 described 
in Chapter 3 instead of the plain average, the global 
optimum 
0
1.0316285

 = − is obtained at the point 
( ) 0.08988, 0.71266 − after 19 iterations. 
Better results can be expectedly obtained by 
increasing the population size. For 100 N = , the best and 
fastest solutions are obtained with 0.85  = after 10 
iterations at the points ( ) 0.08982, 0.71264 − and 
( ) 0.08983, 0.71264 − respectively starting from the 
points ( ) 2.5, 2.5 − and ( ) 2.5, 2.5 − , corresponding both 
to the minimum function value 
0
1.0316285

 = − . For 
N IP Iterate 
1
x 
2
x 
0
 
1000 
0 4 -0.0897507 0.7126449 -1.0316285 
1 5 -0.0898749 0.7127146 -1.0316285 
100 
0 13 -0.0897433 0.7126887 -1.0316285 
1 11 -0.0898545 0.7126102 -1.0316285 
20 
0 15 -0.0897623 0.7127175 -1.0316285 
1 19 -0.0898786 0.7126648 -1.0316285 
Table 2: Camelback’s optimal solutions for 0.85  =  
and starting at ( ) 2.5, 2.5 . 

Design Optimization Average-Based Algorithm Informatica 44 (2020) 23–33 27 
 
1000 N = and 0.85  = , and starting from the point 
( ) 2.5, 2.5 −− , the optimum solution 

= x 
( ) 0.08975, 0.71264 − , 
0
1.0316285

 = − has been 
achieved after 4 iterations. 
Table 2 shows the optimal solutions achieved for 
different N population sizes and different average 
choices on Eq. (5), 
0 IP =
 standing for plain average and 
1 IP =
 meaning weighted average as written on step 4 of 
the algorithm described on Chapter 3, using 
0.85  =
 
and with the starting point located at 
( ) 2.5, 2.5
. 
We may note that is the population size 20 N = the 
one that needs the smallest number of function 
evaluations ( ) 20 15  . 
4.2 Rosenbrock’s function 
Another classical test function in unconstrained 
optimization is the Rosenbrock’s function, which two-
dimensional form (Moré, Garbow, Hillstrom, 1981; 
Rosenbrock, 1960; Yang, 2008) is 
 ( ) ( )
2
2
2
0 1 2 1
1 100 x x x  = − + − (9) 
This function also referred to as the Valley or 
Banana function due to the shape of its contour lines, is a 
popular test problem for gradient-based optimization 
algorithms. The function turns out to be quite challenging 
to find its minimum point by numerical methods. 
Its global optimum point is ( ) 1.0, 1.0

= x that gives 
the optimum cost of 
0
0.0

= . The function is unimodal 
and its analytical solution can be obtained 
straightforwardly by partial differentiation. The 
numerical solution, however, poses a particular 
challenge. The solution lies inside a very deep, narrow, 
banana shaped valley. The valley causes a lot of 
troubling for nonlinear programming search algorithms. 
Using a population of 1000 samples within the 
design space 
12
10.0 , 10.0 xx −   , a plain average, the 
factor 0.85  = and a starting point ( ) 0.0, 0.0 = x 
corresponding to 
0
1.0000000 = , the present algorithm 
achieves the optimum point ( ) 1.00000, 1.00000

= x and 
the corresponding 
0
0.0000000

= after 18 iterations. 
However, the point ( ) 1.00051, 1.00087 = x 
corresponding to 
0
0.0000029

= was obtained after 9 
iterations. If 100 N = , the same optimum point above is 
obtained after 38 iterations. For 20 N = , the algorithm 
converges towards the same optimum point after 2604 
iterations. One may say that is the population of 100 
samples that gives the shortest number of function 
evaluations ( ) 100 38  . 
4.3 Michalewicz’s function 
The third unconstrained optimization problem uses the 
Michalewicz’s function in its two-dimensional form 
 ( ) ( )
2
20
2
0
1
sin sin
ii
i
x i x 
=

 = −


 (10) 
with 
12
0, xx   . 
The function is tricky; it has several local minimum 
values and several flat areas which make the one global 
minimum value hard to find numerically. Its global 
minimum is 
0
1.8012980

 = − at the point 

= x ( ) 2.20319, 1.57049 . 
Using a population with size 20 N = samples within 
the design space 
12
0.0 , 4.0 xx  , a plain average, the 
factor 0.85  = , and a starting point ( ) 2.0, 2.0 = x 
corresponding to 
0
0.3701513  = − , the present 
algorithm achieves the optimum point 
( ) 2.20281, 1.57079

= x and the corresponding cost 
value 
0
1.8013037

 = − after 24 iterations. However, 
after 13 iterations the algorithm achieves the point 
( ) 2.20258, 1.56925 = x corresponding to the objective 
value 
0
1.8012055  = − . Changing the population size to 
100 N = , the global minimum 
0
1.8013039

 = − is 
achieved after 10 iterations at the point 
( ) 2.20289, 1.57091

= x . Changing now the population 
size to 1000 N = , the global minimum 
0
1.8013039

 = − is achieved after 8 iterations at the 
point ( ) 2.20292, 1.57076

= x . The smallest number of 
function evaluations ( ) 20 24  is obtained for 20 N = . 
4.4 Welded beam design 
The welded beam design problem is well studied in the 
context of single-objective optimization. A beam A needs 
to be welded on another beam B and must carry a certain 
load P as shown in Fig. 2. The welded beam is designed 
for minimum fabrication cost subject to constraints on 
 
t 
h 
P 
b 
l 
L 
A 
B 
 
Figure 2: Welded beam structure. 

28 Informatica 44 (2020) 23–33 J.B. Cardoso et al. 
 
shear stress τ, bending stress in the beam σ, buckling 
load on the bar 
c
P , end deflection of the beam δ, cost of 
the weld and beam A materials, and side constraints 
(Ragsdell, Phillips, 1975; Rao, 1996). One wants to find 
four design parameters: thickness of the beam b, width of 
the beam t, length of the weld l, and thickness of the 
weld h. 
In order of formulating the problem in a standard 
form, let ( )
1 2 3 4
, , , x x x x  x ( ) ,,, h l t b = be the design 
vector. Then, our problem may be described as 
 
( ) ( )
( )
2
0 1 2 1 2 3 3 4 2
1 max
2 max
3 1 4
2
4 1 1 3 3 4 2
5 max
6
1
4
12
min
..
10
10
10
5 1 0
10
10
0.125 2
0.1 2
0.1 , 10
c
C C x x C x x L x
st
xx
C x C x x L x
PP
x
x
xx



  + + +
  − 
  − 
  − 
   + + − 

  − 
  − 



 (11) 
where 
3
1
0.10471$/ in C = is the cost per unit volume of 
the weld material, 
3
2
1$ / in C = is the labor cost per unit 
weld volume, 
3
3
0.04811$ / in C = is the cost per unit 
volume of the beam B, 
max
13,600 psi  = , 
max
30,000 psi  = and 
max
0.25 in  = . The other 
parameters are defined as: 
22
1 1 2 2 2
xR      = + + , 
( ) 1 1 2
2 P x x  = , 
2
M R J  = , ( )
2
2 M P L x =+ , 
( )
2
2
2 1 3
0.5 R x x x = + + , 
( )
3
34
6PL x x  = , 
( )
33
34
4PL E x x  = , 
( )
( )
2
2
1 2 2 1 3
2 6 3 J x x x x x

= + +

, 
( ) ( ) ( )
23
3 4 3
4.013 6 1 0.5 0.25
c
P E L x x x L E G

=−

, 
6
30 10 psi E= , 
6
12 10 psi G= , 6,000 lb P = , 
14 in L = 
By using mathematical programming, (Rao, 1996) 
presents the optimum cost function 
0
2.3810

= 
corresponding to the design point 

= x 
( ) 0.2444, 6.2177, 8.2915, 0.2444 . A better nonlinear 
programming solution has been achieved in (Andrei, 
2013) by adding GAMS created nonlinear model: 
0
1.72485

= , ( ) 0.206, 3.470, 9.037, 0.206

= x . 
The lowest optimal solution known so far by the 
N IP Iterate 
1
x 
2
x 
3
x 
4
x 
0
 
5000 
0 
16645 0.205729 3.470519 9.036630 0.205730 1.724858 
268 0.205468 3.476687 9.038588 0.205723 1.725573 
32 0.205436 3.477702 9.034963 0.205964 1.726863 
16 0.204191 3.530001 9.042473 0.205769 1.731815 
1 
11144 0.205726 3.470586 9.036639 0.205730 1.724864 
303 0.204957 3.488440 9.035478 0.205797 1.726386 
57 0.205409 3.475931 9.042488 0.205808 1.726697 
19 0.204431 3.523721 9.041501 0.205707 1.730704 
1000 
0 
9434 0.205727 3.470567 9.036656 0.205730 1.724866 
27 0.204332 3.495317 9.051999 0.205779 1.729056 
19 0.204359 3.503654 9.084193 0.205596 1.734414 
1 
10419 0.205730 3.470530 9.036634 0.205730 1.724866 
1335 0.205707 3.470647 9.038382 0.205760 1.725373 
522 0.205451 3.477778 9.038218 0.205742 1.725777 
37 0.205635 3.479632 9.039730 0.205804 1.727052 
21 0.206227 3.492924 9.013144 0.206816 1.732874 
100 
0 
15080 0.205727 3.470617 9.036695 0.205729 1.724877 
1596 0.205693 3.470650 9.039433 0.205731 1.725315 
205 0.205585 3.477111 9.031481 0.206261 1.728674 
177 0.206150 3.503281 9.041299 0.206170 1.734152 
1 
27689 0.205724 3.470646 9.036589 0.205731 1.724869 
672 0.204555 3.495600 9.037673 0.205735 1.726630 
300 0.205584 3.488843 9.039643 0.205969 1.729467 
141 0.202780 3.547630 9.027128 0.206246 1.732925 
20 
0 
11462 0.205731 3.470480 9.036694 0.205735 1.724909 
2265 0.205546 3.475157 9.036914 0.205765 1.725515 
714 0.207129 3.459350 8.988936 0.208278 1.736547 
1 
22011 0.205726 3.470606 9.036623 0.205730 1.724868 
2788 0.205607 3.474374 9.036784 0.205731 1.725223 
115 0.202686 3.587231 9.037897 0.205726 1.736020 
Table 3: Welded beam optimization solutions. 

Design Optimization Average-Based Algorithm Informatica 44 (2020) 23–33 29 
 
authors is given in (Kayham, Ceylan, Ayvaz, Gurarslan, 
2010): ( ) 0.205830, 3.468338, 9.036624, 0.205730

= x 
and 
0
1.724717

= . 
The optimization results achieved with the present 
algorithm are shown in Table 3 for different population 
sizes N and different average choices on Eq. (5). The first 
row for each choice combination of N and IP represents 
the optimum at the convergence of the algorithm. The 
following rows are the results at intermediary iterations. 
For all these design points all the constraints are 
satisfied. 
The best minimum cost value obtained in the present 
article is 
0
1.724858

= , corresponding to the point 
( ) 0.205729, 3.470519, 9.036630, 0.205730

= x . 
One may observe that the algorithm solutions 
compare very well with the best solution presented 
above, even for earlier iterates of the algorithm. We may 
also observe that convergence is faster when the plain 
average is used in Eq. (5). 
4.5 Pressure vessel design 
The pressure vessel design problem has been proposed in 
(Kannan, Kramer, 1994). It is one of the most used test 
problems for validating optimization algorithms. The 
problem is to find the optimal design of a compressed air 
storage tank (Fig. 3) with a working pressure of 1000 psi 
and a minimum capacity volume of 
3
min
1,296,000in V = . 
The pressure vessel is composed of a cylindrical 
shell capped at both ends by hemispherical heads. 
Let the design variables be 
1 s
xT  the thickness of 
the shell, 
2 h
xT  the thickness of the heads, 
3
xR  the 
inner radius and 
4
xL  the length of the cylindrical 
shell. The variables 
1
x and 
2
x should be integer 
multiples of 0.0625 in. The objective is to minimize the 
manufacturing cost (material, welding and forming costs) 
of the pressure vessel (Sandgren, 1990), subjected to 
constraints on volume capacity and in accordance with 
respective ASME codes. The mathematical model of the 
problem is: 
 
( )
2
0 1 3 4 2 3
22
1 4 1 3
1 3 1
2 3 2
3 min
23
3 4 3
12
34
min 0.6224 1.7781
3.1661 19.84
..
0.0193 1 0
0.00954 1 0
10
43
0.0625 , 99 0.0625
10 , 200
x x x x x
x x x x
st
xx
xx
VV
V x x x
xx
xx

  + +
+
  − 
  − 
  − 
=+
  

 (12) 
The analytical optimum for this problem is 
calculated in the Annex A as 
0
6059.714335

= at 
( )
*
0.8125, 0.4375, 42.0984456, 176.6365958 = x with 
the first and third constraints being active. 
The optimal results achieved with the present 
algorithm are shown in Table 4 for different population 
sizes N and different average choices on Eq. (5). For all 
these solutions there is no violation of the constraints. 
The first constraint is nearly active at the optimal point 
for all the different population sizes; it takes values in the 
interval 
75
1
0.3666 10 0.1215 10
−  −
−     −  . 
The results for the present algorithm compare well 
with the analytical solution. Again, the plain average of 
the Eq. (5) gives origin to faster convergence. 
4.6 Tension/Compression spring design 
The tension/compression spring design optimization 
problem is described in (Arora, 1989). The goal is to 
minimize the weight of a tension/compression spring 
(Fig. 4) subject to constraints on minimum deflection, 
shear stress, surge frequency, limits on outside diameter 
and side constraints. The design variables to be 
considered are the wire diameter d, the mean coil 
diameter D and the number n of active coils. 
Let us set up the vector of design variables as 
( )
1 2 3
,, x x x  x ( ) ,, d D n  . 
 
L 
R 
R 
T
h T
s 
 
Figure 3: Pressure vessel. 
N IP Iterate 
1
x 
2
x 
3
x 
4
x 
0
 
1000 
0 4143 0.81250 0.43750 42.09843 176.63712 6059.7231 
1 13441 0.81250 0.43750 42.09843 176.63678 6059.7158 
500 
0 8285 0.81250 0.43750 42.09843 176.63701 6059.7212 
1 11909 0.81250 0.43750 42.09844 176.63673 6059.7168 
200 
0 4173 0.81250 0.43750 42.09838 176.63742 6059.7227 
1 5077 0.81250 0.43750 42.09843 176.63704 6059.7222 
Table 4: Pressure vessel optimal solutions. 
 
d 
D 
P P 
 
Figure 4: Tension-compression spring. 

30 Informatica 44 (2020) 23–33 J.B. Cardoso et al. 
 
The problem may be formulated as 
 
( )
( )
( ) ( )
( )
( )
( )
2
0 3 1 2
34
1 2 3 1
2 3 4
2 2 1 2 2 1 1
2
1
2
3 1 2 3
4 1 2
1
2
3
min 2
..
1 71785 0
4 12566
1 5108 1 0
1 140.45 0
1.5 1 0
0.05 2
0.25 1.3
2 15
x x x
st
x x x
x x x x x x
x
x x x
xx
x
x
x
  +
  − 

  − − +

−
  − 
  + − 



 (13) 
The analytical solution for this problem is presented 
in the Annex B. The minimum weight of the spring is 
achieved as 
0
0.012665232

= at the point 
( )
*
0.051690, 0.356740, 11.287642 = x . At the optimum, 
the constraints 
1
 and 
2
 are active, and 
3
4.054

 = − , 
4
0.7277

 = − . 
The minimum objective function value obtained in 
(Arora, 1989) by nonlinear programming is 
0
0.012679

= corresponding to the optimum point 
( ) 0.051699, 0.35695, 11.289 x

= . The best result known 
so far by the authors is given in (Cagnina, Esquivel, 
Coello, 2008) as 
0
0.012665

= , with 
( )
*
0.051583, 0.354190, 11.438675 = x . 
The optimum results achieved with the present 
algorithm are shown in Table 5, for 0 IP = , different 
sizes N of the population and plain average selection on 
Eq. (5). For all these solutions there is no violation of the 
constraints, being practically active the first two 
constraints 
1
 and 
2
 . The last two constraints have 
values within the intervals 
3
4.08 4.04

−    − and 
4
0.734 0.720

−    − . 
Again, the present algorithm results compare well with 
the analytical solution. 
5 Concluding remarks 
This article presents an average concept algorithm to 
solve various optimization problems which include 
typical benchmark functions unconstrained problems and 
structural engineering test design constrained problems. 
To evaluate the performance of the present algorithm, 
numerical applications are conducted and the results are 
compared to the results obtained analytically and/or to 
the best ones achieved by other optimization methods. 
The analytical solutions for two of the constrained 
problems, namely the pressure vessel design and the 
tension-compression spring design problems, are 
determined in the present article. The solutions found by 
the proposed algorithm compare well with those results. 
We may conclude that the algorithm finds the global 
solution or a near-global solution in each problem tested. 
The Six-Hump Camelback function, for example, has 6 
local minimal points; however, the algorithm converges 
to the global minima. 
Characterizing the principal advantage of the 
algorithm one should emphasize the good balance 
between the accuracy of the solutions it achieves and its 
rare simplicity. 
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N 1000 500 200 100 
0
 0.012665 0.012666 0.012668 0.012671 
1
x 0.051718 0.051775 0.051308 0.052127 
2
x 0.357409 0.358778 0.347629 0.367321 
3
x 11.248693 11.169669 11.842695 10.694988 
Iterate 10063 11690 24818 13899 
Table 5: Tension-compression spring optimal solutions. 

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Programming in Mechanical Design Optimization. 
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and Optimization: estimation, simulation and 
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the many are smarter than the few and how 
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(43). 
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Lunch Theorems for Optimization. IEEE 
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Algorithms. Luniver Press. 
 
Annex A: Pressure Vessel Classical 
Design Optimization 
Let us assume initially the variables 
1
x
 and 
2
x
 are 
continuous and later on make the convenient correction 
to table (multiple of 0.0675) values. 
The cost function 
0

 decreases monotonically with 
1
x
 or 
2
x
 decrease. The constraint 
1

 is the only 
constraint that increases as 
1
x
 decreases, and the 
constraint 
2

 is the only constraint that increases as 
2
x
 
decreases. Then, 
1

 and 
2

 provide respectively lower 
bounds for 
1
x
 and 
2
x
, and these variables may be 
minimized out as 
 
13
23
0.0193
0.00954
xx
xx
= 

=

 (A1) 
Substituting these relationships into the original 
problem, we have 
 
( )
23
0 3 4 3
23
3 3 4 3
34
min 0.01319166 0.024353275
..
1296000 4 3 0
10 , 200
x x x
st
x x x
xx

  +
  − − 

 (A2) 
where the upper bars on the cost and constraint symbols 
mean we are determining by now the solution for all 
variables continuous. The Lagrangian function for this 
problem is 
 
( ) ( )
( ) ( )
0 3 3 3 3 3 3
4 4 4 4
10 200
10 200
L x x
xx
  

−+
−+
=  +  − − + − −
− + −
(A3) 
where 
3
 , 
3

−
, 
3

+
, 
4

−
, 
4

+
 are the Lagrange 
multipliers for the constraint 
3
 and side constraints. 
The necessary Karush-Kuhn-Tucker conditions (Arora, 
1989) for the problem (A.2) may now be set as 
( )
( )
( ) ( )
2
3 3 4 3
2
3 3 4 3 3 3
22
4 3 3 3 4 4
23
3 3 3 3 4 3
3 3 3 4 4
2
3 3 4
2 0.01319166 3 0.024353275
2 2 0
0.01319166 0
1296000 4 3 0
10 0, 200 0, 3, 4
, , , , 0
1296000 4
k k k k
L x x x x
x x x
L x x x
x x x
x x k
xx
   
   
   

    

−+
−+
−+
− + − +
    + 
− + − + =
   − − + =
   − − =

− = − = =

  − − ( )
3
3
34
30
10 , 200
x
xx
 

 (A4) 

32 Informatica 44 (2020) 23–33 J.B. Cardoso et al. 
 
Let us now search the different combinations of 
Lagrange multipliers: 
1. 
34
0, 0 
−−
   
3 4 3
10, 1288669.6 xx = =  = (violated) 
34
0, 0 
−+
   
3 4 3
10, 200, 1228979.4 xx = =  = (violated) 
Then, whatever the value of 
3
 , must have 
3
0 
−
= 
2. 
3
0  = 
2.1 
3
0 
−
=  
3
0 
+
 (from the 1
st
 condition, then 
violating the 5
th
 one) 
2.2 
4
0 
−
=  
4
0 
+
 (from the 2
nd
 condition, then 
violating the 5
th
 one) 
3. 
3
0    
3
0 =  
( ) ( )
2
4 3 3
1296000 4 3 x x x  =− 
3.1 
3 4 4
0   
+ − +
= = =  
3
0.01319166  = 
(from the 2
nd
 condition) and 
3
0 x = (after 
substituting 
4
x and
4
 into the 1
st
 one) 
3.2 
3 4 4
0, 0   
+ − +
= =   
4
200 x = (from the 4
th
 
condition), 
( )
23
3 4 3
1296000 4 3 0 x x x  − − =  
3
40.3196187244 x = 
( )
2
3 4 3
3
2
3 4 3
2 0.01319166 +3 0.024353275
22
0 004663057579 0
x x x
x x x
.



=
+
=
22
4 3 3 3
0 01319166
2 369851195 0
x . x
.
  
+
=−
=
 
All the optimality necessary conditions are 
satisfied, then 
13
23
3
4
0.0193 0.778168646
0.00954 0.384649165
40.3196187244
200
xx
xx
x
x
== 

==


=


=

 
is candidate local optimum point for the 
assumed continuous variables. 
3.3 
3 4 4
0, 0   
+ + −
= =   
4
10 x = (from the 4
th
 
condition), 
( )
23
3 4 3
1296000 4 3 0 x x x  − − =  
3
65.2252326139 x = 
( )
2
3 4 3
3
2
3 4 3
2 0.01319166 +3 0.024353275
22
x x x
x x x



=
+
0 005698937505 0 .= 
22
4 3 3 3
0 01319166
20 04674872 0
. x x
.
  
−
=−
= − 
 
(violates 5
th
 condition) 
3.4 
4 4 3
0, 0   
− + +
= =  
3
200 x =  
4
256.3534264 0 x = −  
3.5 
4 3 4
0, 0, 0   
− + +
=  
34
200 xx ==  
3
57347062.87 0  = −  
(contradicts point 3: 
3
0 = ) 
3.6 
4 3 4
0, 0, 0 
++−
=   
34
200, 10 xx == 
 
3
33470958.7 0  = −  
(contradicts point 3: 
3
0 = ) 
Testing now the second-order sufficient conditions 
for the only point 
1 2 3 4
( , , , ) x x x x  x , determined in 3.3, 
satisfying the necessary conditions, one may use the so-
called bordered Hessian (Luenberger, 1984) calculated at 
that point: 
( )
3 3 3 4
2 2 2
3 3 3 3 3 3 4
2 2 2
3 4 3 4 4
0
,
0 71095.91969 5107.198124
71095.91969 232.2341915 0.117553254
5107.198124 0.117553254 0
xx
x x x L x L x x
x L x x L x
      

        

        

−− 

= − −

 −−

B
 
As nm − for the problem (A.2) is 2 1 1 −= one has 
to calculate the last principal minor ( ) det B . Since its 
value is negative, its sign is coincident with 
( ) ( ) 11
m
sign sign − = − . Hence, the Hessian of L is 
positive-definite and the point x is a minimum point. 
Now, rounding up the values of 
1
x and 
2
x to the 
table values, we have 
1
2
0.8125
0.4375
x
x


 =


=


 
then determine the other two variables as 
 
3
min 0.8125 0.0193, 0.4375 0.00954
42.0984456
x

=
=
 
From the two first constraints 
3
0.8125 0.0193 x  , 
3
0.4375 0.00954 x  , i.e., the constraint 
1
 is active at 
the optimum, and 
4
176.6365958 x

= 
from the condition of 
3
0 =  
( ) ( )
2
4 3 3
1296000 4 3 x x x  =− 
Therefore, the analytical optimum point for the 
pressure vessel design problem is 
( ) 0.8125, 0.4375, 42.0984456, 176.6365958 =
*
x 

Design Optimization Average-Based Algorithm Informatica 44 (2020) 23–33 33 
 
giving the minimum optimum cost 
0
6059.714335

= . 
At the optimum, the constraints have 
13
0

 =  = and 
2
0.082013323

 = − . 
Annex B: Tension-Compression 
Spring Classical Design Optimization 
Again, let us firstly to analyze the monotonicity of the 
problem (Papalambros, Wilde, 1988). One should 
observe the constraint 
1
 is critical with respect to the 
design variable 
3
x . The cost function 
0
 increases 
monotonically in the variable 
3
x , and there is exactly one 
constraint, the constraint 
1
 , whose monotonicity with 
respect to 
3
x is opposite from that of the objective. Then, 
1
 is active and bounds 
3
x from below: 
 
43
3 1 2
71785 x x x = (B1) 
Substituting the relationship (B.1) into the objective 
function and into the constraint 
3
 we get 
 
2 6 2
0 1 2 1 2
3
3 2 1
2 71785
1 0.001956536881 0
x x x x
xx
  = +


  − 


 (B2) 
Substituting the lower and upper bounds of 
2
x into 
the constraint 
3
 we have that 
1
0.25 0.078790891 x  , 
1
1.3 0.136503503 x  ; then the range of the design 
variable 
1
x can be set up as 
1
0.05 0.136503503 x  
If one uses now the upper bounds of 
1
x and 
2
x in 
the constraint 
4
 , 0.136503503 1.3 1.5 +, it is obvious 
that this constraint is always inactive, not playing any 
role into the optimization problem. 
Studying now the monotonicity of the objective 
expressed in (B.2) with respect to the design variable 
2
x , 
the minimum of 
0
 is given as 
2 6 5
0 2 1 1 2
2 2 71785 0 x x x x     −  =  
4
3
21
71785 xx = 
since 
2 2 6 4
0 2 1 2
6 71785 0 x x x       . Thus, 
0
 
decreases monotonically in 
2
x increase for 
4
3
21
0.25 71785 xx  and increases monotonically in 
2
x increase for 
4
3
12
71785 1.3 xx  . For example, 
within the range 
4
3
21
0.25 71785 xx  , the function 
0
 decreases in the variable 
2
x increase, achieving the 
minimum at 
2
0.765545910 x = for a prescribed 
1
0.05 x = , and increases the value of the minimum point 
at 
2
x as 
1
x increases. For 
1
0.074378786 x  the function 
0
 is a decreasing function all along the feasible 
domain of 
2
x , 
2
0.25 1.3 x  . 
The constraint 
2
 may be expressed as 
( )
2
2 1 2 1
4 1 0 x C x x C x + − −  , 
2
1
2.460062647 12566 Cx =− 
This constraint increases monotonically in 
2
x 
increase; then its monotonicity with respect to 
2
x is 
opposite from that of 
0
 for 
4
3
21
0.25 71785 xx  and 
the constraint 
2
 is critical providing an upper bound 
for 
2
x : 
( )
( )
2
21
8 1 14 1 x x C C C = − + + + 
The reduced problem may now be expressed as 
 
( )
( )
2 6 2
0 1 2 1 2
2
21
12
min 2 71785
..
8 1 14 1
0.05 0.136503503, 0.25 1.3
x x x x
st
x x C C C
xx
  +
= − + + +
   
 (B3) 
The optimum point can be determined easily by 
using a unidimensional search in 
1
x
, with the active 
constraint 
2

 determining 
2
x
. The variable 
3
x
 is 
calculated by using the relationship (B.1) after solving 
the problem (B.3). 
The optimum value of the objective function is 
obtained as 
0
0.012665232

=
 at the point 
( ) 0.051690,0.356740328,11.28764160 
*
x
. At the 
optimum, the constraints have the values 
1
0

=
, 
2
0

=
, 
3
4.05383024

 = −
, 
4
0.727713114

 = −
. 

34 Informatica 44 (2020) 23–33 J.B. Cardoso et al.