EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 4, 2020, 964-976
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

The Component Weighted Median Absolute Deviations
Problem

Vedran Novoselac

Department of Mathematics, Mechanical Engineering Faculty in Slavonski Brod,
University of Slavonski Brod, Trg Ivane Brlić Mažuranić 2, 35000 Slavonski Brod, Croatia

Abstract. This paper considers the problem of robust modeling by using the well-known Least
Absolute Deviation (LAD) regression. For that purpose, the approximation function is designed
and analyzed, which is based on a certain component weight of the Weighted Median of Data. It
is shown that the proposed approximation function is a piecewise constant function with finitely
many pieces with respect to the model parameter. Thereby, an investigation of regions of constant
values of the approximation function is conducted. It is established that the designed model based
on the Component Weighted Median Absolute Deviations estimates an optimal model parameter
on a finite set, which describes the corresponding regions. Furthermore, the specified restriction of
the approximation function is observed and analyzed, in order to examine the observed problem.

2020 Mathematics Subject Classifications: 26E60, 26A15, 62J99

Key Words and Phrases: Weighted median, approximation function, robust regression

1. Introduction

This paper deals with robust regression modeling, and thus the well known Least
Absolute Deviations (LAD) regression model is considered. In general the LAD regression
involves finding estimates which minimize a sum of the residuals’ absolute values, which has
important applications in many fields, including statistics and numerical analysis [1, 7, 9].

In that sense, the approximation function F : Rm+1 → R is modeled, which considers
a certain weight of the Weighted Median of Data as a variable. It is well known that the
Weighted Median of Data is a robust estimator, which has a great number of applications
in many fields of applied research like statistics, data analysis, outlier detection, image
processing, etc. [2, 3, 8]. Considering the approximating of a model, the L1 norm error
model function is defined

∆(w) =

n∑
j=1

|yj − F (xj ;w)|,

DOI: https://doi.org/10.29020/nybg.ejpam.v13i4.3839

Email address: Vedran.Novoselac@sfsb.hr (V. Novoselac)

https://www.ejpam.com 964 c© 2020 EJPAM All rights reserved.



V. Novoselac / Eur. J. Pure Appl. Math, 13 (4) (2020), 964-976 965

where Y = {yj ∈ R : j ∈ {1, . . . , n}} presents the observed target data (dependent vari-
ables), X = {xj ∈ Rm : j ∈ {1, . . . , n}} the feature data (independent variables), and
w > 0 the model parameter [6]. The L1 norm based models are often not trivial to solve,
and thereby an investigation of the Component Weighted Median Function (CWMF)
F : Rm+1 → R is conducted, where it is shown that the restricted approximation function
F |D, D = z×R+, z ∈ Rm, is a piecewise constant function with finitely many pieces with
respect to the parameter w > 0. According to that, the minimization of the Component
Weighted Median Absolute Deviation (CWMAD) model ∆ is analyzed, where it is shown
that the minimization of ∆, i.e.

min
w>0

∆(w)

can be derived on the finite set, which describes the regions of constant values of the
restricted CWMF. Furthermore, the properties of the CWMAD model are presented un-
der the assumption of the specified restriction of the CWMF, which considers that the
observation yj is contained in the independent variable xj on the observed component.

The paper is organized in several sections and a subsection, where in Section 2 the
Weighted Median of Data is presented. In Section 3, the approximation function
F : Rm+1 → R is presented and analyzed. Afterwards, in Section 4, the CWMAD problem
is defined, where it is shown that the minimization of the ∆ can be conducted on the finite
set. Then, in Subsection 4.1, the properties of ∆ are presented, assuming the specified re-
striction of the CWMF. In Section 5, the numerical examples are given in order to present
the performance of the CWMAD problem, where unequal dimensions of the independent
variables are also observed. Finally, in Section 6, the conclusion is given.

2. The Weighted Median of Data

Let us denote a data vector z = (z1, . . . , zm) ∈ Rm, m ∈ N, and a positive vector of
weights w = (w1, . . . , wm) ∈ Rm+ . In this situation, we may define the function f : R→ R
as

f(u) =
m∑
i=1

wi|zi − u|, (1)

which in that sense presents the LAD problem. Considering the minimization of the
function f , which implies that the LAD regression can be observed as a problem of de-
termining the appropriate global minimum of f . Furthermore, the convexity properties
of the function f , can also be taken into consideration of the observed problem [4, 5]. In
that sense, we may conclude that the global minimum of f always exists, which indicates
the existence of the LAD solution [10], and thus it can be written that

med(w, z) = argmin
u∈R

f(u),

where med(w, z) is called the Weighted Median of Data [4, 8]. In general, the global min-
imum of f is not always unique, what confirms the well-known LAD regression properties
of a possibility of a multiple solution existence.



V. Novoselac / Eur. J. Pure Appl. Math, 13 (4) (2020), 964-976 966

The next theorem presents the problem of determining the weighted median of data
med(w, z), which involves the problem of finding the global minimum of the function f ,
which in general is not unique.

Theorem 1. Let z ∈ Rm, m ∈ N, be a data vector with a corresponding weight vector
w ∈ Rm+ , and let z(1) ≤ z(2) ≤ . . . ≤ z(m) be observations in an ascending order. Then it
holds for the defined set

L =
{
` :
∑̀
i=1

w(i) ≤
W

2

}
, ` ∈ {1, . . . ,m}, W =

m∑
i=1

wi, (2)

that:

(a) if L = ∅, then med(w, z) = z(1);

(b) if L 6= ∅, then for ν = maxL, it follows that:

(i) if
ν∑
i=1

w(i) <
W
2 , then med(w, z) = z(ν+1);

(ii) if
ν∑
i=1

w(i) = W
2 , then med(w, z) = (1− λ)z(ν) + λz(ν+1), λ ∈ [0, 1].

Proof. The function f : R→ R, defined by (1), is a piecewise linear function, so let us
denote slopes κ`, ` ∈ {0, . . . ,m}, which correspond to each interval

〈−∞, z(1)〉, 〈z(1), z(2)〉, . . . , 〈z(m−1), z(m)〉, 〈z(m),∞〉.

In this situation, it follows that

κ0 = −W, κm = W, (3)

and for ` ∈ {1, . . . ,m− 1}, it follows that

κ` = 2
∑̀
i=1

w(i) −W = κ`−1 + 2w(`). (4)

(a) Let us consider the case when L = ∅. Then it holds that

∑̀
i=1

w(i) >
W

2
, ∀` ∈ {1, . . . ,m}.

By using (3) and (4), we may conclude that κ0 < 0 < κ`, ∀` ∈ {1, . . . ,m}, which
implies that f is decreasing at 〈−∞, z(1)〉, and increasing at 〈z(1),∞〉. Thereby, we
may conclude that f reaches its global minimum at z(1), i.e. med(w, z) = z(1).

(b) Let us consider the case when L 6= ∅. This means that there exists ν = maxL. By
using (4), we may conclude that κν ≤ 0, while κ` > 0, ∀` > ν.



V. Novoselac / Eur. J. Pure Appl. Math, 13 (4) (2020), 964-976 967

(i) Let us consider the case when κν < 0. This means that f decreases at
〈−∞, z(ν+1)〉, and increases at 〈z(ν+1),∞〉. Therefore, we may conclude that f
reaches its global minimum at z(ν+1), i.e. med(w, z) = z(ν+1).

(ii) Let us consider the case when κν = 0. This implies that f is decreasing
at 〈−∞, z(ν)〉, is constant on [z(ν), z(ν+1)], and is increasing at 〈z(ν+1),∞〉.
Thereby, this implies that f reaches its global minimum at interval [z(ν), z(ν+1)],
i.e. med(w, z) = (1− λ)z(ν) + λz(ν+1), λ ∈ [0, 1].

Remark 1. If the problem of the weighted median of data is observed for weights, which
are all set to one, i.e. w1 = · · · = wm = 1, then the global minimum of the corresponding
function f is called the Median of Data, and is denoted as med(z).

3. The Component Weighted Median Function

Let us denote the component weighted model vector

wk = (1, . . . , w, . . . , 1), w > 0,

where k-th component is observed as a variable, while all other components are set to
one. In that sense, the Component Weighted Median Function (CWMF) F : Rm+1 → R
is defined as

F (x;w) = med(wk,x).

The performance of the constructed function is studied through Theorem 1 by observ-
ing the parameter w > 0, and thus the CWMF restriction F |D will be observed, where
D = z × R+, z ∈ Rm. In order to carry out some properties of F |D, a position p ∈ P,
P = {q : z(q) = zk}, q ∈ {1, . . . ,m}, of k-th component in ordered observation will be
taken into consideration. Considering that, the two cases can be carried out: (I) a weight
w is included into the sum; (II) a weight w is not included into the sum, which determines
the set L from Theorem 1 defined by (2).

(I) The first case considered a situation when w = w(p), p ∈ P, is included into the sum,
i.e. p ≤ ` and thus ∑̀

i=1

w(i) = `− 1 + w.

Now it can be written that

L =
{
` : ` ≤ `−(w)

}
, `−(w) =

m+ 1− w
2

.

(II) The second case is when p ≥ ` + 1, p ∈ P, which means that a weight w = w(p) is
not included into the sum, i.e. ∑̀

i=1

w(i) = `.



V. Novoselac / Eur. J. Pure Appl. Math, 13 (4) (2020), 964-976 968

In this situation

L =
{
` : ` ≤ `+(w)

}
, `+(w) =

m− 1 + w

2
.

The next theorem presents that the restricted CWMF is a piecewise constant function,
and thereby the regions of constant values are investigated.

Theorem 2. The function F |D : D → R is a piecewise constant function with respect to
a parameter w > 0.

Proof.

(I) Let us consider the first case. Following Theorem 1, it can be concluded that

{z(p), . . . , z(dm+1
2
e)} ⊆ {F (z, w) : p ≤ `} ⊆ [z(p), z(dm+1

2
e)].

Considering the inequality

ν−i < `−(w) < ν−i + 1, z(ν−i +1) ∈ {z(p), . . . , z(dm+1
2
e)}, (5)

it holds that ν−i = maxL, w ∈ Ii, where an interval Ii presents the solution of the
observed inequality (5) for the corresponding ν−i . This implies, by the statment
(b)-(i) of Theorem 1, that F (z;w) = z(ν−i +1), w ∈ Ii. In this situation, let us define

the set Ψ− in a descending order as

Ψ− = {ν−i : ν−1 > · · · > ν−s }, z(ν−i +1) ∈ {z(p), . . . , z(dm+1
2
e)}.

Then, the observed inequality (5) generates the intervals Ii for each ν−i :

I1 = 〈0, w+
1 〉, . . . , Ii = 〈w−i , w

+
i 〉, . . . , Is = 〈w−s ,+∞〉,

where w−i = m− 1− 2ν−i , and w+
i = m+ 1− 2ν−i . The situation when the equality

is observed, i.e.
ν−i = `−(w),

which corresponds to the case (b)-(ii) of Theorem 1. In this situation, it holds that
F (z;w+

i ) = (1− λ)z(ν−i ) + λz(ν−i +1), λ ∈ [0, 1], where ν−i ∈ Ψ− \ {ν−s }.

(II) Let us now consider the second case. Analogously, it can be concluded that

{z(bm+1
2
c), . . . , z(p−1)} ⊆ {F (z, w) : p ≥ `+ 1} ⊆ [z(bm+1

2
c), z(p)].

Then, the inequality

ν+i < `+(w) < ν+i + 1, z(ν+i +1) ∈ {z(bm+1
2
c), . . . , z(p−1)},



V. Novoselac / Eur. J. Pure Appl. Math, 13 (4) (2020), 964-976 969

is observed in order to indicate when F (z;w) = z(ν+i +1). Again, we defined the set

Ψ+, but now in an ascending order, i.e.

Ψ+ = {ν+i : ν+1 < · · · < ν+s−1}, z(ν+i +1) ∈ {z(bm+1
2
c), . . . , z(p−1)}.

The results are obtained at intervals

I1 = 〈0, w+
1 〉, . . . , Ii = 〈w−i , w

+
i 〉, . . . , Is−1 = 〈w−s−1, w

+
s−1〉,

where w−i = −m+1+2ν+i , and w+
i = −m+3+2ν+i . The situation when the equality

ν+i = `+(w)

is observed, generates that F (z;w−i ) = (1 − λ)z(ν+i ) + λz(ν+i +1), λ ∈ [0, 1], where

ν+i ∈ Ψ+ \ {ν+1 }.

Remark 2. The second case (II) always converts to the first case (I) when
w > |m + 1 − 2p|. This is the situation when w = w(p) starts to be included into the
sum (3), which immediately generates the interval Is for the second case (II) in Theorem
2, i.e.

Is = 〈w−s ,+∞〉,

so that F (z;w) = z(ν+s +1) = z(p), w ∈ Is, and singleton case when

F (z;w−s ) = (1− λ)z(ν+s ) + λz(ν+s +1),

where w−s = w+
s−1 = |m+ 1− 2p|, and ν+s = p− 1.

Remark 3. The case when the restricted CWMF is constant, i.e. ImaF |D = {zk} is
singleton, implies that F (z;w) = zk, w ∈ I1 = R+.

As it is shown, the restricted CWMF is a piecewise constant function with finitely
many pieces. Therefore, according to Theorem 2, Remark 2 and 3, the restricted CWMF
can be written as a finite linear combination

F (z;w) =
2s∑
t=2

αsgn
t χAt

(w), sgn =

{
+, p > m+1

2 ;

−, p ≤ m+1
2 ,

(6)

where

α±t =

{
z(ν±i +1), t = 2i;

(1− λ)z(ν±i ) + λz(ν±i +1), t = 2i± 1;
At =

{
Ii, t = 2i;
{w∓i }, t = 2i± 1,

and χA presents the indicator function, which is defined as

χA(w) =

{
1, w ∈ A;
0, w /∈ A.



V. Novoselac / Eur. J. Pure Appl. Math, 13 (4) (2020), 964-976 970

The next figure presents the performance of the restricted CWMF, where the dashed
line presents the situation for the first case (I), and the solid line for the second case (II).
According to Remark 2, it can be seen that the first case (I) also starts to appear for the
second case (II) when w > |m+1−2p|. The Figure 1(a) presents F |D, D = z×R+, z ∈ Rm,
(m = 8), when λ = 0 is applied, where the red marked graph denotes the situation when
p > m+1

2 (p = 8), and blue when p ≤ m+1
2 (p = 1). Analogously, Figure 1(b) presents

F |D, D = z × R+, z ∈ Rm, (m = 9), when λ = 1 is applied. It can be seen that in this
situation, the regions of the constant values of F |D shift, what is caused by an odd number
of observations.

(a) � = 0

1 3 5 7

F jD

w

w
+
1 = w

�

2 w
+
2 = w

�

3 w
+
3 = w

�

4 w
+
4 = w

�

5

z(1)

z(2)

z(3)

z(4)

z(5)

z(6)

z(7)

z(8)
zk = z(8)

zk = z(1)

1

(b) � = 1

2 4 6 8

F jD

w

w
+
1 = w

�

2 w
+
2 = w

�

3 w
+
3 = w

�

4 w
+
4 = w

�

5

z(1)

z(2)

z(3)

z(4)

z(5)

z(6)

z(7)

z(8)

z(9)
zk = z(9)

zk = z(1)

1

(I)

(II)

1

Figure 1: The restricted CWMF

Remark 4. The restricted CWMF follows its monotonicity according to the position of
the observed component in ordered observation, i.e.

F (z;w) =

{
monotonically increasing, p > m+1

2 ;

monotonically decreasing, p ≤ m+1
2 .

The next corollary considers the restricted CWMF when λ ∈ {0, 1} is taken into
consideration.

Corollary 1. It holds for F |D : D → R that:

(a) when λ = 0, then:

(i) if F |D decreases, then it is right continuous;

(ii) if F |D increases, then it is left continuous.

(b) when λ = 1, then:

(i) if F |D decreases, then it is left continuous;

(ii) if F |D increases, then it is right continuous;



V. Novoselac / Eur. J. Pure Appl. Math, 13 (4) (2020), 964-976 971

Proof. Considering (6) for λ ∈ {0, 1}, the restricted CWMF can be written as

F (z;w) =

s∑
t=1

αsgn
t χAt

(w),

where α±t = z(ν±t +1), and At can be presented as left-open intervals:

A1 = 〈0, w+
1 ], . . . , At = 〈w−t , w

+
t ], . . . , As = 〈w−s ,+∞〉,

or right-open intevals:

A1 = 〈0, w+
1 〉, . . . , At = [w−t , w

+
t 〉, . . . , As = [w−s ,+∞〉,

and thus the statements of the corollary are proven.

Remark 5. It follows that F (z;w) = z(p) = zk, when w > |m+ 1− 2p∗|, where

p∗ =

{
min{p−, m+1

2 }, p > m+1
2 ;

max{p+, m+1
2 }, p ≤ m+1

2 ,

such that
p+ = maxP, p− = minP, P = {q : z(q) = zk}.

This means that the restricted CWMF converges to the k-th component of z ∈ Rm, i.e.

lim
w→+∞

F (z;w) = zk.

4. The LAD regression with the CWMF

In this section we present the LAD regression model, which considers the CWMF as
an approximation function, whose value is used to predict the outcome of a dependent
variable. It is well known that the L1 norm has been widely used to make robust models,
which is useful in preventing model misspecifications, which are often caused by outliers
[7]. The problem is considered as the L1 norm error model function

∆(w) =
n∑
j=1

|yj − F (xj ;w)|,

where Y = {yj ∈ R : j ∈ {1, . . . , n}} presents the dependent variables, X = {xj ∈ Rm : j ∈
{1, . . . , n}} the independent variables, and w > 0 the model parameter. In this situation,
the robust regression model, which considers the Component Weighted Median Absolute
Deviations (CWMAD), is constructed.

Considering the restricted CWMF as a piecewise constant function with finitely many
pieces (6), the minimization problem of ∆ can be conducted on a finite set

A = {at : at ∈ At}, t ∈ {2, 3, . . . , 2s}, s = max
j∈{1,...,n}

sj , (7)



V. Novoselac / Eur. J. Pure Appl. Math, 13 (4) (2020), 964-976 972

where sj denotes a number of intervals of F |Dj , Dj = xj×R+. In this case, it follows that

min
w>0

∆(w) = min
a∈A

∆(a),

which implies that the global minimum of ∆ satisfies that

∆(w∗) = min
w>0

∆(w), w∗ ∈ A∗ =
⋃
t∈T

At,

where T = {t : ∆(at) = min
a∈A

∆(a)}.

4.1. The CWMAD properties

In this subsection, the properties of the CWMAD are presented with respect to the
specified restriction of the CWMF. The restriction F |Dj

is constructed in such a way that
the k-th component of an independent variable is equal to a dependent variable for each

observation, i.e. Dj = xj × R+, x
(j)
k = yj , where xj = (x

(j)
1 , . . . , x

(j)
m ) ∈ Rm. Thereby, the

residual functions rj : R+ → R of the CWMAD

rj(w) = yj − F |Dj
(x;w),

are studied in order to determine the global minimum of ∆, i.e. the optimal model
parameter.

In the next lemma, the properties of the residuals rj are briefly listed.

Lemma 1. It holds for the residual functions rj : R+ → R that:

(a) rj is a monotonically piecewise constant function;

(b) rj(w
∗) = min

w>0
|rj(w)|, w∗ ∈ A∗j , where

A∗j =

{
[w∗j ,+∞〉 , if rj is right continuous;

〈w∗j ,+∞〉, if rj is left continuous or constant,

such that w∗j = |m+ 1− 2p∗j |.

Proof.

(a) According to (6), F |Dj
is a monotonically piecewise constant function, which accord-

ing to Remark 5, reaches k-th component x
(j)
k = yj at its last piece. Considering

that, we may conclude that the residuals are also monotonically piecewise constant
functions, which can be written as

rj(w) =

2sj∑
t=2

α
(j)
t χAt

(w), α
(j)
t = yj − α

sgnj
t , sgnj =

{
+, pj >

m+1
2 ;

−, pj ≤ m+1
2 ,



V. Novoselac / Eur. J. Pure Appl. Math, 13 (4) (2020), 964-976 973

where sj presents an interval number of F |Dj
, pj ∈ Pj = {q : x

(j)
(q) = x

(j)
k }, and

α
(j)
t =


yj − x(j)(ν±i +1)

, t = 2i;

yj − F (xj ;w
∓
i ), t = 2i± 1;

0 , t ≥ 2sj .

(b) Considering the previous statement (a), the residuals rj reach zero value at their
least piece. Thereby, the minimization problem

min
w>0
|rj(w)|,

reaches its global minimum at a right-unbounded interval, which we denote as A∗j .
Considering F |Dj

when it fulfills the statement (a)-(i), or (b)-(ii) of Corollary 1, we
may conclude that rj is right continuous, and thus A∗j is closed from the left side by
w∗j = |m− 1 + 2p∗j |, where according to Remark 5, it follows that

p∗j =

 min{p−j ,
m+1
2 }, pj >

m+1
2 ;

max{p+j ,
m+1
2 }, pj ≤ m+1

2 ,

so that
p+j = maxPj , p−j = minPj , Pj = {q : x

(j)
(q) = x

(j)
k }.

Otherwise, if F |Dj
is constant, or λ ∈ 〈0, 1〉, then A∗j is a right-unbounded interval

which is open from the left by w∗j .

Theorem 3. It holds for ∆: R+ → R+, defined as

∆(w) =

n∑
j=1

|rj(w)|,

that:

(a) ∆ is a monotonically decreasing piecewise constant function;

(b) ∆(w∗) = min
w∈R

∆(w), w∗ ∈ A∗ =

n⋂
j=1

A∗j .

Proof.

(a) Considering Lemma 1 (a), it follows that |rj | is a monotonically decreasing piecewise
constant function, which converges to zero, and thus we can write that

∆(w) =

2s∑
t=2

αt χAt
(w), s = max

j∈{1,...,n}
sj , αt =

n∑
j=1

|α(j)
t |.

Knowing that the sum of the monotonically decreasing functions is also decreasing,
we may conclude that the statement (a) is proven.



V. Novoselac / Eur. J. Pure Appl. Math, 13 (4) (2020), 964-976 974

(b) Considering the Lema 1 (b), we can conclude that ∆ reaches the global minimum
when all residuals rj are zero. Because all A∗j are right-unbounded intervals, it holds
that the global minimum of the observed problem is contained in the intersections
of all A∗j .

5. Numerical examples

In this section, the numerical examples for the generalized CWMAD problem are
presented, where the independent variables are considered to be with unequal dimensions,
i.e. X = {xj ∈ Rmj : j ∈ {1, . . . , n}}, mj ∈ N, and thus with the different component
weighted model vector of the CWMF. In this situation, the L1 norm error model function
is defined as

∆(w) =
n∑
j=1

|yj − Fj(xj ;w)|,

where the functions Fj : Rmj+1 → R present the CWMF defined as

Fj(x;w) = med(wkj ,x), 1 ≤ kj ≤ mj .

In the next table, the numerical examples are presented in order to show the perfor-
mance of ∆ (λ = 0 is observed), together with the results of the generalized CWMAD
problem. The optimal results are obtained by taking into account the appropriate finite
set defined by (7). In that sense, the finite set must be constructed to present the regions
of the corresponding model function ∆, which contains the finite pieces of the constant
values. For that purpose, the approximating finite set à (⊇ A) is constructed, which
presents the regions of the model function ∆ for each numerical example presented in
Table 1, i.e.

à = {0.5i : i ∈ {1, . . . , 2M}}, M = max
j∈{1,...,10}

mj ,

and thus
min
a∈Ã

∆(w) = min
w>0

∆(w).

The next figure presents the performance of ∆, where Figures 2(a),(b), and (c) cor-
respond to the numerical examples (a), (b), and (c) from Table 1. It can be seen in
Figure 2(b) that the example (b) from Table 1, is constructed according to the specified
restriction, where the dependent variable is contained on the observed component for each
independent variable (see Subsection 4.1). According to Theorem 3, it follows that the
observed model function ∆ is monotonically decreasing, and attains its global minimum
at its least piece.



V. Novoselac / Eur. J. Pure Appl. Math, 13 (4) (2020), 964-976 975

Table 1: Numerical examples

No. (a) (b) (c)

j xj yj kj xj yj kj xj yj kj

1 (4; 4; 4; 8; 2; 5; 5; 9) 1 4 (3; 0; 8; 5; 9; 0; 6; 7; 2) 0 2 (6; 5; 0; 9) 0 4

2 (5; 8; 4; 5; 2; 2; 6; 8) 4 3 (2; 2; 6; 0; 7; 3; 9; 4; 1) 3 6 (7; 6; 9; 7; 3; 9; 7) 6 5

3 (3; 6; 3; 7; 0; 3; 6; 0) 3 8 (3; 8; 9; 8; 8; 6; 9; 8; 4) 8 8 (9; 4; 3) 5 2

4 (4; 5; 8; 4; 1; 6; 1; 4) 8 2 (4; 4; 8; 4; 5; 3; 3; 7; 9) 8 3 (4; 9) 3 1

5 (2; 2; 5; 8; 5; 9; 4; 6) 4 4 (0; 4; 3; 8; 2; 1; 5; 7; 3) 4 2 (6; 7; 1; 6) 0 4

6 (7; 1; 4; 7; 5; 0; 8; 8) 6 2 (7; 7; 8; 5; 8; 0; 4; 6; 7) 6 8 (9; 7; 4; 5; 3) 1 4

7 (2; 8; 1; 5; 8; 0; 0; 8) 5 3 (1; 7; 0; 1; 0; 4; 0; 5; 7) 1 4 (2; 5; 2; 7; 0; 0; 2) 4 6

8 (5; 1; 6; 7; 5; 5; 6; 9) 2 7 (9; 7; 6; 0; 9; 6; 7; 0; 6) 7 7 (0; 0; 7; 4; 2; 6) 0 2

9 (7; 8; 8; 5; 5; 1; 2; 9) 4 6 (2; 7; 6; 7; 0; 1; 7; 2; 6) 7 7 (4; 2) 7 2

10 (6; 6; 7; 8; 7; 9; 9; 4) 6 8 (6; 1; 0; 8; 2; 3; 0; 6; 4) 2 5 (0; 4; 1; 6) 9 3

�(w�) 26 0 34

A� h5; 7] h6;+1i f3g

1

(a)

1 3 5 7

26

28

30

32

34

36

�

w

(b)

2 4 6 8

0

5

10

15

20

�

w

(c)

1 2 3 4 5 6 7 8

34

36

38

40

42

44

�

w

1

Figure 2: The model function ∆

6. Conclusion

It is shown that the designed CWMAD model estimates the optimal model parameter
at the regions of constant values of the restricted approximation CWMF. Furthemore,
the restricted CWMF is studied and analyzed, where the assumption of the specified
restriction is considered in order to examine the CWMAD properties. Considering the
numerical experiments for the observed problem, the generalized CWMAD problem is
considered, where it is shown that the optimal parameter model is detected by designing
the correct discrete optimization model.

Acknowledgements

This work has been fully supported by Mechanical Engineering Faculty in Slavonski
Brod under the internal research project Generalization of Arithmetic Means and Their
Applications.

We are grateful to Željka Rosandić who has linguistically corrected and refined this
paper.



REFERENCES 976

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