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GEODETSKI VESTNIK | letn. / Vol. 67 | št. / No. 2 |
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ABSTRACT IZVLEČEK 
KLJUČNE BESEDE KEY WORDS
special-purpose geodetic network, precision, reliability, 
tolerances, alternative network design solutions, multi-
criteria compromise ranking, VIKOR method, optimal 
network design
namenska geodetska mreža, natančnost, zanesljivost, tolerance, 
alternativne rešitve zasnove mreže, večkriterijska kompromisna 
lestvica, metoda VIKOR, optimalna zasnova mreže
Darko Anđić, Radovan Đurović | UPORABA METODE VIKOR PRI ISKANJU OPTIMALNE REŠITVE ZA ZASNOVO GEODETSKE MREŽE ZA POSEBNE NAMENE | THE USE OF VIKOR METHOD IN FINDING 
THE OPTIMAL SOLUTION FOR A SPECIAL-PURPOSE GEODETIC NETWORK DESIGN  | 196-212 |
UDK: 528.23 
Klasifikacija prispevka po COBISS.SI: 1.01
Prispelo: 7. 2. 2023
Sprejeto: 18. 4. 2023
DOI: 10.15292/geodetski-vestnik.2023.02.196-212
SCIENTIFIC ARTICLE
Received: 7. 2. 2023
Accepted: 18. 4. 2023
Darko Anđić, Radovan Đurović
UPORABA METODE VIKOR 
PRI ISKANJU OPTIMALNE 
REŠITVE ZA ZASNOVO 
GEODETSKE MREŽE ZA 
POSEBNE NAMENE
THE USE OF VIKOR METHOD 
IN FINDING THE OPTIMAL 
SOLUTION FOR A SPECIAL-
PURPOSE GEODETIC 
NETWORK DESIGN
This study deals with the use of VIKOR method in finding 
the optimal solution for a special-purpose geodetic network 
design. It is about a multi-criteria compromise ranking 
method developed in 1986. Based on previously defined 
criteria functions related to precision and reliability, four 
acceptable alternative solutions were firstly established. Then, 
with the aim of finding the optimal one, those solutions were 
subjected to VIKOR method. The geodetic micro-network 
used in this study was simulated as a trilateration one 
and consisted of six control points discretizing a structural 
geometry as well as four, five or six reference points (pillars), 
depending on what alternative solution it was about. The 
optimal network design should have ensured the detection of 
deviations of the control points' positions in accordance with 
a priori introduced tolerances and constraints. It turned out 
the approach based on VIKOR method is a very efficient tool 
in tasks involving multiple conflicting requirements referred 
to a geodetic engineer and in the cases when the decision 
must be made in a short term.
Študija obravnava uporabo metode VIKOR pri iskanju 
optimalne rešitve za načrtovanje namenske geodetske 
mreže. Metoda ponuja večkriterijski kompromisni 
način rangiranja, ki je bil razvit leta 1986. Na podlagi 
predhodno opredeljenih kriterijskih funkcij, povezanih z 
natančnostjo in zanesljivostjo, so bile najprej postavljene 
štiri sprejemljive alternativne rešitve, ki smo jih nato v 
iskanju optimalne podvrgli metodi VIKOR. Geodetska 
mikromreža, uporabljena v študiji, je bila simulirana kot 
trilateracijska in je sestavljena iz šestih kontrolnih točk, ki 
diskretizirajo strukturno geometrijo, ter štirih, petih ali 
šestih referenčnih točk (stebrov), odvisno od tega, za katero 
alternativno rešitev je šlo. Optimalna zasnova omrežja 
bi morala zagotoviti zaznavanje odstopanj položajev 
kontrolnih točk v skladu z vnaprej uvedenimi tolerancami 
in omejitvami. Izkazalo se je, da je pristop, ki temelji na 
metodi VIKOR, zelo učinkovito orodje pri nalogah, ki 
vključujejo več nasprotujočih si zahtev, naloženih geodetu, 
in ko je treba odločitev sprejeti v kratkem času.
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1 INTRODUCTION
Many authors have dealt with the geodetic network optimization issue in their papers so far. It is com-
mon knowledge that, when it comes to optimization approaches in that sense, we have four well-known 
geodetic network design orders, and those are (Grafarend, 1974): the Zero-Order Design (ZOD), the 
First-Order Design (FOD), the Second-Order Design (SOD) and the Third-Order Design (THOD). 
This classification of design orders is conditional, because it also has its own shortcomings, which are 
mainly related to the interpenetration and connection of individual designs when solving problems. 
Namely, each task of the optimal designing of geodetic networks is solved individually, or with a partial 
solving of another task (e.g. in the SOD, the design matrix A is varied, which partially solves the FOD, 
etc). However, optimization tasks according to the above division are not discussed here.
An extensive and detailed review of all geodetic network optimization procedures can be found in 
Тамутис (1979) and Grafarend and Sansò (1985). Besides, there are many other recent studies dealing 
with the optimization of geodetic networks, and some of them are: Bagherbandi et al (2009), Doma and 
El Shoney (2011), Dwivedi and Dikshit (2013), Pachelski and Postek (2016), Postek (2021). 
In this study, the authors present a multi-criteria optimization method that is used with the aim of 
presenting the decision-making process with the special-purpose geodetic networks in a completely 
new way. It is about VIKOR (in serbian: VIšekriterijumsko KOmpromisno Rangiranje/Rešenje; in 
english: Multi-Criteria Optimization and Compromise Ranking /Solution) method that was de-
veloped by Opricović (1986). It has been widely used in non-geodetic problems (see, for example, 
Opricović and Tzeng (2004), Nikolić et al (2010), Kuo and Liang (2011), Nisel (2014), Chatterjee 
and Chakraborty (2016), Jokić et al (2019), etc). Consequently, a curiosity regarding its application 
in geodetic network designing has arisen. Such an approach significantly differs from the ZOD, FOD, 
SOD and THOD, beacause it involves a greater number of conflicting criteria. In other words, the 
main characteristic of this new approach is, actually, the presence of more opposing requirements. 
Some criteria functions that represent drawbacks or losses should be minimzed, while the remaing 
ones that present benefits or gains should be maximized. That is the main issue that makes the entire 
optimization process more complex.
The use of VIKOR method in designing of a special-purpose geodetic network is representatively shown 
through this study.
2 METHODS AND MATERIALS
This section presents the theoretical basis of VIKOR method as well as characteristics of the system to 
be optimized herein.
2.1 Theoretical background of VIKOR method
Basically, this method was developed on the basis of compromise programming elements and it starts 
from the ''boundary'' forms of  Lp-metric. The metric is written as follows (Opricović, 1998):
 { }
1
1( , ( )) [ ( )] ,  1 
n pp
p i iiL F F x f f x p
∗ ∗
== − ≤ < ∞∑  (1)
Darko Anđić, Radovan Đurović | UPORABA METODE VIKOR PRI ISKANJU OPTIMALNE REŠITVE ZA ZASNOVO GEODETSKE MREŽE ZA POSEBNE NAMENE | THE USE OF VIKOR METHOD IN FINDING 
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and it represents the distance between the ''ideal'' point F ∗(f1
∗, ..., fn
∗) and the point F(x) = (f1(x), ..., fn(x)) 
in the space of criteria functions. The ''ideal'' point is determined from n values of criteria functions by 
means of the following equation:  
 fi 
∗ = extj  fij ,   (i, j) ∈ {1, 2, ..., nc} × {1, 2, ..., na}, (2)
where extj (j is the ordinal number of alternative) stands for the maximum if i-th criterion function 
represents a benefit or gain, or minimum if the same function represents a drawback or loss.
Since in practice the values fij are often given in different units, a transformation is introduced to avoid 
the unit-variety problem. This transformation gives the following corresponding dimensionless values:
  ,   ( , ) {1,2,..., } {1,2,..., }.i ijij c a
i i
f f
d i j n n
f f
∗
∗ −
−
= ∈ ×
−
 (3)
Now, for the alternative aj, we can write the ''boundary'' forms of the  Lp–metric as follows:
  
1 1   ,   1
n n i ij
j i ij ii i
i i
f f
S w d w p
f f
∗
∗ −= =
−
= = =
−
∑ ∑  (4)
 max( ) max   ,   ,i ijj i ij i
i i i i
f f
R w d w p
f f
∗
∗ −
 −
 = = = ∞
 − 
 (5)
with fi 
∗ and fi 
- which correspond to the best and worst alternative of the system, respectively, and wi 
representing the weight of the criterion function fij (i.e. the decision-maker preference regarding the i-th 
crtiterion), whereby Σni=1wi = 1.
If Rj = R
- for two or more j-indices, the following modification is introduced:
 ,mod
max
100 100
j j
j j
j j j
S RS R
R R R
− −−
= + = + . (6)
The essence of VIKOR method is to calculate the Qj value for all alternatives and then separate the one 
which the minimal value (the minimal distance from the ''ideal'' point) corresponds to. This measure 
for multi-criteria ranking of the j-th alternative is calculated as follows:
 Qj = v⋅QSj + (1 - v)⋅QRj, (7)
where v is the weight of the strategy of fulfilling most of the criteria (it is chosen by a decision maker, 
but VIKOR method assumes the value of 0.50).
In Eq. (7) we have
 
min
max min
j j
j j
j
j j
jj
S SS S
QS S SS S
∗
− ∗
−−
= = −−
 and (8)
    
min
max min
j j
j j
j
j j
jj
R RR R
QR R RR R
∗
− ∗
−−
= = −−
 (9)
Darko Anđić, Radovan Đurović | UPORABA METODE VIKOR PRI ISKANJU OPTIMALNE REŠITVE ZA ZASNOVO GEODETSKE MREŽE ZA POSEBNE NAMENE | THE USE OF VIKOR METHOD IN FINDING 
THE OPTIMAL SOLUTION FOR A SPECIAL-PURPOSE GEODETIC NETWORK DESIGN  | 196-212 |
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that are, respectively, the measure of deviation expressing the requirement for maximum group benefit 
(serves to form the first ranking list) and measure of deviation expressing the requirement for minimizing 
the maximal distance of an alternative from the ''ideal'' alternative (serves to form the second ranking 
list). These two measures, combined as given in Eq. (7), provide the third, compromise ranking list.
In multi-criteria ranking using VIKOR method, alternative aj is better in total (according to all criteria) 
than alternative ak if Qj(v = 0.50) < Qk(v = 0.50). However, it is still not enough to consider aj the best 
alternative. Namely, to consider it the best one, it has to be first-ranked on the compromise ranking list 
and fulfill two conditions given below (Opricović, 1998).
Condition C1: The first-ranked alternative (a') on the compromise ranking list for v = 0.50 has to have 
a ''sufficient advantage'' over the alternative from the next position (a"), what means that
 Q(a") - Q(a') ≥ DQ = min(0.25;1/(na - 1)), (10)
where DQ is the ''sufficient advantage'' threshold which is equal to 0.25 when the number of alterna-
tives is less than six.
Condition C2: The first-ranked alternative (a') on the compromise ranking list for v = 0.50 has to have 
a ''sufficiently stable'' first position meaning that it also fulfills at least one of the following requirements: 
(1) it is first-ranked on the first ranking list (according to QS); (2) it is first-ranked on the first ranking list 
(according to QR); (3) it is first-ranked on the third ranking list (according to Q, for v = 0.25 and v = 0.75).
Conclusions are made as follows:
 – If the first-ranked alternative on the compromise list fulfills both conditions (C1 and C2), it 
is considered the only and best solution;
 – If the first-ranked alternative on the compromise list does not fulfill only the condition C2, it 
is considered not ''sufficiently'' better than the second-ranked alternative, and then a set of com-
promise solutions is formed so that it includes only the first- and second-ranked alternative;
 – If the first-ranked alternative on the compromise list does not fulfill only the condition C1 or 
both conditions (C1 and C2), it is considered not ''sufficiently'' better than the second-ranked 
alternative and any other alternative (ak) on the list that fulfills
 Q(ak) - Q(a') < DQ = min(0.25;1/(na - 1)), (11)
and then a set of compromise solutions is formed so that it includes the first-, second-ranked 
and any other alternative for which the above inequality is valid.
The results of VIKOR method are the three ranking lists (formed according to QS, QR and Q values) 
and the alternative with ''sufficient advantage'' as well as, at the same time, ''sufficiently stable'' first 
position (if both conditions, C1 and C2, are fulfilled) or a set of compromise solutions (when one or 
both conditions, C1 and C2, are not fulfilled). On the basis of these results, the final solution is adopted.
2.2 Description of the system to be optimized in the study
For the purpose of the study herein, a four-variant special-purpose geodetic network was simulated. 
Each of the four variants represents an individual acceptable solution that fulfills all pre-set constrains 
(see subsection 2.2.2).
Darko Anđić, Radovan Đurović | UPORABA METODE VIKOR PRI ISKANJU OPTIMALNE REŠITVE ZA ZASNOVO GEODETSKE MREŽE ZA POSEBNE NAMENE | THE USE OF VIKOR METHOD IN FINDING 
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Namely, it is about a trilateration control network that should serve as a base for detecting deviations 
of the projected from the marked structure pillars' positions. The nework consists of four, five or six 
reference points (pillars), and six points representing the structure pillars' centers (i.e. the points whose 
positions are controlled). The change in the number of reference points and lengths to be measured 
basically alternates the network design (solution). In this way, four alternative solutions for the geodetic 
control network (hereinafter: alternatives) were established. They are denoted as A1, A2, A3 and A4.
2.2.1 Mathematics of the Least Squares Method used in this study
In this study, we use the well-known principles of the LS (Least Squares) method (for a detailed insight 
into the method, see e.g. Perović (2005)), by means of the following matrices: A – design matrix, P – 
weight matrix, N – normal equation coefficient matrix.
Design n-by-u matrix (A) is formed based on the following:
 
R R R R S S S S
R R R R S S S S
R R R R S S S S
R R R R S S S
1 1 1 1 6 6
1;1 1;1 ;1 ;1 1;1 1;1 6;1 6;1
1;2 1;2 ;2 ;2 1;2 1;2 6;2 6;2
1; 1; ; ; 1; 1; 6
                                        
m m
m m
n u
n n m n m n n n
y x y x y x y x
B A B A B A B A
B A B A B A B A
B A B A B A B
× =A
 
 
 
         
 
m m
S; 6;n nA
 
 
 
 
  
 
, (12)
with r(An×u) = u - d = u - 3, where we have the following denotations:
R1, ..., Rm – reference points (pillars), with m equal to 4, 5 or 6, depending on the alternative;
S1, ..., S6 – six points representing the centers of the structure pillars;
n – the number of the lengths to be measured in the alternative;
u – the number of the unknown parameters (i.e. the coordinates of the geodetic network points) in the 
alternative;
The coefficients which figure in matrix A are taken from the corresponding correction equations for 
measured lengths. The correction equation for the measured length between points j and k is: 
 ,0 ,    k k kj j j
k k
jk j jk j kj k kj k j jD D Dv B y A x B y A x f f D d= + + + + = − , (13)
and, in addition to the mentioned coefficients, it contains unknown parameters in the form of differential 
increments of the coordinates Yj, Xj, Yk and Xk, as well as the absolute term representing the difference 
between approximate and measured length (not used herein). The coefficients are calculated as follows:
 ,0
0
sin
k
j k
jk j kj
j
D
B By ν
 ∂
 = = − = −∂ 
 
, (14)
 ,0
0
cos
k
j k
jk j kj
j
D
A Ax ν
 ∂
 = = − = −∂ 
 
, (15)
whereby the index ''0'' is introduced to indicate the approximate value obtained from the approximate 
coordinates Yj,0, Xj,0, Yk,0 and Xk,0 (see subsection 2.2.4). Index j in Eqs. (13), (14) and (15) represents a 
Darko Anđić, Radovan Đurović | UPORABA METODE VIKOR PRI ISKANJU OPTIMALNE REŠITVE ZA ZASNOVO GEODETSKE MREŽE ZA POSEBNE NAMENE | THE USE OF VIKOR METHOD IN FINDING 
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reference point (pillar), and index k can be represent of a reference point or a structure pillar's center. 
The symbol ν is introduced to denote the bearing.
Weight matrix (P) is a diagonal n-by-n matrix, with the general term calculated from the adopted disper-
sion coefficient a priori (with index ''0'') and the length mensurement variance obtained in accordance 
with a manufacturer's declaration, and it is formed as follows:
 { } { }
1 2
2 2 2
0 0 0
1 2 2 2 2,  , ..., ,  , ..., .
n
n n i n
D D D
diag P diag P P P diag
σ σ σ
σ σ σ×
  = = =  
  
P  (16) 
Normal equation coefficient u-by-u matrix (N) is calculated using the following equation: 
 T( ) .u u n u n n n u× × × ×=N A P A  (17)
On the basis of the metrices given by Eqs. (12), (16) and (17), we can calculate the remaining matrices 
which are used in the study. Those matrices are: the generalized inverse of the singular matrix N (i.e. 
cofactor matrix for the unknown parameters' estimates), the cofactor matrix for the measured length 
values' estimates, the cofactor matrix for the measured length corrections' estimates and the redundancy 
matrix.
Pseudoinverse of the matrix N is extracted from the inverse of the singular matrix N, that is previously 
extended by a chosen datum constraint matrix in the following way:
 
T
T
1
3 3
3 3 3 3 3 3
( )
,
( )
u u u u u u
u u
− + +
× × × ×
+
× × × ×
  
 =       
N B N B
B 0 B 0
 (18)
or calculated as:
 T T1 13 3 3 3( ( ) ) ( ( ) ) ,u u u u u u u u u u
+ − −
× × × × × × ×= + +N N B B N N B B  (19)
where the datum constraint matrix has the following form (provides the minimal trace of the cofactor 
matrix for the unknown parameters' estimates, but only for the reference points):
 
R R R R S S S S
T
R R R R
1 1 1 1 6 6
1 1
                                
1/ 0 1/ 0 0     0      0        0
0 1/ 0 1/ 0     0      0        0
0     0      0        0m m
y x y x y x y x
m m
m m
ξ η ξ η
 
 
=  
 − − 
 
B
 
 
 
 
m m
 (20)
In Eq. (20), for the denotations in the third row of the matrix, and taking into account that 
j ∈ {R1, ..., Rm}, we write:
 ,0 0
2 2
,0 0 ,0 01 1
,
( ) ( )
j
j m m
j jj j
X X
Y Y X X
ξ
= =
−
=
− + −∑ ∑
 (21)
 ,0 0
2 2
,0 0 ,0 01 1
,
( ) ( )
j
j m m
j jj j
Y Y
Y Y X X
η
= =
−
=
− + −∑ ∑
 (22)
Darko Anđić, Radovan Đurović | UPORABA METODE VIKOR PRI ISKANJU OPTIMALNE REŠITVE ZA ZASNOVO GEODETSKE MREŽE ZA POSEBNE NAMENE | THE USE OF VIKOR METHOD IN FINDING 
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whereby Yj,0, and Xj,0 represent the approximate coordinates, while Y
__
0 and X
__
0 are the figure centroid 
coordinates.
The cofactor matrix for the measured length values' estimates, the cofactor matrix for the measured length 
corrections' estimates and the redundancy matrix are, respectively, calculated as follows:
 T TxlQ AN A AQ A x ˆ ˆ ˆ( is the vector of  unknown parameters' estimates),
+= =  (23)
 v lQ P Q
1
ˆˆ ,−= −  (24)
 ˆ .= vR Q P  (25) 
2.2.2 Precision and reliability constraints which were pre-set 
As the constraints which should have been fulfilled in the case of each design being an alternative for 
the subject geodetic control network, the following nine were chosen:
 R R1.5 ,   { 1, ..., },  {4,5,6},j
j
A
j m m
B
≤ ∀ ∈ ∈  (26)
 S S2.2 ,   { 1, ..., 6},s
s
A
s
B
≤ ∀ ∈  (27)
 mm R R, 1.0  ,   { 1, ..., },  {4,5,6},p j j m mσ ≤ ∀ ∈ ∈  (28)
 mm S S, 1.5  ,   { 1, ..., 6},p s sσ ≤ ∀ ∈  (29)
 ˆ( ) 0.20 ,   {1,2,..., },iii ii Dr r i n= = ≥ ∀ ∈vQ P  (30)
 * 7.64  ,   {1,2,..., },
ii DG i nσ≤ ∀ ∈  (31)
 0(1 ) 0.80 ,   {1,2,..., } ,i i nβ− ≥ ∀ ∈  (32)
 S S S S S S S S S S S S S S1, 2 2, 3 4, 5 5, 6 1, 4 2, 5 3, 6(1 ) 0.80 ,   { , , , , , , } ,cs r r cs cs cs cs cs cs csβ− ≥ ∈  (33)
 S S(1 ) 0.80 ,   { 1, ..., 6},ccp s sβ− ≥ ∈  (34)
where the following, previously unmentioned, denotations are present:
 –
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
2 2
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) 0
1
( ( ) 4 )
2 j s j s j s j s j s j s j s j s j s j sj s y y x x x x y y x y
A Q Q Q Q Qσ= + + − +  is the semi-major 
axis of the standard error ellipse for the point j, i.e. s; s, devided by square root of the Chi-square 
distribution quantile for two degrees of freedom and significance level α.
 –
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
2 2
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) 0
1
( ( ) 4 )
2 j s j s j s j s j s j s j s j s j s j sj s y y x x x x y y x y
B Q Q Q Q Qσ= + − − +  is the semi-minor 
axis of the standard error ellipse for the point j, i.e. s, devided by square root of the Chi-square di-
stribution quantile for two degrees of freedom and significance level α.
 –
( ) ( ) ( ) ( )ˆ ˆ ˆ ˆ, ( ) 0 j s j s j s j sp j s y y x xQ Qσ σ= +  is the standard positional error of the point j, i.e. s;
 – rii is the redundancy coefficient for the measurement i;
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 – 0 01 1 /2*
ii D
ii
t t
G
r
β α σ− −
+
=  is the detectable marginal gross-error value in the measurement i (proba-
bilities 1 - β0 and α0 are, respectively, the test power and significance level for one-dimensional 
hypotheses in data snooping test – for 1 - β0 = 0.80, α0 = 0.01 and rii = 0.20, ∀i = {1, 2, ..., n} we 
have Gi
∗ ≈ 7.64σDi; for details about the test, see Baarda (1968));
 – σDi [mm] = a[mm]  b[mm/km] Di [km] is the measured length standard (a and b are empirical coefficients 
provided by manufacturer)
 – (1 - β0)i = normsdist (7.64√
_
rii - t1-α0/2) is the test power in detecting gross error limit value in the 
measurement i;
 –
T
normsdist 1 /2
ˆ0 ,
(1 ) cscs r
pq pq pq
d
t αβ
σ
−
 
 − = − 
 
 xh Q h
 is the test power in detecting minimal devi-
ation of the projected from the marked structure pillar span of 5mm(= dcs), with the vector 
hTpq = (Bpq  Apq  -Bpq  -Apq), pq ∈ {S1S2,S2S3,S4S5,S5S6,S1S4,S2S5,S3S6}, the matrix ˆ , pqxQ  that 
represents the submatrix (of the cofactor matrix x̂Q ) related to the coordinates of points p and q; 
and the significance level α = 0.05; 
 –
T T
normsdist ˆ 1 /2
0
(sin cos )( )(sin cos )
(1 )
tab
ccp s s s s s s
ccp s
d
t α
θ θ θ θ
β
σ −
 
 − = −  
 
xH Q H  is the test 
power in detecting minimal deviation of the projected from the marked structure pillar's center 
position of 4mm(= dccp), where αtab = 0.02367 (derived from the tabular value for non-centrality 
paremeter used in the test for congruence of 2D point position, for adopted test power of 0.80 and 
figure rank equal to 2), θ s is the standard error ellipse azimuth angle obtained from 
tg2 2 ˆ ˆ ˆ ˆ ˆ ˆ/ ( )s s s s s sQ Q Qs x y x x y yθ = − , and Hs is a 2-by-u matrix that separates the cofactor submatrix ˆ ,sxQ  , 
related to the coordinates of the point s, from the cofactor matrix x̂Q .
Remark: The thresholds in inequalities (26) and (27) are introduced on the basis of the fact that, in real 
conditions, i.e. in praxis, it is very difficult, almost impossibly, to establish such a geodetic network's 
configuration that will provide ratio values that are significantly smaller than the corresponding intro-
duced ones. 
2.2.3 Criteria functions used in the application of VIKOR method in the study
The authors introduced eight criteria functions here. Those functions are the following:
1) Deviation of the average redundancy coefficient from the optimal value: 
 1
( )
0.40n uopt
n r
f r r
n
×−= − = −
A  (35)
 with f = n - r(An×u) = n - u + d = n - u + 3 d.f;
2) Sum of cofactors for the unknown coordinates' estimates of the structure pillars centers: 
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 S
S S S
6
ˆ ˆ ˆ ˆ2 1( ) ,   { 1, ..., 6};s s s sy y x xsf Q Q s== + ∈∑  (36)
3)  Deviation of the average ratio of semi-major axis to semi-minor axis from the optimal value for the 
structure pillar's center: 
 S
S S S
6
3 1
1
1 ,   { 1, ..., 6};
6
s
s
sopt
AA A
f s
B B B=
    = − = − ∈    
     
∑  (37)
4) Ratio of the maximal to minimal semi-major axis in the part related to the structure pillars centers: 
  S S4
max
 ,   { 1, ..., 6};
min
s
s
s
s
A
f s
A
= ∈  (38)
5) Average standard positional error of the structure pillar's center: 
 S
S S S
6
5 ,1
1
 ,   { 1, ..., 6};
6p p ss
f sσ σ== = ∈∑  (39)
6) Difference between the minimal value obtained for the test power in detecting minimal deviation (in 
this study dsc =  5mm) of the projected from the marked structure pillar span and the corresponding 
threshold value: 
 6 min(1 ) (1 ) min(1 ) 0.80,cs r cs cs r
r r
f β β β= − − − = − −threshold  (40)
 where r ∈ {csS1,S2,csS2,S3,csS4,S5,csS5,S6,csS1,S4,csS2,S5,csS3,S6};
7) Difference between the minimal value obtained for the test power in detecting minimal deviation 
(in this study dccp =  4mm) of the projected from the marked structure pillar's center position and 
the corresponding threshold value: 
 7 min(1 ) (1 ) min(1 ) 0.80,ccp s ccp ccp s
s s
f β β β= − − − = − −threshold  (41)
 with s ∈ {S1, ...,S6};
8) Difference between the average value obtained for the test power in detecting gross error limit value 
in the measurement and the corresponding threshold value: 
 8 0 0 01
1
1 (1 ) (1 ) 0.80 ,   {1,2,..., }.n iif i nn
β β β=
 = − − − = − − ∈ 
 
∑threshold  (42)
The first five functions represent losses so they should be minimized. On the contrary, the last three 
functions represent gains to be maximized.
Because the main emphasis was placed on the precision and reliability of the considered geodetic control 
network, as well as the negligible differences between the costs of geodetic and other works related to the 
four alternatives are present, no criteria related to price were directly introduced. Even though, a price 
criterion is indirectly involved through the first criterion function (see Eq. (35)).
2.2.4 Main characteristics of the simulated multivariant geodetic network used in the study 
In this subsection, the main characteristics of each established alternative (previously denoted as A1, 
A2, A3 and A4) are shown.
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Approximate coordinates (Y and X) of the reference points R1, R2, R3 and R4 are the same for all the 
four alternatives, but those of the reference points R5 and R6 differ (see Table 1). Of course, approximate 
coordinates of the structure pillars centers, S1, S2, ..., S6, are common for all alternatives (see Table 2).
Table 1:  Approximate values of the reference points' coordinates for all the four alternatives.
Reference point Coordinate Approximate values of coordinates for each alternative
A1 A2 A3 A4
R1* Y [m]
X [m]
990.000
1030.000
990.000
1030.000
990.000
1030.000
990.000
1030.000
R2* Y [m]
X [m]
1220.000
1041.000
1220.000
1041.000
1220.000
1041.000
1220.000
1041.000
R3* Y [m]
X [m]
1172.000
1229.000
1172.000
1229.000
1172.000
1229.000
1172.000
1229.000
R4* Y [m]
X [m]
1015.000
1220.000
1015.000
1220.000
1015.000
1220.000
1015.000
1220.000
R5 Y [m]
X [m]
1097.000
1157.000
1091.000
1127.000
927.000
1125.000
no value
no value
R6 Y [m]
X [m]
1095.000
1099.000
no value
no value
1300.000
1117.000
no value
no value
* Reference points with coordinates common for all the four alternatives
Table 2:  Approximate values of the structure pillar centers' coordinates common for all the four alternatives.
Coordinate Approximate values of coordinates for each structure pillar's center
S1 S2 S3 S4 S5 S6
Y [m] 1070.000 1070.000 1070.000 1110.000 1110.000 1110.000
X [m] 1070.000 1130.000 1190.000 1070.000 1130.000 1190.000
Relevant data for this study related to the established alternatives are shown in Table 3.
Table 3:  Values for main design characteristics and criteria functions  for all the four  alternatives (with denotations introduced 
previously).
Main design characteristics and 
criteria functions
Values for main design characteristic and criteria functions for each alternative 
A1 A2 A3 A4
 m + 6 12 11 12 10
 n 42 38 47 32
u 24 22 24 20
d 3 3 3 3
 f = n - u + d 21 19 26 15
 f1 0.10 0.10 0.15** 0.07*
 f2 9.9360 mm
2* 10.5944 mm2 10.8042 mm2 11.5296 mm2 **
 f3 0.24 0.15* 0.61** 0.33
 f4 1.18* 1.40** 1.32 1.39
 f5 1.29 mm* 1.32 mm 1.34 mm 1.38 mm**
 f6 0.927099** 0.927104 0.932376 0.941131*
 f7 0.922113* 0.838633 0.829618 0.810279**
 f8 0.985734 0.984275** 0.991219* 0.984607
*   Best criterion value
** Worst criterion value
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It should be mentioned that all criteria function values given in Table 3 were obtained using the length 
measurement standard of σi = 1mm + 1.5mm/km. Di [km], which, for the four readings in a series gives 
σDi = σi / √
_
4 = 0.5mm + 0.75mm/km. Di [km], i ∈ {1, 2, ..., n}, what was used in all calculations.
All four alternative designs with the associated standard error ellipses are shown in figures 1, 2, 3 and 4.
Figure 1: Alternative A1 geodetic control network with standard error ellipses.
Figure 2: Alternative A2 geodetic control network with standard error ellipses.
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NFigure 3: Alternative A3 geodetic control network with standard error ellipses.
Figure 4: Alternative A4 geodetic control network with standard error ellipses.
3 RESULTS
Based on the values for the eight criteria functions given in Table 3, we can establish initial ranking lists 
of alternatives. Those lists are presented in Table 4.
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Table 4: Ranking lists of alternatives based on the criteria functions’ values from Table 3.
Position on the ranking 
list
Ranking lists of alternatives for each criterion function
f1 f2 f3 f4 f5 f6 f7 f8
First A4 A1 A2 A1 A1 A4 A1 A3
Second A2 A2 A1 A3 A2 A3 A2 A1
Third A1 A3 A4 A4 A3 A2 A3 A4
Fourth A3 A4 A3 A2 A4 A1 A4 A2
In the continuation of the presentation, all relavant results regarding the use of VIKOR method in the 
study are shown. Those results were obtained after an four-approach ranking. Each individual ranking 
approach involved a set of eight weights, one for each criterion function.
As can be spotted by looking at Table 5, the ratio of weights by approaches are as follows: 3:4:4:3:10:15:15:12 
(for the Ranking approach I); 4:7:7:5:7:5:5:5 (for the Ranking approach II); 3:10:5:3:10:10:10:7 (for 
the Ranking approach III); and 4:10:9:5:10:6:6:5 (for the Ranking approach IV).  
Table 5: Sets of criteria weights for all the four ranking approaches.
Criteria weight Sets of criteria weights for each ranking approach
Ranking approach I Ranking approach II Ranking approach III Ranking approach IV
w1 0.045 0.089 0.052 0.073
w2 0.061 0.156 0.172 0.182
w3 0.061 0.156 0.086 0.164
w4 0.045 0.111 0.052 0.091
w5 0.152 0.156 0.172 0.182
w6 0.227 0.111 0.172 0.109
w7 0.227 0.111 0.172 0.109
w8 0.182 0.111 0.121 0.091
As previously said, the criteria functions  f1, f2, f3, f4 and f5 are minimized, while f6,  f7 and f8 are maximized.
Dimensionless criterion function values dij, i ∈ {1, 2, ..., 8}, j ∈ {1, 2, 3, 4}, are shown in Table 6. It 
is easy to conclude that, for the alternative A1, the second, fourth, fifth and seventh criterion functions 
have the minimal values (0), and the sixth criterion function has the maximal value (1). Analogously, the 
corresponding conclusions can be drawn regarding the remaining three alternatives (A2, A3 and A4). 
Table 6: Dimensionless criterion function's values dij (i-th criterion function and j-th alternative).
Dimensionless 
criterion function
Values for the dimensionless criterion function for each alternative 
A1 A2 A3 A4
d1j 0.3701 0.3701 1 0
d2j 0 0.4131 0.5448 1
d3j 0.2078 0 1 0.3981
d4j 0 1 0.6622 0.9519
d5j 0 0.3876 0.5660 1
d6j 1 0.9996 0.6240 0
d7j 0 0.7465 0.8271 1
d8j 0.7899 1 0 0.9521
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In tables 7, 8, 9 and 10 the main results of the use of VIKOR method in the ranking approaches I, II, 
III and IV, respectively, are presented.
Table 7: Main results after using VIKOR method in the Ranking approach I.
Measure Main results after using VIKOR method – Ranking approach I 
A1 A2 A3 A4
Sj 0.4003 0.7247 0.5847 0.6799  S
∗ 0. 4003
S- 0.7247
Rj 0.2273 0.2272 0.1880 0.2273  R
∗ 0.1880
R- 0.2273
Rj,mod 0.2290 0.2272 0.1880 0.2318  R
∗
mod 0.1880
R-mod 0.2318
QSj 0 1 0.5685 0.8619
QRj,mod 0.9362 0.8948 0 1
Qj,mod(v = 0.50) 0.4681 0.9474 0.2842 0.9309
Qj,mod(v = 0.25) 0.7022 0.9211 0.1421 0.9655
Qj,mod(v = 0.75) 0.2341 0.9737 0.4263 0.8964
Table 8: Main results after using VIKOR method in the Ranking approach II.
Measure Main results after using VIKOR method – Ranking approach II 
A1 A2 A3 A4
Sj 0.2641 0.5737 0.6520 0.6957  S
∗ 0.2641
S- 0.6957
Rj 0.1111 0.1111 0.1556 0.1556  R
∗ 0.1111
R- 0.1556
Rj,mod 0.1111 0.1111 0.1605 0.1610  R
∗
mod 0.1111
R-mod 0.1610
QSj 0 0.7173 0.8988 1
QRj,mod 0 0 0.9912 1
Qj,mod(v = 0.50) 0 0.3587 0.9450 1
Qj,mod(v = 0.25) 0 0.1793 0.9681 1
Qj,mod(v = 0.75) 0 0.5380 0.9219 1
Table 9:  Main results after using VIKOR method in the Ranking approach III.
Measure Main results after using VIKOR method – Ranking approach III 
A1 A2 A3 A4
Sj 0.3048 0.6307 0.6139 0.7157  S
∗ 0.3048
S- 0.7157
Rj 0.1724 0.1723 0.1426 0.1724  R
∗ 0.1426
R- 0.1724
Rj,mod 0.1737 0.1723 0.1426 0.1778  R
∗
mod 0.1426
R-mod 0.1778
QSj 0 0.7931 0.7522 1
QRj,mod 0.8834 0.8440 0 1
Qj,mod(v = 0.50) 0.4417 0.8185 0.3761 1
Qj,mod(v = 0.25) 0.6626 0.8313 0.1880 1
Qj,mod(v = 0.75) 0.2209 0.8058 0.5641 1
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Table 10:  Main results after using VIKOR method in the Ranking approach IV.
Measure Main results after using VIKOR method – Ranking approach IV 
A1 A2 A3 A4
Sj 0.2418 0.5448 0.6568 0.7110  S
∗ 0.2418
S- 0.7110
Rj 0.1091 0.1091 0.1636 0.1818  R
∗ 0.1091
R- 0.1818
QSj 0 0.6459 0.8846 1
QRj 0.0006 0 0.7501 1
Qj(v = 0.50) 0.0003 0.3229 0.8174 1
Qj(v = 0.25) 0.0004 0.1615 0.7838 1
Qj(v = 0.75) 0.0001 0.4844 0.8510 1
Based on the data from tables 7, 8, 9 and 10, the following ranking lists are established: (a) Ranking 
approach I (the first ranking list: A1, A3, A4, A2; the second ranking list: A3, A2, A1, A4; the com-
promise ranking list for v = 0.50: A3, A1, A4, A2); (b) Ranking approach II (the first ranking list: A1, 
A2, A3, A4; the second ranking list: A1, A2, A3, A4; the compromise ranking list for v = 0.50: A1, A2, 
A3, A4); (c) Ranking approach III (the first ranking list: A1, A3, A2, A4; the second ranking list: A3, 
A2, A1, A4; the compromise ranking list for v = 0.50: A3, A1, A2, A4); and (d) Ranking approach IV 
(the first ranking list: A1, A2, A3, A4; the second ranking list: A2, A1, A3, A4; the compromise ranking 
list for v = 0.50: A1, A2, A3, A4).
Considering the results given in Table 7 as well as the established ranking lists for the Ranking ap-
proach I, it can be said that the first alternative on the compromise ranking list, A3, does not have a 
''sufficient advantage'' over the alternative A1, which is the second ranked on the same list (condition 
C1 not fulfilled). On the other hand, alternative A3 has a ''sufficient advantage'' over the alternative 
A4, which is the third on the compromise ranking list. Besides, alternative A3 is the first ranked on 
the QR ranking list, so condition C2 is fulfilled. Taking into account this facts, one can conclude that, 
in the case of the Ranking approach I, we have the set of compromise solutions that contains alternatives 
A3 and A1. 
When it comes to the results obtained using Ranking approach II (Table 8), we have alternative A1 as 
the first-ranked one on the compromise ranking list and it has a ''sufficient advantage'' over the second-
ranked alternative, A2 (condition C1 fulfilled). Alternative A1 has a ''sufficiently stable'' first position, 
because it is also the first-ranked one on the compromise lists for v = 0.25 and v = 0.75. In addition, 
it is also first-ranked on QS and QR ranking list, so condition C2 is fully met. Thus, when applying the 
Ranking approach II, we get alternative A1 as the best solution. 
After applying the Ranking approach III (the corresponding results are shown in Table 9), it turned out 
that we have again the set of compromise solutions that involved alternatives A3 and A1. Here, the conclu-
sions were analogous to those obtained using the Ranking approach I.
Looking at Table 10, we can conclude that if apply the Ranking approach IV, we again get alternative A1 
as the best solution.
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On the basis of the previously stated, the following conclusion can be made: the optimal solution for 
the local geodetic control network design is the one related to alternative A1. Although, it would not 
be wrong to declare the set of compromise solutions that contains alternatives A1 and A3 as the solution. 
4 DISCUSSION
This study showed that the use of VIKOR method in designing a special-purpose geodetic network is 
very desirable when an engineering surveying professional has to make a decision in a sea of conflicting 
requirements, especially those related to precision and reliability. What is the most important, such a 
multi-criteria optimization approach is an efficient tool that gives the best solution for complex geodetic 
systems.
On the basis of the results obtained, the best alternative was adopted as the solution for the design of 
the considered geodetic network in the study. What is very interesting is the fact that, before using the 
VIKOR algorithm, this alternative was the best according to only four criteria. It was ranked third ac-
cording to internal reliability (the first criterion in the study) and, even, fourth  (the last one on the list) 
according to the test power in detecting minimal deviation (adopted herein) of the projected from the 
marked structure pillar span.
All of the above leads to the conclusion that the best solution, i.e. compromise solution, does not al-
ways have to be the best one according to all or most of introduced criteria. On the contrary, it may be 
the worst one according to some criteria, but it will be the best in the compromise. Thus, that's where 
VIKOR method is, we dare say, the best ally in solving problems.
At the very end, the authors propose the use of VIKOR method whenever there is a need to make a 
decision in a complex engineering system involving a number of conflicting requirements that are dif-
ficult to achieve all at the same time.
Literature and references:
Baarda, W. (1968). A Testing Procedure for Use in Geodetic Networks. Delft: Netherlands 
Geodetic Commission, Publications on Geodesy, New Series, 2 (5).
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Darko Anđić, Ph.D. 
Cadastre and State Property Administration, 
Department of State Survey, Real Estate Cadastre
and Geodetic Information System, 
Duke Stanka Radonjića Blvd No 1, 
81000 Podgorica, Montenegro 
e-mail: andjic.darko@gmail.com 
Assoc. Prof. Radovan Đurović, Ph.D. 
University of Montenegro, Faculty of Civil Engineering 
Džordža Vašingtona Blvd bb, 81000 Podgorica, Montenegro 
e-mail: radovandj@ucg.ac.me
Anđić D., Đurović R. (2023). The use of  VIKOR Method in Finding the Optimal Solution for a Special-Purpose Geodetic Network Design. 
Geodestski vestnik, 67 (2), 196-212. 
DOI: https://doi.org/10.15292/geodetski-vestnik.2023.02.196-212
Darko Anđić, Radovan Đurović | UPORABA METODE VIKOR PRI ISKANJU OPTIMALNE REŠITVE ZA ZASNOVO GEODETSKE MREŽE ZA POSEBNE NAMENE | THE USE OF VIKOR METHOD IN FINDING 
THE OPTIMAL SOLUTION FOR A SPECIAL-PURPOSE GEODETIC NETWORK DESIGN  | 196-212 |