Multi-criteria analysis or multi-criteria optimization, as a method and technique for evaluation and decision making, has been developed in different scientific disciplines, such as operational researches, statistics, management science, computer science (Ivanovic, 1973; Saaty, 1982; Mardani et al., 2015a; Mardani et al., 2015b; Brans and Vincke, 1985; Roberts and Goodwin, 2002). Multi-criteria decision analysis has been and continues to be applied in various fields of expertise, such as finance, logistics, transportation, marketing, public services, energy management, environmental planning (Dong and Xu, 2016; Huang et al., 2011; Stanujkic et al., 2017; Mardani et al., 2015a, 2015b; Turskis and Zavadskas, 2011; Zavadskas et al., 2017; Hashemkhani Zolfani et al., 2013). In particular, in economics (Zavadskas et al., 2016a, 2016b) and in engineering (Zavadskas et al., 2016c; Šaparauskas et al., 2011; Zavadskas et al., 2014).

Today, many books are also available, in which an overview can be found of methods adjusted for various fields of application, since it is known that some of the methods can be more easily applied (or, they are better adjusted) to specific areas. Let us mention some of the published books (Zeleny, 1982; Figueira et al., 2005; Hwang and Yoon, 1981; Triantaphyllou, 2000), among the recently published are (Huber et al., 2019; Kaliszewski et al., 2016; Köksalan et al., 2011; Lee and Yang, 2017). See also conference proceedings (Sforza and Sterle, 2017).

Therefore, it should be noted that the most important factor in decision making, i.e. in choosing one of the alternatives or creating a preference order of alternatives, is, in fact, the decision maker. Namely, the decision maker has a decisive role in setting the weight coefficients for the criteria.

The criteria on the basis of which a decision is made are often conflicting; for some problems they are partially conflicting, and sometimes even completely conflicting. Also, they can be very diverse in their essence – sometimes those are cost values, sometimes various physical measurements, sometimes probabilities, sometimes subjective estimations (usually those of the decision maker) given in different possible scales, which are often created for a specific problem. Hence, we have measurement units, which are incomparable according to specific criteria.

We could say that the decision maker has the often-difficult task of comparing possible decisions.

The essential problem of a multi-criteria analysis can be shown in a simple manner through a decision-making matrix (Table 1): which contains n criteria C1,…,Cn , m alternatives A1,…,Am and values xij , i=1,…,m , j=1,…,n , where alternative Ai assumes according to criterion Cj . We also know, due to the nature of the problem, that criteria belong to the maximization type (bigger is better) or minimization type (smaller is better).

When it comes to solving real problems, it is obvious that almost never there is an alternative, which is optimal according to all of the criteria (an ideal alternative). Because of that, it is usually necessary to find a compromise, i.e. a “good” compromise or an alternative that will satisfy in the best manner possible, in a given situation, conditions set by the decision maker.

In order to achieve that, a valorization of criteria is performed, i.e. their importance is determined. Most commonly, the importance of criteria is established with weight coefficients (positive numbers whose sum equals one).

In almost all methods of multi-criteria analysis, the initial Table 1 is transformed into a numeric table, i.e. values according to criteria, which have a linguistic form, and which are translated, with the help of suitable transformation scales for linguistic expressions, into numbers. Thus, we can assume that Table 1 consists of numbers. The next step is to obtain, with Table 1 converted into numbers, a normalized table – Table 2, in which all values are unnamed numbers and all criteria are translated into a maximization type.

Hence, values aij are normalized, i.e. 0⩽aij⩽1 and we also have, for example, that, aik⩽ajk , 1⩽i⩽m , 1⩽k⩽n , 1⩽j⩽m .

This means that alternative Ai is weaker than alternative Aj according to criterion Ck .

Depending on how we perform the transformation of Table 1 into Table 2 and on how we now perform the evaluation of alternatives, from Table 2, by using weight coefficients of criteria whose sum equals one, we have a very large number of different models of multi-criteria analyses.

Of course, the above description does not include all available methods, an example is the method given in Žižović et al. (2016).

Descriptions of different types of normalizations can be found in the following papers: (Celen, 2014; Jahan and Edwards, 2015; Milani et al., 2005).

Descriptions of different methods for finding weight coefficients can be found in the following papers: (Zavadskas and Podvezko, 2016; Krylovas et al., 2017; Keshavarz Ghorabaee et al., 2016).

In the paper (Žižović et al., 2017), the notion of a dominant criterion is defined: Criterion Cj is dominant for solving multi-criteria problems if the value of alternatives according to this criterion is extremely low; then there is no solution for that problem, or the solution obtained has an extremely small (weak) importance. (“Let Cj be the criterion which is the most dominant for the solution of multi-criteria problems, meaning that if its performance values of all alternatives are extremely low then problem should not be treated (it has no solution), or obtained solution of the problem has weak importance”.)

In the paper (Žižović, 2018), a multi-criteria method is also given which disables the option for an alternative with a low value of the dominant criterion to obtain high value based on “good” values according to other criteria.

A new method of multi-criteria analysis is also given here, where an alternative with a low value of the dominant criterion has to have low general value as well, regardless of the values according to other criteria. That is to say that, the value of the dominant criterion, which is several times (more than two) lower than some other alternatives, cannot be compensated with higher values according to other criteria (those approximately similar can be compensated).

Let us assume that our starting point is the multi-criteria method given by the decision making matrix in Table 2, where the dominant criterion is C1 . Other criteria, C2,…,Cn , are sorted according to their importance.

For the arbitrary alternative Ap , 1⩽p⩽m we observe the comparison value in relation to other alternatives according to the formula: (1) V(Ap)=ap1·K1+λ·ap1·(ap2·K2+ap3·K3+⋯+apn·Kn), where api are values which alternative Ap has according to criterion Ci ( i=1,…,n ); Ki is the weight coefficient of criterion Ci ( i=1,…,n ); λ – is a negative number, which determines the importance relation between the first dominant criterion and other criteria.

The comparison of alternatives Ap and Aq is performed according to the following rules:

Rule no. 1: Alternative Ap is better than alternative Aq if V(Ap)>V(Aq) V({A_{q}})$]]>;

Rule no. 2: If V(Ap)=V(Aq) then Ap is better than Aq if ap1>aq1 {a_{q1}}$]]>;

Rule no. 3: If V(Ap)=V(Aq) and if ap1=aq1 then Ap is better than Aq if ap2>aq2 {a_{q2}}$]]>;

Rule no. 4: If V(Ap)=V(Aq) and ap1=aq1 and ap2=aq2 then Ap is better than Aq if ap3>aq3 {a_{q3}}$]]>.

This proceeding continues up to the last comparison.

It is obvious that alternatives Ap and Aq would remain incomparable only if all the values are equal, i.e. if ap1=aq1 , ap2=aq2 , …, apn=aqn .

In formula (1), parameter λ has the important role of determining the influence of the first criterion in relation to other criteria in the model (or vice versa), or, in other words, determining the links and relations of the influence of the first criterion and other criteria on the defining of the ranking order of alternatives and, hence, decision making.

In order to have a better overview of this parameter and to make it easier for the decision maker to make his choice, let us notice, first, that for the arbitrary alternative Axwith characteristicsax1,ax2,…,axn according to formula (1) we get that (2) V(Ax)=ax1·k1+ax1·λ·(ax2·k2+ax3·k3+⋯+axn·kn)⩾ax1·k1.

This is obvious from the condition which every alternative has to fulfil, and, of course, from the condition of parameter λ being set.

Let us just stress that this value is also realistically obtained for a2=a3=⋯=an=0 or for λ=0 .

The case of λ=0 brings this method of multi-criteria analysis down to the familiar lexicographical order which is, basically, linear (the only case when two alternatives are not modified is when they have equal values according to all the criteria).

Value V(Ax) can be limited on the upper side as well [alternative: can be bounded from above], that is to say, we get that (3) V(Ax)⩽ax1·(k1+λ·(1−k1)). This can be easily proven: V(Ax)=ax1·k1+ax1·λ·(ax2·k2+ax3·k3+⋯+axn·kn). If we know that 0⩽axi⩽1 , i=1,…,n then V(Ax)⩽ax1·k1+ax1·λ·(k2+k3+⋯+kn)=ax1·k1+ax1·λ·(k1+k2+k3+⋯+kn−k1)=ax1·k1+ax1·λ·(1−k1)=ax1·(k1+λ·(1−k1)). [Alternative: this upper bound is attained for:] The theoretical and the value on the right side can be reached for ax2=ax3=⋯=axn=1 .

In the expression: (4) aAx1·k1⩽V(Ax)⩽ax1·(k1+λ·(1−k1)) with the choice of parameter λ we can pre-determine how large the right side would be in relation to the left side, if we observe the ratio of the right and the left side: [ax1·(k1+λ·(1−k1))]/(ax1·k1)=[k1+λ·(1−k1)]/k1=t. Here, it is obvious that t>0 0$]]> (it is assumed that k1>0 0$]]>) and this number can be pre-determined. Hence, if we pre-determine, as a requirement, the relation between the maximum possible value and the measurement for an alternative according to the participation of the dominant criterion in that value, we can easily calculate the value for parameter λ: (5) λ=(k1·(t−1))/(1−k1) for which we obtain the desired ratio.

Aside from this relation, it could be interesting for the decision maker to also have an estimate of the size of the difference of the maximal value for the alternative and the participation of the dominant criterion in that measurement compared to the contribution of the dominant criterion: (6) θ=[ax1·(k1+λ·(1−k1))−ax1·k1]/(ax1·k1)=λ·(1−k1)/k1.

Here, we have the possibility of θ being greater than one – then the participation of other criteria in the maximum value is greater than the participation of the dominant criterion; that of θ being less than one – then the participation of other criterion in the maximum value is lower than the participation of the dominant criterion; and, finally, that θ=1 – here we have the equal participation of other criteria and the dominant criterion in the alternatives measuring. It is obvious that here, as well, we can easily get from (6) that (7) λ=θ·k11−k1 and expressions (5) and (6) can be used in an equal manner for calculating the role of parameter λ and the influence it has on the connection in the participation of the first and other criteria in the measuring of the alternative preference order.

We find that expression (7) is more suitable for calculation, and the quotient θ is more suited for understanding; and as an illustration of values which were obtained, Table 3 provides examples of values calculated for parameter λ (rounded up to four decimals) depending on the choice of weight coefficient for the dominant criterion and quotient θ.

N.B.: When solving multi-criteria problems with this method, it is useful to observe the maximum value in the column corresponding to the dominant criterion C1 . Let us mark it with a1max and compare values ai1 (k1+(1−k1)) , i=1,…,m .

With amax·k1 .

In cases when ai1(k1+(1−k1))<amax·k1. Alternative Ai can be left out of the model, because it will be weaker than the alternative in which we have the highest value according to the dominant criterion.

Medical tourism is very popular in the world of today. On one hand, we have people from countries with less developed medical systems going to those with more advanced health-care systems, looking for treatment for health problems which cannot be dealt with in their own countries.

On the other hand, we have people from highly developed countries, where medical care is expensive, going to countries where medical care is at the same quality level, but for a significantly lower price.

Today, the Republic of Serbia is a country whose medical staff performs high-quality medical services, but for a significantly lower price than those in developed countries. This is most prominent when it comes to dental care and cosmetic surgery, as well as some other medical specialities. Aside from the medical staff, hotel and restaurant owners, other people in the tourism economy are also interested in providing these services. The medical staff has been primarily interested in providing these services to our Gastarbeiter (people temporarily working in other countries). It was only recently that websites began to appear offering those services for foreign citizens as well. Because of the interest that tourism industry has here, we will perform the evaluation of the websites of several dental clinics targeting foreign citizens as well.