Introduction

In the decision-making problem, the aim is to find the suitable alternative or alternatives from the set of feasible alternatives. This problem can occur in many different real life circumstances, including economics, education, military, medical sciences, logistics, etc. Therefore, one can find numerous studies related to the decision-making problems in the literature.

In recent years, since the complexity of the problem grows, it is almost impossible for a decision maker to consider all the relevant factors of the problem (Dey et al. 2017). Additionally, most of the decision-making problems occur in an environment where multiple experts must contribute in order to reach a final decision. Therefore, there is a need for new techniques, which require the contribution of multiple decision makers whose experiences, attitudes, knowledge are not same or similar (Ölçer and Odabaşi 2005; Cabrerizo et al. 2013). In literature, these kinds of problems are known as the Group Decision Making (GDM) problem (Lu and Ruan 2007).

As mentioned above, in GDM, decision makers may come from different fields, which means they are not supposed to be homogenous in terms of their knowledge, skills, and characteristics. This results in a variation in their attitudes, motivations, and acceptance of the common problem. That requires different approaches than the regular decision-making problem with the single decision maker. In these approaches, the main problem is to find how much a decision maker should contribute to the final solution, in other words, what should be the weight of each decision maker. If the weights of decision makers are not taken into account in GDM, this can cause improper and mistaken results that may not be compensated in the final solution (Mianabadi et al. 2008). Therefore, in current years, how to derive the weights of decision makers become a challenging and interesting research subject for the researchers.

In literature, the techniques for deriving the decision makers’ weights are grouped into three classes; the subjective techniques, the objective techniques and the combination of these two (Saaty 1980; Ramanathan and Ganesh 1994). In subjective techniques, a supervisor, who evaluates the decision makers, assigns weights to each of them depending on his or her assessment about the decision makers or, these weights are obtained by mutual evaluations of each decision maker which may be accepted as a more democratic way than the first one. On the other side, in objective methods, researchers use numerical methods to determine the weights of decision makers. All the data in hand are used in objective methods and it aims to be more objective in all aspects when compared to subjective methods. In the combined method, these two weights are integrated into one weight, which reflects the effects of both objective and subjective weights.

Kabak and Ervural (2017) state that only in 41% of selected studies, weights of experts are taken into account. It is also stated that, in most of the studies, subjective methods are used. On the other hand, even though it is not as popular as criteria weighting methods, there are some studies that utilize the objective methods in order to determine the weights of decision makers. For instance, Wang et al. (2015) measure the similarity degrees of two partial rankings given by two decision makers and use this measure to find the relative weights of decision makers. In this study, a larger weight is assigned to a decision maker with more similarity to other decision makers. In a similar study, Yue (2011) finds the distance measure of each decision maker to the aggregated solution in interval-valued intuitionistic fuzzy GDM. Pang et al. (2017) use a technique depending on the consensus value, to find the weights of decision makers. In their study, authors develop a mathematical model to maximize the consensus degree, by altering the weights of decision makers. In a similar study, instead of using the consensus value, Zhang and Xu (2014) use two different consistency indexes. Liu et al. (2015) utilize a two level (experts and cluster level) method to find the weights of decision makers in complex multi attribute large GDM problem in a linguistic environment where there are more than 20 decision makers.

In above literature and other related studies in GDM, decision makers chose to express their preferences using different preference formats such as crisp value (real numbers), interval numbers, fuzzy sets (e.g. hesitant fuzzy sets, Type-2 fuzzy sets, intuitionistic fuzzy sets etc.), rankings, linguistics, and other data types. However, most of the approaches in the literature are independent of the data format.

Even though decision making problem can occur in many different real life circumstances including economics, education, military, medical sciences and logistics, the concentration on multi-criteria decision making in the healthcare industry is quite new. The methodologies implemented in medical and health care decision making analysis are provided in the literature (Stang et al. 1988). Among these, most of them prefer using AHP in health care and medical decision making analysis, specifically, evaluation and selection of medical treatments and therapies, for organ transplant eligibility and allocation decisions (Dolan et al. 1993, 1989; Liberatore and Nydick 2008). Our study provides a case study on a medical decision making, specifically, selection of a suitable anesthesia method to apply in the surgery is a multi-criteria decision making which involves conflicting criteria and several alternatives, to illustrate how the combined relative weights of decision makers (doctors or experts) are derived.

We will apply the Analytic Hierarchy Process (AHP) based method in order to derive the relative weights of decision makers depending on the geometric cardinal consensus index (GCCI), and we will combine it with the subjective weights of decision makers provided by a supervisor. Information about the method is provided in the following section. For the details of the approach, readers can refer to the study proposed by Blagojevic et al. (2016).

The rest of the paper is structured as follows. The second section presents the methodology proposed in the study. Next, an illustrative example is provided to show the implementation of the proposed method. The paper ends with our conclusion and suggestions for future research.

Methodology

In this section, before introducing the basis of the weight determination method provided by Blagojevic et al. (2016), we provide some information about AHP and GCCI, which is a well-known consensus index of AHP that is used in the proposed approach.

AHP and Geometric Consistency Index

AHP requires a problem that can be represented as a hierarchy. In this hierarchy, the main goal of the problem should be placed at the top (Level 1), while alternatives (Level 3) are placed at the bottom as presented in Fig. 1.

Fig. 1
figure 1

Hierarchy of the problem

Full size image

Here, the main goal is the overall objective of the problem that must be satisfied by more than one criterion (Level 2). In AHP, preferences of decision makers among the alternatives and criteria are determined by pairwise comparison with respect to the scale in Table 1 provided by Saaty (1980).

Table 1 Importance scale proposed by (Saaty 1980)
Full size table

As it can be easily seen on Table 1, if two elements are considered to have equal importance and contribute same to the upper level, a value of “1” is given to the related element (aij) of the following matrix A. In the same manner, a value of “9” is given if one has extreme importance in pairwise comparison.

$$A = \left[ {\begin{array}{*{20}c} {a_{11} } & \cdots & {a_{1n} } \\ \vdots & \ddots & \vdots \\ {a_{n1} } & \cdots & {a_{nn} } \\ \end{array} } \right]$$

The constructed matrix must be symmetric and reciprocal, which means all elements of the matrix must be greater than “0”, aij = 1/aji and all main diagonal elements must be “1”. Saaty (1980) also proves that, for this matrix, the eigenvector method, which is based on squaring and normalization of row sums, can be used to derive the required priority vector, for both consistent and inconsistent evaluation of the decision maker.

However, there are some other prioritization methods used in the literature [combining different prioritization methods in AHP synthesis] including least squares method (LSM) proposed by Crawford and Williams (1985). In this study, it is suggested that the priorities can be calculated by the following non-linear mathematical model (Eqs. 1 and 2):

$$\mathop {\hbox{min} }\limits_{w} \sum\limits_{i = 1}^{n} {\sum\limits_{i < j}^{n} {\left( {\ln a_{ij} - \left( {\ln w_{i} - \ln w_{j} } \right)} \right)^{2} } }$$
(1)

subject to:

$$w_{i} > 0,\quad \sum\limits_{i = 1}^{n} {w_{i} = 1,\quad i = 1, \ldots ,n} .$$
(2)

They also prove that the solution of the model is equal to the geometric means of the rows of matrix A and simply can be calculated by the following Eq. 3.

$$w_{i} = \frac{{\sqrt[n]{{\prod\nolimits_{j = 1}^{n} {a_{ij} } }}}}{{\sum\nolimits_{i = 1}^{n} {\left( {\sqrt[n]{{\prod\nolimits_{j = 1}^{n} {a_{ij} } }}} \right)} }}$$
(3)

In this study, we use LSM based index like the study of Blagojevic et al. (2016), mainly because, while eigenvector method produces different group priority vector for both AIJ (Aggregation of individual judgments-group acts like a unit) and AIP (Aggregation of individual priorities-group acts separately), LSM produces same group priority vector in both situations (Barzilai and Golany 1994).

After finding the final decision by aggregation procedures, the consistency index must be measured to check the consistency of the preferences. In literature, there are several consistency measures for AHP. One of the comprehensive studies about the subject is proposed by Aguaron and Moreno-Jiménez (2003). They have developed the geometric consistency index (GCI) in Eq. 4 in order to measure individual consistency:

$${\text{GCI}}(A) = \frac{2}{(n - 1)(n - 2)}\sum\limits_{i < j}^{n} {\left( {\ln a_{ij} - \ln w_{i} + \ln w_{j} } \right)^{2} }$$
(4)

Matrix A can be called as fully consistent if GCI (A) is equal to 0. It is acceptable inconsistent if GCI (A) < 0.31 for n = 3 and GCI (A) < 0.35 for n = 4. For larger n values, GCI must be less than 0.37.

In literature, consensus indexes including the GCCI, are computed by measuring the distance of decision makers’ preference matrix to group aggregated preference matrix (Dragincic et al. 2015; Srdjevic et al. 2015). For AHP, using the idea behind GCI, GCCI is developed by (Dong et al. 2010) and provided in Eq. 5.

$${\text{GCCI}}\left( {A^{(k)} } \right) = \frac{2}{(n - 1)(n - 2)}\sum\limits_{i < j\;i,j = 1}^{n} {\left( {\ln \left( {a_{\imath j}^{(k)} } \right) - \ln \left( {w_{i}^{(g)} } \right) + \ln \left( {w_{j}^{(g)} } \right)} \right)^{2} }$$
(5)

Since the GCCI is derived from GCI, LSM method must be used for prioritization in AHP in order to calculate the GCCI, as it is provided by Blagojevic et al. (2016). Here, it can be stated that a decision maker is totally in cardinal consensus if he/she has a value of 0 for the GCCI, and cardinal consensus for a decision maker decreases when the GCCI value increases.

A Method for Deriving Combined Weights

As already mentioned in the previous sub-section, in order to find the GCCI, LSM prioritization method must be applied to obtain the decision makers’ priority vectors from the preference matrices of each decision maker. Therefore, in the first step, LSM must be applied to each decision maker’ preference matrix.

In the second step, these preference matrices must be aggregated into group decision and aggregated priority vector must be determined. As in the first stage, the aggregated priority vector must be obtained by using LSM.

By using Eq. 5. GCCI for each decision maker must be computed in the third step.

In the fourth step, the objective weight of each decision maker will be computed by the following Eq. 6 as it is proposed by (Blagojevic et al. 2016). Since cardinal consensus for a decision maker decreases when the GCCI value increases, reciprocals of GCCI values will be used in order to find the objective weight of kth decision maker (W k0 ).

$$w_{0}^{k} = \frac{{\left( {{\text{GCCI}}^{k} } \right)^{ - 1} }}{{\sum\nolimits_{k = 1}^{d} {\left( {{\text{GCCI}}^{k} } \right)^{ - 1} } }}$$
(6)

In the last step, decision makers’ objective weights \(w_{0}^{k}\) are integrated with the subjective weights \(w_{s}^{k}\) provided by a supervisor with a predefined proportion of α in order to obtain the final combined weights \(w_{c}^{k}\), using Eq. 7.

$$w_{c}^{k} = \alpha \left( {w_{0}^{k} } \right) + (1 - \alpha )\left( {w_{s}^{k} } \right)$$
(7)

An Illustrative Example

There are several approaches to handle multi-criteria problems in the literature. A variety of decision making approaches and tools are available to support health care and medical decision making. Hancerliogullari et al. (2017) study preferences of surgeons for anesthesia method selection, which is the first in the literature, where multi-criteria decision making tools, fuzzy Analytic Hierarchy Process and fuzz, Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), are used to evaluate the anesthesia methods for a surgery. Nevertheless, the objective and subjective weights of the decision makers were not taken into account in that study.

Therefore, in order to show the application of the proposed method, we provide an example of medical decision making. Medical doctors face different alternatives while selecting an appropriate anesthesia method to apply in the surgical procedures. The decision is complex since there are several factors affecting the operations. The alternatives of anesthesia methods performed include general anesthesia, local anesthesia, and sedation. Selection of a suitable anesthesia method and deciding the appropriate method is a real concern since an inappropriate method may threaten patients’ lives and lead to a loss of resources and time.

In this study, we provide an illustrative example to derive the combined relative weights of three decision makers (DM1, DM2, DM3), specifically three medical doctors, depending on the GCCI and subjective weights provided by a supervisor. The alternatives to anesthesia methods, general anesthesia (A1), local anesthesia (A2) and sedation (A3), are determined by the decision makers. As stated in the Blagojevic et al. (2016), we assume that the inconsistency of decision makers is greater than 0.31 since n = 3. Other calculations are expressed in the following steps below:

  • Step 1. Preference matrices of decision makers are provided in Table 2. Decision makers’ priority vectors from the preference matrix of each decision maker are calculated by the LSM method (geometric means of the rows of the matrix) formulation provided in Eq. 3.

    Table 2 The preference matrices of each decision maker with priority vectors
    Full size table

  • Step 2. Preference matrix of each decision maker given in Step 1 must be aggregated into group decision and aggregated priority vector must be determined. The aggregated priority vector must be obtained by using LSM. Since we do not assign a weight to decision makers at the beginning, we assume that they are equally important. Therefore, in this step, we use subjective weights while aggregating the decision makers’ preference matrix. Table 3 summarizes the group priority vector values for each alternative.

    Table 3 Group priority vector
    Full size table

  • Step 3. By using Eq. 5. geometric cardinal consensus index for each decision maker is computed in the third step, and the results are provided in Table 4.

    Table 4 GCCI, objective, subjective and combined weights for each decision maker
    Full size table

  • Step 4. The objective weight of each decision maker is computed by using Eq. 6, and the results are shown in Table 4.

  • Step 5. Decision makers’ objective weights \(w_{0}^{k}\) are integrated with the subjective weights \(w_{s}^{k}\) provided by a supervisor and combined weights \(w_{c}^{k}\) are calculated, using Eq. 7. The values of the combined weights are provided in Table 4. Here, \(\alpha\), which shows the proportion of the impact of subjective and objective weights, is taken 0.6. Subjective weights are assumed to be equal, 0.33 for each decision maker.

After having found the combined weights for each decision makers, we determine the best alternative by using Table 2, which is the preference matrices of each decision maker with priority vectors, and apply the weighted product method (Yoon and Hwang 1995). When the weighted product method is used, weights become exponents associated with each attribute value. This method requires that all ratings be greater than 1 due to exponent property. Otherwise, when an attribute has fractional ratings, as in this study, all ratings in that attribute are multiplies by 10 m to meet the following requirement, given in Eq. 8. By applying the given formula, the order of the alternatives is determined, as given in Table 5. As a result, A1, general anesthesia, is determined to be the anesthesia method based on the decision makers’ preferences and combined weights calculated by these preferences.

Table 5 Order of the alternatives
Full size table
$$V_{i} = \prod\limits_{j} {\left( {x_{ij} } \right)^{{w_{j} }} }$$
(8)

Conclusion

The primary objective of the decision makers is to make the best decision among a set of different alternatives. The complexity of the situation grows as multiple decision makers involved in decision making process. Since each decision maker may have different background and perspective, their approaches might be different from each other on the same problem, which is specifically called GDM problem in the literature, and more comprehensive techniques are needed. Such decision making problems take place on several occasions; especially military operations, medical sciences, economics, etc. Here, the primary problem is to determine how much a decision maker should contribute to the final solution (i.e., the weight of each decision maker).

In literature, the techniques for deriving the decision makers’ weights are grouped into three classes; the subjective techniques, the objective techniques and the combination of these two. The main aim of this study is to propose a combined method to determine decision makers’ weights in decision making environment. The contribution of this study is twofold. First, the suggested method integrates the AHP based methodology depending on the geometric cardinal consensus index proposed by Blagojevic et al. (2016) and the subjective weights of decision makers provided by a supervisor. Second, an application in a medical decision making is provided; the methodology is used to derive the pediatric surgeons’ weights for the first time. We derive the combined relative weights of the medical doctors who face with different alternatives while selecting an appropriate anesthesia method among 3 alternatives, general anesthesia, local anesthesia, and sedation, to apply in the surgical procedures. In this study, the alternatives and decision makers’ opinions were considered. As a future research, a multi-attribute group decision making problem can be studied where criteria will also be taken into account. Additionally, different consistency indices can be utilized other than GCCI and results can be compared. Moreover, further collaborative studies can be conducted with the other decision makers in the health care industry.