1 Introduction

Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is a rather straightforward technique in dealing with multi-criteria or multi-attribute decision making (MCDM/MADM) problems that have cardinal information (Hwang and Yoon 1981), and it is popularly practiced in the Asia-Pacific region (Shih et al. 2007). Behind the use of this technique, there exist many assumptions that are sometimes hard to meet. One case occurs when there are often insufficient metric data for an evaluation. If there exists ordinal information in making a decision, which is easier to obtain, then we only require partial information from decision makers (DMs). Ordinal information is argued to be more reliable and available than cardinal (continuous) information (Larichev 1992), but few works in the literature directly address ordinal or qualitative information in the TOPSIS process without considering fuzzy or interval values. This study deals with the problem of ordinal information that exists in group TOPSIS, which is a generalization of TOPSIS in a group decision making environment. The differentiated decision power in group TOPSIS is addressed as well. Hence, we expect a coherent and complete procedure for a mixed-data multiple-criteria group decision support with differentiated decision power.

The basic concept of TOPSIS is in a straight line. It originates from the concept of a displaced ideal point from which the compromised solution has the shortest distance (Belenson and Kapur 1973; Zeleny 1974). Hwang and Yoon (1981) recommended that the ranking of alternatives is based on the shortest distance from the (positive) ideal solution (PIS) and the farthest distance away from the negative ideal solution (NIS) or anti-ideal solution in an n-dimensional Euclidean space. TOPSIS simultaneously considers these distances to both PIS and NIS and then ranks a preference order on account of their relative closeness, which is a combination of these two distance measures. Although there are multiple versions, the core thinking of the technique is that the distance function represents the DMs’ preferences or utilities.

Because MCDM is a practical tool for selecting and ranking a number of alternatives, its applications are numerous. TOPSIS is deemed as a major analytic technique and has been successfully applied in numerous areas (Shih et al. 2007). Behzadian et al. (2012) collected 266 papers since 2000 and inferred that there are nine major application areas of TOPSIS: (1) supply chain management and logistics, (2) design, engineering and manufacturing systems, (3) business and marketing management, (4) health, safety, and environment management, (5) human resource management, (6) energy management, (7) chemical engineering, (8) water resources management, and (9) other topics. Of these TOPSIS applications, most are combined with other techniques, e.g., fuzzy sets (Langroudi et al. 2013), interval information (Tsaur 2011), analytic hierarchy/network process (AHP/ANP) (Torfi et al. 2010), entropy methods (Hung and Chen 2010), and mathematical programming (Garcia et al. 2010).

Since group TOPSIS is a major interest in the literature, we have to examine TOPSIS used in a group decision environment. Most research works involve fuzzy or interval data. Yue (2013) recently organized 15 papers for preference aggregation in group TOPSIS. Eleven of them are related to the manipulation of fuzzy or interval things, and only a few deal with crisp data. Two debates have arisen if a paper involved with fuzzy or interval data has a technical advantage compared to traditional approaches or if it can better match the decision making environment. The former is out of the scope of this study, while for the latter some scholars have found clues on the issue of the decision making environment in the past. Eckenrode (1965) mentioned that ranking by order is not only the easiest, but also is the most reliable. Kirkwood and Sarin (1985) also supported the argument for the ease of reaching an agreement with some kind of confidence. This situation could commonly happen in a group with a small number of DMs. Hence, from a practical aspect, this study investigates an uncovered area, i.e., group TOPSIS with partial information. Decision power is also posted for a possible better formulation for group decision support. Both issues motivate the present research.

The paper is organized as follows. In the next section the literature survey lists some basic works on ordinal information for decision making and decision power in multi-criteria group decision making. Section 3 focuses on the proposed integrated model for group TOPSIS in a step-by-step fashion. We then give illustrative examples with sensitivity analyses. Section 4 provides some discussions with their limitations. The final section draws concluding remarks.

2 Literature Survey

Many works have utilized TOPSIS as a tool for decision making, but few have presented the essential steps for tackling ordinal or qualitative information in a crisp domain. Most studies treated this information as pseudo-metric data. Hwang and Yoon (1981) suggested that a qualitative criterion could be converted into an interval scale by using a bipolar scale. For instance, decision makers might choose a 10-point scale and calibrate the qualitative attribute into assigned values. For the benefit criteria, 10 points might be the maximum value and 0 points the minimum value for performance measures, with the reverse direction of values measuring the cost criteria. Traditional TOPSIS thus could deal with both qualitative and quantitative criteria. Goh et al. (1996) utilized a 1- to 9-point scale for the level of achievement of the qualitative attribute in a robot selection problem. Parkan and Wu (1997) used a 5-point Likert scale for a process selection problem. No matter which scale is adopted, even for a 0–1.0 normalized scale (Rao 2006), the designed scales or categories are often believed to represent an interval level of measurement, and the above scales seem to be assigned in a subjective way. For instance, one conservative DM might think number 7 is good enough for evaluating the benefit criteria, whereas the other might assign 9 to the same case in a group decision environment. There should be a way for a more objective evaluation for group decision making. Herein the linguistic variables are commonly thought to have the same meanings for representing performances of the qualitative criteria.

In AHP, pairwise comparisons through a 9-point fundamental scale are used to manage tangible (objective, quantitative) and non-tangible (subjective, qualitative) factors (Saaty 1980), but most works employ the process of pairwise comparisons to obtain weights for criteria, e.g., Shyur and Shih (2006), in order to support TOPSIS. It appears that no work in the literature has used AHP to handle ordinal information for the TOPSIS process. Many other MCDM techniques have been proposed for dealing with qualitative criteria. Hinloopen et al. (1983) listed 16 examples of such methods. Voogd (1983) classified some of these into two categories: pure qualitative data and mixed qualitative and quantitative data. The former was further divided into frequency approaches and scaling models. The latter listed the geometric scaling model and three other analytical mixed-data evaluation techniques integrated into the EVAMIX (multicriteria EVAlation with MIXed qualitative and quantitative data) model (Voogd 1983). For mixed data, a debate arose as to whether the conversion of qualitative measures into quantitative form is worth it or not (Larichev et al. 1995) and concluded that some quantitative weighting estimates introduce inaccurate representations of a decision maker’s preference.

In a similar vein, Xu and Lamond (2001) suggested a ranking procedure based on the distance between partial pre-orders without the need of conversion. Nonetheless, when handling mixed data within existing techniques, conversion seems unavoidable due to coherence with the quantitative data. For example, simple multi-attribute rating technique (SMART) allows decision makers (individual or group) to quantify qualitative information that can be categorized through performance scores (Edwards 1977). Measuring attractiveness by a categorical based evaluation technique (MACBETH) employs a non-numerical interactive questioning procedure, requesting a qualitative judgment about the difference of attractiveness, in order to build a quantitative value (Bana e Costa and Chagas 2004). Hence, a conversion of ordinal information to cardinal information in TOPSIS seems similarly possible.

Transforming ordinal information calls for further consideration. For instance, qualitative concordance analysis (Van Delft and Nijkamp 1977) utilizes a nominal scale as the basis for evaluation. The numerical interpretation method relaxes the limited number of ordinal levels for a wide range of representation (Voogd 1983). EVAMIX uses a signum function for measuring the dominance of any two alternatives on any criterion and weights these dominances together with cardinal evaluations (Voogd 1982). According to the guidelines of the MCDM methods by Guitouni and Martel (2006), EVAMIX and TOPSIS are grouped into the category of a single synthesizing criterion. Consequently, the essence of the process of EVAMIX should be able to help TOPSIS handle ordinal information, by proposing a coherent, simple, and operational procedure. While EVAMIX concentrates on the distance of the difference among the performance measures, TOPSIS reasons the distance between the one performance measure and PIS/NIS to be DMs’ preference. Table 1 summarizes the comparisons of AHP, EVAMIX, and TOPSIS. Note that herein we do not take into account stochastic dominance (Lahdelma et al. 2003), which can be left for future study. Moreover, Xu et al. (2014) proposed a distance-based aggregation approach for group decision making with interval preference orderings, which minimizes the weighted arithmetic averaging operator of the interval preference of decision makers. It has no close relation with TOPSIS.

Table 1 Comparison of the characteristics of three common MCDM techniques
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Another issue about ordinal information on criteria weights has not been discussed in detail. Many MCDM techniques, especially in compensatory models, require information about the relative importance of each criterion (Hwang and Yoon 1981). Elicitation of criteria weights can roughly be classified as direct and indirect estimations (Horsky and Rao 1984). The former is the main interest due to its straightforward support of TOPSIS. Barron and Barrett (1996) examined four approximate rank-order weights—rank order centroid (ROC), rank sum, rank reciprocal, and equal weights—and found that ROC weights are the most accurate and efficacious based on their designed simulation study. Alfares and Duffuaa (2009) reviewed past works that inferred criteria weights from ranks and proposed a method to determine the relative weights for any set of criteria. Although there are many other methods for assigning a weight from ordinal information, e.g., distance-metric methodology by Jones and Mardle (2004) and ordered weighting averaging (OWA) method by Ahn (2011), this study chooses ROC weights, because they need much less input from DMs as concluded by Olson and Dorai (1992). However, we shall carry out an extra sensitivity analysis on weights to check the model’s stability on ranks.

The third interest relates to group aggregation with differentiated decision power in multi-criteria group decision. Decision power is defined as the potential ability of a person or a group to exercise control or influence on another person or group for the final decision (Griffin and Moorhead 2013). Though there are many forms of this power, e.g., legitimate power or expertise power, it can be accounted for in the political or business environment based on the number of votes or the percentages of shares (Korhonen 1997; Widerén 1994). The decision power of the person can thus be quantified as his or her relative weight in that group based on his or her position, expertise, percentages of shares, etc. A differentiated decision power can then be identified. Ramanathan and Ganesh (1994) classified the approaches of assigning this weight for decision power as the supra DM approach and the participatory approach. The former assigns weights by a supra DM, which is usually at the top level of the organization. For example, Tsui and Wen (2014) investigated a selection issue for green suppliers in a TFT–LCD company, and the company considers the weights of four stokehold departments as 36, 27, 27, and 10 %, respectively, which is determined by one key department being the supra DM. The latter approach mentions that there exist different expertises in a group, which can be quantified by interperson comparisons or discussions among the group members to obtain the their power. This expertise power can then be simply interpreted as the weight of a member in the group utility function.

For group aggregation, social choice aims to combine individual opinions, preferences, interests, or welfares to reach a collective decision, but there is no procedure for aggregating individual ranking into group ranking without violating some reasonable assumptions (Arrow 1963). From a practical aspect, the aggregation of a group’s ranking is still possible if compromise and agreement exist in the group, as it is reached through feedback and discussion among group members (Bezerra et al. 2014). We observe that many techniques can facilitate a compromise or an agreement (Hwang and Lin 1987). Baucells and Sarin (2003) developed a group utility function from additive individuals’ utility functions over the multiple attributes, thus providing the foundation for applying MCDM in a group decision environment. Group preference is generally aggregated by taking the mean operation of individuals’ preferences.

Ramanathan and Ganesh (1994) utilized the geometric mean and the weighted arithmetic mean for AHP and also took advantage of another hierarchy for deriving members’ weights. Forman and Peniwati (1998) discussed several ways to aggregate group information for AHP and proposed weighted arithmetic and geometric means for aggregating individual judgments and individual priorities. Bernasconi et al. (2014) further examined empirical properties of group preference aggregation methods employed in AHP. On the other hand, Barzilai and Lootsma (1997) proposed a power relation and group aggregation in multiplicative AHP and SMART, however, their approach has drawn debate over its behavior realism (Korhonen 1997). Van den Honert (2001) developed two models for obtaining group members’ weights of multiplicative AHP and SMART, respectively. For TOPSIS, Yue (2013) collected 15 works of preference aggregation in the group TOPSIS whereby most of them utilize arithmetical and geometric means for group aggregation. Since it seems not many works explicitly discussed decision power in the group, Baucells and Sarin (2003) offered some materials for further tackling group TOPSIS with differentiated decision power. We next suggest a mixed-data model for group TOPSIS with differentiated decision power.

3 Proposed Model

We first concentrate on the process of ordinal information and then deal with group aggregation with decision power in group TOPSIS.

3.1 Ordinal Information Process

There are two types of cardinal information involved in group TOPSIS: performance of alternatives on criteria and the weights of criteria. Most efforts are given toward measuring the performance on qualitative criteria. For the second part, centroid or ROC weights will be chosen herein for assigning weights to criteria due to their simplicity.

When there is ordinal information of performance measures on criteria, a simple process is proposed for group TOPSIS. This process considers the dominance of any alternative, by pairwise comparisons on the performance measure of this alternative with other alternatives, and sums up the dominances of this alternative with the others as accumulated dominances.

We present the decision matrix \({{\mathbf {D}}}^{k}\), \(k = 1\), ..., K, representing the kth DM as follows:

(1)

where \(A_{i}\) denotes the alternative i, \(i = 1, {\ldots }\), m; \(X_{j}\) represents the criterion j, \(j = 1, {\ldots }\), n; and the number of quantitative criteria p and the number of qualitative criteria q are related by \(p+q=n\). The element of \({{\mathbf {D}}}^{k}\), \(x_{ij}^k \), indicates the performance rating of alternative \(A_{i}\) with respect to attribute \(X_{j}\) by DM k, \(k = 1, {\ldots }, K\). Note that there should be K decision matrices for K members of the group.

To further process the performance information of alternatives on qualitative criteria, we express the following qualitative decision matrix, which is a submatrix of the decision matrix \({{\mathbf {D}}}^{k}\):

(2)

where there are q qualitative criteria, and their subscripts count from \(p+1\) to n.

Since the ordinal information of performance measures on criterion j is by a partial order set, we can compare the orders of the performance measures. Here, the preference order of \(x_{1j}^k \), \(x_{2j}^k ,{\ldots }, x_{ij}^k , {\ldots }\), and \(x_{mj}^k \) is represented by a function of their performances as \(o(x_{1j}^k )\), \(o(x_{2j}^k ),{\ldots }, o(x_{ij}^k),{\ldots }\), and \(o(x_{mj}^k )\), respectively, on criterion j for DM k. It is noted that the orders of the performance measures are predefined in the partial order set, and they can be linguistic or semantic terms, or levels of achievement. Thus, a comparison can be made.

Based on the presentation of the ordinal information process in EVAMIX, we introduce a signum function to count the dominance of the order comparison on each qualitative criterion for simplicity. Comparing any two orders of the alternatives, if one is preferred to the other, then the function will give 1; if one is indifferent to the other, then the function will give 0; if one is inferior to the other, then the function will give \(-\)1.

$$\begin{aligned} s_j^k \left( o\left( x_{ij}^k \right) ,o\left( x_{i^{\prime }j}^k \right) \right) =\left\{ {{\begin{array}{l@{\quad }l} 1,&{}\hbox { if }o\left( x_{ij}^k \right) \succ o\left( x_{i^{\prime }j}^k \right) , \\ 0,&{}\hbox { if }o\left( x_{ij}^k \right) \approx o\left( x_{i^{\prime }j}^k \right) , \\ -1,&{}\hbox { if }o\left( x_{ij}^k \right) \prec o\left( x_{i^{\prime }j}^k \right) . \\ \end{array} }} \right. \quad \forall i \hbox { and } i^{\prime }, i \ne i^{\prime }, \end{aligned}$$
(3)

where both i and \(i^{\prime } = 1, {\ldots }, m\) for m alternatives on criterion j. Note that there is no need to distinguish the criteria of benefits and costs in the above expression since the comparison is based on the orders that are the preferences in DMs’ minds.

The dominance measure \(s_j^k (o(x_{ij}^k ),o(x_{i^{\prime }j}^k ))\) is a comparison of the performance orders of any two alternatives \(A_{i}\) and \(A_{i^{\prime }}\), respectively, on criterion j for DM k. Thus, we illustrate a dominance matrix \({{\mathbf {S}}}_j^k \) by the following pairwise comparisons on criterion j.

(4)

The accumulated dominance measure \(t_{ij}^k \) is the summation of the comparison of the performance of alternative \(A_{i}\) to all other alternatives \(A_{i^{\prime }}\) on criterion j.

$$\begin{aligned} t_{ij}^k =\sum _{i{\prime }=1}^m {s_j^k } \left( o\left( x_{ij}^k \right) ,o\left( x_{i{\prime }j}^k \right) \right) , \quad i \ne i^{\prime }, \forall j. \end{aligned}$$
(5)

Thus, an accumulated dominance vector on criterion j by DM k is:

$$\begin{aligned} {{\mathbf {T}}}_j^k ={\begin{array}{c} {A_1 } \\ {A_2 } \\ \vdots \\ {A_i } \\ \vdots \\ {A_m } \\ \end{array} }\left[ {{\begin{array}{c} {t_{1j}^k } \\ {t_{2j}^k } \\ \vdots \\ {t_{ij}^k } \\ \vdots \\ {t_{mj}^k } \\ \end{array} }} \right] _{m\times 1}. \end{aligned}$$
(6)

We next obtain an adjusted performance value with respect to each criterion by the following expression within the maximum and minimum values of the above vector.

$$\begin{aligned} q_{ij}^k =\frac{t_{ji}^k -\min _{i=1}^m \left[ t_{ji}^k \right] +\delta }{\max _{i=1}^m \left[ t_{ji}^k \right] -\min _{i=1}^m \left[ t_{ji}^k \right] +\delta }. \end{aligned}$$
(7)

Here, we add a small value \(\delta \) for normalization in order to avoid the zero performance for further computation. In traditional TOPSIS, a small value \(\delta \) is equal to 1 for the worst value of the benefit criteria (Hwang and Yoon 1981). A normalized value is then obtainable by linear or vector normalization as done by TOPSIS.

The adjusted performance values can be linear or vector normalized so that the process is coherent with the traditional TOPSIS procedure. For vector normalization, the equation is:

$$\begin{aligned} r_{ij}^k =\frac{q_{ij}^k }{\sqrt{\sum _{i=1}^m {q_{ij}^k } }}. \end{aligned}$$
(8)

Here, \(i=1,{\ldots }, m\); \(j=1,{\ldots }\), n; \(k=1,{\ldots }, K\). In such a way, the ordinal information on the qualitative criteria can be transformed into cardinal information and be integrated into the traditional TOPSIS process. Here, Eq. (8) provides a consistent form of information as shown at Step 2 of Shih et al. (2007).

It is quite common for DMs to only provide ordinal information on the weight of each criterion. We choose ROC weights, an approximating weight set, because of their simplicity (Edwards and Barron 1994). Given the ranks of criteria are \(w_{1}=w_{2} = {\ldots }= w_{n}>\) 0 for n criteria, the weight of \(j^{\mathrm{th}}\) criterion can thus be calculated asfollows.

$$\begin{aligned} w_j (\hbox {ROC})=\frac{1}{m}\sum _{i=j}^m {\left( {\frac{1}{i}} \right) } , \end{aligned}$$
(9)

where \(j = 1,{\ldots }, n\) by a descending rank order and \(\sum \nolimits _{j=1}^n {w_j }= 1\).

Note that the weights of the criteria depend on their ranks, and their values have no difference for any member in the group. Compared to Step 4 of Shih et al. (2007), we can re-define the weight \(w_{j}=w_j^1 =w_j^2={\ldots }=w_j^k \), for criterion \(j = 1,{\ldots }, n\) and for each DM \(k = 1,{\ldots }, K\).

Edwards and Barron (1994) offered ROC weights by the number of criteria up to 16. For the case of four criteria, their weights are 0.5208, 0.2708, 0.1458, and 0.0625, respectively, in a descending ranked order and without a tie. If the second and the third ranks are a tie, then based on the concept of corner points with duplicated points at the tie, the modified weights are 0.4792, 0.2292, 0.2292, and 0.0625, respectively, after modifying Eq. (9). These default weights are considered for further operation. As the extra routine has now been outlined, we will integrate the operation into the proposed procedure in Sect. 3.3.

3.2 Group Aggregation with Differentiated Decision Power

The influence or power of one member on the other in a group or an organization is this paper’s interest. The power is usually implicitly expressed in a group and can be many types (Griffin and Moorhead 2013). However, it is uneasy to quantify the power. As Baucells and Sarin (2003) proposed a group utility function, group preference can be aggregated from individuals’ utility functions. Following the same stream, a differentiated power can be thought of as another kind of weight to the individual’s utility function. In group TOPSIS, Shih et al. (2007) considered group measures or utility, \(\overline{S_i^+ } \) and \(\overline{S_i^-} \), as aggregates of the separation measures or individuals’ utilities of each DM, \(S_i^{k+}\) and \(S_i^{k-} , k = 1,{\ldots }, K\), respectively, by taking the geometric or arithmetical means of these values without considering differentiated decision power. A simplified modification is suggested by the following aggregation operations on two separation measures of the group:

$$\begin{aligned} \overline{S_i^+ }= & {} \left( u^{1}\oplus S_i^{1+} \right) \otimes \ldots \otimes \left( u^{K}\oplus S_i^{K+} \right) , \quad \hbox { for alternative } i, \hbox { and } \end{aligned}$$
(10)
$$\begin{aligned} \overline{S_i^- }= & {} \left( u^{1}\oplus S_i^{1-} \right) \otimes \ldots \otimes \left( u^{K}\oplus S_i^{K-} \right) , \quad \hbox { for alternative } i. \end{aligned}$$
(11)

Here, \(u^{k}, k = 1,{\ldots }, K\), is the decision power of DM k by his or her relative weight, and \(\sum _{k=1}^K {u^{k}} = 1\), which can be obtained through a predetermined way (Ramanathan and Ganesh 1994). The greater the value of the weight is, the more power the decision maker has in the group. A DM with greater weight also means that he or she seems to possess more influence than others on the final decision of the group.

There are many choices in the operation of internal aggregation, e.g., weighted arithmetic mean, weighted geometric mean, or other modifications. If we take a weighted arithmetic mean of all individual measures, then the group measures, Eqs. (10) and (11), from PIS and NIS are:

$$\begin{aligned} \overline{S_i^+}= & {} \left( \sum \limits _{k={1}}^K {u^{k}} S_i^{k+} \right) \bigg /K,\quad \hbox { for alternative } i, \hbox { and } \end{aligned}$$
(12)
$$\begin{aligned} \overline{S_i^- }= & {} \left( \sum \limits _{k={1}}^K {u^{k}} S_i^{k-} \right) \bigg /K,\quad \hbox { for alternative } i, \end{aligned}$$
(13)

where \(i = 1,{\ldots }, m; k = 1,{\ldots }, K\). If the weighted geometric mean is considered, then the group measures from PIS and NIS are:

$$\begin{aligned} \overline{S_i^+}= & {} \prod _{k=1}^K {\left( S_i^{k+} \right) ^{u^{k}}},\quad \hbox { for alternative } i, \hbox { and } \end{aligned}$$
(14)
$$\begin{aligned} \overline{S_i^-}= & {} \prod _{k=1}^K {\left( S_i^{k-} \right) ^{u^{k}}} ,\quad \hbox { for alternative } i. \end{aligned}$$
(15)

In the following the final ranking is determined by the group relative closeness, as a group preference, of the ith alternative \(A_{i}\):

$$\begin{aligned} \overline{C_i^*} =\frac{\overline{S_i^- } }{\overline{S_i^+ } +\overline{S_i^- } }, \quad i = 1,{\ldots }, m, \end{aligned}$$
(16)

where \(0 \le \overline{C_i^*} \le 1\). The larger the index value is, the better the rank of the alternative will be.

We note that all efforts attempt to obtain the composited separation measures, including ordinal information and group information aggregation. Thus, the proposed procedure is uniformly coherent with the traditional TOPSIS steps and keeps a straightforward procedure.

3.3 Proposed Procedure

Based on the developments of the above two sections, we modify the procedure of group TOPSIS of Shih et al. (2007) as follows.

Step 1 :

Construction of mixed-data decision matrix individually, Eq. (1).

(1.1) Acquisition of cardinal and ordinal performance information.

(1.2) Transformation of the ordinal performance to be the adjusted

performance, Eqs. (3)–(7).

Step 2 :

Construction of normalized decision matrix individually.

Step 3 :

Determination of PIS/NIS individually.

Step 4 :

Assignment of weights to the criteria individually.

(4.1) Elicitation of the weights from the importance of the criteria.

(4.2) Processing the importance numerically directly from comparisons or ranks.

Step 5 :

Determination of decision power for all group members.

(5.1) Assignment of members’ weights to all members by a supra decision maker or by participants under an agreement.

Step 6 :

Calculation of separation measure from PIS/NIS for the group, Eqs. (10)–(15).

Step 7 :

Calculation of relative closeness for the group, Eqs. (16).

Step 8 :

Ranking and selection of preferable alternative(s).

(8.1) Ranking of all alternatives.

(8.2) Examination of the preferable alternative(s) and screening them out if the alternative(s) are questionable.

(8.3) Selection of preferable alternative(s).

We can see that one major difference is Step 5 being added for processing the decision power among the group members. Some minor modifications are also shown at Steps 1, 4, and 8. In addition, an assumption is implicitly presented in the procedure that all group members are willing to obtain a compromised or consensus resolution. If the group members here do not agree with the resolution, then extra effort is needed to reach a consensus by some techniques of group decision making (Hwang and Lin 1987), e.g., the Delphi technique. Some decision support systems can also facilitate such a consensus, e.g., Shih et al. (2004).

4 Illustrative Examples

Two group decision examples are illustrated as follows. Example 1 demonstrates the example of fighter selection by various combinations of the parameters. Example 2 presents a case of personnel recruitment. We also show the original fighter selection problem (Hwang and Yoon 1981) with ordinal information on subjective criteria in “Appendix” for ease of tracing the proposed procedure.

Table 2 Evaluation data of the fighter selection example
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Table 3 The comparison of the ranks with group decision under vector normalization for the fighter selection example with equal decision power
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Example 1

A fighter selection problem (Hwang and Yoon 1981)

Four alternatives are evaluated by six criteria for the selection. Table 2 shows the detailed information after adding one DM. The different evaluations between two DMs are on the last two criteria, which are subjective with different level ordinal performances. The first decision maker keeps the original five levels of measurement, and the second one only uses three levels of measurement. These measurements are handled by Eqs. (3)–(8). Here, we assume equal decision power for both DMs and employ the weighted arithmetical mean to aggregate the preferences of both DMs in which decision power \(u^{1}=u^{2}= 0.5\) for Eqs. (12) and (13). The rank results are tabled by three sets of weights on the criteria—the given weights from the original example, equal weights, and the ordinal weights from ROC—with vector normalization on performance measures as shown in Table 3. In its three sub-tables, each has 18 combinations of ranks that are classified by different group aggregations, i.e., geometric and arithmetical means and Borda function, Minkowski’s \(L_{p}\) metric parameters (p=1, 2, and \(\infty \)), and by different weighting steps, whereby before the PIS/NIS calculation as traditional TOPSIS (Hwang and Yoon 1981) is marked as “before”, and after the PIS/NIS calculation whose weights are given to the calculation of separation measures (Shih et al. 2007) is marked as “after”.

We see that the ranks are rather stable in terms of the three weight sets under the aggregation by geometric and arithmetical means, but the results from the Borda function seem to show not much discrimination ability here, as 14 cases out of 18 have ties. This could be one advantage of the internal aggregation of TOPSIS instead of the external aggregation by the Borda function. In addition, aside from the case of \(p= \infty \), the ranks are stable under the same combinations across the three weight sets.

Table 4 Sensitivity analysis on the differentiated decision power of the two DMs in the fighter selection example
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To understand the effects of the weights on the criteria and other weights on decision power, we execute two sensitivity analyses. The former deals with the weight change on the criterion of maneuverability, whose weight is 0.3611 from ROC, with the range of \(\pm \)40 %. Under the conditions of considering geometric mean aggregation, weighting the step after PIS/NIS calculation, and metric parameter \(p=2\), the ranks of alternatives are \(A_{3}\succ A_{1}\succ A_{4}\succ A_{2}\) and are unchanged within the range. The ranks from the Borda function are \(A_{3}\approx A_{1}\succ A_{4}\succ A_{2}\), within the range \(-\)10 to 40 %, and their ranks switch to the previous ranks during the range \(-\)20 to \(-\)40 %. The latter inspects the effect of the weight changes on decision power in which DM#1 holds weights of 0.9 decreasing to 0.1. The basic condition of the combinations is to maintain the same as the former. The ranks of alternatives are \(A_{3}\succ A_{1}\succ A_{4}\succ A_{2}\), within the range of weights from 0.8 decreasing to 0.1, and the ranks of alternatives are \(A_{1}\succ A_{3}\succ A_{4}\succ A_{2}\) with the weights being 0.9 as shown in Table 4. With this small example, the presented results of the proposed model are stable.

Example 2

A personnel recruitment problem.

A semiconductor corporation is extending its business by selling its newly-developed light-emitting diode (LED) components to lighting and other manufacturers of LED products. The sales and marketing department of the corporation initializes a recruitment of two sales engineers to promote the new business. After a screening process on the applications, 15 candidates remain on the final list for an interview in the second stage. The interview evaluation includes four dimensions with the items they belong to in the corresponding parentheses: knowledge (management and specialty), skills (communication, expressiveness, problem-solving, and analytic), behavior (team-work and customer-oriented), and personal characteristics (vigor, confidence, compressive resistance, and integrity). Since these 12 items are difficult to evaluate by cardinal information, e.g., 0–100 scores, interval values are used with some limited levels for ease of evaluation. The human resources department (HR) wants to introduce a new evaluating system to fit the company’s needs. For ease of manipulation, HR sets up ordinal data by five levels for the interview evaluation, and our proposed approach satisfies the firm’s needs. Table 5 lists the evaluation sheet of the 15 candidates on the 12 items by two managers from the employing unit (EU) and HR, respectively. The weights of the evaluating items or criteria are tested by the given criteria weights from AHP, equal weights, and ROC weights.

Table 5 The interview evaluation of 15 candidates of the case company
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For a group decision, it is not easy to define the decision power for both departments to reflect their expertise. The final decision relies heavily upon the evaluation of the EU, because the candidates will work for that unit later, but the opinion of HR is mainly as a reference for the decision makers. However, HR does keep veto power on the final decision if any preferable candidate has an unethical or unacceptable behavior in the past (at Step 8 of the proposed procedure). In our case, they agree on EU obtaining a weight of 0.8 and HR obtaining the rest after a long and tedious discussion, as mentioned at Step 5.

Table 6 presents a comparison of group ranks for the case from \(18\times 3\) combinations, i.e., three weight sets for criteria, weighted geometric or arithmetic means for differentiated decision power, \(L_{p}\) metric parameters (\(p=1, 2\), and \(\infty \)), and a weighting step before or after PIS/NIS calculation, and all combinations are under vector normalization and EU is assigned the decision power of 0.8. We also list the results of external aggregation by the Borda function for comparison, and the differentiated decision power is inapplicable to the function.

Table 6 The results of group ranks under different combinations for the human recruitment case with differentiated decision power
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Based on the shown results, Candidate \(A_{11}\) should be the best one without much debate, but the second one shows a little bit of diversity. Candidate \(A_{5}\) or \(A_{10}\) could be the second best, while Candidate \(A_{5}\) is selected due to stable ranks among the different combinations. Furthermore, we see some unexpected results with metric parameter \(p=\infty \) under aggregation by the geometric mean. This is caused by the zero value of separation measures, i.e., the alternatives are at PIS or NIS, and the geometric mean does not work well. In addition, the metric parameter \(p=\infty \) also gives us slightly inconsistent results. This manipulation takes an extreme value from multiple dimensions, and the other decision information is almost neglected. It easily makes the candidates less discriminated under full ordinal information. We would suggest to avoid taking metric parameter \(p=\infty \) as a choice in using TOPSIS evaluation. From the same table, we find that the results aggregated from the Borda function have many ties, which are not good for our evaluation.

Table 7 Sensitivity analysis on the differentiated decision power of the two DMs (EU and HR)
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To illustrate the effect of differentiated power on the final decision, we execute a sensitivity analysis on the unequal decision weights for both departments using Eqs. (12) and (13) and consider the weighted arithmetical mean for the aggregation in which EU takes a weight from 0.5 (equally shared decision power between EU and HR) to 0.95 (major decision by EU). The results in Table 7 are based on ordinal weighting on the criteria, \(L_{p}\) metric \(p=2\), and vector normalization. Although there are 15 candidates, the ranks of the leading two candidates are unchanged. The case demonstrates that our proposed approach is stable for the problem. In the end, the case company is satisfied with the proposed model for making better judgments without using cardinal scores.

5 Discussions

Based on the analyses of the two examples, we now offer some discussions on ordinal information, score conversion, group aggregation, and decision power, with some limitations.

5.1 Ordinal Information

The ordinal information provided for decision making has no need for assumptions on the equality of the ordinal intervals or levels and the units of measurement (Coombs 1950). The processed information can include some defined categories in a partial order set for the performance measure. We think the proposed model is rather flexible as shown in Example 1 with ordinal data of three levels and five levels. The settings of different levels do not also generate distinguishable rank results in the example.

One might argue why not just directly transfer the ordinal information into cardinal information as in Hwang and Yoon (1981). In fact, such an assignment would incur a subjective measure in which the values and their intervals might not be represented very well. It seems that they force partial information to be complete before the TOPSIS process. This transformation could be distorted due to a scaling effect or inappropriate assignment. In addition, such a subjective measure should be serious in a group decision making environment due to the multiple heterogeneous preferences. Different attitudes of DMs, e.g., conservative or liberal, will make the subjective measure more complicated. It is worth it to make direct use of ordinal information at the cost of extra computation.

The concept of approximating weights based on ranks is an attempt to divert the difficulties of weight elicitation in MCDM problems. The basis in using such weights is that DMs can differentiate among rank positions by reflecting their preferences (Ahn and Choi 2012), and the ordinal weights are applicable after the establishment of these weak preference relations on the criteria. For various ordinal weights, Barron and Barrett (1996) examined four approximate rank-order weights and found that ROC weights are the most accurate. Roberts and Goodwin (2002) classified weight elicitation methods as three categories—direct rating, point allocation, and ranking—and pointed out that ROC weights are appropriate to use as a substitute for point allocation methods, but the ROC weights do not provide the best approximation to the original weights in direct rating methods. On this point, they proposed rank order distribution (ROD) weights for a better approximation, and ROD weights will be close to the rank sum weights as the numbers of criteria increase. Ahn (2011) also confirmed that ROC weights result in the highest performance in MCDM evaluation. Though there are some debates on which one is the best, we divert the debates by executing extra sensitivity analyses on ROC weights to show the stability on rankings of our proposed model.

5.2 Score Conversion

In Sect. 3, Eqs. (7) and (8) are derived for converting an ordinal performance measure into a cardinal performance measure based on the frequency of dominating and dominated cases of any qualitative criterion by pairwise comparison among alternatives. It is common to count the frequency of the superior and inferior situations as EVAMIX. EVAMIX utilizes two weighting steps: the first one posts criterion weights into dominating and dominated cases for obtaining dominance scores; and the second one uses the weights of qualitative and quantitative criteria to combine the corresponding standardized dominance scores into the total standardized dominance scores for the final ranking. On the other hand, the core of TOPSIS is to count PIS/NIS by a distance function. We try to keep the core in the TOPSIS extension so that the conversion process on an ordinal measure is rather simple. Following the TOPSIS procedure, Eq. (8) can be thought of a vector normalization process. After the converted data of qualitative criteria have the same style as that of the quantitative criteria with vector normalization, PIS/NIS becomes obtainable for further use.

5.3 Group Aggregation

When using MCDM techniques in group decision making, there will be an extra consideration on how to aggregate the group preference. It is assumed that the group can achieve an agreement or work in a cooperative environment so that a consensus can be reached (Bezerra et al. 2014). This study takes advantage of Baucells and Sarin’s work (2003) for the development, and some common operators can be employed. The operator of arithmetic mean has a sense of an equal distance to all members in the group, whereas the operator of geometric mean considers an equal ratio to all members (Medhi 2006). The latter is a better fit to normalized results and is especially good for AHP (Forman and Peniwati 1998). After checking the results of both examples, we perceive that rank ties frequently occur by external aggregation of the Borda function. It seems that the discrimination ability of the external aggregation in TOPSIS is questionable if we want to have a total differentiated rank. This can be considered as one advantage for internal aggregation of TOPSIS in our proposed model. Though the geometric mean is rather common for the aggregation in AHP (Ramanathan and Ganesh 1994), it appears to not work well for the case with full ordinal performance measures, because some measures are easy to reach the points of PIS or NIS. Thus, the generated zero value in the denominator undermines the relative closeness.

There are various issues on aggregating individual preferences from a technical aspect. We do not provide an operation that is involved in the preference variations among a large number of group members, as it is difficult to reach an agreement. Please check the contents of Huang and Li (2012) for details.

5.4 Decision Power

There indeed exists differentiated decision power in the real world, such as business and political organizations, yet formulating the power with differentiation is a big issue. The power is usually implicitly expressed in a group with many types (Griffin and Moorhead 2013). Since there is no explicit clue from the area of organizational behavior, we refer to Ramanathan and Ganesh’s work (1994) and extend it to the group utility function of Baucells and Sarin (2003) to aggregate individuals’ utility functions with differentiated decision power. Other elegant ways to formulate the decision power could be left for future study. In addition, it is possible that there exist conflicts or regrets regarding decision power, meaning a consensus cannot be reached. Extra work is needed for consensus facilitation.

Ramanathan and Ganesh (1994) also indicated two approaches for assigning weights of the decision power. In our case study, the participatory approach to determine weights is feasible by interpersonal pairwise comparison or discussions among the group members. The features of the job statement for the candidate position can be thought of a basis for predetermining the weights of the power among different units. It is worth noting that social choice functions, e.g., the Borda function, are usually not involved in differentiated decision power.

For the personnel recruitment problem, HR keeps a veto power on the final decision if the target candidate has any unethical or unacceptable behavior in the past. If his or her bad record cannot be found in the screening process, then HR will demonstrate its veto power later, even during the probation period or later on. This situation cannot be covered in our model. Moreover, for the selection of senior managers the reference check plays a vital role on the final decision. That means that our model can be quite applicable to junior positions, whose decisions heavily rely on candidates’ performances.

6 Concluding Remarks

This study has proposed a decision support model to manage the mixed-data model with differentiated decision power for group TOPSIS. We integrate ordinal and cardinal information with differentiated weights into the traditional TOPSIS steps based on DMs’ distance-based utilities. The suggested model maintains a straightforward procedure like the original TOPSIS does. The illustrated examples demonstrate that our approach is useful in a cooperative group decision making environment. From the viewpoint of group aggregation, the proposed model with internal aggregation has an edge over the Borda function for external aggregation in group TOPSIS, achieving better discrimination ability. Sensitivity analysis also demonstrates the robustness of the model on alternative ranks.

After testing 54 combinations of different parameters on three weight sets on the criteria, group aggregation, two different weighting steps, and three \(L_{p}\) metric parameters for group TOPSIS, the fighter selection example performs quite stable with two sets of rankings, but the personnel recruitment case with \(p=\infty \) for the \(L_{p}\) metric gives slight different rank results when dealing with full ordinal performance information. In addition, the recruitment case with full ordinal data shows us that aggregating group preferences by geometric mean seems inadequate, because it has little ability to process a zero value under metric \(p=\infty \). Though there exists a slight distortion on using vector normalization of TOPSIS (Kaliszewski et al. 2014), we do not notice its effect on the final ranks.

The core concept of the proposed decision model with mixed data and differentiated decision power can be extended to other cardinal MCDM techniques in a group decision environment. In this study we have not dealt with stochastic dominance in group TOPSIS or with incomplete decision information. We also do not take into account the heterogeneous preferences for a large number of DMs as well as conflicts or regrets regarding decision power or weights on the criteria. These types of problems could be left for future study.