1. Introduction
Over the last two decades, to alleviate business pressure and focus on core competencies, firms are increasingly utilizing not only their own technologies, but also external resources and technologies [
1,
2,
3,
4], which is referred as open innovation (OI). Later, the OI phenomenon rapidly spread into many industries, and has received more and more attention [
5,
6,
7,
8]. As a form of OI, outsourcing has become a key component for business performance and has quickly expanded into the field of public services. Government public service outsourcing aims to enlist private organizations into the public service sector through a competitive market mechanism to compensate for government failures, reduce administrative costs, and improve service quality and efficiency. At the same time, considering that the contractors, who belong to the public sector and are the users of public services, can better understand the demands of the public, the government public service outsourcing can also build harmonious relationships among the government, the PSOs, and the public through the intermediary of the supplier. That is, by outsourcing public services to the service suppliers, the government can enhance both the economic effectiveness and the public satisfaction in line with the service target.
Driven by friendly policies, public service outsourcing has been rapidly promoted, and has gradually expanded from basic service areas, such as public infrastructure and solid waste management [
9], to high-level public service areas, such as online IT service [
10], energy supply [
11], medical care [
12], and social security [
13]. As outsourced contractors, private organizations have the obligation to produce high-quality and efficient public services. The internal management level, scale, experience, and service capability of the contractor can directly determine the quality of public services, select the most efficient contractors, and standardize the contractors. When purchasing services and selecting the most appropriate service providers, the key to guaranteeing the quality of public services is to screen and inspect the service capabilities of contractors.
Regarding different service outsourcing providers as alternatives and their performances as criteria, respectively, the public service outsourcer (PSO) selection issue can then be identified as a typical multi-criteria decision-making (MCDM) problem [
14]. The first and most basic step to solving the multi-attribute decision-making problem is to gather the assessment information given by the decision makers (DMs) about the alternatives with respect to the criteria. Due to the variability of environments and the complexity of people’s cognition, uncertainty has become one of the characteristics of the decision-making field. In addition, as the selection of PSOs involves qualitative criteria, such as user experience and reliability, it is difficult for DMs to express evaluation information with accurate real numbers. Therefore, reasonably describing the expression of uncertain information has caused extensive concern from researchers. In this case, the fuzzy set [
15] arose at the historic moment and has been continuously extended to different forms, such as type-2 fuzzy sets [
16], intuitionistic fuzzy sets [
17], and hesitant fuzzy sets [
18]. Compared with precise values, quantitative linguistic information can more accurately depict the complicacy and fuzziness in the evaluation process. For instance, when assessing the user experience of a service provider, qualitative descriptions like “good” and “perfect” are commonly used rather than crisp numbers. From this perspective, various different linguistic representation models have been proposed to express the evaluation information of DMs, such as hesitant fuzzy linguistic term sets (HFLTSs) [
19], the 2-tuple linguistic mode [
20], and probability linguistic term sets [
21].
Although various linguistic representation models have been applied to outsourcer selection, the abovementioned linguistic models cannot exactly depict the evaluation information given by experts, such as “between a little bad and very bad” and “between entirely moderate and a little good”. Therefore, to better describe such linguistic evaluation information, based on the concept of HFLTSs, Gou et al. [
22] further introduced the definition of double hierarchy hesitant linguistic term sets (DHHLTSs), which are made up of two hierarchy linguistic term sets (LTSs) in which the internal hierarchy LTS is used to make a further description or elaborate explanation for every LT contained in the external hierarchy LTS. DHHFLTSs are pretty useful tools in managing situations where DMs deliberate between several complex LTs when presenting their views on alternatives in the course of decision-making; this has become a project of widespread concern to researchers. Recently, Gou et al. [
23] presented a number of distance and similarity measures of double hierarchy hesitant fuzzy linguistic (DHHFL) elements (DHHFLEs) and investigated a method for tackling the MCDM problems based on these measures. Taking the interactive features among criteria and the incomplete preference information into consideration, Liu et al. [
24] developed a DHHFL mathematical programming means to handle the multi-attribute group decision-making problems. Gou et al. [
25] proposed a consensus-reaching process for large-scale group decision making, in which the preference relations are described in the form of DHHFLEs. Krishankumar [
26] constructed a novel decision framework in the context of DHHFLTS and proposed a new hybrid aggregation operator to aggregate the expert’s preference under the DHHFL environment.
The MCDM method plays an important role in MCDM and directly affects the accuracy and rationality of decision-making results. A great number of scholars have meticulously conducted in-depth research on outsourcer selection from this perspective; many classical decision-making methods and their extensions were applied to tackle the problem of outsourcer selection [
27,
28,
29,
30]. Among the methods applied to the problem of outsourcer selection, the elimination and choice translating reality (ELECTRE) method is a representative outranking method, originally proposed by Benayoun et al. [
31], which has a distinct advantage in tackling the issue with non-compensatory criteria. After being meticulously probed and extended by a range of specialists, the method further evolved into the ELECTRE family, including ELECTRE I, ELECTRE II, ELECTRE III, and so on. Among them, the ELECTRE I method is suitable for constructing a partial prioritization and for choosing a set of promising alternatives [
32], the ELECTRE III method aims to distribute each potential alternative into one of the categories in a previously defined family [
33]. Compared with them, the ELECTRE II method [
34] is a more pragmatic approach for handling the problem of ranking alternatives from the best to the worst [
35,
36], which is more suitable for selecting the outsourcer in practical applications. In recent years, the ELECTRE II method has attracted wide attention from academia and has been extended into various forms under different fuzzy environments [
37,
38,
39,
40]. For example, Wan et al. [
41] proposed the interval 2-tuple linguistic ELECTRE II to handle situations with non-compensatory criteria. Liao et al. [
42] introduced two distinct ELECTRE II methods to deal with MCDM problems in line with the hesitant fuzzy linguistic terms set. Additionally, Lin et al. [
43] investigated a new ELECTRE II method under the probabilistic linguistic environment. Nevertheless, the ELECTRE II method has not been extended to the DHHFL environment.
Based on the above literature review and analysis, we found the gaps in and insufficiency of the existing research. This paper is devoted to revising the comparison rule of DHHFLEs, extending the ELECTRE II method to the DHHFL environment, and further proposing a new ELECTRE II method combined with a novel weight-determination method. We implement our research through the following three points:
Considering the hesitation and complexity of experts’ cognition in the problem of outsourcer selection, the DHHFLTS is introduced to express the evaluation information of experts for a given service provider. In addition, we found that the existing comparison method for DHHFLEs may provide erroneous results of comparisons in some cases. Hence, we proposed the concept of a hesitant deviation degree for DHHFLEs and developed a new comparison method for DHHFLEs based on this.
Inspired by the power average (PA) operator, we defined the support degree of the DHHFLEs and proposed a new weight determination method to determine the expert’s weight for each criterion based on this, which can not only weaken the influence of extreme values on the results, but also reflects the expert’s authority for each criterion.
The classical ELECTRE II method is extended to the DHHFL environment and then an improved ELECTRE II method is proposed, which has a distinct advantage in tackling issues with non-compensatory criteria, in order to select the most desirable PSOs. In this method, the proposed comparison method for DHHFLEs is used to identify the magnitude relationship between two alternatives under each criterion.
The rest of the article is organized as follows:
Section 2 firstly reviews the corresponding concepts of the DHHFLTS and introduces the theoretical basis of the ELECTRE methods.
Section 3 defines the concept of the hesitant deviation degree and further proposes a new comparison approach for DHHFLEs.
Section 4 develops a new weight-determination method for experts with respect to each criterion and constructs a novel DHHFL–ELECTRE II method based on the new weight-determination method and the comparison method.
Section 5 applies the novel ELECTRE II to pick out the e-government outsourcing provider.
Section 6 carries out a comparative analysis to illustrate the effectiveness and strengths of our proposed method. Finally,
Section 7 comes to a conclusion for this paper and indicates interesting directions for the following research work.
4. The Improved Expected-HDD-ELECTRE II Method
4.1. Problem Characterization
Consider an MCDM that involves
m alternatives expressed by
, a group of experts denoted by
, and n attributes expressed as
. The weight vectors of experts and criteria are completely unknown, owing to the information asymmetry of the real problem. To accurately describe their opinions, the experts take the form of DHHFLE
to express their evaluations of the i-th option with respect to the criterion j. Then, the decision matrix of the expert
is given as follows:
4.2. Determining the Expert’s Weight with Respect to Each Attribute
Among the existing weight determination methods, one of the familiar reference values that they rely on is the distance or similarity between assessments and the mean values, which are vulnerable to extreme values and may lead to inaccurate results. In addition, the research fields of experts might be so various that there are some differences in the knowledge, skills, and experiences of each expert. An expert may be an expert in certain attributes, but not in other attributes. Considering these two points, we present a novel approach to determine the expert’s weight regarding each attribute based on the degrees of support to each other. Then the steps are as follows.
Step 1: To facilitate subsequent calculations, we combine all of the decision matrices
provided by each expert and convert them into other matrices regarding each attribute, which can be denoted as shown below.
in which the
-column vector in
is exactly the duplicate of the
-column vector in
.
Step 2: Obtain the support degrees of from .
Inspired by the PA operator, the support degrees from
to
can be defined as follows:
Obviously, the smaller the distance between and , the greater the support degree from to .
Step 3: Calculate the total support degree
of
as
Apparently, a greater indicates a better similarity to others. From the compromise point of view, the DM whose evaluation value is close to the evaluation values of other experts, i.e., who enjoys a great total support degree, would have a big weight.
Then, the total support degree matrix can be given as below:
Step 4: Taking the sum of elements in each column of the matrix
, we can obtain the overall support degree of each expert regarding the criterion
by the following equation.
Step 5: Obtain each expert’s weight regarding the criterion
by the following equation.
Apparently, and .
In conclusion, the novel weight-determination method for experts has two features or advantages: On one hand, this method, combined with the PA operator, leads to the definition of the total support degree to eliminate the influence of extreme values, which is more rational than other methods based on the average values or ideal solutions. On the other hand, the method aims to build a convergent position: For each expert, the weight may vary with different attributes, and the higher the total support degree from other experts, the greater the weight of this expert under this attribute. In this way, the unreasonable deviation caused by the expert’s knowledge, skills, experience, and other factors can be avoided, and the decision process would be more precise and reasonable.
4.3. Obtaining the Criterion’s Weight Based on the Optimization Model
After getting each DM’s weight with respect to each attribute, an overall matrix
can be given as follows by aggregating all of the decision matrices
.
In the present world, considering that decision making is always accompanied by uncertain settings and inconsistent opinions, most decisions are made from the perspective of compromise. Based on this point, Ju [
45] presented a weight-determination method for criteria whose weights are completely unknown based on the optimization model and ideal solutions. Then, we apply it to the DHHFL environment; the steps are given as follows:
Step 1: Obtain the DHHFL’s positive ideal solution (DHHFL-PIS) and the DHHFL’s negative ideal solution (DHHFL-NIS) by Equations (24) and (25), respectively.
where
,
,
.
Step 2: Calculate the Hamming distance
Step 3: Calculate the Hamming distance .
Step 4: Establish a multiple objective optimization model and solve it to obtain the weight vector of the criteria.
From the perspective of compromise, the greater the values of
and
are, the higher priority the alternate option
will be. In other words, an appropriate weight vector
should make the value of
and
as large as possible. Hence, a multiple objective optimization model can be established as below:
To solve the optimization model and obtain the most desirable weight vector, a Lagrange function with a Lagrange multiplier
can be given as:
Taking the partial derivative of
with respect to
and
and setting them equal to zero, respectively, we can obtain the following equation:
Settling the simultaneous equations above, we can obtain the weight of each criterion as follows:
In real practical terms, the convention assumes that the criteria weights add up to 1. Therefore, we make a further normalization for the criteria weight obtained by Equation (33) as follows:
By using the calculated criterion’s weight to weight the matrix
, we can get the following final decision matrix:
where
.
4.4. The Expected-HDD-Based ELECTRE II Method
On the basis of the new comparison method for DHHFLEs and the weight-determination methods for experts and criteria, an improved ELECTRE II method is proposed to rank the given alternatives in the decision making with DHHFL information. In our method, on one hand, the improved comparison method for DHHFLEs is used to identify the magnitude relationship between two alternatives under each criterion, which can make the division of concordant, indifferent, and discordant sets more accurate. On the other hand, the novel weight-determination method for experts can eliminate the effects of both extremes and the experts’ experience, which can provide more accurate information for the following steps of the method and makes the decision result more convincing.
4.4.1. Concordant, Indifferent, and Discordant Sets
In the traditional ELECTRE II method, by means of the comparisons of each pair of options on the attributes, the attributes of distinct alternate options can be grouped into three mutually exclusive collections: The concordant set indicating those attributes under which is superior to , the indifference set indicating those attributes under which is indiscriminate with and the discordance set denoting those attributes under which is inferior to , in which represents the performance of the option as regard to the criterion .
In the following, based the proposed comparison method for DHHFLEs, the expected value and HDD of the DHHFLEs are denoted together to distinguish the different alternatives under a certain criterion, which makes the division of attributes more sophisticated and reasonable. In view of the notions of the expected value and the HDD function for DHHFLEs, the DHHFL concordant set (DHHFLCS) and DHHFL discordant set (DHHFLDS) can be subdivided into the DHHFL strong, moderate, and weak concordant (discordant) sets, respectively.
For any pair of options and , the DHHFLCS of them is a collection of criteria under which is superior to . It can be subdivided into three types, as shown below:
- (1)
The strong DHHFLCS
:
- (2)
The moderate DHHFLCS
:
- (3)
The weak DHHFLCS
:
These three DHHFLCSs indicate three distinct and decreasing extents of the concordance for the assertion “ is at least as good as ”. Obviously, the distinction between and is caused by the HDD. A lower HDD denotes a high stability degree of the DHHFLE; therefore, compared with , represents a higher degree of concordance. According to Definition 4, a conclusion can easily be drawn that is superior to as long as . In other words, the expected value plays a more important role than the HDD when comparing two DHHFLEs. Hence, indicates a higher degree of concordance than .
In the same manner, the DHHFLDS indicates a collection of criteria under which is inferior to . It can be subdivided into three types, as shown below:
- (1)
The strong DHHFLDS
:
- (2)
The moderate DHHFLDS
:
- (3)
The weak DHHFLDS
:
In addition to the abovementioned conditions, there still stands a case where
and
; the set of corresponding subscripts of these criteria is called the indifferent set, and can be defined as follows:
4.4.2. The DHHFL Concordance and Discordance Index
In line with the abovementioned DHHFLCS and DHHFLDS, for the assertion “the
is at least as good as the
”, the DHHFL concordance index (DHHFLCI) can be obtained by
where
is the weight of the criterion
.
,
,
and
are the given position weights for the strong DHHFLCS, moderate DHHFLCS, weak DHHFLCS, and DHHFL indifferent set, respectively. The DHHFLCI
indicates the weighted superior degree of the alternative
over the alternative
, and
. Then, we can construct the DHHFL concordant matrix as follows:
Similarly, the DHHFL discordance index (DHHFLDI) of DHHFLDS can be given as
where
is the weight of the criterion
.
,
and
are the given attitude weights for the strong DHHFLDS, moderate DHHFLDS, and weak DHHFLDS, respectively.
denotes the Hamming distance between
and
, which can be obtained by Equation (6). The DHHFLDI
represents the weighted inferior degree of the alternative
over the alternative
, and
.
Likewise, the DHHFL discordant matrix can be established as:
4.4.3. Determine the Outranking Relations
When the DHHFL concordant matrix and the DHHFL discordant matrix are determined, we can define the critical value
by taking the mean value of all the elements in the DHHFL concordant matrix, which can be expressed by
. Then, making the comparisons between the elements in the DHHFL concordant set and the critical value
, respectively, the DHHFL concordant Boolean matrix can be obtained as follows:
where the element
can only be 0 or 1, and when
,
; otherwise,
.
means that the alternative
outranks the alternative
from the perspective of concordance.
Similarly, a marginal value can be given by the average of all of the elements in the DHHFL discordant matrix, presented by
. Then, the discordant Boolean matrix is defined as below:
where the element
can only be 0 or 1, and when
,
; otherwise,
.
indicates that the alternative
is not inferior to the alternative
from the perspective of discordance.
Then, to show the overall outranking relations between the alternatives, a comprehensive Boolean matrix can be defined through making a multiplication of the Boolean matrices and , which can be expressed as . Obviously, only when and are both assigned of a value 1, , which means that the option is strictly outranking to the option .
4.4.4. Draw the Outranking Graph
According to the comprehensive Boolean matrix obtained in the last section, an outranking graph can be depicted to concisely show the outranking relations between alternatives and to help the DM choose the most desirable alternative. In the graph, is a set of vertices and each vertex represents an alternative, indicates the set of directed edges connecting the vertices, and each arc shows the outranking relation between the two alternatives at either end of it. There are two situations of the directed arcs between two alternatives and : (1) The directed edge is from to , which indicates that outperforms ; (2) when the two alternatives and are incomparable, there is no arc between the two corresponding vertices.
In general, the symmetric positions in the matrix
should be complementary, as the relation of strictly outranking cannot be mutual. In other words, there should be and merely is one arc between any two vertices in the outranking graph. However, owing to the subjectivity of the threshold selection, the symmetric positions in the comprehensive Boolean matrix
are not always complementary, which makes it impossible to compare the two schemes. To handle this issue, Lin et al. [
36] proposed the following method to revise the Boolean matrices.
Let be a comprehensive matrix obtained by the multiplication of the Boolean matrices and in component form. When the elements or , which implies that the elements in the corresponding positions in and satisfy the condition: and , or and . Hence, there is no arc between the two alternatives in the outranking graph, which means that they are incomparable. In such cases, we set , which indicates that weakly outperforms when and .
4.4.5. The Decision-Making Procedures
On the basis of the above description, the procedure of the presented expected-HDD-based ELECTRE II method can be given as follows:
Step 1: Obtain the decision matrices provided by the invited experts. Set the attitude (position) weights of the strong, moderate, and weak DHHFLCS, the DHHFL indifference set, and the strong, moderate, and weak DHHFLDS, represented as , , , , , and , respectively.
Step 2: Calculate the DM’s weight
regarding each attribute by means of the method defined in
Section 3.2.
Step 3: Calculate the criteria’s weight based on the optimization model and obtain the overall decision matrix .
Step 4: Compute the expected value and HDD of the DHHFLEs in the final matrix
as
where
is the count of elements in
.
Step 5: Construct the strong, moderate, and weak DHHFLCS by Equations (35)–(37), the strong, moderate, and weak DHHFLDS by Equations (38)–(40), and the DHHFL indifferent set by Equation (41).
Step 6: Calculate the DHHFLCI of any pair of alternatives by Equation (42) to establish the DHHFL concordant matrix .
Calculate the DHHFLDI of any pair of alternatives by Equation (43) to establish the DHHFL discordant matrix .
Step 7: Calculate the critical value as and construct the DHHFL concordant Boolean matrix .
Calculate the marginal value as and construct the DHHFL discordant Boolean matrix .
Step 8: Establish the comprehensive Boolean matrix and then depict the outranking graph based on the matrix.
If there is no arc between two vertices in the graph, locate the corresponding symmetrical positions in where the elements may be not complementary. Then, revise the elements of the same positions in the matrices , , and , and further redraw the outranking graph with the arcs of weak outranking.
To make the decision-making processes of the proposed expected-HDD-based ELECTRE II method clearer, a flowchart is summarized, as shown in
Figure 2.