A large group linguistic Z-DEMATEL approach for identifying key performance indicators in hospital performance management

Applied Soft Computing Journal xxx (xxxx) xxx

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A large group linguistic Z-DEMATEL approach for identifying key performance indicators in hospital performance management ∗

Shan Jiang a , Hua Shi a , Wanlong Lin b , , Hu-Chen Liu c a

School of Management, Shanghai University, Shanghai 200444, PR China Shanghai No.3 Rehabilitation Hospital, 100 JiaoCheng Road, Shanghai 200436, PR China c College of Economics and Management, China Jiliang University, Hangzhou 310018, China b

article

info

Article history: Received 8 April 2019 Received in revised form 13 October 2019 Accepted 28 October 2019 Available online xxxx Keywords: Hospital management Key performance indicator DEMATEL method Linguistic Z-number Large group decision-making

a b s t r a c t In hospital management, performance measurement is of vital importance for improving healthcare service quality. The performance of a healthcare organization is often influenced by numerous indicators, and it is unrealistic to manage them all due to the restriction of resources. In addition, the performance measurement for improvement relates to the benefits of many departments, and it is necessary for large number of experts with different backgrounds to participate in the evaluation process of healthcare indicators. In response, this study develops a large group evaluation approach using linguistic Z-numbers and decision-making trial and evaluation laboratory (DEMATEL) to determine key performance indicators (KPIs) for hospital management. For this approach, the complex and uncertain interrelation evaluations among indicators are given by experts using linguistic Znumbers. An extended DEMATEL method is proposed to determine KPIs based on the cause and effect relationships of performance indicators. Finally, a case study in a rehabilitation hospital is presented to illustrate the effectiveness and usefulness of the proposed large group linguistic ZDEMATEL approach. The results indicate that incidents/errors, accidents/adverse events, nosocomial infection, nursing technology pass rate, and length of stay are KPIs for the given application. © 2019 Elsevier B.V. All rights reserved.

1. Introduction In China, the deepening of medical reform and the formation of a diversified medical treatment patterns have brought challenges to different types of hospitals [1]. These challenges require managers to pay more attention to the operational efficiency of hospitals. Performance measurement plays a central role in hospital management and healthcare performance assessment has concerned more and more by the public [2–4]. An efficient and rational healthcare performance measurement system can improve medical service quality, reduce costs, optimize service processes, and achieve optimal resource allocation [5,6]. As evidence of the achievement of organizational goals, a growing number of scholars focus on the improvement of hospital management using performance indicators and many national projects have been initiated to evaluate hospital performance [7,8]. However, current healthcare performance measurement frameworks usually contain a lot of indicators, and it is not realistic to improve all of them due to the constraint of hospital resources [9]. Therefore, it is essential to determine a limited number of key performance ∗ Corresponding author. E-mail addresses: [email protected] (W. Lin), [email protected] (H.C. Liu).

indicators (KPIs) to measure and monitor the performance of a healthcare organization [10–12]. Generally, the KPIs for healthcare performance evaluation were derived via prepared questionnaires and expert interviews [10,12,13], failing to considering the interrelationships of indicators [14]. The decision-making trial and evaluation laboratory (DEMATEL) method, originally developed by the Geneva Research Center of the Battelle Memorial Institute [15], is an effective technique in extracting the relationships as well as the interdependence intensity among system elements. It is able to visualize the structure of complex systems through matrices and digraphs [16,17]. By using the DEMATEL, we can establish an association matrix between influencing factors, and classify them into causal and effect groups according to their influences on a special system [18]. In recent years, the DEMATEL method has been adopted by researchers to identity critical factors in various areas due to its simplicity and effectiveness [19–21]. Thus, the DEMATEL also is of great value in examining the interdependence among correlated indicators and identifying KPIs for healthcare performance improvement. In the KPI determination process, it is often difficult for an expert to give an accurate numerical value on the degree of influence between indicators. Due to the increasing complexity of healthcare systems, experts are accustomed to employ linguistic

https://doi.org/10.1016/j.asoc.2019.105900 1568-4946/© 2019 Elsevier B.V. All rights reserved.

Please cite this article as: S. Jiang, H. Shi, W. Lin et al., A large group linguistic Z-DEMATEL approach for identifying key performance indicators in hospital performance management, Applied Soft Computing Journal (2019) 105900, https://doi.org/10.1016/j.asoc.2019.105900.

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S. Jiang, H. Shi, W. Lin et al. / Applied Soft Computing Journal xxx (xxxx) xxx

terms in their assessments [11,22]. Moreover, they may not familiar to all the given performance indicators because of the limited work experience and knowledge level. That is, the reliability of different experts’ assessments for the direct influence between each pair of indicators may be different. Based on Z-numbers [23] and linguistic term sets, the concept of linguistic Z-numbers was presented by Wang et al. [24] to describe both fuzziness and randomness of uncertain linguistic information. For a linguistic Znumber, the two components (i.e., restriction and reliability measure) of a Z-number are represented with linguistic expressions. Compared to other linguistic computing models, the linguistic Z-numbers can not only represent decision-making information more flexibly and comprehensively, but also avoid the distortion and loss of original information effectively [24,25]. Therefore, it is promising to apply the linguistic Z-numbers to represent various interrelation evaluations of healthcare performance indicators given by domain experts. Against the above discussions, in this paper, we aim to develop an integrated evaluation approach with the combination of linguistic Z-numbers and DEMATEL method for analyzing the interrelationships among indicators and identifying healthcare KPIs under the large group context. The main contributions of this paper are summarized as follows: First, we introduce the linguistic Z-numbers for dealing with the vagueness and uncertainty of experts’ uncertain assessments on the direct relations between indicators. Second, we aggregate the evaluations of large-scale experts by a maximizing consensus method to improve group consistency. Third, an extend DEMATEL method is utilized to analyze the causal interdependence among indicators and find out KPIs for improving healthcare system. Finally, we apply the proposed large group linguistic Z-DEMATEL approach to a rehabilitation hospital, and verify it by sensitivity analysis of the results and comparative analysis with other relevant methods. This remaining part of this paper is organized as follows: Section 2 presents a literature review of related works, focusing on healthcare performance management and DEMATEL method application. Section 3 describes the basic concepts of linguistic Z-numbers briefly. In Section 4, we propose the large group linguistic Z-DEMATEL methodology to identify KPIs for hospital management. In Section 5, an empirical case is presented to illustrate the effectiveness and superiority of the proposed approach. Section 6 discusses and summarizes the managerial implications of the present work. Finally, we conclude our research and outline future directions in Section 7. 2. Literature review 2.1. Healthcare performance management In the past decades, performance management in healthcare organizations has become an active research topic received great attention from academics. For instance, Behrouzi and Ma’aram [2] provided a flexible method for assisting private hospitals to identify and rank feasible and relevant performance measures under the balanced scorecard perspectives. Ali et al. [8] used a SERVQUAL gap model to measure the quality of service in Indian commercial hospitals. Cinaroglu and Baser [3] explored the relationship between effectiveness and health outcome indicators to improve healthcare performance by using a path analytic model. Gu and Itoh [10] conducted two questionnaire surveys to extract performance measures for dialysis facility management in the Japanese context, and Gu and Itoh [13] investigated the factors behind professional views of indicator usefulness and their important characteristics to design KPIs for hospital management. Núñez et al. [12] defined five categories of KPIs based on an indepth field study and expert opinions to monitor and manage

the performance of emergency departments. Peixoto et al. [4] applied principal component and cluster analysis techniques for the performance measurement of Federal university hospitals of Brazil. Soysa et al. [5] developed a performance scoring system to assess the overall strategic performance of healthcare nonprofit organizations in Australasia. In addition, Kahraman [14] analyzed the performance indicators of healthcare service in a research hospital by an integrated approach of fuzzy analytical network process (ANP) and DEMATEL. Si et al. [11] reported a performance indicator identification and assessment framework, which combines evidential reasoning method, interval 2-tuple linguistic variables and DEMATEL technique, for hospital management. In [26], the authors evaluated the service quality of Turkish hospitals by using fuzzy sets, analytic hierarchy process (AHP), and technique for order performance by similarity to ideal solution (TOPSIS). 2.2. Application of MCDM methods in healthcare There are also some studies which applied MCDM methods in the healthcare and hospital fields. In this aspect, Chen [27] introduced a compromising model based on interval-valued Pythagorean fuzzy sets and VIKOR (VlseKriterijumska Optimizacija I Kompromisno Resenje) to multiple criteria decision analysis for hospital-based post-acute care. Zhu et al. [28] proposed a hybrid MCDM framework using 2-tuple DEMATEL technique and fuzzy VIKOR method for elective admission control in a Chinese public hospital. Liu et al. [29] presented a failure mode and effect analysis approach combining interval-valued intuitionistic fuzzy sets with the multi-attributive border approximation area comparison (MABAC) method and used it for healthcare risk analysis in a clinical radiation department. Aung et al. [30] applied AHP and ANP methods to evaluate the medical waste management practices of hospitals, and Shi et al. [31] utilized cloud model theory and MABAC method for assessing healthcare waste treatment technologies from a multiple stakeholder perspective. Rouyendegh et al. [32] measured the efficiency of public healthcare institutions in Turkey by a hybridization method that integrates fuzzy AHP and data envelopment analysis (DEA). Torkzad and Beheshtinia [33] evaluated criteria that affect hospital service quality in Iranian public hospitals using the hybrid MCDM methods based on TOPSIS and ELECTRE (ELimination and Choice Expressing REality). In addition, Mardani et al. [34] conducted a literature review on the researches employing decision-making and fuzzy sets to address healthcare and medical problems and classified the selected studies into nine application areas (e.g., environmental sustainability, waste management, and service quality). 2.3. Application of the DEMATEL method In the literature, the DEMATEL method has been applied in many fields to interpret the interrelationships among factors and visualize them in cause and effect groups. For example, Zhang et al. [35] used the DEMATEL to found out critical risk factors of the Sponge City public-private capitalship projects in China. Ahmad Alinejad et al. [36] applied the DEMATEL to investigate the key success factors of logistics provider enterprises in Iran. Abdullah and Zulkifli [37] employed an interval type2 fuzzy DEMATEL method for developing causal relationships of knowledge management criteria. Bhatia and Srivastava [38] adopted a gray DEMATEL approach to evaluate the external barriers to implement e-waste remanufacturing in Indian context. Ding and Liu [22] proposed a 2-dimension uncertain linguistic DEMATEL model for recognizing critical success factors in emergency management, and Han and Deng [39] suggested a

Please cite this article as: S. Jiang, H. Shi, W. Lin et al., A large group linguistic Z-DEMATEL approach for identifying key performance indicators in hospital performance management, Applied Soft Computing Journal (2019) 105900, https://doi.org/10.1016/j.asoc.2019.105900.

S. Jiang, H. Shi, W. Lin et al. / Applied Soft Computing Journal xxx (xxxx) xxx

fuzzy evidential DEMATEL method to identify critical success factors in emergency system. Namjoo and Keramati [40] analyzed causal dependencies of composite service resilience in cloud manufacturing system using resource-based theory and DEMATEL method. Yadegaridehkordi et al. [41] studied the significant factors influencing big data adoption on manufacturing companies’ performance using a hybrid approach of DEMATEL and adaptive neuro-fuzzy inference system. For a comprehensive review on the DEMATEL technique and its various applications, one can refer to [17]. The above literature review shows that although lots of efforts have been spent on healthcare performance measurement and improvement, only a few of those studies have quantitatively investigated the interactions among healthcare performance indicators. Besides, the DEMATEL and its many extensions have been employed to identify dependent relationships among factors and determine critical ones in different complex systems. However, the current DEMATEL methods are inefficient at expressing the reliability of decision makers’ cognitive information and unable to handle the assessment data from a large number of decision makers. To close these gaps, this paper extends the DEMATEL method with linguistic Z-numbers and establishes a large group linguistic Z-DEMATEL approach for evaluating the contextual relationships of healthcare performance indicators and identifying KPIs for hospital management.

To cluster the experts’ evaluation matrices, a clustering threshold is determined as follows:

λ =

n×n

(

ij

)

ij

is the linguistic Z-number

provided by expert Ek on the direct influence between indicators Fi and Fj based on the linguistic term sets S = {s0 , s1 , . . . , s2g } and S ′ = {s′0 , s′1 , . . . , s′2g }. Next, the procedure of the proposed large group linguistic Z-DEMATEL approach for the identification of KPIs in hospital management is detailed. Stage 1: Clustering experts into subgroups based on the similar degree method In this stage, the large number of experts are clustered into small subgroups by using the method of similar degree [42,44]. Step 1: Compute the similarity degree between experts’ evaluation matrices. The similarity degree( between ) the linguistic direct influencing matrices Z k and Z q , SD Z k , Z q , can be computed by SD Z k , Z q = 1 − d Z k , Z q = 1 −

(

)

(

)

1 n×n

n n ∑ ∑ (

q

d zijk , zij .

)

( k,q=1,2,...,m,k̸ =j

)

min

k,q=1,2,...,m,k̸ =j

SD Z k , Z q

(

) )

,

where 0(≤ λ ≤)1. If SD Z k , Z q ≥ λ, we can place Z k and Z q into the same cluster. In this way, we can divide the m evaluation matrices given The number of by experts into M clusters Ch (h = 1, 2, . . . , M ). ∑ M experts in the hth cluster is lh (lh ≥ 2), satisfying h=1 lh = m. Stage 2: Aggregating clusters by the maximizing consensus approach This stage is to aggregate all the linguistic direct influencing matrices Z k (k = 1, 2, . . . , m) in each cluster to form an overall direct influencing matrix. Step 3: Construct the cluster direct influencing matrix Y h . Considering that the similarity degrees between experts’ evaluation matrices in each cluster are considerably high, the cluster ( ) direct influencing matrix Y h = yhij is calculated by the n×n arithmetic mean as lh 1 ∑

=

lh

zijk .

(3)

k=1

Step 4: Obtain the importance weight of each cluster wh . The maximizing consensus approach [45] is a popular used method to derive expert weights based on consensus scheme [46,47]. According to this method, if the consensus level of the cluster direct influencing matrix Y h is greater than that of the ( ) matrix Y u = yuij , then more weight should assign to the n×n cluster Ch . Thus, the following constrained optimization model can be constructed for deriving the weights of clusters: max F (wh ) = M

n

M ( ∑

1 n × n × (M − 1)

h=1 n

⎞

∑ ∑∑(

1 − d yhij , yij ⎠ wh

(

)) u

(4)

u=1,u̸ =h i=1 j=1

s. t .

⎧ M ⎪ ∑ ⎪ ⎪ ⎨ wh = 1 h=1

⎪ ⎪ w ∈W ⎪ ⎩ h wh ≥ 0

where W is the partial known weighting information regarding clusters. By solving the above model, the cluster weights wh (h = 1, 2, . . . , M ) can be obtained. Step 5: Establish the overall direct influencing matrix Z. By applying ( )the LZPWG operator, the overall direct influencing matrix Z = zij n×n is computed by zij = LZPWG y1ij , y2ij , . . . , y(M = ij ( ))

( ) ⎞ ⎛ wh 1+T yhij M ( )) ( ∑ ∏ ( h ) M w 1+T yh ⎟ ⎜ ∗−1 ⎜ ∗ ij ⎠, ⎝f ⎝ f φij h=1 h ⎛

h=1

(1)

( ( )) wh 1+T yhij ( ( )) ∑M w 1+T yhij h h=1 h ij

⎛

i=1 j=1

Step 2: Cluster experts into subgroups according to the clustering threshold.

SD Z k , Z q −

(

max

3

)

(2)

3. The proposed KPI evaluation approach

the kth expert, where zijk = Aφ k , Bϕ k

2

SD Z k , Z q

(

min

k,q=1,2,...,m,k̸ =j

+

yhij

In this section, we develop a large group linguistic Z-DEMATEL approach to analyze the interdependence of healthcare performance indicators on each other and identify KPIs for hospital management. This model consists of three stages: (1) Clustering experts based on the similarity degree between individual evaluation matrices; (2) Aggregating clusters by using a maximizing consensus approach; (3) Identifying KPIs with an extended DEMATEL method. The flowchart of the large group linguistic Z-DEMATEL model being proposed is shown in Fig. 1. Suppose that F = {F1 , F2 , . . . , Fn } is a set of healthcare performance indicators to be analyzed by a large group of experts E = {E1 , E2 , . . . , Em } (m > 20). Note that the group decisionmaking problem can be treated as a large group decision-making problem if (the) number of experts involved exceeds 20 [42,43]. be the linguistic direct influencing matrix of Let Z k = zijk

3

g∗

−1

M ⎜∏ ∗ ( ) ⎝ g ϕ

(5)

⎞⎞ ⎟⎟ ⎠⎠ ,

h=1

Please cite this article as: S. Jiang, H. Shi, W. Lin et al., A large group linguistic Z-DEMATEL approach for identifying key performance indicators in hospital performance management, Applied Soft Computing Journal (2019) 105900, https://doi.org/10.1016/j.asoc.2019.105900.

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Fig. 1. Flowchart of the proposed linguistic Z-DEMATEL model.

T yhij =

( )

M ∑

M ∑

Sup yhij , ylij =

(

)

Step 8: Acquire the total influencing matrix T. Based on the normalized ( ) direct influencing matrix X , the total influencing matrix T = tij n×n is derived by

1 − d yhij , ylij ,

(

)

h= 1 h̸ =u

h= 1 h̸ = u

is the average matrix of the M clusters. where Y l = yij n×n Stage 3: Identifying KPIs by an extended DEMATEL method In this stage, the classical DEMATEL technique is extended in the context of linguistic Z-numbers to interpret the interrelationships of healthcare performance indicators for identifying KPIs. Step 6: Obtain the crisp direct influencing Z ′. ( matrix ) The crisp direct influencing matrix Z ′ = zij′ is determined n×n by computing the score value of each element in the overall direct influencing matrix Z. That is, ′

( )

∗

∗

∗

zij = S zij = f

(

)

Aφij × g

∗

Bϕij ,

)

(

Z′ s

,

s = max

(7)

⎧ ⎨

max

⎩1≤i≤n

n ∑ j=1

zij′ , max

1≤j≤n

⎫ n ⎬ ∑ zij′

i=1

)

p→∞

⎭

.

(8)

(9)

where I is an identity matrix and tij indicates the full direct and indirect influence exerted from Fi to Fj . Step 9: Construct the influential relation diagram (IRD). This step is to calculate the sum of rows R and the sum of columns C from the total influencing matrix T. That is

⎛ R = (ri )n×1 = ⎝

n ∑

⎞ tij ⎠

j=1

(6)

where f and g are the possible linguistic scale functions defined in Section 4. Step 7: Calculate the normalized direct influencing matrix ( ) X. The normalized direct influencing matrix X = xij n×n is calculated through Eqs. (7)–(8). X =

T = lim X + X 2 + X 3 + · · · + X p = X (I − X )−1 ,

(

( l)

( ( )

C = cj

n×1

=

n ∑ i=1

,

(10)

.

(11)

n×1

) tij 1×n

The ri represents the total influence that indicator Fi exerts to the rest of the indicators while the cj represents the total influence that indicator Fj receives from all the other indicators. Let i = j and i, j = 1, 2, . . . , n; the importance degree (ri + ci ) and net effect degree (ri − ci ) can be computed for each performance indicator to construct an IRD. Step 10: Analyze the structure of performance indicators to identify KPIs. The horizontal axis values (ri + ci ) (i = 1, 2, . . . , n) named as ‘‘prominence’’ indicate the strength of influences that are given

Please cite this article as: S. Jiang, H. Shi, W. Lin et al., A large group linguistic Z-DEMATEL approach for identifying key performance indicators in hospital performance management, Applied Soft Computing Journal (2019) 105900, https://doi.org/10.1016/j.asoc.2019.105900.

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and received of the indicators. The larger the value of (ri + ci ), the greater the overall prominence of the indicator Fi . Similarity, the vertical axis values (ri − ci ) (i = 1, 2, . . . , n) named as ‘‘relation’’ show the net effect that contributed by the indicators. If ri − ci > 0, then the indicator Fi has a net influence on the other indicators and can be classified into a cause group; if ri − ci < 0, then the indicator Fi is a net effect of the other indicators and should be classified into an effect group. By visualizing the relationships among the n healthcare performance indicators, the KPIs which have great effect on other indicators or have complicated relationship with other indicators can be identified.

5

(3) The third one is based on the prospect theory:

f3 (θi ) = θi =

⎧ α g − (g − i)α ⎪ ⎪ ⎨ (0 ≤ i ≤ g ) , α 2g

(15)

β β ⎪ ⎪ ⎩ g + (i − g) (g + 1 ≤ i ≤ 2g ) , β

2g

where α and β are parameters in the interval [0, 1]. According to experiments, it can be obtained that α = β = 0.88 [52]. 4.2. Linguistic Z-numbers The linguistic Z-numbers were proposed by Wang et al. [24] to represent human cognitive information more flexibly and accurately.

4. Related concepts In this section, some basic concepts related to linguistic scale functions and linguistic Z-numbers are introduced.

Definition 2 ([24]). Let X{ be a universe } of expression, S = } s0 , s1 , . . . , s2g and S ′ = s′0 , s′1 , . . . , s′2g ′ be two linguistic term

{

sets. Then, a linguistic Z-number set Z in X is defined as follows: 4.1. Linguistic scale functions Z =

x, Aφ(X ) , Bϕ(X ) |x ∈ X ,

{(

)

}

(16)

Let S = s0 , s1 , . . . , s2g be a finite and completely ordered linguistic term set with odd cardinality, where g is a nonnegative integer and si represents a possible value for a linguistic variable. Normally, S should satisfy the following properties [22,48]:

which Aφ (x) is a fuzzy restriction on the values that the uncertain variable x can be taken, and Bϕ (x) is a reliability measure of the Aφ (x) .

(1) The set S is ordered: si ≤ sj if and only if i ≤ j; (2) The negation operator is: neg(si ) = sj , satisfying j = 2g − i.

Definition 3 ([24]). Let zi = Aφ (i) , Bϕ (i) and zj = Aφ (j) , Bϕ (j) be two linguistic Z-numbers; f ∗ and g ∗ be two different linguistic scale functions. The operational rules of linguistic Z-numbers are given as below: ⎛ ⎞

{

}

(

In the process of aggregation operation, a continuous set is often adopted to preserve all provided information. Thus, Xu [49] extended the discrete linguistic term set S to a continuous form ˜ S = {si |i ∈ [0, τ ]}, in which si > sj if i > j, and τ (τ > 2g ) is a large positive integer. If si ∈ S, then si is an original linguistic term; Otherwise, si is a virtual linguistic term. Generally, virtual linguistic terms only appear in the operation process to avoid information loss and distortion. To express semantics flexibly and impose original information effectively, linguistic scale functions [50] were proposed to assign different semantic values to linguistic terms under different situations. Definition 1 ([50]). Let S = s0 , s1 , . . . , s2g be a linguistic term set. If θi ∈ [0, 1] is a numeric value, then the linguistic scale function is a mapping from si to θi shown as follows:

{

}

f : si → θi (i = 0, 1, . . . , 2g ) ,

(12)

where 0 ≤ θ0 ≤ θ1 ≤ · · · ≤ θ2g and f is a strictly monotonically increasing function with respect to the subscript i.

i 2g

(0 ≤ i ≤ 2g ) .

(

)

S (zi ) = f ∗ Aφ(i) × g ∗ Bϕ(i) .

(

)

(

)

(17)

The accuracy function of zi is represented as A (zi ) = f ∗ Aφ(i) × 1 − g ∗ Bϕ(i)

(

)

(

(

(

.

(18)

)

(

)

(1) zi is strictly greater than zj , zi > zj , if Aφ(i) > Aφ(j) and Bϕ(i) > Bϕ(j) ; (2) zi is great than zj , zi ≻ zj , if S(zi ) > S(zj ) or S(zi ) = S(zj ) and

A (zi ) > A zj ; ( ) (3) zi equals zj , zi ∼ zj , if S(zi ) = S(zj ) and A (zi ) = A zj ; ( ) (4) zi is less than zj , zi ≺ zj , if S(zi ) = S(zj ) and A (zi ) < A zj or S(zi ) < S(zj ) .

( )

Definition 6 ([24]). Let zi = Aφ(i) , Bϕ(i) and zj = Aφ(j) , Bϕ(j) , f ∗ and g ∗ be two different linguistic scale functions. Then the distance between them is defined as

(

where the parameter a can be obtained in the interval [1.36, 1.4] [51].

))

Definition 5 ([24]). Let zi = Aφ(i) , Bϕ(i) and zj = Aφ(j) , Bϕ(j) be two arbitrary linguistic Z-numbers. The comparison method for the linguistic Z-numbers is given as follows:

(13)

(14)

)

Definition 4 ([24]). Let zi = Aφ(i) , Bϕ(i) be a linguistic Z-number. The score function S (zi ) of zi is computed by

(2) The second one is based on the exponential scale:

⎧ ⎪ ag − ag −i ⎪ ⎨ (0 ≤ i ≤ g ) , 2ag − 2 f2 (θi ) = θi = g i−g ⎪ ⎪ ⎩ a + a − 2 (g + 1 ≤ i ≤ 2g ) . g 2a − 2

(

( ∗( ) ( )) ∗ −1 f Aφ (i) + f ∗ Aφ (j) , ⎟ ⎜ f ( ∗ ) ⎠; (1) zi ⊕ zj = ⎝ ∗ ∗ ∗ ∗−1 f (Aφ (i) )×g (Bϕ (i) )+f (Aφ (j) )×g (Bϕ (j) ) g f ∗ (Aφ (i) )+f ∗ (Aφ (j) ) ( −1 ( ) ( )) (2) λzi = f ∗ λf ∗ Aφ(i) , Bϕ(i) , whereλ ≥ 0; ( −1 ( ( )λ ) ∗−1 ( ∗ ( )λ )) (3) ziλ = f ∗ f ∗ Aφ(i) ,g g Bϕ(i) , here λ ≥ 0.

There are three types of established linguistic scale functions used in the literature: (1) The first one is based on the subscript function. f1 (θi ) = θi =

)

d zi , zj

(

=

)

(

)

)

) |f ∗ (φ (i)) × g ∗ (ϕ (i)) − f ∗ (φ (j)) × g ∗ (ϕ (j))| . (19) ∗ ∗ ∗ ∗ 2 + max {|f (φ (i)) − f (φ (j)) |, g (ϕ (i)) − g (ϕ (j))|} 1

(

Please cite this article as: S. Jiang, H. Shi, W. Lin et al., A large group linguistic Z-DEMATEL approach for identifying key performance indicators in hospital performance management, Applied Soft Computing Journal (2019) 105900, https://doi.org/10.1016/j.asoc.2019.105900.

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Table 1 Performance indicators used in the case study. Dimension

Indicator

Description

Social benefit

Patient Satisfaction (F1 ) Patient complaint (F2 ) Patient medical expenses (F3 )

Satisfaction with healthcare service [10,12] Overall complaints about healthcare service [10] Per capita medical expenses for patients [9,12]

Quality

Incidents/Errors (F4 ) Accidents/Adverse events (F5 ) Nosocomial infection (F6 ) Nursing technology pass rate (F7 ) Percentage of readmissions (F8 ) Mortality/Death (F9 )

Incidents/errors occurred in healthcare treatment process [10,11] Accident/adverse events occurred in healthcare treatment process [11,12] Nosocomial infection in a hospital [10–12] Number of occurrences of acne and the rate of accompanying care [12] Ratio of readmissions within 40 days of discharge [9,10] Mortality/death in a healthcare organization [9–11]

Operating efficiency

Length of stay (F10 ) Bed occupancy Ratio (F11 ) Waiting time (F12 )

Time that the patient pass in hospital from the entrance to the exit [10,12] Average percentage occupancy of hospital beds [12] Total of time that a patient wait for an initial rehabilitation service [10–12]

Financial status

Net profit margin (F13 )

Total operating revenue-total operating expenses/total operating revenue

Development ability

Staff satisfaction (F14 ) Employee turnover (F15 )

Number of staffs expressed satisfaction [9,10] Turnover rate of employees leaving the hospital [11,12]

Definition 7. Let zi = Aφ(i) , Bϕ(i) (i = 1, 2, . . . , n)be a collection of linguistic Z-numbers, w = (w1 ,∑ w2 , . . . , wn )be the weight n vector of zi satisfying wi ∈ [0, 1] and i=1 wi = 1. Then, the linguistic Z-number power weighted geometric (LZPWG) operator is defined as:

(

)

(

( )) ( ))

(

( )) ( ))

w1 1+T z1 w2 1+T z2 ∑n ∑n w 1+T zi w 1+T zi i=1 i i=1 i z1 z2 wn 1+T z2 ∑n w 1+T zi i=1 i zn

LZPWG (z1 , z2 , . . . , zn ) =

(

(

⊗··· ⊗ (

∑n

⊗ (

(

( )) ( ))

, (

(20)

Sup zi , zj and Sup zi , zj = 1 − i=1 i ̸ = j ) ( d zi , zj is the support degree for zi and zj . The aggregated value by the LZPWG operator is also a linguistic Z-number, and where T (zi ) =

⎛ LZPWG (z1 , z2 , . . . , zn ) = ⎝f

∗−1

)

)

⎛ ⎞ w 1+T z n ∏ ( ) ∑n i (w (1(+Ti )) z ∗ ( i )) ⎠ , ⎝ f Aφ(i) i=1 i i=1

g

∗−1

⎞⎞ ⎛ w 1+T z n ∏ ( ) ∑n i (w (1(+Ti )) z ∗ ( )) i i ⎠⎠ . (21) ⎝ g Bϕ(i) i=1 i=1

5. Illustrative example In this section, we consider the identification of KPIs in a rehabilitation hospital as an example to illustrate the feasibility and effectiveness of the proposed large group linguistic Z-DEMATEL approach. 5.1. Background description The rehabilitation hospital is a second-level public hospital located in Shanghai, China. It provides the services of medical treatment, teaching, scientific research, rehabilitation, prevention, health care and nursing. Up to now, the hospital has 10 rehabilitation specialists, 54 rehabilitation doctors and 132 rehabilitation therapists. The purpose of this case is to help the rehabilitation hospital to determine KPIs for performance management, and in turn verify the validity of our proposed approach. In [9], a set of indicators frequently used in previous studies was identified via a systematic review of literature. Based on these initial indicators, we conducted an interview with mangers of the rehabilitation hospital to make usefulness rating of each indicator for hospital management. Moreover, they were also asked to list other important indicators specific to the rehabilitation hospital

performance measurement. By removing meaningless indicators and adding other useful indicators, 15 performance indicators, denoted as {F1 , F2 , . . . , F15 }, were considered as important indicators for the rehabilitation hospital management, which are summarized in Table 1. To determine interrelationships among the performance indicators, 32 healthcare professionals within the rehabilitation hospital were invited to give the direct influence between each pair of indicators through questionnaire. Excluding ineffective questionnaires due to incomplete or invalid information, the evaluation data of 25 experts {E1 , E2 , . . . , E25 } were utilized in the following analysis. Descriptive statistics of these respondents are displayed in Table 2. All the experts have professional knowledge in the healthcare domain and have worked in relevant fields for more than three years. The 15 performance indicators were assessed by experts using the following two linguistic term sets for the direct relation between indicators and the reliability of evaluation value, respectively.

⎧ s0 = No influence, s1 = Very low influence, ⎪ ⎪ ⎪ s2 = Low influence, ⎨ S = s3 = Medium influence, s4 = High influence, ⎪ ⎪ ⎪ ⎩ s5 = Very high influence, s6 = Extremely high influence } {′ s0 = Uncertain, s′1 = Slightly uncertain, ′ ′ ′ s2 = Medium, s3 = Slightly sure, S = s′4 = Sure

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

Based on the questionnaire survey, the linguistic evaluations of the 25 experts on the direct relation between indicators were obtained. ( ) For example, the linguistic direct influencing matrix Z k = zijk given by E1 is presented in Table 3. 15×15

5.2. Application Next, the proposed large group linguistic Z-DEMATEL approach is applied based on the collected data and experts’ assessments. Note that f ∗ = f3 (φi ) and g ∗ = f1 (ϕi ) in the calculations below. Step 1: By Eq. (1), the similarity degrees ( )between experts’ linguistic direct influencing matrices SD Z k , Z q (k, q = 1, 2, . . . , 25) are computed. As an example, the similarity degree between Z 1 and Z 2 is computed as below: SD Z 1 , Z 2 = 1 − d Z 1 , Z 2

(

)

d Z 1, Z 2 =

(

)

(

1 15 × 15

)

15 15 ∑ ∑ (

d zij1 , zij2 =

i=1 j=1

)

1 225

× 56.56 = 0.251

Please cite this article as: S. Jiang, H. Shi, W. Lin et al., A large group linguistic Z-DEMATEL approach for identifying key performance indicators in hospital performance management, Applied Soft Computing Journal (2019) 105900, https://doi.org/10.1016/j.asoc.2019.105900.

S. Jiang, H. Shi, W. Lin et al. / Applied Soft Computing Journal xxx (xxxx) xxx Table 2 Characteristics of the survey experts.

7

model is established:

Characteristics

n

%

Profession

Doctor Nurse Manager

10 6 9

40 24 36

Age

<29 years old 30–39 years old 40–49 years old 50–60 years old

5 7 8 5

20 28 32 20

Work experience

3-9 years 10-20 years >20 years

5 10 10

20 40 40

max F⎧ (wh ) = 0.8719w1 + 0.8580w2 + 0.8292w3 0.5 ≤ w1 ≤ 0.7; 0.05 ≤ w3 ≤ 0.1; ⎪ ⎨ w3 ≤ w2 ≤ w1 ; w1 ≤ 9w3 ; s. t . ⎪ ⎩ w1 + w2 + w3 = 1; w1 , w2 , w3 ≥ 0. By solving the above model, the weight of each cluster is yield as: w1 = 0.700, w2 = 0.222, and w3 = 0.078. Step 5: Applying ( ) Eq. (5), we obtain the overall direct influencing matrix Z = zij 15×15 as shown in Table 5. ′ ( ′ )Step 6: By Eq. (6), the crisp direct influencing matrix Z = is established as presented in Table 6. zij 15×15

Thus, SD Z 1 , Z = 1 − 0.251 = 0.749. Step 2: Based on Eq. (2), the clustering threshold is computed as λ = 0.771. Then, according to the introduced clustering method, all the experts are divided into three subgroups as below:

(

) 2

C1 = {E1 , E4 , E7 , E8 , E9 , E10 , E11 , E13 , E15 , E18 , E19 , E20 , E21 , E22 , E23 , E24 , E25 } C2 = {E2 , E3 , E5 , E6 , E12 , E14 } ; C3 = {E16 , E17 } . Step 3: Using Eq. (3), the cluster direct influencing matrices Y h (h = 1, 2, 3) are obtained and Table 4 indicates the direct influencing matrix of the first cluster. Step 4: Based on the experience of experts, the known weighting information of the clusters was acquired as: W = {0.3 ≤ w1 ≤ 0.7, 0.05 ≤ w3 ≤ 0.1, w3 ≤ w2 ≤ w1 , w1 ≤ 9w3 }. Then, by using Eq. (4), the following constrained optimization

( )

Step 7: The normalized direct influencing matrix X = xij 15×15 is constructed by using Eqs. (7)–(8) and listed [ in ] Table 7. Step 8: The total influencing matrix T = tij 15×15 is obtained via Eq. (9) and shown in Table 8. Step 9: By using Eqs. (10)–(11), the sum of rows R and the sum of columns C of the matrix T are calculated. Consequently, the importance degree (ri + ci ) and the net effect degree (ri − ci ) are calculated. The results are displayed in Table 9. Based on the data, the IRD of the 15 indicators is drawn as shown in Fig. 2. Step 10: Analyze the structure of performance indicators to identify KPIs. From Fig. 2, it can be observed that the 15 indicators are divided into a cause group and an effect group according to the values of ri − ci (i = 1, 2, . . . , 15). The indicators contained in the cause group are F4 , F5 , F6 , F7 , F8 , and the effect group include the indicators F1 , F2 , F3 , F9 , F10 , F11 , F12 , F13 , F14 , F15 . The indicators with bigger values of ri − ci will have higher influence on the other indicators and are assumed to have higher priority. Thus, F7 , F6 , F4 and F5 can be considered as KPIs. Besides, we can find

Table 3 Linguistic direct influencing matrix of the first expert. Indicators

F1

F2

F3

F4

F5

F6

F7

F8

F9

F10

F11

F12

F13

F14

F15

F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15

/ (s6 ,s’4 ) (s5 ,s’4 ) (s6 ,s’4 ) (s6 ,s’4 ) (s6 ,s’4 ) (s5 ,s’4 ) (s6 ,s’4 ) (s5 ,s’4 ) (s5 ,s’4 ) (s4 ,s’4 ) (s5 ,s’4 ) (s6 ,s’3 ) (s3 ,s’3 ) (s3 ,s’3 )

(s5 ,s’4 ) / (s6 ,s’4 ) (s6 ,s’4 ) (s6 ,s’4 ) (s5 ,s’2 ) (s6 ,s’4 ) (s5 ,s’2 ) (s4 ,s’3 ) (s5 ,s’3 ) (s4 ,s’2 ) (s5 ,s’2 ) (s5 ,s’3 ) (s3 ,s’2 ) (s3 ,s’1 )

(s2 ,s’3 ) (s3 ,s’4 ) / (s4 ,s’3 ) (s5 ,s’2 ) (s6 ,s’4 ) (s5 ,s’4 ) (s6 ,s’2 ) (s2 ,s’3 ) (s6 ,s’4 ) (s2 ,s’2 ) (s2 ,s’3 ) (s0 ,s’4 ) (s1 ,s’3 ) (s2 ,s’2 )

(s2 ,s’3 ) (s2 ,s’2 ) (s1 ,s’4 ) / (s5 ,s’4 ) (s5 ,s’2 ) (s4 ,s’4 ) (s2 ,s’3 ) (s2 ,s’2 ) (s3 ,s’3 ) (s3 ,s’2 ) (s1 ,s’4 ) (s0 ,s’4 ) (s3 ,s’2 ) (s3 ,s’2 )

(s1 ,s’4 ) (s1 ,s’3 ) (s0 ,s’2 ) (s5 ,s’4 ) / (s5 ,s’4 ) (s4 ,s’3 ) (s3 ,s’3 ) (s2 ,s’4 ) (s3 ,s’2 ) (s2 ,s’4 ) (s1 ,s’3 ) (s2 ,s’2 ) (s2 ,s’2 ) (s3 ,s’4 )

(s2 ,s’2 ) (s1 ,s’4 ) (s2 ,s’3 ) (s4 ,s’4 ) (s4 ,s’3 ) / (s5 ,s’2 ) (s2 ,s’2 ) (s1 ,s’4 ) (s5 ,s’3 ) (s5 ,s’2 ) (s3 ,s’3 ) (s2 ,s’1 ) (s1 ,s’0 ) (s2 ,s’3 )

(s0 ,s’4 ) (s2 ,s’3 ) (s3 ,s’2 ) (s5 ,s’4 ) (s6 ,s’4 ) (s6 ,s’4 ) / (s2 ,s’4 ) (s2 ,s’3 ) (s3 ,s’2 ) (s3 ,s’2 ) (s0 ,s’4 ) (s2 ,s’3 ) (s2 ,s’3 ) (s1 ,s’4 )

(s1 ,s’3 ) (s2 ,s’4 ) (s0 ,s’3 ) (s5 ,s’4 ) (s6 ,s’4 ) (s6 ,s’3 ) (s6 ,s’4 ) / (s2 ,s’2 ) (s2 ,s’3 ) (s2 ,s’2 ) (s1 ,s’4 ) (s1 ,s’3 ) (s1 ,s’2 ) (s1 ,s’0 )

(s2 ,s’2 ) (s3 ,s’3 ) (s4 ,s’2 ) (s5 ,s’4 ) (s6 ,s’4 ) (s5 ,s’4 ) (s6 ,s’3 ) (s6 ,s’4 ) / (s3 ,s’2 ) (s2 ,s’2 ) (s1 ,s’3 ) (s0 ,s’4 ) (s1 ,s’3 ) (s1 ,s’2 )

(s2 ,s’3 ) (s4 ,s’2 ) (s5 ,s’2 ) (s5 ,s’4 ) (s5 ,s’3 ) (s6 ,s’4 ) (s5 ,s’3 ) (s5 ,s’4 ) (s3 ,s’3 ) / (s5 ,s’3 ) (s5 ,s’1 ) (s4 ,s’0 ) (s2 ,s’3 ) (s0 ,s’3 )

(s6 ,s’4 ) (s5 ,s’3 ) (s4 ,s’2 ) (s5 ,s’3 ) (s5 ,s’4 ) (s5 ,s’2 ) (s6 ,s’4 ) (s5 ,s’4 ) (s5 ,s’3 ) (s6 ,s’4 ) / (s5 ,s’4 ) (s5 ,s’4 ) (s1 ,s’2 ) (s1 ,s’3 )

(s2 ,s’3 ) (s3 ,s’2 ) (s2 ,s’4 ) (s4 ,s’3 ) (s5 ,s’4 ) (s3 ,s’2 ) (s5 ,s’4 ) (s1 ,s’3 ) (s1 ,s’4 ) (s5 ,s’2 ) (s5 ,s’3 ) / (s4 ,s’2 ) (s2 ,s’3 ) (s3 ,s’2 )

(s5 ,s’4 ) (s6 ,s’4 ) (s5 ,s’4 ) (s6 ,s’4 ) (s6 ,s’3 ) (s5 ,s’3 ) (s6 ,s’3 ) (s4 ,s’2 ) (s5 ,s’3 ) (s6 ,s’4 ) (s6 ,s’4 ) (s4 ,s’2 ) / (s5 ,s’1 ) (s3 ,s’2 )

(s4 ,s’1 ) (s2 ,s’2 ) (s2 ,s’2 ) (s3 ,s’2 ) (s3 ,s’2 ) (s2 ,s’3 ) (s2 ,s’3 ) (s1 ,s’2 ) (s1 ,s’3 ) (s2 ,s’2 ) (s1 ,s’4 ) (s2 ,s’2 ) (s4 ,s’3 ) / (s4 ,s’3 )

(s2 ,s’3 ) (s1 ,s’2 ) (s2 ,s’4 ) (s2 ,s’3 ) (s3 ,s’2 ) (s1 ,s’4 ) (s2 ,s’2 ) (s2 ,s’1 ) (s1 ,s’3 ) (s3 ,s’2 ) (s0 ,s’4 ) (s1 ,s’3 ) (s4 ,s’1 ) (s6 ,s’4 ) /

Table 4 Direct influencing matrix of the first cluster. Indicators

F1

F2

F3

F4

F5

F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15

/ (s5.471 ,s’3.882 ) (s4.647 ,s’3.353 ) (s5.765 ,s’4.000 ) (s5.824 ,s’3.824 ) (s5.294 ,s’3.529 ) (s5.235 ,s’3.529 ) (s4.059 ,s’2.824 ) (s4.647 ,s’3.235 ) (s4.176 ,s’3.294 ) (s3.647 ,s’2.647 ) (s4.588 ,s’3.059 ) (s1.765 ,s’2.882 ) (s2.118 ,s’2.000 ) (s2.000 ,s’3.118 )

(s5.471 ,s’3.941 ) / (s4.941 ,s’3.353 ) (s5.412 ,s’3.706 ) (s5.824 ,s’3.824 ) (s5.118 ,s’3.235 ) (s5.000 ,s’3.471 ) (s4.294 ,s’2.471 ) (s4.118 ,s’2.588 ) (s4.412 ,s’2.941 ) (s3.647 ,s’2.353 ) (s4.765 ,s’3.176 ) (s1.176 ,s’2.882 ) (s2.059 ,s’2.529 ) (s2.059 ,s’2.412 )

(s2.000 ,s’3.000 ) (s1.647 ,s’3.235 ) / (s4.529 ,s’3.235 ) (s5.235 ,s’3.176 ) (s5.176 ,s’3.118 ) (s5.118 ,s’3.294 ) (s5.529 ,s’3.412 ) (s1.941 ,s’2.647 ) (s5.647 ,s’3.824 ) (s3.059 ,s’2.118 ) (s0.824 ,s’3.000 ) (s1.412 ,s’2.529 ) (s0.529 ,s’3.176 ) (s0.412 ,s’3.353 )

(s1.235 ,s’3.118 ) (s1.529 ,s’2.765 ) (s1.059 ,s’2.941 ) / (s5.000 ,s’3.471 ) (s4.647 ,s’3.176 ) (s5.000 ,s’3.529 ) (s2.824 ,s’2.588 ) (s1.706 ,s’2.765 ) (s1.706 ,s’2.294 ) (s1.118 ,s’2.765 ) (s0.412 ,s’3.294 ) (s0.824 ,s’2.765 ) (s2.529 ,s’2.353 ) (s2.353 ,s’2.882 )

(s1.353 ,s’3.235 ) (s1.353 ,s’3.000 ) (s1.176 ,s’2.882 ) (s5.176 ,s’3.588 ) / (s4.647 ,s’3.471 ) (s5.059 ,s’3.000 ) (s2.706 ,s’2.588 ) (s2.000 ,s’2.471 ) (s2.000 ,s’2.529 ) (s1.176 ,s’2.824 ) (s0.588 ,s’3.294 ) (s0.765 ,s’3.059 ) (s2.294 ,s’2.059 ) (s2.294 ,s’2.765 )

... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

F13

F14

F15

(s4.706 ,s’3.176 ) (s4.882 ,s’3.471 ) (s4.235 ,s’3.118 ) (s5.294 ,s’3.471 ) (s5.353 ,s’3.588 ) (s5.529 ,s’3.294 ) (s5.647 ,s’3.588 ) (s4.647 ,s’2.882 ) (s4.235 ,s’2.176 ) (s5.706 ,s’3.765 ) (s5.235 ,s’3.529 ) (s2.882 ,s’2.412 ) / (s3.471 ,s’2.471 ) (s3.235 ,s’2.059 )

(s2.941 ,s’2.353 ) (s2.824 ,s’2.706 ) (s0.824 ,s’2.882 ) (s3.000 ,s’2.647 ) (s2.765 ,s’2.765 ) (s2.647 ,s’2.647 ) (s2.765 ,s’2.471 ) (s1.059 ,s’3.000 ) (s1.471 ,s’2.235 ) (s1.176 ,s’2.765 ) (s1.059 ,s’3.059 ) (s1.235 ,s’2.882 ) (s4.471 ,s’3.412 ) / (s5.059 ,s’3.647 )

(s2.118 ,s’2.941 ) (s2.706 ,s’2.647 ) (s0.765 ,s’3.235 ) (s2.588 ,s’2.353 ) (s2.706 ,s’2.529 ) (s2.412 ,s’2.471 ) (s2.294 ,s’2.118 ) (s1.059 ,s’3.000 ) (s1.176 ,s’2.647 ) (s1.000 ,s’2.588 ) (s0.529 ,s’2.647 ) (s1.000 ,s’2.706 ) (s4.824 ,s’2.706 ) (s5.824 ,s’3.824 ) /

Please cite this article as: S. Jiang, H. Shi, W. Lin et al., A large group linguistic Z-DEMATEL approach for identifying key performance indicators in hospital performance management, Applied Soft Computing Journal (2019) 105900, https://doi.org/10.1016/j.asoc.2019.105900.

8

S. Jiang, H. Shi, W. Lin et al. / Applied Soft Computing Journal xxx (xxxx) xxx

Fig. 2. Influential relation diagram of the 15 indicators.

Table 5 The overall direct influencing matrix Z. Indicators

F1

F2

F3

F4

F5

...

F13

F14

F15

F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15

/ (s5.442 ,s’3.832 ) (s4.906 ,s’3.391 ) (s5.577 ,s’3.922 ) (s5.691 ,s’3.724 ) (s5.356 ,s’3.557 ) (s5.352 ,s’3.593 ) (s3.969 ,s’2.762 ) (s4.753 ,s’3.157 ) (s4.202 ,s’3.120 ) (s3.830 ,s’2.747 ) (s4.713 ,s’3.184 ) (s1.529 ,s’2.842 ) (s2.280 ,s’2.113 ) (s2.175 ,s’3.076 )

(s5.479 ,s’3.959 ) / (s4.813 ,s’3.117 ) (s5.438 ,s’3.600 ) (s5.765 ,s’3.801 ) (s5.272 ,s’3.239 ) (s5.002 ,s’3.515 ) (s4.208 ,s’2.515 ) (s4.309 ,s’2.598 ) (s4.475 ,s’3.028 ) (s3.571 ,s’2.458 ) (s4.798 ,s’3.219 ) (s1.263 ,s’2.828 ) (s1.990 ,s’2.669 ) (s2.045 ,s’2.367 )

(s2.071 ,s’3.036 ) (s1.824 ,s’2.997 ) / (s4.671 ,s’3.039 ) (s5.202 ,s’3.041 ) (s5.124 ,s’3.080 ) (s4.905 ,s’3.191 ) (s5.521 ,s’3.438 ) (s2.496 ,s’2.522 ) (s5.567 ,s’3.756 ) (s2.933 ,s’2.187 ) (s0.998 ,s’2.794 ) (s1.400 ,s’2.547 ) (s0.000 ,s’2.958 ) (s0.486 ,s’3.283 )

(s0.000 ,s’3.155 ) (s1.588 ,s’2.833 ) (s1.073 ,s’2.993 ) / (s5.190 ,s’3.586 ) (s4.906 ,s’3.197 ) (s4.866 ,s’3.349 ) (s3.183 ,s’2.519 ) (s2.323 ,s’2.572 ) (s1.589 ,s’2.240 ) (s1.115 ,s’2.488 ) (s0.464 ,s’3.067 ) (s0.826 ,s’2.758 ) (s2.443 ,s’2.199 ) (s2.417 ,s’2.675 )

(s1.351 ,s’3.037 ) (s1.513 ,s’2.842 ) (s1.059 ,s’3.085 ) (s5.202 ,s’3.478 ) / (s4.716 ,s’3.238 ) (s5.044 ,s’3.103 ) (s2.799 ,s’2.523 ) (s2.706 ,s’2.515 ) (s2.098 ,s’2.416 ) (s1.291 ,s’2.677 ) (s0.708 ,s’3.179 ) (s0.833 ,s’2.831 ) (s2.361 ,s’2.078 ) (s2.369 ,s’2.424 )

... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

(s4.536 ,s’2.962 ) (s4.627 ,s’3.190 ) (s4.358 ,s’3.003 ) (s5.174 ,s’3.277 ) (s5.322 ,s’3.484 ) (s5.376 ,s’3.352 ) (s5.494 ,s’3.376 ) (s4.353 ,s’2.617 ) (s4.390 ,s’2.333 ) (s5.719 ,s’3.722 ) (s5.202 ,s’3.277 ) (s2.844 ,s’2.345 ) / (s3.068 ,s’2.193 ) (s2.933 ,s’2.066 )

(s2.181 ,s’2.527 ) (s2.331 ,s’2.643 ) (s0.797 ,s’2.877 ) (s2.126 ,s’2.713 ) (s1.931 ,s’2.827 ) (s2.331 ,s’2.559 ) (s2.499 ,s’2.538 ) (s0.997 ,s’3.072 ) (s1.551 ,s’2.336 ) (s0.000 ,s’2.797 ) (s0.000 ,s’2.811 ) (s0.000 ,s’2.917 ) (s4.381 ,s’3.438 ) / (s5.153 ,s’3.601 )

(s1.926 ,s’2.918 ) (s2.176 ,s’2.696 ) (s0.673 ,s’3.001 ) (s1.670 ,s’2.354 ) (s1.948 ,s’2.548 ) (s2.056 ,s’2.514 ) (s2.109 ,s’2.219 ) (s1.132 ,s’2.838 ) (s1.160 ,s’2.803 ) (s0.979 ,s’2.693 ) (s0.688 ,s’2.599 ) (s0.000 ,s’2.598 ) (s4.846 ,s’2.790 ) (s5.728 ,s’3.839 ) /

Table 6 ′ The crisp direct influencing matrix Z . Indicators

F1

F2

F3

F4

F5

F6

F7

F8

F9

F10

F11

F12

F13

F14

F15

F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15

0.000 0.832 0.646 0.881 0.861 0.756 0.763 0.424 0.580 0.504 0.409 0.579 0.222 0.221 0.312

0.867 0.000 0.581 0.781 0.895 0.674 0.685 0.407 0.430 0.521 0.346 0.597 0.190 0.254 0.230

0.297 0.268 0.000 0.547 0.622 0.618 0.607 0.761 0.281 0.841 0.270 0.154 0.186 0.000 0.095

0.000 0.228 0.175 0.000 0.731 0.609 0.632 0.327 0.273 0.180 0.151 0.085 0.129 0.241 0.292

0.215 0.220 0.179 0.711 0.000 0.589 0.611 0.302 0.295 0.239 0.183 0.130 0.133 0.223 0.261

0.189 0.000 0.213 0.592 0.499 0.000 0.471 0.339 0.288 0.368 0.310 0.179 0.000 0.000 0.176

0.227 0.183 0.249 0.543 0.589 0.595 0.000 0.410 0.208 0.214 0.189 0.143 0.000 0.000 0.193

0.244 0.262 0.280 0.720 0.780 0.778 0.709 0.000 0.200 0.168 0.267 0.149 0.124 0.000 0.000

0.136 0.176 0.000 0.857 0.941 0.789 0.717 0.676 0.000 0.219 0.212 0.202 0.113 0.000 0.144

0.313 0.358 0.537 0.676 0.625 0.635 0.656 0.809 0.271 0.000 0.694 0.277 0.277 0.000 0.166

0.360 0.391 0.409 0.472 0.579 0.405 0.647 0.577 0.432 0.940 0.000 0.387 0.300 0.000 0.000

0.294 0.341 0.203 0.308 0.381 0.288 0.576 0.315 0.265 0.388 0.377 0.000 0.000 0.000 0.000

0.517 0.569 0.503 0.666 0.734 0.716 0.742 0.438 0.393 0.866 0.670 0.284 0.000 0.278 0.255

0.257 0.281 0.131 0.271 0.263 0.272 0.283 0.169 0.185 0.000 0.000 0.000 0.579 0.000 0.728

0.271 0.273 0.117 0.197 0.239 0.245 0.220 0.174 0.175 0.146 0.104 0.000 0.524 0.895 0.000

F10 has a ri + ci score as high as 3.073 and its ri − ci value is slightly below 0. It means that although F10 is an effect indicator, it can greatly affect the overall system. From this point, F10 is also a KPI. Therefore, the rehabilitation hospital should pay more attention to these five KPIs to monitor and improve its healthcare performance systematically.

5.3. Sensitivity analysis To explore the influence of cluster weights on the obtained results, a sensitivity analysis is carried out by changing the weights of clusters. In the analysis, four cases are considered and the different weights assigned to clusters are as follows: Case 0: w1 = 0.700, w2 = 0.222, w3 = 0.078; Case 1: w1 = 0.6, w2 = 0.2, w3 = 0.2;

Please cite this article as: S. Jiang, H. Shi, W. Lin et al., A large group linguistic Z-DEMATEL approach for identifying key performance indicators in hospital performance management, Applied Soft Computing Journal (2019) 105900, https://doi.org/10.1016/j.asoc.2019.105900.

S. Jiang, H. Shi, W. Lin et al. / Applied Soft Computing Journal xxx (xxxx) xxx

9

Table 7 The normalized direct influencing matrix X. Indicators

F1

F2

F3

F4

F5

F6

F7

F8

F9

F10

F11

F12

F13

F14

F15

F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15

0.000 0.095 0.074 0.101 0.098 0.086 0.087 0.048 0.066 0.058 0.047 0.066 0.025 0.025 0.036

0.099 0.000 0.066 0.089 0.102 0.077 0.078 0.047 0.049 0.060 0.040 0.068 0.022 0.029 0.026

0.034 0.031 0.000 0.063 0.071 0.071 0.070 0.087 0.032 0.096 0.031 0.018 0.021 0.000 0.011

0.000 0.026 0.020 0.000 0.084 0.070 0.072 0.037 0.031 0.021 0.017 0.010 0.015 0.028 0.033

0.025 0.025 0.020 0.081 0.000 0.067 0.070 0.035 0.034 0.027 0.021 0.015 0.015 0.026 0.030

0.022 0.000 0.024 0.068 0.057 0.000 0.054 0.039 0.033 0.042 0.035 0.020 0.000 0.000 0.020

0.026 0.021 0.028 0.062 0.067 0.068 0.000 0.047 0.024 0.025 0.022 0.016 0.000 0.000 0.022

0.028 0.030 0.032 0.082 0.089 0.089 0.081 0.000 0.023 0.019 0.031 0.017 0.014 0.000 0.000

0.016 0.020 0.000 0.098 0.108 0.090 0.082 0.077 0.000 0.025 0.024 0.023 0.013 0.000 0.016

0.036 0.041 0.062 0.077 0.072 0.073 0.075 0.093 0.031 0.000 0.079 0.032 0.032 0.000 0.019

0.041 0.045 0.047 0.054 0.066 0.046 0.074 0.066 0.049 0.108 0.000 0.044 0.034 0.000 0.000

0.034 0.039 0.023 0.035 0.044 0.033 0.066 0.036 0.030 0.044 0.043 0.000 0.000 0.000 0.000

0.059 0.065 0.058 0.076 0.084 0.082 0.085 0.050 0.045 0.099 0.077 0.032 0.000 0.032 0.029

0.029 0.032 0.015 0.031 0.030 0.031 0.032 0.019 0.021 0.000 0.000 0.000 0.066 0.000 0.083

0.031 0.031 0.013 0.023 0.027 0.028 0.025 0.020 0.020 0.017 0.012 0.000 0.060 0.102 0.000

Table 8 The total influencing matrix T. Indicators

F1

F2

F3

F4

F5

F6

F7

F8

F9

F10

F11

F12

F13

F14

F15

F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15

0.063 0.152 0.136 0.230 0.234 0.212 0.216 0.145 0.131 0.139 0.110 0.114 0.060 0.054 0.078

0.150 0.062 0.126 0.212 0.229 0.196 0.201 0.138 0.113 0.136 0.100 0.113 0.055 0.055 0.068

0.075 0.074 0.048 0.158 0.169 0.162 0.162 0.154 0.079 0.149 0.079 0.053 0.045 0.018 0.040

0.030 0.053 0.049 0.066 0.146 0.129 0.132 0.082 0.062 0.058 0.047 0.032 0.032 0.042 0.055

0.055 0.056 0.053 0.147 0.075 0.133 0.135 0.084 0.068 0.068 0.054 0.039 0.034 0.041 0.053

0.046 0.029 0.052 0.125 0.118 0.060 0.112 0.083 0.062 0.077 0.063 0.041 0.016 0.013 0.038

0.052 0.048 0.057 0.122 0.130 0.126 0.063 0.090 0.055 0.061 0.051 0.038 0.016 0.014 0.041

0.060 0.063 0.068 0.157 0.166 0.160 0.154 0.056 0.062 0.067 0.066 0.045 0.032 0.016 0.026

0.050 0.055 0.039 0.176 0.188 0.167 0.160 0.131 0.041 0.071 0.062 0.051 0.032 0.018 0.043

0.083 0.090 0.112 0.182 0.182 0.175 0.179 0.167 0.085 0.070 0.128 0.071 0.058 0.021 0.051

0.087 0.093 0.098 0.159 0.174 0.148 0.176 0.143 0.100 0.167 0.055 0.083 0.059 0.019 0.032

0.063 0.069 0.056 0.102 0.112 0.097 0.129 0.085 0.063 0.084 0.074 0.026 0.017 0.012 0.020

0.118 0.125 0.122 0.205 0.218 0.206 0.212 0.147 0.111 0.178 0.139 0.082 0.037 0.058 0.071

0.058 0.061 0.044 0.089 0.091 0.088 0.089 0.060 0.051 0.038 0.030 0.022 0.084 0.021 0.103

0.061 0.062 0.045 0.083 0.090 0.087 0.085 0.063 0.052 0.055 0.042 0.023 0.080 0.116 0.027

Table 9 Influences given and received for each indicator. Indicators

R

C

R+C

R−C

F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15

1.051 1.092 1.106 2.212 2.320 2.146 2.204 1.629 1.135 1.419 1.101 0.831 0.659 0.518 0.747

2.073 1.953 1.466 1.015 1.097 0.934 0.965 1.198 1.284 1.654 1.594 1.010 2.030 0.927 0.971

3.124 3.046 2.572 3.227 3.417 3.080 3.169 2.827 2.419 3.073 2.694 1.841 2.689 1.445 1.718

−1.023 −0.861 −0.359 1.197 1.223 1.212 1.239 0.431 −0.149 −0.236 −0.493 −0.178 −1.370 −0.409 −0.224

Case 2: w1 = 0.2, w2 = 0.6, w3 = 0.2; Case 3: w1 = 0.2, w2 = 0.2, w3 = 0.6. Case 0 indicates the original weights of the clusters yielded by the maximizing consensus method. The other cases show different cluster weights for possible situations, in which one cluster is assigned the highest weight while other cluster weights are kept fixed proportionally so that they add up to 1. Based on the importance index proposed by Liu et al. [53], the ranking results of the 15 indicators determined in the four cases are shown in Fig. 3. From Fig. 3, we can find that the ranking orders of the 15 indicators are greatly affected by the cluster weights. Except the top indicator F5 , the rest indicators have inconsistent rank orders

in the four cases. Bedsides, in Case 0, F4 is the second most important indicator when the weight of C1 is the highest and the weight of C3 is the lowest. In Case 1, F6 is at the second position where the weight of C1 is the highest whereas the weights of C2 and C3 are relatively low. In contrast, F7 ranks second in Case 2 and F14 becomes the second most important indicator in Case 3. The sensitivity analysis shows that the weights of clusters can have a big influence on the final ranking of healthcare performance indicators. Therefore, in practical situations, it is of significance to determine suitable cluster weights for the identification of reasonable and reliable KPIs. 5.4. Comparison analysis In this section, a comparative analysis is conducted to demonstrate the effectiveness of our proposed large group linguistic Z-DEMATEL approach. The classical DEMATEL [54], the fuzzy DEMATEL [55], and the 2-tuple DEMATEL [56] methods are selected and applied for the above case study. Based on the importance index of [53], the priority rankings for the 15 indicators by using the four methods are displayed in Fig. 4. As we can see, the top four indicators and last five indicators determined by all these methods are exactly the same. They are (F4 , F5 , F6 , F7 ) and (F3 , F9 , F12 , F14 , F15 ), respectively. Therefore, the effectiveness of the proposed linguistic Z-DEMATEL approach is verified. In addition, the results obtained by the proposed model and other three methods still have some differences. For example, the most important indicator by the proposed model is different from the ones by the other three methods. The specific ranking positions of the top four indicators generated by the proposed model and the other methods are inconsistent. In addition, according to the classical DEMATEL, the fuzzy DEMATEL, and the 2-tuple

Please cite this article as: S. Jiang, H. Shi, W. Lin et al., A large group linguistic Z-DEMATEL approach for identifying key performance indicators in hospital performance management, Applied Soft Computing Journal (2019) 105900, https://doi.org/10.1016/j.asoc.2019.105900.

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Fig. 3. Ranking results of sensitivity analysis.

Fig. 4. Comparative results of the proposed model against three other methods.

DEMATEL, F2 is assumed to be a more important than F1 . However, the result of the proposed model shows that F1 has a higher priority compared with F2 . The reasons for these differences in the ranking results can be listed as follows: First, the classical DEMATEL method uses crisp values, the fuzzy DEMATEL method uses fuzzy numbers and the 2-tuple DEMATEL method uses 2tuple linguistic variables for evaluating the interrelationships of indicators. Compared with the proposed model, these methods cannot consider the reliability of experts’ evaluation information. Second, only five experts are involved in the three compared methods, which may lead to a lack of precision in the final ranking result. Third, comparing with the simple weighted average used in the listed methods, the LZPWG operator is adopted in our proposed model to aggregate the individual evaluation information of experts. As a result, the correlations of different experts’ evaluations can be taken into account in the information aggregation process. Therefore, the comparison analysis shows that a more accurate and reasonable result can be achieved by combining linguistic Z-numbers and DEMATEL method for the identification of healthcare KPIs in the large group environment.

6. Research implications With increased pressure from the public and government, healthcare institutions in China need to be efficient, attract customers, increase profitability, and continuously improve service quality. The presented large group linguistic Z-DEMATEL approach is utterly useful for hospital managers to determine KPIs to monitor and improve healthcare performance. The literature review indicated that the developed KPI evaluation model is unique in its integration of linguistic Z-numbers and the DEMATEL method in the large group environment. In summary, the new model for identifying KPIs has the following advantages in area of hospital performance management. First, based on linguistic Z-numbers, the proposed model can flexibly characterize the complex interrelation evaluations of indicators as well as describe the reliability of evaluation information. It is not only a more comprehensive reflection of experts’ judgments but also more in line with expression habits. Second, by using the cluster analysis method, the proposed model can obtain the direct influence assessments between indicators when the number of experts is large. This can reduce the influence of unfair and biased judgments given by the participators and obtain more reasonable interrelation evaluation data. Finally, with

Please cite this article as: S. Jiang, H. Shi, W. Lin et al., A large group linguistic Z-DEMATEL approach for identifying key performance indicators in hospital performance management, Applied Soft Computing Journal (2019) 105900, https://doi.org/10.1016/j.asoc.2019.105900.

S. Jiang, H. Shi, W. Lin et al. / Applied Soft Computing Journal xxx (xxxx) xxx

the DEMETAL method, the inter-relationships and cause–effect relationships among indicators can be captured in determining KPIs for hospital management. As the real-world performance measurement involves many interactive indicators, the proposed model can effectively deal with such scenarios. Therefore, this study provides practical and theoretical guidance for healthcare organizations to identify central and influential indicators to improve their performance gradually with limited resources. 7. Conclusions This study presented a large group linguistic Z-DEMATEL approach to identify KPIs for hospital performance improvement. The proposed approach was initiated by considering the complex evaluations of experts based on linguistic Z-numbers. Subsequently, the large group experts were clustered via a similarity measure-based clustering method. Then, an overall direct influencing matrix was established by applying a maximizing consensus method. Finally, the KPIs for hospital management are determined through analyzing the interdependence among indicators with an extended DEMATEL method. An illustrative example was presented to verify the validity and applicability of our proposed KPI evaluation approach. The results reveal that the model being proposed can help managers to determine a limited number of essential indicators to monitor and manage the performance of hospitals systematically. Although this study was thorough, there are opportunities for future research. First, the initial direct influencing information used in this study was given by healthcare experts subjectively, which may be affected by their professional knowledge and bounded rationalities. In the future study, it is suggested to develop an improved DEMATEL method based on both subjective experience and objective data for the identification of KPIs. Second, hospital performance management involves many stakeholders, such as government, patients, and hospital practitioners. However, this study is limited to the hospital respondents. The future study could be separate the different voices from different stakeholders in determining KPIs. In addition, only a rehabilitation hospital example was provided in this study to illustrate the proposed approach. To offer more objective information on the applicability of the new KPI evaluation model, future studies could employ case studies of different types of hospitals and thus prove its effectiveness and usefulness for hospital performance management. Declaration of competing interest No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.asoc.2019.105900. Acknowledgments The authors are very grateful to the respected editor and the anonymous referees for their insightful and constructive comments, which helped to improve the overall quality of the paper. This work was partially supported by the National Natural Science Foundation of China (Nos. 61773250 and 71671125) and the Program for Shanghai Youth Top-Notch Talent. References [1] Z. Chen, Launch of the health-care reform plan in China, Lancet 373 (9672) (2009) 1322–1324.

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