Inpatient admission assessment in West China Hospital based on hesitant fuzzy linguistic VIKOR method

Abstract

Classified diagnose and treatment (CDT) is a hot research topic in hospital management in China. As an essential way to implement CDT, the inpatient admission assessment involves different kinds of vague and uncertain information. The hesitant fuzzy linguistic term set provides a new technology to depict the vague and subjective evaluation information of the experts in hospital. In this paper, we apply the hesitant fuzzy linguistic VIKOR (HFL-VIKOR) method to the inpatient admission assessment process in which all the evaluation information over the patients are expressed as linguistic term sets or linguistic expressions. Firstly, the index of CDT (ICDT) of a patient is introduced to describe the degree to which the patient conforms to the classified diagnose and treatment principle. Based on these indices of different patients, the priorities of inpatient admission are assigned to the patients to ensure that the scarce medical resources be distributed to the most appropriate patients in the view of CDT. The numerical example in the West China Hospital reveals that the HFL-VIKOR method is a feasible and efficient methodology to solve the inpatient admission problem for the purpose of the classified diagnose and treatment.

Keywords

Hospital management HFL-VIKOR classifed diagnose and treatment inpatient admission assessment hesitant fuzzy linguistic term set

1 Introduction

Chinese medical institutions are divided into three tiers, which are aiming at the common and frequently occurring diseases, the more complex diseases, the difficult miscellaneous diseases, respectively. The actual situation, however, is different from the ideal situation. As a developing country, China is experiencing a long period of transformation, and the citizens’ livelihood related domains, such as healthcare, have been paid more and more attentions. In a short time, the contradiction between supply and demand on medical resources could not be eased. According to the statistic of National Health and Family Planning Commission of China (NHFPC) shown in Table 1, the overall utilization rate of beds in China’s hospital has been contained and the utilization rate of beds in high level medical institution is already beyond 100 percent. This situation makes the contradiction between supply and demand constantly acute. In this case, the higher level hospitals will be blocked by patients with low intensity, and thus could not launch medical research on urgent diseases. Besides, the low utilization rate of hospital beds in lower level medical institutions shows that the lower medical institutions have the capacity to admit inpatients with common, frequently occurring disease or even complex disease. This phenomenon occurs due to the increasing investment in medical resources and the inclination of patients to cure their disease, even if common cough, in high level medical institutions. This trend makes the medical resources in lower level medical institutions comparatively idle while in higher level medical institutions overusing. Therefore, the policy of classified diagnose and treatment (CDT) has been proposed to alleviate this plight.

Generally speaking, the aim of CDT is to select the most appropriate medicare facilities to implement diagnose and treatment according to the patients’ condition and status. The common principles to implement CDT can be summarized as follows:

distributing more medical resources to the lower level hospitals;

offering different ratio of medical health insurance for patients in different medical levels;

scheduling medical resources in higher level hospitals more rationally, such as elective inpatient admission.

The advantage of principle 1 is that it balances the contradiction between demand and supply on medical resources, while the disadvantage is that it is time consuming. Principle 2 reduces the demand on high level medical service efficiently, but this measure only restrains the citizens in middle and low income classes and thus is essentially unfair. Assessing every patient from a comprehensive view such as necessity and then assigning priority to the one of high degree of inpatient admission, could efficiently and fairly distribute medical resources in overcrowd high level hospitals. In this way, the citizens with low fitness degree of CDT principle will spend more time in waiting, and thus have to tend to lower level hospitals spontaneously, whilst the patients with serious disease could obtain the medical resources quickly and easily. Therefore, the inpatient admission assessment becomes an essential way to implement CDT.

Up to now, a wide range of techniques have been employed to inpatient admission assessment, such as the Coxian phase type model [9], the Markov chain based model [7, 8], the particle swarm optimization based model [10], the semi-closed migration network based model [15], the simulation based model [6, 30] and the queuing model [3 , 29]. Generally, all of these models can be viewed as objective programming models with distinct objective functions and various constraints such as the earliest and latest hospitalization period, the sex, the length of stay, the pathology, the contagiousness, the single and double rooms, the incompatibility between pathologies and the contagious patients [27]. However, these principles have obvious drawbacks for Chinese healthcare management: their indicators do not meet the national conditions of China. In many countries, the inpatient admission only focuses on the economic benefit and operational efficiency. In China, however, hospitals should also follow the policy from government, which considers the comprehensive utility of the whole nation. As an important implementation approach of CDT, the inpatient admission process should consider the fitness degree of patients for CDT principle. To achieve this goal, we propose a new index of CDT (ICDT) to measure the closeness degree between these patients and CDT principle. The emergency degree (ED), the necessity to hospitalize (NH) and the value of pathology (VP) are considered as constraints or objectives of inpatient admission model for CDT. Due to the transformation from vague and imprecise information to precise numeric data, the CDT index can be taken as a useful criterion to instruct the inpatient admission problem, which is an innovation for solving inpatient admissionproblem.

Besides, the inpatient admission assessment for CDT problem consists of many vague and imprecise information, in which natural language is the best way to describe the status of each alternative patient. However, the natural language is limited to qualitative analysis. How to transform qualitative information to precise and quantitative data has been a big problem for many years. The hesitant fuzzy linguistic term set (HFLTS) [19, 26] offers a robust way to convert qualitative information to precise and quantitative data and empirical research conforms that its transformation is valid.

Assessing alternative patients is a multiple criteria decision making (MCDM) problem, and it involves many conflicting criteria. For the fuzzy MCDMproblem with conflict and non commensurable criteria, Opricovic [25] introduced the fuzzy VIKOR method which solves problems in fuzzy circumstances where both criteria and weights could be fuzzy sets. In his model, the triangular fuzzy numbers were employed to represent imprecise numerical quantities. As for the situation where the decision makers’ opinions are represented as trapezoidal fuzzy numbers, Shemshadi et al. [28] proposed the fuzzy VIKOR method which extracted and deployed the objective weights of the criteria based on Shannon entropy concept, and then applied the method into the supplier selection problem. To evaluate the website quality of professional accounting firms, Chow and Cheng [4] developed a hybrid approach which combines the fuzzy analytic network process (FANP) and the fuzzy VIKOR together. Liao and Xu [18] introduced the hesitant fuzzy VIKOR method and used it to evaluate the domestic airlines’ service quality. Zhang and Wei [35] employed the hesitant fuzzy VIKOR method to help the enterprise’s board of directors evaluate the candidate large projects. Based on the Shapley value based L_p - metric, Zhang and Wei [36] introduced an extended hesitant fuzzy VIKOR method to deal with the correlative MCDM problem under hesitant fuzzy circumstances. Liao et al. [23] proposed the HFL-VIKOR method which is a great innovation of traditional VIKOR for its robustness to handle fuzzy information in MCDM problem.The HFL-VIKOR method is very useful in handling the qualitative MCDM problems especially when the criteria conflict with each other and the evaluation values of the alternatives are represented as linguistic expressions or simple linguistic terms. All the above mentioned extended forms of VIKOR methods are based on different types of quantitative information thus can not be implemented in qualitative situations. As HFLTS is the newest form of linguistic representation model and more flexible in depicting the human cognitions, the HFL-VIKOR, which combines the HFLTS and the VIKOR method together, can describe the real preferences of decision makers and reflect their uncertainty, hesitancy and inconsistency, eliminate the vague and imprecise information from natural language, and assess each alternative patient according to conflicting criteria at the same time [31–33 , 37].

Therefore, applying the HFL-VIKOR method in inpatient admission assessment could efficiently solve the inpatient admission assessment problem. West China hospital (WCH) is one of the most influential and powerful hospital in China, and it also suffers the plight of inefficiency. Considering such situation, we applied the HFL-VIKOR method to WCH’s inpatient admission assessment problem, and achieved valid distinction results for each patient. Based on the previous work, further work is to employ the distinction results as constrains or decision objectives in WCH’s inpatient admission problem.

The remainder of this paper is organized as follow: Section 2 describes the theory of HFL-VIKOR method. Section 3 applies the HFL-VIKOR method to WCH’s patient assessment to obtain the rank of patients and ICDT in WCH. In Section 4, we conclude the advantage of this method in patient assessment and discuss the study in future work.

2 Methodology

2.1 Hesitant fuzzy linguistic term set

Fuzzy linguistic term set has been applied to handle fuzzy information since the fuzzy theory rose. However, for more complicated decision making problems, few attributes could be described by only one single term. The more suitable way is to present comprehensive linguistic expressions. In this case, Rodríguez et al. [26] introduced the concept of hesitant fuzzy linguistic term set, which can be used to elicit several linguistic terms for a linguistic variable. By this method, we could describe an attribute as “between medium and high”, without hesitating “high” or “medium” to choose.

However, according to the definition of Rodríguez et al. [26], the subscripts of the HFLTSs are not symmetric, and this may lead to some counterintuitive results. For example, for a linguistic term set S={s₀ = none, s₁=verylow, s₂=low, s₃ = medium, s₄ = high, s₅ = veryhigh, s₆ = perfect}, according to the operational law, we have s₂ ⊕ s₃ = s₅, which means the aggregated result of linguistic terms “low” and “medium” is “very high”. This is not coincident with our intuition. Therefore, Liao et al. [19] redefined the hesitant fuzzy linguistic term set to overcome this plight.

Definition 1. [19] Let x_i ∈ X, i = 1, 2, ⋯ , N, be fixed and S = {s_t|t = - τ, . . . , -1, 0, 1, . . . , τ} be a linguistic term set. A hesitant fuzzy linguistic term set (HFLTS) in X, H_S, is in mathematical terms of $H_{S} = {< x_{i}, h_{S} (x_{i}) > | x_{i} \in X}$ (1) where function h_S (x_i) : X → S defines the possible membership grades of the element x_i ∈ X to the set A ⊂ X and for every x_i ∈ X, the value of h_S (x_i) is represented by a set of some values in the linguistic term set S and can be expressed as h_S (x_i) ={ s_{φ
_l} (x_i) |s_{φ
_l} (x_i) ∈ S, l = 1, …, L (x_i) } with φ_l ∈ {- τ, . . . , -1, 0, 1, . . . , τ} being the subscript of a linguistic term and the L (x_i) being the number of linguistic terms in h_S (x_i). For convenience, h_S (x_i) is called the hesitant fuzzy linguistic element (HFLE) and H_S is the set of all HFLEs.

Example 1. Here we just consider a simple example that the head administrator in inpatient admission service center of West China hospital, which is established to assess patients and distribute medical resource rationally, evaluates the emergency degree of three patients applying for hospitalization, represented as x₁, x₂ and x₃. Since this criterion is qualitative, it is impossible to give crisp values but only linguistic terms. The emergency degree of these patients can be taken as a linguistic variable. The linguistic term set of the emergency degree can be set up as:

S={s_-3= very non urgent, s_-2=non urgent, s_-1=a little non urgent, s₀=medium, s₁=a little urgent, s₂=urgent, s₃=very urgent}

where H_S (x) = {<x₁ ; h_S (x₁)>,<x₂ ; h_S (x₂)>,<x₃ ; h_S (x₃)>} with h_S (x₁) = {s₁, s₂, s₃}, h_S (x₂) = {s_-2, s_-1, s₀} and h_S (x₃) = {s₃}.

Although the HFLTS can be used to elicit several linguistic values for a linguistic variable, it is still not similar to the human way of thinking and reasoning. Thus, Rodríguez et al. [26] further proposed a context-free grammar to generate simple but elaborated linguistic expressions that are more similar to the human expressions and can be easily represented by means of HFLTS. The grammar G_H is a 4-tuple (V_N,V_T,I,P) where V_N is a set of nonterminal symbols, V_T is the set of terminals symbols, I is the starting symbol, and P is the production rules that are defined in an extended Backus-Naurform [2].

Definition 2. [26] Let S be a linguistic term set, G_H be a context-free grammar. The element of G_H = (V_N,V_T,I,P) are defined as follows:

V_N = {< primary term >, < composite term >, < unary relation >, < binary relation >, < conjunction >};

V_T = {lower than, greater than, at least, at most, between, and, s_-τ,…,s_-1,s₀,s₁,…,s_τ};

I ∈ V_N

P = {I:: = <primary term>|<composite term>

<primary term>:: = s_-τ | ⋯ | s_-1 | s₀ | s_- | ⋯ | s_τ

<unary relation>:: = lower than | greater than

<binary relation>:: = between

<conjunction>:: = and }

Considering that HFLTS is not similar to the human way of thinking and reasoning, a context-free grammar G_H was introduced to generate simple but elaborated linguistic expressions, ll, which are more similar to humans expressions and can be semantically represented by HFLTS. In addition, a transformation function E_{G
_H} was proposed to transform the expressions ll into the HFLTS H_S.

Definition 3. [26] Let E_{G
_H} be a function that transforms linguistic expressions ll ∈ S_ll, obtained by using G_H, into the HFLTS H_S. S is the linguistic term set used by G_H, and S_ll is the expression domain generated by G_H:

$E_{G_{H}} : S_{ll} \to H_{S} ∥$ (2)

The linguistic expression generated by G_H using the production rules are converted into HFLTS by means of the following transformations:

E_{G
_H} (s_t) = {s_t|s_t∈ S} ;

E_{G
_H} (at most s_m) = {s_t|s_t∈ S and s_t ≤ s_m} ;

E_{G
_H} (lower than s_m) = {s_t|s_t∈ S and s_t < s_m} ;

E_{G
_H} (at least s_m) = {s_t|s_t∈ S and s_t ≥ s_m} ;

E_{G
_H} (great than s_m) = {s_t|s_t∈ S and s_t > s_m} ;

E_{G
_H} (between s_m and s_n) = {s_t|s_t ∈ S and s_m ≤ s_t ≤ s_n} .

With the transformation function E_{G
_H} defined as Definition 3, it is easy to transform the initial linguistic expressions into HFLTS.

Example 2. Emergency degree is an important criterion to evaluate a patient waiting for hospitalization admission. An precise describe for this criteria is somehow unnatural, but HFLTS is appropriate for this situation for that it is more similar to the thinking method of human. There are three patients in the list and the description in each criteria are given in Table 2.

According to the E_{G
_H} defined in Definition 3, the HFLTS judgment matrix could be obtained as follow:

$ℍ = [\begin{matrix} {s_{- 1}, s_{0}, s_{1}, s_{2}} \\ {s_{2}, s_{3}} \\ {s_{0}, s_{1}, s_{2}} \end{matrix}]$ (3)

In order to extend the VIKOR method into the HFL-VIKOR, two phrases are indispensible, i.e., finding out the ideal solution and defining the particular measure. Firstly, how to find out the ideal solution for a HFL-MCDM problem is investigated. For this purpose, the MAX operator and MIN operator for HFLTSs have to be defined. Motivated by the score function and the variance function of HFS [20], Liao et al. [23] defined the score function and the variance function of H_S. By employing these two functions, a kind of partial order structure could be constructed among HFLTSs.

Definition 4. For a HFLTS H_S =∪ _{s_{δ
_l}∈H_S} { s_{δ
_l}|l = 1, …, # H_S }, where # H_S is the number of linguistic terms in H_S,

$ρ (H_{S}) = \frac{1}{# H_{S}} \sum_{s_{δ_{l}} \in H_{S}} s_{δ_{l}} = s_{\frac{1}{# H_{S}} \sum_{1}^{# H_{S}} δ_{l}}$

is called the score function of H_S.

Definition 5. [23] For a HFLTS H_S =∪ _{s_{δ
_l}∈H_S} { s_{δ
_l}|l = 1, …, # H_S }, where # H_S is the number of linguistic terms in H_S,

$σ (H_{S}) = \frac{1}{# H_{S}} \sqrt{\sum_{s_{δ_{l}}, s_{δ_{k}} \in H_{S}} {(s_{δ_{l}} - s_{δ_{k}})}^{2}} = s_{\frac{1}{# H_{S}} \sqrt{\sum_{s_{δ_{l}}, s_{δ_{k}} \in H_{S}} {(δ_{l} - δ_{k})}^{2}}}$

is called the variance function of H_S.

The relationship between the score function and the variance function is similar to the relationship between mean and variance in statistics. Thus, for two HFLTSs $H_{S}^{1}$ and $H_{S}^{2}$ , the following approach can be used to compare any two HFLTSs:

If $ρ (H_{S}^{1}) > ρ (H_{S}^{2})$ , then,

$H_{S}^{1} > H_{S}^{2}$ ;

Else if $ρ (H_{S}^{1}) = ρ (H_{S}^{2})$ , then,

if $σ (H_{S}^{1}) < σ (H_{S}^{2})$ , then $H_{S}^{1} > H_{S}^{2}$ ;

else if $σ (H_{S}^{1}) = σ (H_{S}^{2})$ , then $H_{S}^{1} = H_{S}^{1}$ .

For a HFLE h_S ={ s_φl|l = 1, 2, …, L }, the linguistic terms in it might be out of order. To simplify the computation, we can arrange the linguistic terms s_{φ
_l} (l = 1, …, L) in any of the following orders: 1) ascending order δ : (1, 2, ⋯ , n) → (1, 2, ⋯ , n) is a permutation satisfying δ_l ≤ δ_l+1, l = 1, …, L; 2) descending order η : (1, 2, ⋯ , n) → (1, 2, ⋯ , n) is a permutation satisfying η_l ≥ η_l+1, l = 1, …, L.

In addition, considering that different HFLEs may have different numbers of linguistic terms, for the sake of operating correctly when comparing or computing with HFLEs, we can extend the short HFLEs by adding some linguistic terms in it till they have same length (for more information, refer to [21, 22]). In this paper, we suppose that the linguistic terms are arranged in ascending order δ, and the short HFLEs are extended by adding the linguistic term $\bar{s} = \frac{1}{2} (s^{+} \oplus s^{-})$ respectively, where s⁺ and s^- are the maximal and minimal linguistic terms in the HFLE h_S, defined as $s^{+} = max_{φ_{l}} {s_{φ_{l}} | l = 1, 2, \dots, L}$ and $s^{-} = min_{φ_{l}} {s_{φ_{l}} | l = 1, 2, \dots, L}$ respectively. For any two linguistic terms s_α, s_β ∈ S and λ ∈ [0, 1], the operational laws were defined as Xu [34]:

s_α ⊕ s_β = s_α+β;

λs_α = s_λα.

2.2 Hesitant fuzzy linguistic VIKOR method

The general procedure of the HFL-VIKOR method to solve the qualitative MCDM problem involves the following steps(for more details, please refer to [21]):

Step 1: Establish the alternatives a_i (i = 1, 2, ⋯ , I), the criteria c_j (j = 1, 2, ⋯ , J), and the weights ω_j of the criteria c_j (j = 1, 2, ⋯ , J).

Step 2: Define the semantics and syntax of the linguistic term sets for each criterion. The expert then gives the linguistic evaluation values for the alternatives over the criteria according to these semantics and syntax in expressions or multiple linguistic terms, denoted as ll_ij.

Step 3: Define the context free grammar G_H, then transform the initial linguistic expressions or the multiple linguistic terms ll_ij into the HFLTS $H_{j}^{i}$ via the transformation function E_{G
_H}.

Step 4: Find out the positive ideal solution and the negative ideal solution. Then calculate the hesitant fuzzy linguistic group utility measure HFLGU_i, the hesitant fuzzy linguistic individual regret measure HFLIR_i and the hesitant fuzzy linguistic compromise measure HFLC_i for each alternative. The equations are given below:

$\begin{matrix} {HFLGU}_{i} & = & {HFLEL}_{1, i} \\ = & \sum_{j = 1}^{J} ω_{j} \frac{d_{ed} (H_{j}^{+}, H_{j}^{i})}{d_{ed} (H_{j}^{+}, H_{j}^{-})} \end{matrix}$ (4)

$\begin{matrix} {HFLIR}_{i} & = & {HFLEL}_{\infty, i} \\ = & max (ω_{j} \frac{d_{ed} (H_{j}^{+}, H_{j}^{i})}{d_{ed} (H_{j}^{+}, H_{j}^{-})}) \end{matrix}$ (5)

$\begin{matrix} {HFLC}_{i} & = & (1 - θ) \frac{{HFLIR}_{i} - {HFLIR}^{+}}{{HFLIR}^{-} - {HFLIR}^{+}} \\ + & θ \frac{{HFLGU}_{i} - {HFLGU}^{+}}{{HFLGU}^{-} - {HFLGU}^{+}} \end{matrix}$ (6)

where ω_j (j=1,2,⋯,J) are the corresponding weights of criteria satisfying 0 ≤ ω_j ≤ 1, j = 1,2,⋯,J, $\sum_{j = 1}^{J}$ ω_j=1, and θ ∈ [0,1]. Besides, HFLGU⁺ = $min_{i}$ HFLGU_i, ${HFLGU}_{i}^{-}$ = $max_{i}$ HFLGU_i, HFLIR⁺ = $min_{i}$ HFLIR_i, HFLIR^- = $max_{i}$ HFLIR_i, and θ is the weight of the strategy of the majority of criteria or the maximum overall utility. d_ed $(H_{j}^{+}$ , $H_{j}^{i})$ and d_ed $(H_{j}^{+}$ , $H_{j}^{-})$ are the hesitant fuzzy linguistic Euclidean distance measures, and its expression is given below:

$d_{ed} (H_{S}^{1}, H_{S}^{2}) = {(\frac{1}{L} \sum_{l = 1}^{L} {(\frac{| δ_{l}^{1} - δ_{l}^{2} |}{2 τ + 1})}^{2})}^{0.5}$ (7)

where $H_{S}^{1} = \cup_{s_{δ_{l}^{1}} \in H_{S}^{1}} {s_{δ_{l}^{1}} | l = 1, \dots, # H_{S}^{1}}$ ( $# H_{S}^{1}$ is the number of linguistic terms in $H_{S}^{1})$ and $H_{S}^{2} = \cup_{s_{δ_{l}^{2}} \in H_{S}^{2}} {s_{δ_{l}^{2}} | l = 1, \dots, # H_{S}^{2}}$ are two HFLTSs with $# H_{S}^{1} = # H_{S}^{2} = L$ .

Step 5: Derive the compromise solution(s). Each compromise solution should be the optimal solution which minimizes the hesitant fuzzy linguistic group utility measure HFLGU_i as well as the hesitant fuzzy linguistic individual regret measure HFLIR_i. Therefore, to select the compromise solution, we should rank HFLGU_i, HFLIR_i and HFLC_i in descending order and obtain three ranking lists. The final optimal solution should be the one that makes those measures attain the minimum values. A compromise solution is the alternative A′ which is ranked the best by measuring HFLC_i in descending order and alsosatisfies:

Acceptable advantage: HFLC_A″ - HFLC_A′ ≥ 1/(m-1), where A″ is the alternative with the second position in the ranking list by HFLC_i.

Acceptable stability in decision making: The alternative A′ should also be the best ranked by HFLGU_i and HFLIR_i.

If one of the conditions is not satisfied, more than one solution is taken as the compromise solution, which consists of:

Alternatives A′ and A″ if only Condition 2 is not satisfied.

Alternatives A′, A″, …, A⁽ⁿ⁾. if Condition 1 is not satisfied, where A⁽ⁿ⁾ is established by the relation HFLC_{A
⁽ⁿ⁾}HFLC_A′ < 1/(m-1) for the maximum n.

The simplified flow chart of the general HFL-VIKOR procedure is shown in Fig. 1. The advantages of the HFL-VIKOR methods can be listed as follows:

Comparing to the traditional VIKOR method and its extensions, the HFL-VIKOR method can handle the MCDM problems in which all the evaluation values of each alternative over the criteria are represent as linguistic expressions or simple linguistic terms.

Comparing to the HFL-TOPSIS method [1, 24] and the symbolic aggregation based method [26], there is no need to do any normalization over the different criteria for HFL-VIKOR method.

3 Apply HFL-VIKOR method to inpatient admission assessment in West China Hospital

West China Hospital (WCH), located in Chengdu, is one of the largest single-site hospitals in the world and a leading medical center of West China, treating complicated and severe cases, especially in the fields of living donor liver transplantation, severe acute pancreatitis, and clinical anesthesia. During the past long time, a great number of patients, which beyond the capacity of WCH, are applying for the hospitalization admission. Among them, a small part have the necessity to applying for the admission of WCH due to their high intensity diseases and distinctive value of pathology; besides, the rest of them are the patients suffer low intensity diseases, such as cough and influenza, which could be cured by low level medical institution readily.

In December 2011, WCH established an inpatient admission service center, for the purpose of optimizing the hospitalization admission process. The inpatient admission service center indeed reduces the trifles of patients for hospitalization matters and improves the efficiency of utilizing hospital resources. Acute contradiction, however, still exists due to the extremely finite medical resource and the considerable demand of citizens who irrationally pursuing higher level medical resource. Therefore, an efficient decision support technique should be employed to assess the patients who apply for hospitalization admission after the preliminary diagnose by doctors in ambulant clinic to guarantee that the scarce medical resources are implemented to the most desirable patients.

In this section, the HFL-VIKOR is employed as a key technique to handle the inpatient admission assessment problem. Not all patients have the necessity to hospitalize, for that the patients’ status varying. A feasible way to solve this problem is to assess the fitness degree of each patient and make sure that admission decisions depend on the numeric assessment results. Under the circumstance of CDT, the assessment results is called ICDT, which will influence the priority in admission to a large extent. Due to the transformation from vague and imprecise information to precise numeric data, ICDT could be a useful criteria to instruct the inpatient admission problem, which is an innovation for inpatient admission problem. In this case, ICDT would be a constraint or an objective of inpatient admission programme problem. We define ICDT as the formula given below:

$ICDT = 1 - HFLC + \frac{1}{I}$ (8)

where I is the amount of patient waiting in the queue. In inpatient admission, HFLGU and HFLIR respectively represent the maximum group utility for the majority and the minimum individual regret for the opponent. Besides, HFLC is the compromise measure of the maximum utility and minimum regret in inpatient admission selection and can be obtained by Formula 4. The smaller the value of the hesitant fuzzy compromise measure HFLC, the fitter the patient for CDT principle.

Huang et al. [14] proposed that the core meaning of the implementation of CDT is to effectively divert patients to the appropriate medical institution to give appropriate treatment, without delay, waste of medical resources and health care funds, and finally construct a ideal environment for both doctors and patients. However, CDT covers many comprehensive domains, and there is no explicit instruction to measure CDT principle. Therefore, based on the research of Li [16] and He et al. [13], and the discipline development prospect of WCH, the criteria of CDT are finally selected as follow:

Emergency degree (ED), describes how emergent the patient need to be hospitalized, is a positive index for CDT. For example, the patient with chronic pharyngitis has a inferior ED compared to the one with acute pendicitis.

Necessity to hospitalize (NH), represents the necessity to let patients to hospitalize rather than stay at home. For example, the patient with extensive skin burn has a superior NH compared to the one with dislocation.

Value of pathology (VP), indicates the potential value to adopt such a patient in the aspect of medical development. For example, a patient with rare tumor is prior to the one with common catagma in the aspect of VP.

The terms of the above criteria vary in different contexts, thus it’s necessary to make sure that every linguistic term in this application be out of ambiguity. Through the interview with doctors in WCH, we determine the linguistic term sets for these three criteria:

S_ED = {s_-3 = very non urgent, s_-2 = non urgent, s_-1 = a little non urgent, s₀ = medium, s₁ = a little urgent, s₂ = urgent, s₃ = very urgent}.

S_NH={s_-3 = very unnecessary, s_-2 = unnecessary, s_-1 = a little unnecessary, s₀ = medium, s₁ = a little necessary, s₂ = necessary, s₃ = very necessary}.

S_VP = {s_-3 = very valueless, s_-2 = valueless, s_-1 = valueless, s₀ = medium, s₁ = a little valuable, s₂ = valuable, s₃ = very valuable}.

With these linguistic term sets and also the context-free grammar shown in Definition 2, officer in the inpatient admission service center in WCH provides descriptions of 50 patients waiting for hospitalisation admission, which are shown in Table 3 (for more details, please refer to Example 1):

The linguistic expressions shown in Table 3 are similar to the human way of thinking and they can reflect the decision makers hesitant cognition intuitively. To assess each patient’s closeness degree to CDT principle, The linguistic expressions must be transformed to the HFLTS judgment matrix, in the form of which the distance between each patient in the aspect of attribute could be calculated. Ignoring the affection of different linguistic term sets on criteria, the transformation function H_{G
_E} given in Definition 3 is employed to obtain the following HFLTS judgment matrix(for more details, please refer toExample 2):

$ℍ = [\begin{matrix} {s_{2}, s_{3}} & {s_{2}, s_{3}} & {s_{- 3}, s_{- 2}} \\ {s_{- 2}, s_{- 1}, s_{0}} & {s_{3}} & {s_{1}, s_{2}} \\ \dots & \dots & \dots \\ {s_{- 2}, s_{- 1}, s_{0}} & {s_{1}, s_{2}, s_{3}} & {s_{0}} \end{matrix}]$ (9)

Based on the semantics of different criteria, we can see that all of the three criteria are benefic criteria, i.e., the higher the value, the better the alternative. According to Definition 4 and Definition 5, we can calculate the score function value and variance function value for each HFLTS. Then, we obtain

max(ED) = {s₂, s₃}, min(ED) = {s_-3}

max(NH) = {s₃}, min(NH) = {s_-3}

max(VP) = {s₃}, min(VP) = {s_-3, s_-2, s_-1, s₀}

$f_{j}^{+} = max_{i} f_{ij}$ and $f_{j}^{-} = min_{i} f_{ij}$ are the best and worst values of a_i over the benefit-type criterion c_j, respectively. In this case, the positive ideal solution for this HFL-MCDM problem is f⁺ = ({s_-3, s_-2, s_-1} , {s₃} , {s_-2, s_-1, s₀}) ^T, whereas the negative ideal solution is f^- = ({s_-3} , {s₃} , {s_-3, s_-2, s_-1, s₀}) ^T.

The upper bound is defined as a n-tuple of the highest degree in each criterion, and the lower bound is similar. In this case, the upper bound and the lower bound are 3-tuples. Especially, for patient P₁, the upper bound and lower bound are (s₃, s₃, s_-2) and (s₂, s₂, s_-3) respectively. Figure 2 depicts the upper bound and lower bound of each patient, and the raw data of each patient can be found in it.

The weight matrix for ED, NH and VP from the head administrator is [0.1, 0.5, 0.4]^T. θ refers to the weight of the strategy of the majority of criteria or the maximum overall utility to generate the HFLC. In this case, θ is set to be 0.5 to avoid losing generality. The HFLGUs, HFLIRs and HFLCs are shown in Fig. 3. This figure record every patient in the way of HFLGUs, HFLIRs and HFLCs, which show strong positive correlation. According to the HFLCs, we obtain the priority rank, shown in Table 4, where M₁ refers to HFL-TOPSIS and M₂ refers to HFL-VIKOR. The results show that the top 5 ranks are (P₂₄, P₄₉, P₂₉, P₃₈, P₂₈) and (P₁₄, P₄₆, P₂₅, P₃₃, P₃₁), and it indicates that HFL-VIKOR is more stable, for that it didn’t prefer to extreme value, such as {s_-3} and {s₃} of P₂₈, shown in Table 5. Besides, the top 5 ICDTs are 1.0200, 0.8940, 0.8853, 0.8826 and 0.8127, respectively.

4 Conclusion

There is a much long period before Chinese medical contradiction between supply and demand be remarkably alleviate. The police of CDT, which values social utility and fairness more, is proposed to ease this contradiction. Among all implement approaches, inpatient admission is the most fair and practical one, therefore need be paid more attention.

In the field of inpatient admission, a feasible way to solve medical resource deficiency is to assess social value according to CDT principle, and the scheduling process should optimize the ICDT or ensure the ICDT attains a certain high level. From the application of HFL-VIKOR in patient assessment in WCH, we can draw a conclusion that, HFL-VIKOR method could efficiently assess the closeness, between the alternative patient and CDT principle, to increase the comprehensive utility of inpatient admission management. This ability mainly lies in the following four advantages of HFL-VIKOR:

Firstly, comparing to the traditional VIKOR method and its extensions, the HFL-VIKOR method is proposed to handle the MCDM problems in which all the evaluation values of each alternative over the criteria are represent as linguistic expressions or simple linguistic terms.

There is no need to do any normalization over the different criteria for HFL-VIKOR method.

Another advantage of the HFL-VIKOR method comparing to the HFL-TOPSIS method is that it derives the compromise solution(s) which consider not only maximizing the group utility for the majority but also minimizing individual regret for the opponent.

In addition, the HFL-VIKOR method is much more convincing than the symbolic aggregation based method [26] because it takes different weights of the criteria into account.

In this paper, ICDT is proposed as measurement for the fitness degree of patients for CDT principle, to assess the priority should be assigned in inpatient admission problem. Different from earlier works, we consider each patient with distinct status, for that different kinds of medical institutions involve different kinds of goals. The proposition of ICDT reveals a new way to solve inpatient admission problem, that is to consider ICDT, which refers to the comprehensive influence, as a important factor when process hospitalization scheduling. Further research will be done by employing ICDT of alternative patients to optimise the inpatient admission process in WCH.

Footnotes

Acknowledgments

The authors would like to thank the editors and the anonymous reviewers for their insightful and constructive comments and suggestions that have led to this improved version of the paper. The work was supported in part by the National Natural Science Foundation of China (No. 71532007, No. 71501135, No. 71131006 and No. 71172197).

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