# A Novel Selection Model of Surgical Treatments for Early Gastric Cancer Patients Based on Heterogeneous Multicriteria Group Decision-Making

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## Abstract

**:**

## 1. Introduction

- (1)
- The evaluation of surgical treatments involves several criteria including subjective and objective criteria. Some scholars have used several objective criteria to evaluate surgical treatments quantitatively [6,11,12]. Subjective criteria such as severity of the side effects and severity of the complications were utilized in Chenabc’s study [9]. Thus, the subjective criteria combined with objective criteria are applied in the index system of the proposed model.
- (2)
- With regard to the partial use of information, it is appropriate to apply fuzzy logic to describe evaluation information regarding surgical treatments. The evaluation information from hospital cases mainly involves crisp numbers and interval numbers. Zhang et al. [13] presented that a neutrosophic set is an effective tool for reflecting the fuzziness in text evaluation because the evaluation information from patients is text information that represents sentiment values, and every sentiment value has not only a certain degree of truth, but also a falsity degree and an indeterminacy degree [14]. So, it needs to be transformed into neutrosophic numbers with positive, medium, and passive values. For example, when asked to assess whether medical equipment would be “good”, from the sentiment value of a patient we may deduce that the membership degree of truth is 0.6, the membership degree of indeterminacy is 0.2, and the membership degree of falsity is 0.2. Pang et al. [15] stated that probabilistic linguistic term sets (PLTS) are more convenient for the DMs to provide their preference as they may have hesitancy among several possible linguistic terms when expressing their evaluation information, so the PLTS need to be applied to express experts’ linguistic terms more accurately. For example, when asked to assess whether a surgical treatment would be appropriate for a particular patient based on the patient’s conditions, an expert may deduce that the probability of “high” is 0.7, the probability of “medium high” is 0.2, and the probability of “medium” is 0.1. Therefore, evaluation information, including crisp numbers, interval numbers, neutrosophic numbers, and probabilistic linguistic labels, needs to be considered in the selection model.
- (3)
- To deal with the priority order of surgical treatments based on heterogeneous MCGDM, a systematic approach need to be used in the proposed model. Shih et al. [16] hold that TOPSIS is a practical and useful technique for the ranking and selection of a number of externally determined alternatives through distance measures, and it has been connected to multiple-criteria decision-making (MCDM) [17]. Lourenzutti et al. [18] and Li et al. [19] presented heterogeneous TOPSIS for multicriteria group decision-making. Thus, the TOPSIS method is applied in the proposed model to solve the surgical treatment selection.

## 2. Literature Review

## 3. Preliminaries

#### 3.1. Interval Numbers

**Definition**

**1.**

**a**=$\left[{a}^{L},{a}^{U}\right]$, where${a}^{L}\le {a}^{U}$, defined on the real line, is called an interval number, as introduced in Tsaur’s study [30]. The values${a}^{L}$and${a}^{U}$stand for the lower and upper bounds of

**a**, respectively.

**Definition**

**2.**

#### 3.2. Neutrosophic Set Theory

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

**Definition**

**9.**

_{−3}= very bad, s

_{−2}= bad, s

_{−1}= slightly bad, s

_{0}= ok, s

_{1}= slightly good, s

_{2}= good, s

_{3}= very good}. The semantic values in Table 1 are computed throughout sentiment analysis by utilizing the software of ‘The R Project for Statistical Computing’. According to different sentiment words, we allocate linguistic variable${s}_{i}$to positive, neutral, and passive values as a neutrosophic number${A}_{i}=<{T}_{i},{I}_{i},{F}_{i}>$,$i=1,\dots ,n$(positive value is T value, neutral value is I, passive value is F). The T value is the average value of positive values, while the I value is 1 if s

_{0}exists or 0 if s

_{0}does not exist, and the F value is the mean of the absolute passive values. For example, for a set$S=\left\{{s}_{-1},{s}_{0},{s}_{1},{s}_{2},{s}_{3}\right\}$, the neutrosophic value is$\langle \frac{{s}_{1}+{s}_{2}+{s}_{3}}{3},{s}_{0},|{s}_{-1}|\rangle $, i.e.,$\langle \frac{0.106066+0.75+0.954594}{3},1,0.10607\rangle $.

#### 3.3. Probabilistic Linguistic Term Sets and Their Basic Concepts

**Definition**

**10.**

**Example**

**1.**

**Definition**

**11.**

**Definition**

**12.**

**Definition**

**13.**

- (1)
- If$\sum _{k=1}^{\#L(p)}{{p}_{i}}^{(k)}}<1$, then by Equation (3), we can compute${\stackrel{.}{L}}_{i}(p)$, i = 1, 2.
- (2)
- If$\#{L}_{1}(p)\ne \#{L}_{2}(p)$, then we add some elements to the one with the smaller number of elements according to Definition 13.

**Example**

**2.**

**Definition**

**14.**

**Definition**

**15.**

**Definition**

**16.**

**Definition**

**17.**

**Definition**

**18.**

**Definition**

**19.**

**Definition**

**20.**

## 4. The Proposed Selection Model of Surgical Treatment for Early Gastric Cancer

#### 4.1. The Establishment of the Early Gastric Cancer Surgery Index System

#### 4.2. The Estimation of Criteria Weights with BWM

**Step****1.**- Determine a set of decision criteria.In this step, we consider the criteria $\left\{{c}_{1},{c}_{2},\dots ,{c}_{n}\right\}$ that should be used to arrive at a decision.
**Step****2.**- Determine the best (e.g., most desirable, most important) and the worst (e.g., least desirable, least important) criterion.
**Step****3.**- Determine the preference of the best criterion over all the other criteria, using a number between 1 and 9. The resulting best-to-others vector would be ${A}_{B}=({a}_{B1},{a}_{B2},\dots ,{a}_{Bn})$ where ${a}_{Bj}$ indicates the preference of the best criterion B over criterion j. It is clear that ${a}_{BB}$ = 1.
**Step****4.**- Determine the preference of all the criteria over the worst criterion, using a number between 1 and 9. The resulting others-to-worst vector would be: ${A}_{W}=({a}_{1W},{a}_{2W},\dots ,{a}_{nW})$ where ${a}_{jW}$ indicates the preference of the criterion j over the worst criterion W. It is clear that ${a}_{WW}$ = 1.
**Step****5.**- Find the optimal weights $({w}_{1}{}^{\ast},{w}_{2}{}^{\ast},\dots ,{w}_{n}{}^{\ast})$.

^{*}are obtained.

^{*}and the corresponding consistency index as follows:

#### 4.3. The Evaluation Matrix of Early Gastric Cancer Surgery

**Step 1.**Obtain real numbers.

**Step 2.**Get the interval numbers.

**Step 3.**Calculate neutrosophic numbers.

**Step 4.**Acquire probabilistic linguistic values.

#### 4.4. The Calculation of Index Weight

**Step 1.**Determine the entropy weight of data index.

**Step 2.**Compute the entropy weight of the interval-valued index.

**Step 3.**Manage the entropy weight of the neutrosophic-valued index.

**Step 4.**Calculate the entropy weight of the probabilistic linguistic-valued index.

**Step 5.**Calculate the weight of the early gastric cancer surgery index.

#### 4.5. TOPSIS and Its Application in Heterogeneous MCGDM

**Step****1.**- Define and normalize the decision matrix $R=({r}_{ij})$.
**Step****2.**- Aggregate the weights to the decision matrix by making ${v}_{ij}={w}_{j}{r}_{ij}$.
**Step****3.**- Define the positive ideal solution (PIS), ${v}_{j}^{+}$, and the negative ideal solution (NIS), ${v}_{j}^{-}$, for each criterion. Usually, ${v}_{j}^{+}=\mathrm{max}\left\{{v}_{ij},\dots ,{v}_{mj}\right\}$ and ${v}_{j}^{-}=\mathrm{min}\left\{{v}_{ij},\dots ,{v}_{mj}\right\}$ for benefit criteria, and ${v}_{j}^{+}=\mathrm{min}\left\{{v}_{ij},\dots ,{v}_{mj}\right\}$ and ${v}_{j}^{-}=\mathrm{max}\left\{{v}_{ij},\dots ,{v}_{mj}\right\}$ for cost criteria.
**Step****4.**- Calculate the separation measures for each alternative.$${S}_{i}^{+}=\sqrt{{\displaystyle \sum _{j=1}^{n}{({v}_{j}^{+}-{v}_{ij})}^{2}}},\text{\hspace{1em}}i=1,2,\dots ,m$$$${S}_{i}^{-}=\sqrt{{\displaystyle \sum _{j=1}^{n}{({v}_{j}^{-}-{v}_{ij})}^{2}}},\text{\hspace{1em}}i=1,2,\dots ,m$$
**Step****5.**- Calculate the closeness coefficients to the ideal solution for each alternative.$$C{C}_{i}=\frac{{S}_{i}^{-}}{{S}_{i}^{-}+{S}_{i}^{+}}$$
**Step****6.**- Rank the alternatives according to $C{C}_{i}$. The bigger $C{C}_{i}$ is, the better alternative ${A}_{i}$ will be.

**Step****1.**- Normalize evaluation matrix R [18].

**Step****2.**- Construct the positive ideal solution (PIS) and the negative ideal solution (NIS) for experts.

**Step****3.**

**Step****4.**- Calculate relative closeness degree of surgical treatments to the PIS for experts.

**Step****5.**- Compute the weights of experts.

**Step****6.**- Compute relative closeness degrees of surgical treatments with respect to the PIS for the group.

**Step****7.**- Rank the surgical treatments by using ${\tau}^{G}({x}_{j})$.

## 5. Empirical Study

#### 5.1. Early Gastric Cancer Surgery Criteria Weight

_{1}to A

_{5}; among them, tumor characteristics $({A}_{1})$ is the most important criterion and medical equipment $({A}_{5})$ is the least important criterion, as determined by experts. The pairwise comparison vectors of the most important and least important criteria are described in Table 4 and Table 5. Table 4 shows that the preference values of the most important criterion $({A}_{1})$ over criterion $({A}_{2})$, criterion $({A}_{3})$, criterion $({A}_{4})$, and criterion $({A}_{5})$ are 3, 4, 6, and 8, respectively. The preference values of criteria $({A}_{1})$, $({A}_{2})$, $({A}_{4})$, and $({A}_{5})$ over the least important criterion $({A}_{5})$ are 8, 7, 6, and 4, respectively, which are described in Table 5. Then, the weight vector of criteria ${w}^{\ast}=({w}_{1}{}^{\ast},{w}_{2}{}^{\ast},{w}_{3}{}^{\ast},{w}_{4}{}^{\ast},{w}_{5}{}^{\ast})$ is computed.

_{1}

^{*}= 0.5333, w

_{2}

^{*}= 0.1778, w

_{3}

^{*}= 0.1333, w

_{4}

^{*}= 0.0889, w

_{5}

^{*}= 0.0667, and ${\xi}^{L\ast}$ = 0. Based on the proposed method, ${\xi}^{L\ast}$ indicates the consistency of the index directly without extra computation. As ${\xi}^{L\ast}$ = 0, we can arrive at complete consistency. So, the criteria weight vector is ${w}^{\ast}=(0.5333,0.1778,0.1333,0.0889,0.0667)$.

#### 5.2. Evaluation Matrix

#### 5.3. Weight of Gastric Cancer Surgery Index

#### 5.4. Selecting Result of Surgical Treatments

_{1}, x

_{2}, x

_{3}, and x

_{4}with respect to the PIS ${x}^{1+}$ for expert ${e}_{1}$ are calculated, respectively, as follows:

_{1}, x

_{2}, x

_{3}, and x

_{4}with respect to the PIS ${x}^{k+}(k=2,3)$ for expert ${e}_{k}(k=1,2,3)$ are calculated, respectively, as follows:

#### 5.5. Sensitivity Analysis

#### 5.6. Comparison Analysis

- (1)
- The proposed model considers both subjective and objective criteria comprehensively in the index system for early gastric cancer, which combines fuzzy theory with quantitative data analysis. This enables the surgical treatment selection to be solved more realistically.
- (2)
- The evaluation information is evaluated from medical records, patient’s sentiment, and experts based on the patient’s conditions, the surgery, and the hospital’s medical status, etc., including crisp numbers, interval numbers, neutrosophic numbers, and probabilistic linguistic term sets; this makes the surgical treatment selection more accurate and reliable.
- (3)
- With the proposed model, the prioritization of alternative surgical treatment methods is determined by using TOPSIS, which is more flexible and simple in solving MGCDM problem [18]. Thus, the proposed selection model of surgical treatments for early gastric cancer patients can provide the most appropriate surgical treatment reliably.

## 6. Conclusions and Future Research

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Evaluation | Very Bad | Bad | Slightly Bad | OK | Slightly Good | Good | Very Good |
---|---|---|---|---|---|---|---|

Sentiment degree | −0.95459 | −0.75 | −0.10607 | 0 | 0.106066 | 0.75 | 0.954594 |

Criteria | Indices | Definition | Index Type |
---|---|---|---|

Tumor characteristics (A_{1}) | Suitability of tumor size (a_{11}) | The degree of suitability that tumor size is suitable for this surgery | Benefit |

Suitability of differentiated degree (a_{12}) | The degree of suitability that tumor differentiation is suitable for this surgery | Benefit | |

Suitability of depth of invasion (a_{13}) | The degree of suitability that the depth of invasion is suitable for this surgery | Benefit | |

Surgical situation (A_{2}) | Complexity of surgery (a_{21}) | The more complicated the operation is, the higher the risk becomes | Cost |

Blood loss (a_{22}) | The amount of bleeding in the surgery | Cost | |

Survival rate (a_{23}) | The survival probability in surgery | Benefit | |

Operating time (a_{24}) | The time spent in surgery | Cost | |

Oncological clearance (a_{25}) | The condition of oncological clearance | Benefit | |

Operative wound (a_{26}) | The wound size of surgery | Cost | |

Surgical outcomes (A_{3}) | Wound infection (a_{31}) | Wound infections after surgery | Cost |

Probability of a cure (a_{32}) | The probability of curing early gastric cancer | Benefit | |

Severity of the complications (a_{33}) | The possibility of complications like wound dehiscence, fever | Cost | |

Severity of the side effects (a_{34}) | The possibility of side effects after surgery | Cost | |

Probability of a recurrence (a_{35}) | The probability of a recurrence because of unsuccessful surgery | Cost | |

Hospital stays (a_{36}) | Length of hospital stay | Cost | |

Recovery time (a_{37}) | The postoperative recovery time | Cost | |

Degree of dysfunction (a_{38}) | The function of gastric system for patients after the surgery | Cost | |

Medical technology (A_{4}) | Medical technical level (a_{41}) | The technical force and medical standards in the hospital | Benefit |

Teamwork Capacity (a_{42}) | The teamwork capacity of medical team | Benefit | |

Medical resources (a_{43}) | Available medical resources in the hospital | Benefit | |

Proficiency (a_{44}) | Skill degree of medical professionals | Benefit | |

Medical equipment (A_{5}) | Advanced equipment (a_{51}) | The performance of medical equipment | Benefit |

Perfection level (a_{52}) | Complete supporting facilities in medical | Benefit | |

Disinfecting technical (a_{53}) | The equipment for disinfection and sterilization | Benefit | |

Emergency facilities (a_{54}) | The perfection of emergency medical facilities | Benefit |

${\mathit{a}}_{\mathit{B}\mathit{w}}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

Consistency Index (max ξ) | 0.00 | 0.44 | 1.00 | 1.63 | 2.30 | 3.00 | 3.73 | 4.47 | 5.23 |

Criteria | A_{1} | A_{2} | A_{3} | A_{4} | A_{5} |
---|---|---|---|---|---|

Best criterion: A_{1} | 1 | 3 | 4 | 6 | 8 |

Criteria | A_{1} | A_{2} | A_{3} | A_{4} | A_{5} |
---|---|---|---|---|---|

Worst criterion: A_{5} | 8 | 7 | 6 | 4 | 1 |

Tumor Characteristics | Tumor Size | Differentiated Degree | Depth of Invasion |
---|---|---|---|

Condition | 6.5 × 4.5 × 2 (cm) | Middle differentiation | Invading serosa |

Indices | Alternatives | |||
---|---|---|---|---|

x_{1} | x_{2} | x_{3} | x_{4} | |

a_{22} | [3.28, 5.36] | [2.58, 5.7] | [25, 80] | [80, 180] |

a_{23} | 0.978 | 0.993 | 0.959 | 0.949 |

a_{24} | [20, 30] | [60, 90] | [270, 302] | [263, 314] |

a_{25} | <0.4280, 0, 0> | <0.6036, 0, 0> | <0.3207, 0, 0> | <0.2671, 0, 0> |

a_{26} | <0.5621, 0, 0> | <0.8523, 0, 0> | <0.8182, 0, 0> | <0.1597, 0, 0> |

a_{35} | 0.0835 | 0.004 | 0.0122 | 0.0135 |

a_{36} | [5.8, 8] | [2, 10] | [9.6, 12] | [8.6, 16.2] |

a_{37} | [13.12, 17.44] | [16.29, 20.06] | [25.69, 31.39] | [41.33, 47.97] |

a_{41} | <0.8112, 0, 0> | <0.6656, 0, 0> | <0.7591, 0, 0> | <0.5927, 0, 0> |

a_{42} | <0.8129, 0, 0> | <0.7995, 0, 0> | <0.7397, 0, 0> | <0.7035, 0, 0> |

a_{43} | <0.6611, 0, 0> | <0.6784, 0, 0> | <0.7706, 0, 0> | <0.7727, 0, 0> |

a_{44} | <0.6469, 0, 0> | <0.6913, 0, 0> | <0.7851, 0, 0> | <0.8134, 0, 0> |

a_{51} | <0.7851, 0, 0> | <0.8626, 0, 0> | <0.7914, 0, 0> | <0.6776, 0, 0> |

a_{52} | <0.6157, 0, 0> | <0.6873, 0, 0> | <0.7133, 0, 0> | <0.7848, 0, 0> |

a_{53} | <0.7648, 0, 0> | <0.7897, 0, 0> | <0.8127, 0, 0> | <0.7876, 0, 0> |

a_{54} | <0.7442, 0, 0> | <0.7533, 0, 0> | <0.7194, 0, 0> | <0.8072, 0, 0> |

Indices | Experts | Alternatives | |||
---|---|---|---|---|---|

x_{1} | x_{2} | x_{3} | x_{1} | ||

a_{11} | e_{1} | {s_{2}(0.5), s_{3}(0.4)} | {s_{3}(0.4), s_{4}(0.3)} | {s_{3}(0.2), s_{4}(0.3), s_{5}(0.5)} | {s_{1}(0.6), s_{2}(0.4)} |

e_{2} | {s_{2}(0.3), s_{3}(0.4)} | {s_{3}(0.4), s_{4}(0.6)} | {s_{3}(0.2), s_{4}(0.3), s_{5}(0.4)} | {s_{1}(0.5), s_{2}(0.3) | |

e_{3} | {s_{2}(0.4), s_{3}(0.5)} | {s_{3}(0.3), s_{4}(0.4)} | {s_{3}(0.3), s_{4}(0.3), s_{5}(0.4)} | {s_{1}(0.6), s_{2}(0.2) | |

a_{12} | e_{1} | {s_{1}(0.3), s_{2}(0.4)} | {s_{1}(0.3), s_{2}(0.2)} | {s_{3}(0.2), s_{4}(0.4), s_{5}(0.4)} | {s_{4}(0.4), s_{5}(0.6)} |

e_{2} | {s_{1}(0.3), s_{2}(0.4), s_{3}(0.2)} | {s_{1}(0.3), s_{2}(0.4)} | {s_{4}(0.4), s_{5}(0.5)} | {s_{3}(0.2), s_{4}(0.4), s_{5}(0.4)} | |

e_{3} | {s_{1}(0.3), s_{2}(0.3), s_{3}(0.2)} | {s_{1}(0.3), s_{2}(0.4), s_{3}(0.1)} | {s_{4}(0.4), s_{5}(0.3), s_{6}(0.2)} | {s_{3}(0.2), s_{4}(0.4), s_{5}(0.3)} | |

a_{13} | e_{1} | {s_{1}(0.3), s_{2}(0.5) } | {s_{1}(0.3), s_{2}(0.7) } | {s_{4}(0.4), s_{5}(0.3), s_{6}(0.3)} | {s_{4}(0.4), s_{5}(0.3), s_{6}(0.2)} |

e_{2} | {s_{1}(0.6), s_{2}(0.4) } | {s_{1}(0.5), s_{2}(0.3) } | {s_{4}(0.4), s_{5}(0.6) } | {s_{4}(0.3), s_{5}(0.4), s_{6}(0.3)} | |

e_{3} | {s_{1}(0.4), s_{2}(0.2), s_{3}(0.1)} | {s_{1}(0.3), s_{2}(0.6) } | {s_{5}(0.3), s_{6}(0.4)} | { s_{5}(0.5), s_{6}(0.2)} | |

a_{21} | e_{1} | {s_{2}(0.1), s_{3}(0.5), s_{4}(0.3)} | {s_{2}(0.1), s_{3}(0.4), s_{4}(0.2)} | {s_{2}(0.1), s_{3}(0.3), s_{4}(0.4)} | {s_{2}(0.2), s_{3}(0.3), s_{4}(0.3)} |

e_{2} | {s_{2}(0.2), s_{3}(0.4), s_{4}(0.3)} | {s_{4}(0.2), s_{5}(0.3), s_{6}(0.1)} | {s_{3}(0.2), s_{4}(0.4), s_{5}(0.1)} | {s_{4}(0.4), s_{5}(0.1)} | |

e_{3} | {s_{4}(0.3), s_{5}(0.2)} | {s_{3}(0.4), s_{4}(0.2), s_{5}(0.3)} | {s_{4}(0.6), s_{5}(0.3)} | {s_{4}(0.3), s_{5}(0.2), s_{6}(0.2)} | |

a_{31} | e_{1} | {s_{0}(0.3), s_{1}(0.3), s_{2}(0.1)} | {s_{0}(0.3), s_{1}(0.3), s_{2}(0.2)} | {s_{1}(0.3), s_{2}(0.3), s_{3}(0.3)} | {s_{1}(0.2), s_{2}(0.3), s_{3}(0.3)} |

e_{2} | {s_{2}(0.1), s_{3}(0.2), s_{4}(0.2)} | {s_{2}(0.2), s_{3}(0.3), s_{4}(0.1)} | {s_{2}(0.1), s_{3}(0.3), s_{4}(0.3)} | {s_{3}(0.3), s_{4}(0.3), s_{5}(0.3)} | |

e_{3} | {s_{0}(0.4), s_{1}(0.3)} | {s_{0}(0.4), s_{1}(0.2)} | {s_{1}(0.3), s_{2}(0.2) } | {s_{2}(0.4), s_{3}(0.3)} | |

a_{32} | e_{1} | {s_{2}(0.1), s_{3}(0.4), s_{4}(0.4)} | {s_{2}(0.1), s_{3}(0.5), s_{4}(0.1)} | {s_{3}(0.1), s_{4}(0.4), s_{5}(0.4)} | {s_{2}(0.2), s_{3}(0.1), s_{4}(0.3)} |

e_{2} | {s_{3}(0.2), s_{4}(0.4), s_{5}(0.3)} | {s_{3}(0.5), s_{4}(0.1), s_{5}(0.3)} | {s_{4}(0.4), s_{5}(0.3), s_{6}(0.2)} | {s_{4}(0.2), s_{5}(0.3), s_{6}(0.3)} | |

e_{3} | {s_{3}(0.4), s_{4}(0.4), s_{5}(0.2)} | {s_{3}(0.6), s_{4}(0.3) } | {s_{4}(0.5), s_{5}(0.4)} | {s_{5}(0.4), s_{6}(0.3)} | |

a_{33} | e_{1} | {s_{1}(0.5), s_{2}(0.1), s_{3}(0.2)} | {s_{1}(0.5), s_{2}(0.2), s_{3}(0.2)} | {s_{1}(0.2), s_{2}(0.3), s_{3}(0.2)} | {s_{1}(0.3), s_{2}(0.2), s_{3}(0.1)} |

e_{2} | {s_{2}(0.1), s_{3}(0.2), s_{4}(0.1)} | {s_{1}(0.5), s_{2}(0.2)} | {s_{2}(0.4), s_{3}(0.2)} | {s_{2}(0.3), s_{3}(0.2), s_{4}(0.4)} | |

e_{3} | {s_{1}(0.6), s_{2}(0.2)} | {s_{2}(0.1), s_{3}(0.2), s_{4}(0.2)} | {s_{2}(0.3), s_{3}(0.2), s_{4}(0.2)} | {s_{4}(0.2), s_{5}(0.4)} | |

a_{34} | e_{1} | {s_{1}(0.3), s_{2}(0.3), s_{3}(0.4)} | {s_{1}(0.5), s_{2}(0.2), s_{3}(0.3)} | {s_{1}(0.4), s_{2}(0.3), s_{3}(0.2)} | {s_{2}(0.2), s_{3}(0.2), s_{4}(0.6)} |

e_{2} | {s_{2}(0.3), s_{3}(0.2), s_{4}(0.4)} | {s_{2}(0.4), s_{3}(0.2), s_{4}(0.1)} | {s_{2}(0.1), s_{3}(0.3), s_{4}(0.4)} | {s_{3}(0.3), s_{4}(0.2), s_{5}(0.2)} | |

e_{3} | {s_{3}(0.3), s_{4}(0.3)} | {s_{3}(0.3), s_{4}(0.1)} | {s_{3}(0.4), s_{4}(0.2)} | {s_{3}(0.2), s_{4}(0.4)} | |

a_{38} | e_{1} | {s_{0}(0.2), s_{1}(0.3), s_{2}(0.1)} | {s_{1}(0.3), s_{2}(0.3)} | {s_{1}(0.2), s_{2}(0.3), s_{3}(0.3)} | {s_{1}(0.2), s_{2}(0.2), s_{3}(0.4)} |

e_{2} | {s_{2}(0.1), s_{3}(0.2), s_{4}(0.3)} | {s_{0}(0.2), s_{1}(0.2), s_{2}(0.2)} | {s_{2}(0.4), s_{3}(0.4)} | {s_{2}(0.2), s_{3}(0.4), s_{4}(0.2} | |

e_{3} | {s_{1}(0.4), s_{2}(0.2)} | {s_{2}(0.2), s_{3}(0.2), s_{4}(0.2} | {s_{3}(0.3), s_{4}(0.2), s_{5}(0.2)} | {s_{2}(0.3), s_{3}(0.6)} |

Indices | Alternatives | |||
---|---|---|---|---|

x_{1} | x_{2} | x_{3} | x_{4} | |

a_{22} | [0.9702, 0.9818] | [0.9683, 0.9857] | [0.5556, 0.8611] | [0, 0.5556] |

a_{23} | 0.659090909 | 1 | 0.227272727 | 0 |

a_{24} | [0.9045, 0.9363] | [0.7134, 0.8089] | [0.0382, 0.1401] | [0, 0.1624] |

a_{25} | <0.4280, 0, 0> | <0.6036, 0, 0> | <0.3207, 0, 0> | <0.2671, 0, 0> |

a_{26} | <0.5621, 0, 0> | <0.8523, 0, 0> | <0.8182, 0, 0> | <0.1597, 0, 0> |

a_{35} | 0 | 1 | 0.896855346 | 0.880503145 |

a_{36} | [0.5062, 0.6420] | [0.3827, 0.8765] | [0.2593, 0.4074] | [0, 0.4691] |

a_{37} | [0.6364, 0.7265] | [0.5631, 0.6604] | [0.3456, 0.4645] | [0, 0.1384] |

a_{41} | <0.8112, 0, 0> | <0.6656, 0, 0> | <0.7591, 0, 0> | <0.5927, 0, 0> |

a_{42} | <0.8129, 0, 0> | <0.7995, 0, 0> | <0.7397, 0, 0> | <0.7035, 0, 0> |

a_{43} | <0.6611, 0, 0> | <0.6784, 0, 0> | <0.7706, 0, 0> | <0.7727, 0, 0> |

a_{44} | <0.6469, 0, 0> | <0.6913, 0, 0> | <0.7851, 0, 0> | <0.8134, 0, 0> |

a_{51} | <0.7851, 0, 0> | <0.8626, 0, 0> | <0.7914, 0, 0> | <0.6776, 0, 0> |

a_{52} | <0.6157, 0, 0> | <0.6873, 0, 0> | <0.7133, 0, 0> | <0.7848, 0, 0> |

a_{53} | <0.7648, 0, 0> | <0.7897, 0, 0> | <0.8127, 0, 0> | <0.7876, 0, 0> |

a_{54} | <0.7442, 0, 0> | <0.7533, 0, 0> | <0.7194, 0, 0> | <0.8072, 0, 0> |

Indices | Alternatives | |||
---|---|---|---|---|

x_{1} | x_{2} | x_{3} | x_{4} | |

a_{11} | {s_{0.95}, s_{1.57}, s_{0}} | {s_{1.4}, s_{2.13}, s_{0}} | {s_{0.72}, s_{1.24}, s_{2.25}} | {s_{0.66}, s_{0.68}, s_{0}} |

a_{12} | {s_{0.38}, s_{0.93}, s_{0.48}} | {s_{0.47}, s_{0.98}, s_{0.13}} | {s_{1.39}, s_{2.02}, s_{1.11}} | {s_{0.95}, s_{2.13}, s_{1.22}} |

a_{13} | {s_{0.51}, s_{0.88}, s_{0.14}} | {s_{0.42}, s_{1.16}, s_{0}} | {s_{1.78}, s_{2.64}, s_{0.65}} | {s_{2.18}, s_{1.8}, s_{1.04}} |

a_{21} | {s_{0.84}, s_{1.14}, s_{0.44}} | {s_{0.86}, s_{0.88}, s_{0.33}} | {s_{0.91}, s_{0.86}, s_{0.38}} | {s_{1.17}, s_{0.53}, s_{0.25}} |

a_{31} | {s_{2.27}, s_{1.83}, s_{0.45}} | {s_{1.92}, s_{1.68}, s_{0.45}} | {s_{1.74}, s_{1.41}, s_{0.63}} | {s_{1.46}, s_{1.25}, s_{0.44}} |

a_{32} | {s_{0.69}, s_{1.57}, s_{1.48}} | {s_{1.32}, s_{1.31}, s_{0.74}} | {s_{1.44}, s_{1.89}, s_{1.19}} | {s_{1.50}, s_{1.65}, s_{1.43}} |

a_{33} | {s_{2.62}, s_{1.01}, s_{0.42}} | {s_{2.38}, s_{1.08}, s_{0.49}} | {s_{1.94}, s_{1.18}, s_{0.48}} | {s_{1.49}, s_{0.82}, s_{0.47}} |

a_{34} | {s_{1.44}, s_{0.95}, s_{0.7}} | {s_{2.34}, s_{0.72}, s_{0.39}} | {s_{1.59}, s_{1.03}, s_{0.55}} | {s_{1.02}, s_{0.84}, s_{0.50}} |

a_{38} | {s_{2.00}, s_{1.60}, s_{0.56}} | {s_{1.93}, s_{1.55}, s_{0.68}} | {s_{1.51}, s_{1.19}, s_{0.47}} | {s_{1.19}, s_{1.50}, s_{0.67}} |

Criteria | Weights | Indices | Weights |
---|---|---|---|

A_{1} | 0.5333 | a_{11} | 0.263315 |

a_{12} | 0.319028 | ||

a_{13} | 0.417657 | ||

A_{2} | 0.1778 | a_{21} | 0.153238 |

a_{22} | 0.023316 | ||

a_{23} | 0.270952 | ||

a_{24} | 0.06846 | ||

a_{25} | 0.194089 | ||

a_{26} | 0.289945 | ||

A_{3} | 0.1333 | a_{31} | 0.205559 |

a_{32} | 0.228812 | ||

a_{33} | 0.172388 | ||

a_{34} | 0.108363 | ||

a_{35} | 0.088117 | ||

a_{36} | 0.003831 | ||

a_{37} | 0.007015 | ||

a_{38} | 0.185916 | ||

A_{4} | 0.0889 | a_{41} | 0.241684 |

a_{42} | 0.26108 | ||

a_{43} | 0.246315 | ||

a_{44} | 0.250921 | ||

A_{5} | 0.0667 | a_{51} | 0.257649 |

a_{52} | 0.231559 | ||

a_{53} | 0.260798 | ||

a_{54} | 0.249994 |

Indices | Experts | Alternatives | |||
---|---|---|---|---|---|

x_{1} | x_{2} | x_{3} | x_{4} | ||

a_{11} | e_{1} | {s_{2}(0.56), s_{3}(0.44), s_{2}(0)} | {s_{3}(0.57), s_{4}(0.43), s_{3}(0)} | {s_{3}(0.2), s_{4}(0.3), s_{5}(0.5)} | {s_{1}(0.6), s_{2}(0.4), s_{1}(0)} |

e_{2} | {s_{2}(0.43), s_{3}(0.57), s_{2}(0)} | {s_{3}(0.4), s_{4}(0.6), s_{3}(0)} | {s_{3}(0.22), s_{4}(0.33), s_{5}(0.45)} | {s_{1}(0.63), s_{2}(0.37), s_{1}(0)} | |

e_{3} | {s_{2}(0.44), s_{3}(0.56), s_{2}(0)} | {s_{3}(0.43), s_{4}(0.57), s_{3}(0)} | {s_{3}(0.3), s_{4}(0.3), s_{5}(0.4)} | {s_{1}(0.75), s2(0.25), s_{1}(0)} | |

a_{12} | e_{1} | {s_{1}(0.43), s_{2}(0.57), s_{1}(0)} | {s_{1}(0.6), s_{2}(0.4), s_{1}(0)} | {s_{3}(0.2), s_{4}(0.4), s_{5}(0.4)} | {s_{4}(0.4), s_{5}(0.6), s_{4}(0)} |

e_{2} | {s_{1}(0.33), s_{2}(0.45), s_{3}(0.22)} | {s_{1}(0.43), s_{2}(0.57), s_{1}(0)} | {s_{4}(0.44), s_{5}(0.56), s_{4}(0)} | {s_{3}(0.2), s_{4}(0.4), s_{5}(0.4)} | |

e_{3} | {s_{1}(0.37), s_{2}(0.37), s_{3}(0.26)} | {s_{1}(0.37), s_{2}(0.5), s_{3}(0.13)} | {s_{4}(0.45), s_{5}(0.33), s_{6}(0.22)} | {s_{3}(0.22), s_{4}(0.45), s_{5}(0.33)} | |

a_{13} | e_{1} | {s_{1}(0.37), s_{2}(0.63), s_{1}(0)} | {s_{1}(0.3), s_{2}(0.7), s_{1}(0)} | {s_{4}(0.4), s_{5}(0.3), s_{6}(0.3)} | {s_{4}(0.45), s_{5}(0.33), s_{6}(0.22)} |

e_{2} | {s_{1}(0.6), s_{2}(0.4), s_{1}(0)} | {s_{1}(0.63), s_{2}(0.37), s_{1}(0)} | {s_{4}(0.4), s_{5}(0.6), s_{4}(0)} | {s_{4}(0.3), s_{5}(0.4), s_{6}(0.3)} | |

e_{3} | {s_{1}(0.57), s_{2}(0.29), s_{3}(0.14)} | {s_{1}(0.33), s_{2}(0.67), s_{1}(0)} | {s_{5}(0.43), s_{6}(0.57), s_{5}(0)} | {s_{5}(0.71), s_{6}(0.29), s_{5}(0)} | |

a_{21} | e_{1} | {s_{4}(0.11), s_{3}(0.56), s_{2}(0.33)} | {s_{4}(0.14), s_{3}(0.57), s_{2}(0.29)} | {s_{4}(0.13), s_{3}(0.37), s_{2}(0.5)} | {s_{4}(0.26), s_{3}(0.37), s_{2}(0.37)} |

e_{2} | {s_{4}(0.22), s_{3}(0.45), s_{2}(0.33)} | {s_{2}(0.33), s_{1}(0.5), s_{0}(0.17)} | {s_{3}(0.29), s_{2}(0.57), s_{1}(0.14)} | {s_{2}(0.8), s_{1}(0.2), s_{1}(0)} | |

e_{3} | {s_{2}(0.6), s_{1}(0.4), s_{1}(0)} | {s_{3}(0.45), s_{2}(0.22), s_{1}(0.33)} | {s_{2}(0.67), s_{1}(0.33), s_{1}(0)} | {s_{2}(0.43), s_{1}(0.28), s_{0}(0.29)} | |

a_{31} | e_{1} | {s_{6}(0.43), s_{5}(0.43), s_{4}(0.14)} | {s_{6}(0.37), s_{5}(0.38), s_{4}(0.25)} | {s_{5}(0.33), s_{4}(0.33), s_{3}(0.34)} | {s_{5}(0.22), s_{4}(0.45), s_{3}(0.33)} |

e_{2} | {s_{4}(0.2), s_{3}(0.4), s_{2}(0.4)} | {s_{4}(0.33), s_{3}(0.5), s_{2}(0.17)} | {s_{4}(0.14), s_{3}(0.43), s_{2}(0.43)} | {s_{3}(0.33), s_{2}(0.33), s_{1}(0.34)} | |

e_{3} | {s_{6}(0.57), s_{5}(0.43), s_{5}(0)} | {s_{6}(0.67), s_{5}(0.33), s_{5}(0)} | {s_{5}(0.6), s_{4}(0.4), s_{4}(0)} | {s_{4}(0.57), s_{3}(0.43), s_{3}(0)} | |

a_{32} | e_{1} | {s_{2}(0.11), s_{3}(0.44), s_{4}(0.45)} | {s_{2}(0.14), s_{3}(0.72), s_{4}(0.14)} | {s_{3}(0.11), s_{4}(0.44), s_{5}(0.45)} | {s_{2}(0.33), s_{3}(0.17), s_{4}(0.5)} |

e_{2} | {s_{3}(0.22), s_{4}(0.45), s_{5}(0.33)} | {s_{3}(0.56), s_{4}(0.11), s_{5}(0.33)} | {s_{4}(0.45), s_{5}(0.33), s_{6}(0.22)} | {s_{4}(0.25), s_{5}(0.37), s_{6}(0.38)} | |

e_{3} | {s_{3}(0.4), s_{4}(0.4), s_{5}(0.2)} | {s_{3}(0.67), s_{4}(0.33), s_{3}(0)} | {s_{4}(0.55), s_{5}(0.45), s_{4}(0)} | {s_{5}(0.57), s_{6}(0.43), s_{5}(0)} | |

a_{33} | e_{1} | {s_{5}(0.62), s_{4}(0.13), s_{3}(0.25)} | {s_{5}(0.56), s_{4}(0.22), s_{3}(0.22)} | {s_{5}(0.29), s_{4}(0.42), s_{3}(0.29)} | {s_{5}(0.5), s_{4}(0.33), s_{3}(0.17)} |

e_{2} | {s_{4}(0.25), s_{3}(0.5), s_{2}(0.25)} | {s_{5}(0.71), s_{4}(0.29), s_{4}(0)} | {s_{4}(0.67), s_{3}(0.33), s_{3}(0)} | {s_{4}(0.33), s_{3}(0.22), s_{2}(0.45)} | |

e_{3} | {s_{5}(0.75), s_{4}(0.25), s_{4}(0)} | {s_{4}(0.2), s_{3}(0.4), s_{2}(0.4)} | {s_{4}(0.42), s_{3}(0.29), s_{2}(0.29)} | {s_{2}(0.33), s_{1}(0.47), s_{1}(0)} | |

a_{34} | e_{1} | {s_{5}(0.3), s_{4}(0.3), s_{3}(0.4)} | {s_{5}(0.5), s_{4}(0.2), s_{3}(0.3)} | {s_{5}(0.45), s_{4}(0.33), s_{3}(0.22)} | {s_{4}(0.2), s_{3}(0.2), 2(0.6)} |

e_{2} | {s_{4}(0.33), s_{3}(0.22), s_{2}(0.45)} | {s_{4}(0.57), s_{3}(0.29), s_{2}(0.14)} | {s_{4}(0.13), s_{3}(0.37), s_{2}(0.5)} | {s_{3}(0.42), s_{2}(0.29), s_{1}(0.29)} | |

e_{3} | {s_{3}(0.5), s_{2}(0.5), s_{2}(0)} | {s3(0.75), s_{2}(0.25), s_{2}(0)} | {s_{3}(0.67), s_{2}(0.33), s_{2}(0)} | {s_{3}(0.33), s_{2}(0.67), s_{2}(0)} | |

a_{38} | e_{1} | {s_{6}(0.33), s_{5}(0.5), s_{4}(0.17)} | {s_{5}(0.5), s_{4}(0.5), s_{4}(0)} | {s_{5}(0.25), s_{4}(0.37), s_{3}(0.38)} | {s_{5}(0.25), s_{4}(0.25), s_{3}(0.5)} |

e_{2} | {s_{4}(0.17), s_{3}(0.33), s_{2}(0.5)} | {s_{6}(0.33), s_{5}(0.33), s_{4}(0.34)} | {s_{4}(0.5), s_{3}(0.5), s_{3}(0)} | {s_{4}(0.25), s_{3}(0.5), s_{2}(0.25)} | |

e_{3} | {s_{5}(0.67), s_{4}(0.33), s_{4}(0)} | {s_{4}(0.33), s_{3}(0.33), s_{2}(0.34)} | {s_{3}(0.43), s_{2}(0.29), s_{1}(0.28)} | {s_{4}(0.33), s_{3}(0.67), s_{3}(0)} |

Different Values of $\mathit{\epsilon}$ | The Ranking of Surgical Treatments | Ranking Orders | |||
---|---|---|---|---|---|

x_{1} | x_{2} | x_{3} | x_{4} | ||

$\epsilon =0$ | 0 | 0.067280513 | 1 | 0.704819514 | ${x}_{1}\prec {x}_{2}\prec {x}_{4}\prec {x}_{3}$ |

$\epsilon =0.3$ | 0 | 0.067280513 | 1 | 0.704819514 | ${x}_{1}\prec {x}_{2}\prec {x}_{4}\prec {x}_{3}$ |

$\epsilon =0.5$ | 0 | 0.067280513 | 1 | 0.704819514 | ${x}_{1}\prec {x}_{2}\prec {x}_{4}\prec {x}_{3}$ |

$\epsilon =0.7$ | 0 | 0.067280513 | 1 | 0.704819514 | ${x}_{1}\prec {x}_{2}\prec {x}_{4}\prec {x}_{3}$ |

$\epsilon =1$ | 0 | 0.067280513 | 1 | 0.704819514 | ${x}_{1}\prec {x}_{2}\prec {x}_{4}\prec {x}_{3}$ |

Different Values of $\mathit{\lambda}$ | The Ranking of Surgical Treatments | Ranking Orders | |||
---|---|---|---|---|---|

x_{1} | x_{2} | x_{3} | x_{4} | ||

$\lambda =0.2$ | 0 | 0.069246088 | 1 | 0.703826118 | ${x}_{1}\prec {x}_{2}\prec {x}_{4}\prec {x}_{3}$ |

$\lambda =0.4$ | 0.131619756 | 0 | 1 | 0.804240841 | ${x}_{2}\prec {x}_{1}\prec {x}_{4}\prec {x}_{3}$ |

$\lambda =0.5$ | 0 | 0.067280513 | 1 | 0.704819514 | ${x}_{1}\prec {x}_{2}\prec {x}_{4}\prec {x}_{3}$ |

$\lambda =0.6$ | 0 | 0.531275419 | 1 | 0.213842101 | ${x}_{1}\prec {x}_{4}\prec {x}_{2}\prec {x}_{3}$ |

$\lambda =0.8$ | 0 | 0.510492538 | 1 | 0.219962121 | ${x}_{1}\prec {x}_{4}\prec {x}_{2}\prec {x}_{3}$ |

Alternatives | Heterogeneous TODIM | Heterogeneous VIKOR | ||||
---|---|---|---|---|---|---|

${\epsilon}_{i}$ | Ranking | ${S}_{i}$ | ${R}_{i}$ | ${Q}_{i}$ | Ranking | |

${x}_{1}$ | 0.131573 | 3 | 0.702564564 | 0.222736429 | 1 | 4 |

${x}_{2}$ | 0.324262 | 2 | 0.62637026 | 0.216910241 | 0.901839 | 3 |

${x}_{3}$ | 1 | 1 | 0.240544451 | 0.037226276 | 0 | 1 |

${x}_{4}$ | 0 | 4 | 0.516377929 | 0.138206602 | 0.570677 | 2 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, D.-P.; He, J.-Q.; Cheng, P.-F.; Wang, J.-Q.; Zhang, H.-Y.
A Novel Selection Model of Surgical Treatments for Early Gastric Cancer Patients Based on Heterogeneous Multicriteria Group Decision-Making. *Symmetry* **2018**, *10*, 223.
https://doi.org/10.3390/sym10060223

**AMA Style**

Li D-P, He J-Q, Cheng P-F, Wang J-Q, Zhang H-Y.
A Novel Selection Model of Surgical Treatments for Early Gastric Cancer Patients Based on Heterogeneous Multicriteria Group Decision-Making. *Symmetry*. 2018; 10(6):223.
https://doi.org/10.3390/sym10060223

**Chicago/Turabian Style**

Li, Dan-Ping, Ji-Qun He, Peng-Fei Cheng, Jian-Qiang Wang, and Hong-Yu Zhang.
2018. "A Novel Selection Model of Surgical Treatments for Early Gastric Cancer Patients Based on Heterogeneous Multicriteria Group Decision-Making" *Symmetry* 10, no. 6: 223.
https://doi.org/10.3390/sym10060223