# Robot Evaluation and Selection with Entropy-Based Combination Weighting and Cloud TODIM Approach

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Basic Concepts

#### 3.1. Cloud Model Theory

**Definition**

**1**

**.**Supposing a qualitative concept T defined on a universe of discourse U, let $x\left(x\in U\right)$ be a random instantiation of the concept T and $y\in [0,1]$ be the certainty degree of x belonging to T, which corresponds to a random number with a stable tendency. Then the distribution of x in the universe U is called a cloud, and the cloud drop is denoted as (x, y).

**Definition**

**2**

**.**The characteristics of a cloud y are depicted by expectation Ex, entropy En, and hyper entropy He. Here, Ex is the center value of the qualitative concept domain, En measures the randomness and fuzziness of the qualitative concept, and He reflects the dispersion of the cloud drops and the uncertain degree of the membership function. Based on the three numerical characteristics, a cloud can be described as $\tilde{y}=\left(Ex,En,He\right)$.

**Definition**

**3.**

- (1)
- ${\tilde{y}}_{1}+{\tilde{y}}_{2}=\left(E{x}_{1}+E{x}_{2},\sqrt{E{n}_{1}^{2}+E{n}_{2}^{2}},\sqrt{H{e}_{1}^{2}+H{e}_{2}^{2}}\right);$
- (2)
- ${\tilde{y}}_{1}\times {\tilde{y}}_{2}=\left(E{x}_{1}E{x}_{2},\sqrt{{\left(E{n}_{1}E{x}_{2}\right)}^{2}+{\left(E{n}_{2}E{x}_{1}\right)}^{2}},\sqrt{{\left(H{e}_{1}E{x}_{2}\right)}^{2}+{\left(H{e}_{2}E{x}_{1}\right)}^{2}}\right);$
- (3)
- $\lambda {\tilde{y}}_{1}^{}=\left(\lambda E{x}_{1},\sqrt{\lambda}E{n}_{1},\sqrt{\lambda}H{e}_{1}\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\lambda >0;$
- (4)
- ${\tilde{y}}_{1}^{\lambda}=\left(E{x}_{1}^{\lambda},\sqrt{\lambda}{\left(E{x}_{1}\right)}^{\lambda -1}E{n}_{1},\sqrt{\lambda}{\left(E{x}_{1}\right)}^{\lambda -1}H{e}_{1}\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\lambda >0$.

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7**

**.**Let ${\tilde{y}}_{i}=\left(E{x}_{i},E{n}_{i},H{e}_{i}\right)\left(i=1,2,\dots ,n\right)$ be a collection of normal clouds in the universe U, and $\omega =\left({\omega}_{1},{\omega}_{2},\dots ,{\omega}_{n}\right)$ be an associated weight vector with ${\omega}_{j}\in \left[0,1\right],and\hspace{0.17em}{\displaystyle {\sum}_{j=1}^{n}{\omega}_{j}}=1$, then the cloud hybrid averaging (CHA) operator is defined as:

#### 3.2. Conversion between Linguistic Terms and Clouds

**Definition**

**8.**

- (1)
- Negation operator: Neg (s
_{i}) = s_{j}such that $j=g-i$; - (2)
- The set is ordered: s
_{i}> s_{j}, if i > j; - (3)
- Max operator: max (s
_{i}, s_{j}) = s_{i}, if ${s}_{i}\ge {s}_{j}$.

**Definition**

**9.**

_{min}, X

_{max}] is an effective domain, then g + 1 basic clouds can be generated corresponding to the linguistic values ${s}_{i}\left(i=0,1,\dots ,g\right)$, which are denoted as ${\tilde{y}}_{0}=\left(E{x}_{0},E{n}_{0},H{e}_{0}\right),$ ${\tilde{y}}_{1}=\left(E{x}_{1},E{n}_{1},H{e}_{1}\right),\dots ,$ ${\tilde{y}}_{g}=\left(E{x}_{g},E{n}_{g},H{e}_{g}\right)$, respectively.

_{min}, X

_{max}]. By applying a golden radio method, seven basic clouds can be produced and their numerical characteristics are shown below:

_{3}can be designated in advance by decision makers.

_{3}= 0.1, six basic clouds can be computed as below for the linguistic term set S [44]:

**Definition**

**10.**

_{i}values in [0, 1] denoting the possible membership degrees of the element ${s}_{i}\in S$ to the set LH.

**Definition**

**11**

**.**Let $S=\left\{{s}_{0},{s}_{1},\dots ,{s}_{g}\right\}$ be a linguistic term set and [X

_{min}, X

_{max}] is an effective domain, the corresponding cloud ${\tilde{y}}_{LH}=\left(E{x}_{LH},E{n}_{LH},H{e}_{LH}\right)$ of the LHFS $LH=\left\{{s}_{i},lh\left({s}_{i}\right)|{s}_{i}\in S\right\}$ can be computed by

## 4. The Proposed Robot Selection Approach

#### 4.1. Determine Robot Assessments

_{k}, where ${d}_{ij}^{k}$ denotes the judgement of alternative A

_{i}against C

_{j}assigned by ${\mathrm{DM}}_{k}$. Because decision makers from different working backgrounds have dissimilar experience and knowledge, they are given different weights ${\lambda}_{k}(k=1,2,\dots ,l$ with ${\sum}_{k=1}^{l}{\lambda}_{k}}=1$) in the robot selection process. Next, the cloud model is implemented to address the decision makers’ linguistic assessments of robots against each criterion.

**Step 1:**Establish the normalized linguistic decision matrix ${R}^{k}={\left({r}_{ij}^{k}\right)}_{m\times n}$

**Step 2:**Obtain the cloud decision matrix ${\tilde{Y}}_{k}$

**Step 3:**Construct the collective cloud decision matrix $\tilde{Y}$

_{i}on criterion C

_{j}, i.e., ${\tilde{y}}_{ij}^{}$, is calculated by

#### 4.2. Determine Criteria Weights

**Step 4:**Determine the subjective criteria weights

_{j}given by decision maker DM

_{k}to indicate its importance in the ranking of robots. The corresponding cloud weights ${\tilde{w}}_{j}^{k}\left(j=1,2,\dots ,n\right)$ are aggregated to find the collective cloud weights $\tilde{w}={\left({\tilde{w}}_{j}\right)}_{1\times n}$ by using the CHA operator. Then, the subjective weight of each evaluation criterion is computed by

**Step 5:**Compute the objective criteria weights

_{j}is the entropy of the projected results of the criterion C

_{j}, which can be obtained by

**Step 6:**Compute the combination criteria weights

#### 4.3. Define the Ranking of Robots

**Step 7:**Compute the relative weight of C

_{j}with respect to the reference criterion C

_{r}by

_{r}is the criterion associated with w

_{r}.

**Step 8:**Determine the domination degree of A

_{i}over A

_{p}under C

_{j}, i.e.,

**Step 9:**Obtain the overall domination degree of A

_{i}over A

_{p}by

**Step 10:**Acquire the global value of alternative A

_{i}over the other alternatives by using the following equation:

## 5. Case Study

#### 5.1. Application

_{1}, A

_{2}, A

_{3}, and A

_{4}are left for further assessment. Besides, seven evaluation criteria are considered for selecting the most appropriate robot: Inconsistency with infrastructure (C

_{1}), Man-machine interface (C

_{2}), Programming flexibility (C

_{3}), Vendor’s service contract (C

_{4}), Supporting channel partner’s performance (C

_{5}), Compliance (C

_{6}), and Stability (C

_{7}). All these criteria are benefit criteria except C

_{1}, which is a cost criterion.

_{1}, DM

_{2},…, DM

_{5}) is established for the evaluation and selection of the most suitable robot. The decision makers’ weights are set as 0.20, 0.30, 0.10, 0.25, and 0.15, respectively, due to their differentiated knowledge and backgrounds. According to the materials and data concerning the considered robots, a seven-point linguistic term set S is adopted by the experts to evaluate the given robots and the criteria importance weights, i.e.,

**Stage 1.**Evaluate the performance of robots

_{1}is cost type, the linguistic ratings of the alternatives about C

_{1}are normalized and listed in Table 3 and the normalized linguistic decision matrix ${R}^{k}={\left({r}_{ij}^{k}\right)}_{4\times 7}\left(k=1,2,\dots ,5\right)$ can be constructed accordingly.

_{3}= 0.1 in the computation.

**Stage 2.**Calculate the criteria weights

_{ij}and E

_{j}are derived by Equations (10) and (11), which are furnished in Table 6, and the objective criteria weights are determined via Equation (9) as: ${w}_{1}^{O}=0.435,{w}_{2}^{O}=0.147,{w}_{3}^{O}=0.082,{w}_{4}^{O}=0.083,{w}_{5}^{O}=0.078,{w}_{6}^{O}=0.065,{w}_{7}^{O}=0.109$.

**Stage 3.**Acquire the ranking orders of alternatives

_{r}can be calculated through Equation (13) as:

_{1}with the top global value is the best robot for the considered case study, and the ranking of the four robots is ${A}_{1}\succ {A}_{4}\succ {A}_{3}\succ {A}_{2}.$

#### 5.2. Sensitivity Analysis

#### 5.3. Comparison Analysis

_{1}. This shows the verification and validation of the proposed approach. However, compared with other robot selection methods, the presented cloud TODIM model has the following merits:

- The performance ratings of robots are evaluated in linguistic expressions and the hesitancy and inconsistency in the decision makers’ evaluations on robots can be well represented. This allows decision makers to define their opinions more realistically and make the robot selection easier to perform.
- Based on the cloud model, the new approach can not only reflect average level but also the vagueness and randomness of the evaluation criteria. Moreover, the aggregation of performance information utilizing the CHA operator can reflect the importance weights of experts and simultaneously minimize the effect of biased assessments on the ranking results.
- We consider both subjective and objective weights of criteria in ranking the alternative robots, and the combination criteria weights are computed directly without the need to determine the weight coefficient between subjective and objective weights in advance. This makes the ranking results more accurate and theoretically reasonable.
- By applying an extended TODIM method, the presented approach takes the behavioral characteristics of decision makers (e.g., reference dependence and loss aversion) into consideration in determining the ranking of robots. Therefore, the robot selection approach proposed in this paper is more realistic in practical applications.

## 6. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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Robots | Experts | Criteria | ||||||
---|---|---|---|---|---|---|---|---|

C_{1} | C_{2} | C_{3} | C_{4} | C_{5} | C_{6} | C_{7} | ||

A_{1} | DM_{1} | s_{0} | {(s_{6}, 0.6, 0.9)} | s_{3} | s_{4} | s_{4} | s_{6} | s_{6} |

DM_{2} | {(s_{0}, 0.7)} | s_{6} | s_{4} | s_{4} | s_{5} | s_{5} | s_{5} | |

DM_{3} | s_{1} | s_{5} | s_{3} | s_{4} | s_{4} | {(s_{5}, s_{6})} | s_{6} | |

DM_{4} | s_{1} | s_{6} | s_{3} | s_{5} | s_{4} | s_{5} | s_{6} | |

DM_{5} | s_{1} | s_{5} | {(s_{3}, 0.8), (s_{4}, 0.9)} | s_{5} | {(s_{5}, 0.6, 0.8)} | s_{6} | {(s_{5}, 0.7), (s_{6}, 0.6, 0.9)} | |

A_{2} | DM_{1} | s_{6} | s_{1} | s_{2} | s_{3} | s_{5} | s_{4} | {(s_{3}, 0.7, 0.9)} |

DM_{2} | s_{6} | s_{2} | s_{1} | s_{2} | s_{5} | s_{3} | s_{2} | |

DM_{3} | {(s_{5}, s_{6})} | {(s_{1}, 0.6, 0.8)} | s_{1} | {(s_{3}, 0.6)} | s_{4} | s_{3} | s_{3} | |

DM_{4} | {(s_{5}, 0.5, 0.8)} | s_{2} | {(s_{2}, 0.8)} | s_{2} | s_{4} | s_{5} | s_{3} | |

DM_{5} | s_{5} | s_{2} | s_{2} | s_{3} | s_{5} | s_{4} | s_{2} | |

A_{3} | DM_{1} | s_{2} | s_{3} | s_{5} | s_{5} | s_{3} | {(s_{4}, 0.5, 0.7)} | s_{3} |

DM_{2} | s_{1} | s_{2} | s_{4} | s_{4} | s_{2} | s_{3} | s_{2} | |

DM_{3} | s_{2} | {(s_{3}, 0.8)} | {(s_{5}, 0.7)} | s_{5} | s_{3} | s_{4} | s_{2} | |

DM_{4} | s_{2} | s_{3} | s_{5} | s_{5} | s_{2} | s_{3} | s_{3} | |

DM_{5} | {(s_{0}, 0.6, 0.8), (s_{1}, 0.7)} | {(s_{4}, 0.3, 0.5, 0.8)} | s_{4} | s_{4} | {(s_{3}, 0.7), (s_{4}, 0.6)} | s_{4} | s_{3} | |

A_{4} | DM_{1} | s_{1} | s_{4} | s_{4} | s_{3} | s_{6} | s_{5} | s_{5} |

DM_{2} | s_{0} | s_{5} | s_{5} | s_{2} | s_{6} | s_{5} | s_{6} | |

DM_{3} | s_{1} | s_{4} | {(s_{4}, s_{5})} | {(s_{2}, 0.4), (s_{3}, 0.7), (s_{4}, 0.4)} | s_{6} | s_{4} | {(s_{6}, 0.6)} | |

DM_{4} | s_{1} | s_{4} | s_{4} | s_{1} | s_{5} | s_{4} | s_{5} | |

DM_{5} | {(s_{0}, 0.7, 0.8)} | s_{5} | s_{5} | s_{2} | s_{5} | s_{5} | s_{5} |

Experts | Criteria | ||||||
---|---|---|---|---|---|---|---|

C_{1} | C_{2} | C_{3} | C_{4} | C_{5} | C_{6} | C_{7} | |

DM_{1} | s_{5} | s_{3} | s_{5} | s_{4} | s_{5} | s_{4} | s_{5} |

DM_{2} | s_{6} | s_{3} | s_{4} | s_{6} | s_{4} | s_{5} | s_{4} |

DM_{3} | s_{5} | s_{4} | s_{4} | s_{4} | s_{4} | s_{5} | s_{5} |

DM_{4} | s_{6} | s_{4} | s_{3} | s_{6} | s_{4} | s_{5} | s_{4} |

DM_{5} | s_{6} | s_{3} | s_{3} | s_{6} | s_{5} | s_{4} | s_{4} |

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

A_{1} | A_{2} | A_{3} | A_{4} | |

DM_{1} | S_{6} | s_{0} | s_{4} | s_{5} |

DM_{2} | {(s_{6}, 0.7)} | s_{0} | s_{5} | s_{6} |

DM_{3} | s_{5} | {(s_{0}, s_{1})} | s_{4} | s_{5} |

DM_{4} | s_{5} | {(s_{1}, 0.5, 0.8)} | s_{4} | s_{5} |

DM_{5} | s_{5} | s_{1} | {(s_{5}, 0.7), (s_{6}, 0.6, 0.8)} | {(s_{6} 0.7, 0.8)} |

Alternatives | Experts | Criteria | ||||||
---|---|---|---|---|---|---|---|---|

C_{1} | C_{2} | C_{3} | C_{4} | C_{5} | C_{6} | C_{7} | ||

A_{1} | DM_{1} | (10, 0.833, 0.424) | (7.5, 0.833, 0.424) | (5, 0.197, 0.1) | (5.96, 0.318, 0.162) | (5.96, 0.318, 0.162) | (10, 0.833, 0.424) | (10, 0.833, 0.424) |

DM_{2} | (7, 0.833, 0.424) | (10, 0.833, 0.424) | (5.96, 0.318, 0.162) | (5.96, 0.318, 0.162) | (7.5, 0.515, 0.262) | (7.5, 0.515, 0.262) | (7.5, 0.515, 0.262) | |

DM_{3} | (7.5, 0.515, 0.262) | (7.5, 0.515, 0.262) | (5, 0.197, 0.1) | (5.96, 0.318, 0.162) | (5.96, 0.318, 0.162) | (8.75, 0.693, 0.352) | (10, 0.833, 0.424) | |

DM_{4} | (7.5, 0.515, 0.262) | (10, 0.833, 0.424) | (5, 0.197, 0.1) | (7.5, 0.515, 0.262) | (5.96, 0.318, 0.162) | (7.5, 0.515, 0.262) | (10, 0.833, 0.424) | |

DM_{5} | (7.5, 0.515, 0.262) | (7.5, 0.515, 0.262) | (4.68, 0.265, 0.135) | (7.5, 0.515, 0.262) | (5.25, 0.515, 0.262) | (10, 0.833, 0.424) | (6.38, 0.693, 0.352) | |

A_{2} | DM_{1} | (0, 0.833, 0.424) | (2.5, 0.515, 0.262) | (4.05, 0.318, 0.162) | (5, 0.197, 0.1) | (7.5, 0.515, 0.262) | (5.96, 0.318, 0.162) | (4, 0.197, 0.1) |

DM_{2} | (0, 0.833, 0.424) | (4.05, 0.318, 0.162) | (2.5, 0.515, 0.262) | (4.05, 0.318, 0.162) | (7.5, 0.515, 0.262) | (5, 0.197, 0.1) | (4.05, 0.318, 0.162) | |

DM_{3} | (1.25, 0.693, 0.352) | (1.75, 0.515, 0.262) | (2.5, 0.515, 0.262) | (3, 0.197, 0.1) | (5.96, 0.318, 0.162) | (5, 0.197, 0.1) | (5, 0.197, 0.1) | |

DM_{4} | (1.63, 0.515, 0.262) | (4.05, 0.318, 0.162) | (3.24, 0.318, 0.162) | (4.05, 0.318, 0.162) | (5.96, 0.318, 0.162) | (7.5, 0.515, 0.262) | (5, 0.197, 0.1) | |

DM_{5} | (2.5, 0.515, 0.262) | (4.05, 0.318, 0.162) | (4.05, 0.318, 0.162) | (5, 0.197, 0.1) | (7.5, 0.515, 0.262) | (5.96, 0.318, 0.162) | (4.05, 0.318, 0.162) | |

A_{3} | DM_{1} | (5.96, 0.318, 0.162) | (5, 0.197, 0.1) | (7.5, 0.515, 0.262) | (7.5, 0.515, 0.262) | (5, 0.197, 0.1) | (3.58, 0.318, 0.162) | (5, 0.197, 0.1) |

DM_{2} | (7.5, 0.515, 0.262) | (4.05, 0.318, 0.162) | (5.96, 0.318, 0.162) | (5.96, 0.318, 0.162) | (4.05, 0.318, 0.162) | (5, 0.197, 0.1) | (4.05, 0.318, 0.162) | |

DM_{3} | (5.96, 0.318, 0.162) | (4, 0.197, 0.1) | (5.25, 0.515, 0.262) | (7.5, 0.515, 0.262) | (5, 0.197, 0.1) | (5.96, 0.318, 0.162) | (4.05, 0.318, 0.162) | |

DM_{4} | (5.96, 0.318, 0.162) | (5, 0.197, 0.1) | (7.5, 0.515, 0.262) | (7.5, 0.515, 0.262) | (4.05, 0.318, 0.162) | (5, 0.197, 0.1) | (5, 0.197, 0.1) | |

DM_{5} | (6.13, 0.693, 0.352) | (3.18, 0.318, 0.162) | (5.96, 0.318, 0.162) | (5.96, 0.318, 0.162) | (3.54, 0.265, 0.135) | (5.96, 0.318, 0.162) | (5, 0.197, 0.1) | |

A_{4} | DM_{1} | (7.5, 0.515, 0.262) | (5.96, 0.318, 0.162) | (5.96, 0.318, 0.162) | (5, 0.197, 0.1) | (10, 0.833, 0.424) | (7.5, 0.515, 0.262) | (7.5, 0.515, 0.262) |

DM_{2} | (10, 0.833, 0.424) | (7.5, 0.515, 0.262) | (7.5, 0.515, 0.262) | (4.05, 0.318, 0.162) | (10, 0.833, 0.424) | (7.5, 0.515, 0.262) | (10, 0.833, 0.424) | |

DM_{3} | (7.5, 0.515, 0.262) | (5.96, 0.318, 0.162) | (6.73, 0.428, 0.218) | (2.5, 0.283, 0.144) | (10, 0.833, 0.424) | (5.96, 0.318, 0.162) | (6, 0.833, 0.424) | |

DM_{4} | (7.5, 0.515, 0.262) | (5.96, 0.318, 0.162) | (5.96, 0.318, 0.162) | (2.5, 0.515, 0.262) | (7.5, 0.515, 0.262) | (5.96, 0.318, 0.162) | (7.5, 0.515, 0.262) | |

DM_{5} | (7.5, 0.833, 0.424) | (7.5, 0.515, 0.262) | (7.5, 0.515, 0.262) | (4.05, 0.318, 0.162) | (7.5, 0.515, 0.262) | (7.5, 0.515, 0.262) | (7.5, 0.515, 0.262) |

**Table 5.**Collective cloud decision matrix $\tilde{Y}$ and Collective cloud weight vector $\tilde{w}$.

Alternatives | Criteria | ||||||
---|---|---|---|---|---|---|---|

C_{1} | C_{2} | C_{3} | C_{4} | C_{5} | C_{6} | C_{7} | |

A_{1} | (8.13, 0.665, 0.339) | (8.66, 0.771, 0.392) | (5.1, 0.234, 0.119) | (6.63, 0.395, 0.201) | (6.09, 0.397, 0.202) | (8.73, 0.678, 0.345) | (8.81, 0.724, 0.368) |

A_{2} | (0.86, 0.661, 0.337) | (3.49, 0.384, 0.195) | (3.41, 0.441, 0.225) | (4.45, 0.260, 0.132) | (6.96, 0.445, 0.227) | (5.97, 0.319, 0.162) | (4.35, 0.271, 0.138) |

A_{3} | (6.25, 0.442, 0.225) | (4.44, 0.271, 0.138) | (6.79, 0.431, 0.220) | (6.91, 0.431, 0.220) | (4.30, 0.271, 0.138) | (4.78, 0.253, 0.129) | (4.77, 0.256, 0.130) |

A_{4} | (7.92, 0.643, 0.327) | (6.49, 0.397, 0.202) | (6.51, 0.411, 0.209) | (3.67, 0.386, 0.197) | (8.78, 0.683, 0.348) | (7.68, 0.445, 0.226) | (7.84, 0.601, 0.306) |

$\tilde{w}$ | (7.92, 0.643, 0.327) | (6.49, 0.397, 0.202) | (6.51, 0.411, 0.209) | (3.67, 0.386, 0.197) | (8.78, 0.683, 0.348) | (7.68, 0.445, 0.226) | (7.84, 0.601, 0.306) |

P_{ij} | C_{1} | C_{2} | C_{3} | C_{4} | C_{5} | C_{6} | C_{7} |
---|---|---|---|---|---|---|---|

A_{1} | 0.350 | 0.375 | 0.234 | 0.305 | 0.233 | 0.323 | 0.342 |

A_{2} | 0.037 | 0.151 | 0.155 | 0.205 | 0.266 | 0.220 | 0.169 |

A_{3} | 0.271 | 0.192 | 0.313 | 0.320 | 0.164 | 0.175 | 0.186 |

A_{4} | 0.341 | 0.282 | 0.297 | 0.170 | 0.337 | 0.283 | 0.303 |

E_{j} | 0.874 | 0.957 | 0.976 | 0.976 | 0.977 | 0.981 | 0.968 |

C_{1} | C_{2} | C_{3} | C_{4} | C_{5} | C_{6} | C_{7} | |
---|---|---|---|---|---|---|---|

${\phi}_{j}\left({A}_{1},{A}_{2}\right)$ | 1.038 | 0.378 | 0.581 | 0.375 | −0.184 | 0.575 | 0.590 |

${\phi}_{j}\left({A}_{1},{A}_{3}\right)$ | 0.741 | 0.632 | −0.501 | −0.240 | 0.331 | 0.559 | 0.644 |

${\phi}_{j}\left({A}_{1},{A}_{4}\right)$ | 0.227 | 0.664 | −0.486 | 0.417 | −0.504 | 0.524 | 0.365 |

${\phi}_{j}\left({A}_{2},{A}_{3}\right)$ | −1.100 | −0.491 | −0.434 | −0.426 | 0.371 | 0.248 | −0.242 |

${\phi}_{j}\left({A}_{2},{A}_{4}\right)$ | −1.045 | −0.421 | −0.446 | 0.494 | −0.496 | −0.365 | −0.540 |

${\phi}_{j}\left({A}_{3},{A}_{4}\right)$ | −0.711 | −0.360 | 0.139 | 0.364 | −0.524 | −0.415 | −0.600 |

${\phi}_{j}\left({A}_{2},{A}_{1}\right)$ | −1.038 | −0.378 | −0.581 | −0.375 | 0.184 | −0.575 | −0.590 |

${\phi}_{j}\left({A}_{3},{A}_{1}\right)$ | −0.741 | −0.632 | 0.501 | 0.240 | −0.331 | −0.559 | −0.644 |

${\phi}_{j}\left({A}_{4},{A}_{1}\right)$ | −0.227 | −0.664 | 0.486 | −0.417 | 0.504 | −0.524 | −0.365 |

${\phi}_{j}\left({A}_{3},{A}_{2}\right)$ | 1.100 | 0.491 | 0.434 | 0.426 | −0.371 | −0.248 | 0.242 |

${\phi}_{j}\left({A}_{4},{A}_{2}\right)$ | 1.045 | 0.421 | 0.446 | −0.494 | 0.496 | 0.365 | 0.540 |

${\phi}_{j}\left({A}_{4},{A}_{3}\right)$ | 0.711 | 0.360 | −0.139 | −0.364 | 0.524 | 0.415 | 0.600 |

$\phi \left({A}_{1},{A}_{2}\right)$ | $\phi \left({A}_{1},{A}_{3}\right)$ | $\phi \left({A}_{1},{A}_{4}\right)$ | $\delta \left({A}_{1}\right)$ | $\xi \left({A}_{i}\right)$ |

3.353 | 2.166 | 1.207 | 6.727 | 1.000 |

$\phi \left({A}_{2},{A}_{1}\right)$ | $\phi \left({A}_{2},{A}_{3}\right)$ | $\phi \left({A}_{2},{A}_{4}\right)$ | $\delta \left({A}_{2}\right)$ | $\xi \left({A}_{2}\right)$ |

−3.353 | −2.073 | −2.820 | −8.246 | 0.000 |

$\phi \left({A}_{3},{A}_{1}\right)$ | $\phi \left({A}_{3},{A}_{2}\right)$ | $\phi \left({A}_{3},{A}_{4}\right)$ | $\delta \left({A}_{3}\right)$ | $\xi \left({A}_{3}\right)$ |

−2.166 | 2.073 | −2.107 | −2.200 | 0.404 |

$\phi \left({A}_{4},{A}_{1}\right)$ | $\phi \left({A}_{4},{A}_{2}\right)$ | $\phi \left({A}_{4},{A}_{3}\right)$ | $\delta \left({A}_{4}\right)$ | $\xi \left({A}_{4}\right)$ |

−1.207 | 2.820 | 2.107 | 3.720 | 0.799 |

Alternatives | Case 1 | Case 2 | Case 3 | Case 4 |
---|---|---|---|---|

A_{1} | 1.000 | 1.000 | 1.000 | 1.000 |

A_{2} | 0.000 | 0.000 | 0.000 | 0.000 |

A_{3} | 0.407 | 0.413 | 0.402 | 0.417 |

A_{4} | 0.832 | 0.837 | 0.824 | 0.825 |

© 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/).

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**MDPI and ACS Style**

Wang, J.-J.; Miao, Z.-H.; Cui, F.-B.; Liu, H.-C.
Robot Evaluation and Selection with Entropy-Based Combination Weighting and Cloud TODIM Approach. *Entropy* **2018**, *20*, 349.
https://doi.org/10.3390/e20050349

**AMA Style**

Wang J-J, Miao Z-H, Cui F-B, Liu H-C.
Robot Evaluation and Selection with Entropy-Based Combination Weighting and Cloud TODIM Approach. *Entropy*. 2018; 20(5):349.
https://doi.org/10.3390/e20050349

**Chicago/Turabian Style**

Wang, Jing-Jing, Zhong-Hua Miao, Feng-Bao Cui, and Hu-Chen Liu.
2018. "Robot Evaluation and Selection with Entropy-Based Combination Weighting and Cloud TODIM Approach" *Entropy* 20, no. 5: 349.
https://doi.org/10.3390/e20050349