# Fuzzy AHP, DEA, and Managerial Analysis for Supplier Selection and Development; From the Perspective of Open Innovation

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## Abstract

**:**

## 1. Introduction

## 2. Methodologies

#### 2.1. Fuzzy Analytical Hierarchy Process

- Step 1:
- A fuzzy matrix is generated from a pairwise comparison matrix based on the comparison ratings given by an expert. The fuzzy matrix ${\tilde{A}}^{h}$ from expert h is represented by Equation (4). ${\tilde{a}}_{ij}^{h}$ = (${b}_{ij}^{h},{c}_{ij}^{h},{d}_{ij}^{h}$) is the fuzzy triangular value that represents the preference of expert h for criterion i over criterion j (or alternative i over alternative j with respect to a criterion). For example, ${\tilde{a}}_{34}^{1}$ = (4, 5, 6) denotes the first expert’s preference for the third criterion over the fourth criterion that is materialized by fuzzy triangular number (4, 5, 6).$${\tilde{A}}^{h}=\left[\begin{array}{ccc}{\tilde{a}}_{11}^{h}& \cdots & {\tilde{a}}_{1n}^{h}\\ \vdots & \ddots & \vdots \\ {\tilde{a}}_{n1}^{h}& \cdots & {\tilde{a}}_{nn}^{h}\end{array}\right]$$
- Step 2:
- With more than one expert in the pairwise comparisons, the average of the preferences of H experts can be calculated by Equation (5).$${\tilde{a}}_{ij}=({a}_{ij1},\text{}{a}_{ij2},\text{}{a}_{ij3})=\frac{{\sum}_{h=1}^{H}{\tilde{a}}_{ij}^{h}}{H}=(\frac{{\sum}_{h=1}^{H}{a}_{ij1}^{h}}{H},\text{}\frac{{\sum}_{h=1}^{H}{a}_{ij2}^{h}}{H},\text{}\frac{{\sum}_{h=1}^{H}{a}_{ij3}^{h}}{H})$$
- Step 3:
- The average matrix is obtained with the average preferences by Equation (6).$$\tilde{A}=\left[\begin{array}{ccc}{\tilde{a}}_{11}& \cdots & {\tilde{a}}_{1n}\\ \vdots & \ddots & \vdots \\ {\tilde{a}}_{n1}& \cdots & {\tilde{a}}_{nn}\end{array}\right]$$
- Step 4:
- The geometric mean ${\tilde{p}}_{i}$ of the fuzzy comparison values of criterion i with all criteria (or alternative i with all alternatives with regard to each criterion) is computed by Equation (7).$${\tilde{p}}_{i}={\left({\prod}_{j=1}^{n}{\tilde{a}}_{ij}\right)}^{1/n}=\{{\left({\prod}_{j=1}^{n}{a}_{ij1}\right)}^{1/n},\text{}{\left({\prod}_{j=1}^{n}{a}_{ij2}\right)}^{1/n},\text{}{\left({\prod}_{j=1}^{n}{a}_{ij3}\right)}^{1/n}\},\text{}i=1,\text{}2,\text{}3,\text{}\dots ,\text{}n$$
- Step 5:
- The fuzzy weight ${\tilde{w}}_{i}$ for criterion i (or fuzzy score ${\tilde{w}}_{i}$ for alternative i with regard to each criterion) is calculated by Equation (8).$${\tilde{w}}_{i}=({w}_{i1},\text{}{w}_{i2},\text{}{w}_{i3})={\tilde{p}}_{i}\otimes ({\tilde{p}}_{1}\oplus {\tilde{p}}_{2}\oplus \dots \oplus {\tilde{p}}_{n}){}^{-1},\text{}i=1,\text{}2,\text{}3,\text{}\dots ,\text{}n$$
- Step 6:
- To defuzzify the fuzzy weight ${\tilde{w}}_{i}$ for criterion i (or the fuzzy score ${\tilde{w}}_{i}$ for alternative i with regard to each criterion), the center of area method [24] is applied using Equation (9).$${Q}_{i}=\frac{\left({w}_{i1}+{w}_{i2}+{w}_{i3}\right)}{3},\text{}i=1,\text{}2,\text{}3,\text{}\dots ,\text{}n$$
- Step 7:
- The last step is to normalize ${Q}_{i}$ by Equation (10).$${R}_{i}=\frac{{Q}_{i}}{{\sum}_{i=1}^{n}{Q}_{i}},\text{}i=1,\text{}2,\text{}3,\text{}\dots ,\text{}n$$

#### 2.2. Data Envelopment Analysis

- ${E}_{k}$ is the efficiency score of the ${k}^{th}$ alternative (supplier), and
- $\epsilon $ is a non-Archimedean.

## 3. Problem Statements

#### 3.1. Publishing Company

#### 3.2. Criteria

#### 3.3. Suppliers

## 4. Results

#### 4.1. The Results of Fuzzy AHP

#### 4.2. The Results of DEA

#### 4.3. Managerial Analysis

#### 4.4. The Second Results of Fuzzy AHP

#### 4.5. The Results for Supplier Development

#### 4.6. Managerial Implications

#### 4.7. Discussions from the Perspective of Open Innovation

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Preference Value | Numeric Value | Fuzzy Numbers |
---|---|---|

Equally preferred | 1 | (1, 1, 1) |

Moderately preferred | 3 | (2, 3, 4) |

Strongly preferred | 5 | (4, 5, 6) |

Very strongly preferred | 7 | (6, 7, 8) |

Extremely preferred | 9 | (9, 9, 9) |

Criteria | C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 |
---|---|---|---|---|---|---|---|---|

Quality (C1) | 1 | 5 | 7 | 5 | 1 | 1 | 3 | 3 |

Price (C2) | 1/5 | 1 | 3 | 3 | 1/3 | 1/3 | 1/3 | 1 |

Delivery (C3) | 1/7 | 1/3 | 1 | 3 | 1/5 | 1 | 1 | 1 |

Flexibility (C4) | 1/5 | 1/3 | 1/3 | 1 | 1/5 | 1/3 | 1/3 | 1 |

Tech. Cap. (C5) | 1 | 3 | 5 | 5 | 1 | 1 | 3 | 3 |

Trust (C6) | 1 | 3 | 1 | 3 | 1 | 1 | 3 | 3 |

Fin. Cap. (C7) | 1/3 | 3 | 1 | 3 | 1/3 | 1/3 | 1 | 3 |

Cust. Ser. (C8) | 1/3 | 1 | 1 | 1 | 1/3 | 1/3 | 1/3 | 1 |

Quality (C1) | S1 | S2 | S3 | S4 | S5 | S6 |
---|---|---|---|---|---|---|

Supplier 1 (S1) | 1 | 3 | 7 | 7 | 5 | 1 |

Supplier 2 (S2) | 1/3 | 1 | 5 | 5 | 1 | 1/3 |

Supplier 3 (S3) | 1/7 | 1/5 | 1 | 1 | 1/3 | 1/5 |

Supplier 4 (S4) | 1/7 | 1/5 | 1 | 1 | 1/3 | 1/7 |

Supplier 5 (S5) | 1/5 | 1 | 3 | 3 | 1 | 1/5 |

Supplier 6 (S6) | 1 | 3 | 5 | 7 | 5 | 1 |

Criteria | C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 |
---|---|---|---|---|---|---|---|---|

Weights | 0.2524 | 0.0748 | 0.0652 | 0.0410 | 0.2274 | 0.1739 | 0.1047 | 0.0603 |

Criteria | C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | |
---|---|---|---|---|---|---|---|---|---|

Suppliers | |||||||||

Supplier 1 | 0.3484 | 0.0461 | 0.3091 | 0.3562 | 0.3261 | 0.4206 | 0.1464 | 0.3786 | |

Supplier 2 | 0.1415 | 0.2173 | 0.0565 | 0.0463 | 0.1479 | 0.1117 | 0.1764 | 0.0881 | |

Supplier 3 | 0.0418 | 0.0294 | 0.2791 | 0.2503 | 0.0390 | 0.0501 | 0.0367 | 0.0823 | |

Supplier 4 | 0.0394 | 0.1143 | 0.0565 | 0.0801 | 0.0355 | 0.0590 | 0.0367 | 0.2019 | |

Supplier 5 | 0.0989 | 0.1504 | 0.2363 | 0.1334 | 0.1065 | 0.0864 | 0.1240 | 0.0823 | |

Supplier 6 | 0.3296 | 0.4422 | 0.0621 | 0.1334 | 0.3447 | 0.2720 | 0.4796 | 0.1666 | |

Total | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

**Table 6.**Overall scores and ranking of the suppliers by the fuzzy analytical hierarchy process (AHP) approach.

Suppliers | Scores | Ranking |
---|---|---|

Supplier 6 | 0.3118 | 1 |

Supplier 1 | 0.3117 | 2 |

Supplier 2 | 0.1344 | 3 |

Supplier 5 | 0.1143 | 4 |

Supplier 3 | 0.0676 | 5 |

Supplier 4 | 0.0598 | 6 |

DMU | OUTPUTS | INPUT | |||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 1 | |

Supplier 1 | 0.348 | 0.046 | 0.309 | 0.356 | 0.326 | 0.421 | 0.146 | 0.379 | 1 |

Supplier 2 | 0.142 | 0.217 | 0.057 | 0.046 | 0.148 | 0.112 | 0.176 | 0.088 | 1 |

Supplier 3 | 0.042 | 0.029 | 0.279 | 0.250 | 0.039 | 0.05 | 0.037 | 0.082 | 1 |

Supplier 4 | 0.039 | 0.114 | 0.057 | 0.080 | 0.036 | 0.059 | 0.037 | 0.202 | 1 |

Supplier 5 | 0.099 | 0.150 | 0.236 | 0.133 | 0.107 | 0.086 | 0.124 | 0.082 | 1 |

Supplier 6 | 0.33 | 0.442 | 0.062 | 0.133 | 0.345 | 0.272 | 0.48 | 0.167 | 1 |

DMU | Objective Value | Efficient | Reference Set |
---|---|---|---|

Supplier 1 | 1 | Yes | - |

Supplier 2 | 0.569 | No | S1(0.151); S6(0.849) |

Supplier 3 | 0.903 | No | S1(1) |

Supplier 4 | 0.652 | No | S1(0.674); S6(0.326) |

Supplier 5 | 0.977 | No | S1(0.728); S6(0.272) |

Supplier 6 | 1 | Yes | - |

Criteria | C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 |
---|---|---|---|---|---|---|---|---|

Weights | 0.2780 | 0.4331 | 0.0580 | 0.0445 | 0.0756 | 0.0571 | 0.0232 | 0.0302 |

Suppliers | Scores | Ranking |
---|---|---|

Supplier 6 | 0.3505 | 1 |

Supplier 1 | 0.2142 | 2 |

Supplier 2 | 0.1631 | 3 |

Supplier 5 | 0.1307 | 4 |

Supplier 4 | 0.0804 | 5 |

Supplier 3 | 0.0609 | 6 |

Suppliers | |||||
---|---|---|---|---|---|

2 | 3 | 4 | 5 | ||

Input | Current | 1 | 1 | 1 | 1 |

Target | 1 | 1 | 1 | 1 | |

Output 1 | Current | 0.142 | 0.042 | 0.039 | 0.099 |

Target | 0.3327 | 0.348 | 0.3421 | 0.3431 | |

Output 2 | Current | 0.217 | 0.029 | 0.114 | 0.15 |

Target | 0.3822 | 0.046 | 0.175 | 0.1537 | |

Output 3 | Current | 0.057 | 0.279 | 0.057 | 0.236 |

Target | 0.0992 | 0.309 | 0.2284 | 0.2418 | |

Output 4 | Current | 0.046 | 0.25 | 0.08 | 0.133 |

Target | 0.1666 | 0.356 | 0.2833 | 0.2953 | |

Output 5 | Current | 0.148 | 0.039 | 0.036 | 0.107 |

Target | 0.3421 | 0.326 | 0.3321 | 0.3311 | |

Output 6 | Current | 0.112 | 0.05 | 0.059 | 0.086 |

Target | 0.2944 | 0.421 | 0.3724 | 0.3804 | |

Output 7 | Current | 0.176 | 0.037 | 0.037 | 0.124 |

Target | 0.4295 | 0.146 | 0.2548 | 0.2368 | |

Output 8 | Current | 0.088 | 0.082 | 0.202 | 0.082 |

Target | 0.199 | 0.379 | 0.3098 | 0.3213 |

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**MDPI and ACS Style**

Diouf, M.; Kwak, C. Fuzzy AHP, DEA, and Managerial Analysis for Supplier Selection and Development; From the Perspective of Open Innovation. *Sustainability* **2018**, *10*, 3779.
https://doi.org/10.3390/su10103779

**AMA Style**

Diouf M, Kwak C. Fuzzy AHP, DEA, and Managerial Analysis for Supplier Selection and Development; From the Perspective of Open Innovation. *Sustainability*. 2018; 10(10):3779.
https://doi.org/10.3390/su10103779

**Chicago/Turabian Style**

Diouf, Maimouna, and Choonjong Kwak. 2018. "Fuzzy AHP, DEA, and Managerial Analysis for Supplier Selection and Development; From the Perspective of Open Innovation" *Sustainability* 10, no. 10: 3779.
https://doi.org/10.3390/su10103779