# Application of Improved Best Worst Method (BWM) in Real-World Problems

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## Abstract

**:**

## 1. Introduction

#### Motivation for the Modification of the Traditional Best Worst Method

## 2. Applications of BWM: A Literature Review

## 3. Improved Best Worst Method (BWM-I)

**Example**

**1.**

**Example**

**2.**

## 4. Case Study: The Application of BWM-I

## 5. Managerial Implications

- By preferring the BWM-I model, authorities can make more accurate decisions.
- Since the weight of each criterion is found according to the opinions of decision-makers, firms can improve their evaluation process through the BWM-I approach.
- Firms can create a better competitive advantage over their business competitors by determining the best alternatives with the BWM-I model.

## 6. Conclusions

- (1)
- Due to non-determinedness and imprecision in data, it is realistic that more than one best and/or worst criterion/criteria with the same significance may appear in experts’ preferences. The BWM-I enables a realistic expression of experts’ preferences irrespective of the number of the best/worst criteria in a set of evaluation criteria.
- (2)
- In case more than one best and worst criterion appear (${m}_{b}>1$ and ${m}_{w}>1$) in the decision-making process, the application of the BWM-I reduces the number of comparisons from 2n-3 (in the traditional BWM) to 2n-5 (in the BWM-I). In that manner, the possibility of making a mistake while conducting a pairwise comparison of the criteria is also reduced, which further exerts an influence on the greater reliability of results.
- (3)
- The flexibility of the BWM-I is expressed in two ways: (1) the possibilities of the realistic processing of experts’ preferences irrespective of the number of the criteria with the same significance (even in the case of the best/worst criteria), and (2) in the case of ${m}_{b}={m}_{w}=1$, the BWM-I transforms into the traditional BWM. This flexibility opens the possibility of applying the BWM-I in complex studies, in which criteria and experts’ preferences differ within the framework of the cluster(s)/group of criteria.

#### Future Research

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Best-to-Others Vector | Others-to-Worst Vector | ||
---|---|---|---|

Best: C2 and C4 | Evaluation | Worst: C5 | Evaluation |

C1 | 2 | C1 | 4 |

C2 | 1 | C2 | 9 |

C3 | 4 | C3 | 2 |

C4 | 1 | C4 | 9 |

C5 | 9 | C5 | 1 |

Main Criteria | Sub-Criteria | Code | Definition | References |
---|---|---|---|---|

Technical (C1) | Efficiency | C11 | How technology is widespread at the regional, national, and international levels. | [57,58,59] |

Reliability | C12 | An energy system’s ability to perform the required functions | [56,58,60] | |

Resource reserves | C13 | The availability of the energy source to generate energy | [58] | |

Technology maturity | C14 | The penetration of a specific technology in the
energy mix at the regional, national, and international levels. | [58,60] | |

Safety of the system | C15 | The security of the workers and the local community | [56] | |

Economic (C2) | Investment cost | C21 | All costs of products and services, except for the costs of labor or the cost of equipment maintenance | [56,58,59,60] |

Operation and maintenance cost | C22 | Operating the energy system adequately, as well as the costs related to the maintenance of the energy system | [56,58] | |

Return of investment | C23 | The time required to recover the investment | [56,58] | |

Energy cost | C24 | The cost of the energy-generating system | [60,63] | |

Operational life | C25 | The period during which the power plant can operate before being decommissioned | [56] | |

R&D cost | C26 | The expenses incurred for the R&D of technological innovations | [65] | |

Social (C3) | Social acceptance | C31 | The opinions of residents, local authorities, and other stakeholders on an energy project | [56,57,58] |

Job creation | C32 | Jobs created per unit of the energy produced | [57,58,61] | |

Social benefits | C33 | The contribution of an energy system to the improvement and advancement of local society | [56,58] | |

Noise | C34 | The noise generated during the lifecycle under
consideration | [62] | |

Visual impact | C35 | The aesthetics of the installations of the energy system | [62] | |

Environmental (C4) | Greenhouse Gas (GHG) Emissions | C41 | Lifecycle GHG emissions (in the equivalent emission of CO2) from technology | [58,61,63] |

Land use | C42 | The area used per unit of the energy produced | [58,59,60,61] | |

Impact on the environment and humans | C43 | The detriment level of the energy facility to humans and nature | [58,59,60,64] | |

Water use | C44 | Water consumed per unit of the energy produced | [60,61] | |

Climate change | C45 | The global warming potential | [57] | |

Risk (C5) | Health risk | C51 | Emissions harmful to human health | [66] |

Accident risk | C52 | Accidents of any type during the lifecycle considered | [57,59,62,66] | |

Economic risk | C53 | The risk financial stakeholders should bear for business in new plants | [60] | |

Political (C6) | Foreign dependency | C61 | The dependency of countries on international legislations | [57,58] |

Compatibility with the national energy policy | C62 | The national energy policy related to renewable energy sources | [58] | |

Compatibility with the public policy | C63 | Voluntary agreements and general codes of conduct in line with national priorities | [64] | |

Government support | C64 | Approving and adapting to renewable energy sources. | [64] |

**Table 3.**The best-to-others (M-BO) and modified others-to-worst (M-OW) vectors of the dimensions/sub-criteria.

Dimensions | |||

Best: C4 | Preference | Worst: C5 and C6 | Preference |

C1 | 3 | C1 | 3 |

C2 | 2 | C2 | 4 |

C3 | 4 | C3 | 2 |

C4 | 1 | C4 | 5 |

C5 | 5 | C5 | 1 |

C6 | 5 | C6 | 1 |

Technical sub-criteria | |||

Best: C14 | Preference | Worst: C12 | Preference |

C11 | 4 | C11 | 2 |

C12 | 7 | C12 | 1 |

C13 | 3 | C13 | 3 |

C14 | 1 | C14 | 7 |

C15 | 2 | C15 | 4 |

Economic sub-criteria | |||

Best: C21, C22 and C24 | Preference | Worst: C23 | Preference |

C21 | 1 | C21 | 4 |

C22 | 1 | C22 | 4 |

C23 | 4 | C23 | 1 |

C24 | 1 | C24 | 4 |

C25 | 3 | C25 | 2 |

C26 | 2 | C26 | 3 |

Social sub-criteria | |||

Best: C31 | Preference | Worst: C34 and C35 | Preference |

C31 | 1 | C31 | 4 |

C32 | 2 | C32 | 3 |

C33 | 3 | C33 | 2 |

C34 | 4 | C34 | 1 |

C35 | 4 | C35 | 1 |

Environmental sub-criteria | |||

Best: C43 and C45 | Preference | Worst: C41 and C44 | Preference |

C41 | 4 | C41 | 1 |

C42 | 2 | C42 | 2 |

C43 | 1 | C43 | 4 |

C44 | 4 | C44 | 1 |

C45 | 1 | C45 | 4 |

Risk sub-criteria | |||

Best: C51 | Preference | Worst: C53 | Preference |

C51 | 1 | C51 | 3 |

C52 | 2 | C52 | 2 |

C53 | 3 | C53 | 1 |

Political sub-criteria | |||

Best: C62 and C63 | Preference | Worst: C64 | Preference |

C61 | 2 | C61 | 2 |

C62 | 1 | C62 | 3 |

C63 | 1 | C63 | 3 |

C64 | 3 | C64 | 1 |

Dimensions/Sub-Criteria | Code | Local Weights | Global Weights | Rank |
---|---|---|---|---|

Technical | C1 | 0.1674 | - | 3 |

Efficiency | C11 | 0.1037 | 0.0174 | 17 |

Reliability | C12 | 0.0586 | 0.0098 | 19 |

Resource reserves | C13 | 0.1584 | 0.0265 | 12 |

Technology maturity | C14 | 0.4278 | 0.0716 | 4 |

Safety of the system | C15 | 0.2514 | 0.0421 | 9 |

Economic | C2 | 0.2823 | - | 2 |

Investment cost | C21 | 0.2372 | 0.0670 | 5 |

Operation and maintenance cost | C22 | 0.2372 | 0.0670 | 5 |

Return of investment | C23 | 0.0545 | 0.0154 | 18 |

Energy cost | C24 | 0.2372 | 0.0670 | 5 |

Operational life | C25 | 0.0897 | 0.0253 | 13 |

R&D cost | C26 | 0.1441 | 0.0407 | 10 |

Social | C3 | 0.1178 | - | 4 |

Social acceptance | C31 | 0.4761 | 0.0561 | 8 |

Job creation | C32 | 0.2893 | 0.0341 | 11 |

Social benefits | C33 | 0.1799 | 0.0212 | 16 |

Noise | C34 | 0.0273 | 0.0032 | 25 |

Visual impact | C35 | 0.0273 | 0.0032 | 25 |

Environmental | C4 | 0.3972 | - | 1 |

GHG Emissions | C41 | 0.0617 | 0.0245 | 14 |

Land use | C42 | 0.2729 | 0.1084 | 3 |

Impact on the environment and humans | C43 | 0.3019 | 0.1199 | 1 |

Water use | C44 | 0.0617 | 0.0245 | 14 |

Climate change | C45 | 0.3019 | 0.1199 | 1 |

Risk | C5 | 0.0176 | - | 5 |

Health risk | C51 | 0.5348 | 0.0094 | 20 |

Accident risk | C52 | 0.2985 | 0.0053 | 23 |

Economic risk | C53 | 0.1667 | 0.0029 | 27 |

Political | C6 | 0.0176 | - | 5 |

Foreign dependency | C61 | 0.1945 | 0.0034 | 24 |

Compatibility with the national energy policy | C62 | 0.3484 | 0.0061 | 21 |

Compatibility with the public policy | C63 | 0.3484 | 0.0061 | 21 |

Government support | C64 | 0.1086 | 0.0019 | 28 |

Criterion Level | C1–C6 | C11–C15 | C21–C26 | C31–C35 | C41–C45 | C51–C53 | C61–C64 |
---|---|---|---|---|---|---|---|

${a}_{BW}$ | 5 | 7 | 4 | 4 | 4 | 3 | 3 |

CI ($\mathrm{max}\xi $) | 2.30 | 3.73 | 1.63 | 1.63 | 1.63 | 1.00 | 1.00 |

CR | 0.27 | 0.08 | 0.22 | 0.22 | 0.55 | 0.21 | 0.21 |

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**MDPI and ACS Style**

Pamučar, D.; Ecer, F.; Cirovic, G.; Arlasheedi, M.A.
Application of Improved Best Worst Method (BWM) in Real-World Problems. *Mathematics* **2020**, *8*, 1342.
https://doi.org/10.3390/math8081342

**AMA Style**

Pamučar D, Ecer F, Cirovic G, Arlasheedi MA.
Application of Improved Best Worst Method (BWM) in Real-World Problems. *Mathematics*. 2020; 8(8):1342.
https://doi.org/10.3390/math8081342

**Chicago/Turabian Style**

Pamučar, Dragan, Fatih Ecer, Goran Cirovic, and Melfi A. Arlasheedi.
2020. "Application of Improved Best Worst Method (BWM) in Real-World Problems" *Mathematics* 8, no. 8: 1342.
https://doi.org/10.3390/math8081342