IDEAS home Printed from https://ideas.repec.org/a/eee/jomega/v123y2024ics030504832300138x.html
   My bibliography  Save this article

An analytical framework for the best–worst method

Author

Listed:
  • Wu, Qun
  • Liu, Xinwang
  • Zhou, Ligang
  • Qin, Jindong
  • Rezaei, Jafar

Abstract

Since the development of the best–worst method (BWM) in 2015, it has become a popular research focus in multi-criteria decision-making. The original optimization problem of the BWM is a nonlinear min–max model that can lead to multiple optimal solutions, while the linear model of the BWM produces a unique solution. The two models need to be solved by optimization software packages. In addition, although the linear model of the BWM can obtain a unique solution, it produces different feasible regions than the nonlinear model of the BWM, and it changes the objective function. This study aims to solve the nonlinear model of the BWM mathematically to obtain the analytical forms of the optimal solutions. First, we transform the original nonlinear model of BWM into an equivalent optimization model driven by the optimally modified comparison vectors. The equivalent BWM provides a solid basis for computing the analytical solutions. Second, for not-fully consistent pairwise comparison systems, we strictly prove that there is only one unique optimal solution with three criteria, and there might be multiple optimal solutions with more than three criteria. We further develop the analytical forms of these unique and multiple optimal solutions and the optimal interval weights. Third, we develop a secondary objective function to select a unique solution for the BWM. The secondary objective function retains all the characteristics of the original nonlinear model of the BWM, and we find the unique solution analytically. Finally, some numerical examples are examined, and a comparative analysis is performed to demonstrate the effectiveness of our analytical solution approach.

Suggested Citation

  • Wu, Qun & Liu, Xinwang & Zhou, Ligang & Qin, Jindong & Rezaei, Jafar, 2024. "An analytical framework for the best–worst method," Omega, Elsevier, vol. 123(C).
    • Handle: RePEc:eee:jomega:v:123:y:2024:i:c:s030504832300138x
    DOI: 10.1016/j.omega.2023.102974
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S030504832300138X
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.omega.2023.102974?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Kheybari, Siamak & Javdanmehr, Mahsa & Rezaie, Fariba Mahdi & Rezaei, Jafar, 2021. "Corn cultivation location selection for bioethanol production: An application of BWM and extended PROMETHEE II," Energy, Elsevier, vol. 228(C).
    2. Aguarón, Juan & Escobar, María Teresa & Moreno-Jiménez, José María, 2021. "Reducing inconsistency measured by the geometric consistency index in the analytic hierarchy process," European Journal of Operational Research, Elsevier, vol. 288(2), pages 576-583.
    3. Hasan, Mostafa & Büyüktahtakın, İ. Esra & Elamin, Elshami, 2019. "A multi-criteria ranking algorithm (MCRA) for determining breast cancer therapy," Omega, Elsevier, vol. 82(C), pages 83-101.
    4. Huang, Beijia & Zhang, Long & Ma, Linmao & Bai, Wuliyasu & Ren, Jingzheng, 2021. "Multi-criteria decision analysis of China’s energy security from 2008 to 2017 based on Fuzzy BWM-DEA-AR model and Malmquist Productivity Index," Energy, Elsevier, vol. 228(C).
    5. Liang, Fuqi & Brunelli, Matteo & Rezaei, Jafar, 2020. "Consistency issues in the best worst method: Measurements and thresholds," Omega, Elsevier, vol. 96(C).
    6. Amenta, Pietro & Lucadamo, Antonio & Marcarelli, Gabriella, 2021. "On the choice of weights for aggregating judgments in non-negotiable AHP group decision making," European Journal of Operational Research, Elsevier, vol. 288(1), pages 294-301.
    7. Kumar, Anish & Mangla, Sachin Kumar & Kumar, Pradeep & Song, Malin, 2021. "Mitigate risks in perishable food supply chains: Learning from COVID-19," Technological Forecasting and Social Change, Elsevier, vol. 166(C).
    8. Mohammadi, Majid & Rezaei, Jafar, 2020. "Bayesian best-worst method: A probabilistic group decision making model," Omega, Elsevier, vol. 96(C).
    9. Govindan, Kannan & Shankar, K. Madan & Kannan, Devika, 2020. "Achieving sustainable development goals through identifying and analyzing barriers to industrial sharing economy: A framework development," International Journal of Production Economics, Elsevier, vol. 227(C).
    10. Kusi-Sarpong, Simonov & Orji, Ifeyinwa Juliet & Gupta, Himanshu & Kunc, Martin, 2021. "Risks associated with the implementation of big data analytics in sustainable supply chains," Omega, Elsevier, vol. 105(C).
    11. Gunjan Yadav & Sachin Kumar Mangla & Sunil Luthra & Suresh Jakhar, 2018. "Hybrid BWM-ELECTRE-based decision framework for effective offshore outsourcing adoption: a case study," International Journal of Production Research, Taylor & Francis Journals, vol. 56(18), pages 6259-6278, September.
    12. Md. Abdul Moktadir & Anil Kumar & Syed Mithun Ali & Sanjoy Kumar Paul & Razia Sultana & Jafar Rezaei, 2020. "Critical success factors for a circular economy: Implications for business strategy and the environment," Business Strategy and the Environment, Wiley Blackwell, vol. 29(8), pages 3611-3635, December.
    13. Bottomley, Paul A. & Doyle, John R., 2001. "A comparison of three weight elicitation methods: good, better, and best," Omega, Elsevier, vol. 29(6), pages 553-560, December.
    14. Ren, Jingzheng & Liang, Hanwei & Chan, Felix T.S., 2017. "Urban sewage sludge, sustainability, and transition for Eco-City: Multi-criteria sustainability assessment of technologies based on best-worst method," Technological Forecasting and Social Change, Elsevier, vol. 116(C), pages 29-39.
    15. Jafar Rezaei, 2020. "A Concentration Ratio for Nonlinear Best Worst Method," International Journal of Information Technology & Decision Making (IJITDM), World Scientific Publishing Co. Pte. Ltd., vol. 19(03), pages 891-907, May.
    16. Ru-xin Nie & Zhang-peng Tian & Chin Kwai Sang & Jian-qiang Wang, 2022. "Implementing healthcare service quality enhancement using a cloud-support QFD model integrated with TODIM method and linguistic distribution assessments," Journal of the Operational Research Society, Taylor & Francis Journals, vol. 73(2), pages 207-229, March.
    17. Rezaei, Jafar, 2016. "Best-worst multi-criteria decision-making method: Some properties and a linear model," Omega, Elsevier, vol. 64(C), pages 126-130.
    18. Rezaei, Jafar, 2015. "Best-worst multi-criteria decision-making method," Omega, Elsevier, vol. 53(C), pages 49-57.
    19. Chunguang Bai & Simonov Kusi-Sarpong & Hadi Badri Ahmadi & Joseph Sarkis, 2019. "Social sustainable supplier evaluation and selection: a group decision-support approach," International Journal of Production Research, Taylor & Francis Journals, vol. 57(22), pages 7046-7067, November.
    20. van de Kaa, Geerten & Janssen, Marijn & Rezaei, Jafar, 2018. "Standards battles for business-to-government data exchange: Identifying success factors for standard dominance using the Best Worst Method," Technological Forecasting and Social Change, Elsevier, vol. 137(C), pages 182-189.
    21. Kannan Govindan & Arash Khalili Nasr & Farzane Karimi & Hassan Mina, 2022. "Circular economy adoption barriers: An extended fuzzy best–worst method using fuzzy DEMATEL and Supermatrix structure," Business Strategy and the Environment, Wiley Blackwell, vol. 31(4), pages 1566-1586, May.
    22. Mi, Xiaomei & Tang, Ming & Liao, Huchang & Shen, Wenjing & Lev, Benjamin, 2019. "The state-of-the-art survey on integrations and applications of the best worst method in decision making: Why, what, what for and what's next?," Omega, Elsevier, vol. 87(C), pages 205-225.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Wu, Qun & Liu, Xinwang & Qin, Jindong & Zhou, Ligang & Mardani, Abbas & Deveci, Muhammet, 2022. "An integrated multi-criteria decision-making and multi-objective optimization model for socially responsible portfolio selection," Technological Forecasting and Social Change, Elsevier, vol. 184(C).
    2. Zsombor Szádoczki & Sándor Bozóki & Patrik Juhász & Sergii V. Kadenko & Vitaliy Tsyganok, 2023. "Incomplete pairwise comparison matrices based on graphs with average degree approximately 3," Annals of Operations Research, Springer, vol. 326(2), pages 783-807, July.
    3. Murad, C.A. & Bellinello, M.M. & Silva, A.J. & Netto, A. Caminada & de Souza, G.F.M. & Nabeta, S.I., 2022. "A novel methodology employed for ranking and consolidating performance indicators in holding companies with multiple power plants based on multi-criteria decision-making method," Operations Research Perspectives, Elsevier, vol. 9(C).
    4. Kheybari, Siamak & Javdanmehr, Mahsa & Rezaie, Fariba Mahdi & Rezaei, Jafar, 2021. "Corn cultivation location selection for bioethanol production: An application of BWM and extended PROMETHEE II," Energy, Elsevier, vol. 228(C).
    5. Amin Vafadarnikjoo & Madjid Tavana & Tiago Botelho & Konstantinos Chalvatzis, 2020. "A neutrosophic enhanced best–worst method for considering decision-makers’ confidence in the best and worst criteria," Annals of Operations Research, Springer, vol. 289(2), pages 391-418, June.
    6. Mi, Xiaomei & Tang, Ming & Liao, Huchang & Shen, Wenjing & Lev, Benjamin, 2019. "The state-of-the-art survey on integrations and applications of the best worst method in decision making: Why, what, what for and what's next?," Omega, Elsevier, vol. 87(C), pages 205-225.
    7. Sisto, Roberta & Fernández-Portillo, Luis A. & Yazdani, Morteza & Estepa-Mohedano, Lorenzo & Torkayesh, Ali Ebadi, 2022. "Strategic planning of rural areas: Integrating participatory backcasting and multiple criteria decision analysis tools," Socio-Economic Planning Sciences, Elsevier, vol. 82(PA).
    8. Liang, Fuqi & Brunelli, Matteo & Rezaei, Jafar, 2020. "Consistency issues in the best worst method: Measurements and thresholds," Omega, Elsevier, vol. 96(C).
    9. Vieira, Fabiana C. & Ferreira, Fernando A.F. & Govindan, Kannan & Ferreira, Neuza C.M.Q.F. & Banaitis, Audrius, 2022. "Measuring urban digitalization using cognitive mapping and the best worst method (BWM)," Technology in Society, Elsevier, vol. 71(C).
    10. Bai, Chunguang & Zhu, Qingyun & Sarkis, Joseph, 2021. "Joint blockchain service vendor-platform selection using social network relationships: A multi-provider multi-user decision perspective," International Journal of Production Economics, Elsevier, vol. 238(C).
    11. Danish Farooq & Sarbast Moslem & Arshad Jamal & Farhan Muhammad Butt & Yahya Almarhabi & Rana Faisal Tufail & Meshal Almoshaogeh, 2021. "Assessment of Significant Factors Affecting Frequent Lane-Changing Related to Road Safety: An Integrated Approach of the AHP–BWM Model," IJERPH, MDPI, vol. 18(20), pages 1-17, October.
    12. Ecer, Fatih & Pamucar, Dragan, 2022. "A novel LOPCOW‐DOBI multi‐criteria sustainability performance assessment methodology: An application in developing country banking sector," Omega, Elsevier, vol. 112(C).
    13. Maghsoud Amiri & Mohammad Hashemi-Tabatabaei & Mohammad Ghahremanloo & Mehdi Keshavarz-Ghorabaee & Edmundas Kazimieras Zavadskas & Arturas Kaklauskas, 2021. "Evaluating Life Cycle of Buildings Using an Integrated Approach Based on Quantitative-Qualitative and Simplified Best-Worst Methods (QQM-SBWM)," Sustainability, MDPI, vol. 13(8), pages 1-28, April.
    14. Burak Can Altay & Abdullah Erdem Boztas & Abdullah Okumuş & Muhammet Gul & Erkan Çelik, 2023. "How Will Autonomous Vehicles Decide in Case of an Accident? An Interval Type-2 Fuzzy Best–Worst Method for Weighting the Criteria from Moral Values Point of View," Sustainability, MDPI, vol. 15(11), pages 1-20, June.
    15. Aziz Naghizadeh Vardin & Ramin Ansari & Mohammad Khalilzadeh & Jurgita Antucheviciene & Romualdas Bausys, 2021. "An Integrated Decision Support Model Based on BWM and Fuzzy-VIKOR Techniques for Contractor Selection in Construction Projects," Sustainability, MDPI, vol. 13(12), pages 1-28, June.
    16. Sadia Samar Ali & Rajbir Kaur & Shahbaz Khan, 2023. "Evaluating sustainability initiatives in warehouse for measuring sustainability performance: an emerging economy perspective," Annals of Operations Research, Springer, vol. 324(1), pages 461-500, May.
    17. Junnan Wu & Xin Liu & Dianqi Pan & Yichen Zhang & Jiquan Zhang & Kai Ke, 2023. "Research on Safety Evaluation of Municipal Sewage Treatment Plant Based on Improved Best-Worst Method and Fuzzy Comprehensive Method," Sustainability, MDPI, vol. 15(11), pages 1-15, May.
    18. Hamzeh Soltanali & Mehdi Khojastehpour & Siamak Kheybari, 2023. "Evaluating the critical success factors for maintenance management in agro-industries using multi-criteria decision-making techniques," Operations Management Research, Springer, vol. 16(2), pages 949-968, June.
    19. Besharati Fard, Moein & Moradian, Parisa & Emarati, Mohammadreza & Ebadi, Mehdi & Gholamzadeh Chofreh, Abdoulmohammad & Klemeŝ, Jiří Jaromír, 2022. "Ground-mounted photovoltaic power station site selection and economic analysis based on a hybrid fuzzy best-worst method and geographic information system: A case study Guilan province," Renewable and Sustainable Energy Reviews, Elsevier, vol. 169(C).
    20. Xiao-Kang Wang & Wen-Hui Hou & Chao Song & Min-Hui Deng & Yong-Yi Li & Jian-Qiang Wang, 2021. "BW-MaxEnt: A Novel MCDM Method for Limited Knowledge," Mathematics, MDPI, vol. 9(14), pages 1-17, July.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:jomega:v:123:y:2024:i:c:s030504832300138x. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/375/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.