{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T03:17:46Z","timestamp":1775704666661,"version":"3.50.1"},"reference-count":40,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,12,25]],"date-time":"2020-12-25T00:00:00Z","timestamp":1608854400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The major drawback of the classic approaches for project appraisal is the lack of the possibility to handle change requests during the project\u2019s life cycle. This fact incorporates the concept of uncertainty in the estimation of this investment\u2019s worth. To resolve this issue, the authors use fuzzy numbers, possibilistic moments of fuzzy numbers and the hybrid (fuzzy statistic) fuzzy estimators\u2019 method in order to introduce a fuzzy possibilistic version of the expanded net present value method (FPeNPV). This approach consists of two factors: the fuzzy possibilistic NPV and the fuzzy option premium. For the estimation of the fuzzy NPV, some basic assumptions are taken into consideration: (1) the opportunity cost of capital, used as the present value interest factor calculated through the weighted average cost of capital (WACC), (2) the equity cost, determined through the possibilistic set-up of the capital asset pricing model CAPM, and (3) the inflation factor, also included in the estimation of the NPV. The fuzzy estimators\u2019 method is used for the computation of the fuzzy option premium. An algorithm of nine major steps leads to the computation of the FPeNPV. This gives the administration the opportunity to adapt to potential changes in the company\u2019s internal and external environments. In this way, the symmetry between the planning and execution phase of a project can be reinstated. The results validate the statement that fuzzy and intelligent methods remain valuable tools to express uncertainty in various scientific areas. Finally, an illustrative example aims at a thorough comprehension of this new approach of the expanded NPV method.<\/jats:p>","DOI":"10.3390\/sym13010027","type":"journal-article","created":{"date-parts":[[2020,12,27]],"date-time":"2020-12-27T20:52:21Z","timestamp":1609102341000},"page":"27","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["Decision Making for Project Appraisal in Uncertain Environments: A Fuzzy-Possibilistic Approach of the Expanded NPV Method"],"prefix":"10.3390","volume":"13","author":[{"given":"Konstantinos A.","family":"Chrysafis","sequence":"first","affiliation":[{"name":"Section of Mathematics and Management, Department of Civil Engineering, School of Engineering, Democritus University of Thrace, 67100 Xanthi, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6519-895X","authenticated-orcid":false,"given":"Basil K.","family":"Papadopoulos","sequence":"additional","affiliation":[{"name":"Section of Mathematics and Management, Department of Civil Engineering, School of Engineering, Democritus University of Thrace, 67100 Xanthi, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Gotze, U., Northcott, D., and Schuster, P. (2015). Investment Appraisal Methods and Models, Springer. [2nd ed.].","DOI":"10.1007\/978-3-662-45851-8"},{"key":"ref_2","unstructured":"Ross, T.J. (2004). Fuzzy Logic with Engineering Applications, John Wiley & Sons Ltd.. [3rd ed.]."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"5977","DOI":"10.1080\/00207540802112660","article-title":"Cost\u2013volume\u2013profit analysis under uncertainty: A model with fuzzy estimators based on confidence intervals","volume":"47","author":"Chrysafis","year":"2009","journal-title":"Int. J. Prod. Res."},{"key":"ref_4","first-page":"106","article-title":"Assessing the probabilistic fuzzy net present value for a capital investment choice using fuzzy arithmetic","volume":"22","author":"Tsao","year":"2005","journal-title":"J. Chin. Inst. Ind. Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1341","DOI":"10.1057\/palgrave.jors.2602139","article-title":"A fuzzy MCDM approach for stock selection","volume":"57","author":"Tsao","year":"2006","journal-title":"J. Oper. Res. Soc."},{"key":"ref_6","unstructured":"Filev, D.P., and Yager, R.R. (2020, December 17). Learning Celibate Fuzzy Models. Available online: http:\/\/www.panix.com\/~yager\/HP\/pubs.html."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1086\/296072","article-title":"On the Estimation of Security Price Volatilities from Historical Data","volume":"53","author":"Garman","year":"1980","journal-title":"J. Bus."},{"key":"ref_8","first-page":"220","article-title":"Evaluating the BOT project of sport facility: An application of fuzzy net present value method","volume":"30","author":"Lin","year":"2013","journal-title":"J. Ind. Prod. Eng."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1490","DOI":"10.1080\/01605682.2018.1495995","article-title":"A systematic fuzzy multi-criteria group decision-making approach for alternatives evaluation","volume":"70","author":"Peng","year":"2019","journal-title":"J. Oper. Res. Soc."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1080\/758526905","article-title":"Estimating the volatility of stock prices: A comparison of methods that use high and low prices","volume":"4","author":"Rogers","year":"1994","journal-title":"Appl. Financ. Econ."},{"key":"ref_11","first-page":"267","article-title":"The expectation-deviation net present value by fuzzy arithmetic for capital investments","volume":"13","author":"Tsao","year":"2010","journal-title":"J. Stat. Manag. Syst."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1155","DOI":"10.1057\/jors.2013.47","article-title":"Three fuzzy goal programming models for index portfolios","volume":"65","author":"Wu","year":"2014","journal-title":"J. Oper. Res. Soc."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1080\/01446190050024824","article-title":"A fuzzy stochastic technique for project selection","volume":"18","author":"Wong","year":"2000","journal-title":"Constr. Manag. Econ."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1007\/s10700-007-9023-9","article-title":"Application of possibility theory to investment decisions","volume":"7","author":"Appadoo","year":"2008","journal-title":"Fuzzy Optim. Decis. Mak."},{"key":"ref_15","first-page":"33","article-title":"Possibilistic moment generating functions of fuzzy numbers with GARCH applications","volume":"6","author":"Appadoo","year":"2010","journal-title":"Adv. Fuzzy Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1016\/S0165-0114(00)00043-9","article-title":"On possibilistic mean value and variance of fuzzy numbers","volume":"122","author":"Carlsson","year":"2001","journal-title":"Fuzzy Sets Syst."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"140","DOI":"10.1016\/j.ijar.2007.06.011","article-title":"American option pricing with imprecise risk-neutral probabilities","volume":"49","author":"Muzzioli","year":"2008","journal-title":"Int. J. Approx. Reason."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1218","DOI":"10.1016\/j.ejor.2005.10.055","article-title":"The solution of fuzzy linear systems by non-linear programming: A financial application","volume":"177","author":"Muzzioli","year":"2007","journal-title":"Eur. J. Oper. Res."},{"key":"ref_19","first-page":"43","article-title":"Investment analyses using fuzzy probability concept","volume":"16","author":"Kahraman","year":"2010","journal-title":"Ukio Technol. Ir Ekon. Vystym."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"391","DOI":"10.3166\/jds.12.391-416","article-title":"Fuzzy Black and Scholes Real Options Pricing","volume":"12","author":"Collan","year":"2003","journal-title":"J. Decis. Syst."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1885","DOI":"10.1016\/j.cor.2011.07.015","article-title":"Fuzzy net present values for capital investments in an uncertain environment","volume":"39","author":"Tsao","year":"2012","journal-title":"Comput. Oper. Res."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1061\/(ASCE)ME.1943-5479.0000296","article-title":"Possibilisic moments for the task duration in Fuzzzy PERT","volume":"31","author":"Chrysafis","year":"2015","journal-title":"J. Manag. Eng."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Chatterjee, K., Zavadskas, E.K., Tamo\u0161aitien\u0117, J., Adhikary, K., and Kar, S. (2018). A Hybrid MCDM Technique for Risk Management in Construction Projects. Symmetry, 10.","DOI":"10.3390\/sym10020046"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Dahooie, J.H., Zavadskas, E.K., Abolhasani, M., Vanaki, A., and Turskis, Z. (2018). A Novel Approach for Evaluation of Projects Using an Interval\u2013Valued Fuzzy Additive Ratio Assessment (ARAS) Method: A Case Study of Oil and Gas Well Drilling Projects. Symmetry, 10.","DOI":"10.3390\/sym10020045"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Gebrehiwet, T., and Luo, H. (2019). Risk Level Evaluation on Construction Project Lifecycle Using Fuzzy Comprehensive Evaluation and TOPSIS. Symmetry, 11.","DOI":"10.3390\/sym11010012"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Le\u015bniak, A., Kubek, D., Plebankiewicz, E., Zima, K., and Belniak, S. (2018). Fuzzy AHP Application for Supporting Contractors\u2019 Bidding Decision. Symmetry, 10.","DOI":"10.3390\/sym10110642"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Rola, P., and Kuchta, D. (2019). Application of Fuzzy Sets to the Expert Estimation of Scrum-Based Projects. Symmetry, 11.","DOI":"10.3390\/sym11081032"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Bolo\u0219, M.-I., Bradea, I.-A., and Delcea, C. (2019). A Fuzzy Logic Algorithm for Optimizing the Investment Decisions within Companies. Symmetry, 11.","DOI":"10.3390\/sym11020186"},{"key":"ref_29","unstructured":"Chrysafis, K.A., Theotokas, I.N., and Lagoudis, I.N. (2020, January 10\u201313). Managing fuel price variability for ship operations via bunkering contracts. Proceedings of the International Association of Maritime Economists (IAME) Conference, \u201cSustainable development of shipping and trading\u201d, Hong Kong, China. ID320."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Zainol Abidin, S.N., Jaaman, S.H., Ismail, M., and Abu Bakar, A.S. (2020). Clustering Stock Performance Considering Investor Preferences Using a Fuzzy Inference System. Symmetry, 12.","DOI":"10.3390\/sym12071148"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Plebankiewicz, E., and Wieczorek, D. (2020). Adaptation of a Cost Overrun Risk Prediction Model to the Type of Construction Facility. Symmetry, 12.","DOI":"10.3390\/sym12101739"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1672","DOI":"10.3390\/sym12101672","article-title":"On Application Oriented Fuzzy Numbers for Imprecise Investment Recommendations","volume":"12","year":"2020","journal-title":"Symmetry"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy Sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. Control."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/0165-0114(78)90029-5","article-title":"Fuzzy sets as the basis for a theory possibility","volume":"1","author":"Zadeh","year":"1978","journal-title":"Fuzzy Sets Syst."},{"key":"ref_35","unstructured":"Klir, G.J., and Yuan, B. (1995). Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice Hall."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"446","DOI":"10.1016\/j.ins.2014.03.131","article-title":"Non-asymptotic fuzzy estimators based on confidence intervals","volume":"279","author":"Sfiris","year":"2014","journal-title":"Inf. Sci."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1041","DOI":"10.1016\/j.mcm.2011.09.029","article-title":"Corporate investment appraisal with possibilistic CAPM","volume":"55","author":"Chrysafis","year":"2012","journal-title":"Math. Computer Model."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1016\/0304-405X(79)90015-1","article-title":"Options pricing: A simplified formula","volume":"7","author":"Cox","year":"1979","journal-title":"J. Financ. Econ."},{"key":"ref_39","unstructured":"Adams, A.T., Bloomfield, D.S.F., Booth, P.M., and England, P.D. (1993). Investment Mathematics and Statistics, Graham & Trotman."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1086\/209650","article-title":"Drift-Independent Volatility Estimation Based on High, Low, Open, and Close Prices","volume":"73","author":"Yang","year":"2000","journal-title":"J. Bus."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/1\/27\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:46:20Z","timestamp":1760179580000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/1\/27"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12,25]]},"references-count":40,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,1]]}},"alternative-id":["sym13010027"],"URL":"https:\/\/doi.org\/10.3390\/sym13010027","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,12,25]]}}}