{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,2]],"date-time":"2026-07-02T00:15:08Z","timestamp":1782951308776,"version":"3.54.5"},"reference-count":54,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,1,19]],"date-time":"2020-01-19T00:00:00Z","timestamp":1579392000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper studies the problem of tangible assets acquisition within the company by proposing a new hybrid model that uses linear programming and fuzzy numbers. Regarding linear programming, two methods were implemented in the model, namely: the graphical method and the primal simplex algorithm. This hybrid model is proposed for solving investment decision problems, based on decision variables, objective function coefficients, and a matrix of constraints, all of them presented in the form of triangular fuzzy numbers. Solving the primal simplex algorithm using fuzzy numbers and coefficients, allowed the results of the linear programming problem to also be in the form of fuzzy variables. The fuzzy variables compared to the crisp variables allow the determination of optimal intervals for which the objective function has values depending on the fuzzy variables. The major advantage of this model is that the results are presented as value ranges that intervene in the decision-making process. Thus, the company\u2019s decision makers can select any of the result values as they satisfy two basic requirements namely: minimizing\/maximizing the objective function and satisfying the basic requirements regarding the constraints resulting from the company\u2019s activity. The paper is accompanied by a practical example.<\/jats:p>","DOI":"10.3390\/e22010121","type":"journal-article","created":{"date-parts":[[2020,1,20]],"date-time":"2020-01-20T04:27:09Z","timestamp":1579494429000},"page":"121","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Linear Programming and Fuzzy Optimization to Substantiate Investment Decisions in Tangible Assets"],"prefix":"10.3390","volume":"22","author":[{"given":"Marcel-Ioan","family":"Bolo\u0219","sequence":"first","affiliation":[{"name":"Department of Finance and Banks, University of Oradea, 410087 Oradea, Romania"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ioana-Alexandra","family":"Bradea","sequence":"additional","affiliation":[{"name":"Department of Informatics and Cybernetics, Bucharest University of Economic Studies, 010374 Bucharest, Romania"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3589-1969","authenticated-orcid":false,"given":"Camelia","family":"Delcea","sequence":"additional","affiliation":[{"name":"Department of Informatics and Cybernetics, Bucharest University of Economic Studies, 010374 Bucharest, Romania"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,1,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1016\/S0965-9978(00)00022-3","article-title":"Computational aspects of linear programming Simplex method","volume":"31","author":"Nguyen","year":"2000","journal-title":"Adv. Eng. Softw."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1016\/0167-6377(93)90045-I","article-title":"A primal-dual simplex method for linear programs","volume":"13","author":"Curet","year":"1993","journal-title":"Oper. Res. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"454","DOI":"10.1016\/S0377-2217(99)00297-0","article-title":"A note on a modified simplex approach for solving bilevel linear programming problems","volume":"126","author":"Scheimberg","year":"2000","journal-title":"Eur. J. Oper. Res."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1016\/0377-2217(93)90327-J","article-title":"A modified simplex approach for solving bilevel linear programming problems","volume":"67","year":"1993","journal-title":"Eur. J. Oper. Res."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1016\/S0167-6377(02)00152-9","article-title":"An asymptotic simplex method for singularly perturbed linear programs","volume":"30","author":"Filar","year":"2002","journal-title":"Oper. Res. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1913","DOI":"10.1016\/j.automatica.2006.06.003","article-title":"Necessary and sufficient conditions for reachability on a simplex","volume":"42","author":"Roszak","year":"2006","journal-title":"Automatica"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1016\/j.csda.2008.06.023","article-title":"Computing -optimal experimental designs using the simplex method of linear programming","volume":"53","author":"Harman","year":"2008","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"830","DOI":"10.1016\/j.applthermaleng.2017.07.045","article-title":"Parametric optimization of steam cycle in PWR nuclear power plant using improved genetic-simplex algorithm","volume":"125","author":"Wang","year":"2017","journal-title":"Appl. Therm. Eng."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1360\/aas-007-0091","article-title":"Hybrid Simplex-improved Genetic Algorithm for Global Numerical Optimization","volume":"33","author":"Ren","year":"2007","journal-title":"Acta Autom. Sin."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/j.jelechem.2003.11.060","article-title":"Application of the hybrid genetic-simplex algorithm for deconvolution of electrochemical responses in SDLSV method","volume":"567","author":"Kasprzyk","year":"2004","journal-title":"J. Electroanal. Chem."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2550","DOI":"10.1016\/j.camwa.2008.05.006","article-title":"Genetic algorithm based on simplex method for solving linear-quadratic bilevel programming problem","volume":"56","author":"Wang","year":"2008","journal-title":"Comput. Math. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1016\/j.jmsy.2013.05.010","article-title":"Methodology for shape optimization of ultrasonic amplifier using genetic algorithms and simplex method","volume":"32","author":"Deibel","year":"2013","journal-title":"J. Manuf. Syst."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1016\/S0925-2312(03)00374-6","article-title":"Support vector machine models in drug design: Applications to drug transport processes and QSAR using simplex optimisations and variable selection","volume":"55","author":"Norinder","year":"2003","journal-title":"Neurocomputing"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"324","DOI":"10.1016\/j.ins.2018.12.026","article-title":"A novel perspective on multiclass classification: Regular simplex support vector machine","volume":"480","author":"Tang","year":"2019","journal-title":"Inf. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1016\/j.apradiso.2005.03.009","article-title":"Prediction of peak-to-background ratio in gamma-ray spectrometry using simplex optimized artificial neural network","volume":"63","author":"Onjia","year":"2005","journal-title":"Appl. Radiat. Isot."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"308","DOI":"10.1016\/j.nima.2006.03.047","article-title":"Simplex optimization of artificial neural networks for the prediction of minimum detectable activity in gamma-ray spectrometry","volume":"564","author":"Onjia","year":"2006","journal-title":"Nucl. Instrum. Methods Phys. Res. Sect. A Accel. Spectrometers Detect. Assoc. Equip."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1961","DOI":"10.1016\/j.fss.2007.05.005","article-title":"Duality results and a dual simplex method for linear programming problems with trapezoidal fuzzy variables","volume":"158","author":"Nasseri","year":"2007","journal-title":"Fuzzy Sets Syst."},{"key":"ref_18","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_19","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1108\/GS-12-2015-0079","article-title":"Adjusting the errors of the GM (1, 2) grey model in the financial data series using an adaptive fuzzy controller","volume":"6","author":"Bolos","year":"2016","journal-title":"Grey Syst. Theory Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1016\/S0375-9601(00)00015-3","article-title":"Slow regularization through chaotic oscillation transfer in an unidirectional chain of Hindmarsh\u2013Rose models","volume":"266","author":"Rabinovich","year":"2000","journal-title":"Phys. Lett. A"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/j.ins.2016.04.047","article-title":"Consistency analysis and group decision making based on triangular fuzzy additive reciprocal preference relations","volume":"361","author":"Wang","year":"2016","journal-title":"Inf. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1016\/j.jhydrol.2018.04.069","article-title":"An interval-valued triangular fuzzy modified multi-attribute preference model for prioritization of groundwater resources management","volume":"562","author":"Ren","year":"2018","journal-title":"J. Hydrol."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Bolos, M.I., Bradea, I.A., and Delcea, C. (2019). Modeling the Performance Indicators of Financial Assets with Neutrosophic Fuzzy Numbers. Symmetry, 11.","DOI":"10.3390\/sym11081021"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1016\/j.cie.2016.10.027","article-title":"A geometric approach for ranking interval-valued intuitionistic fuzzy numbers with an application to group decision-making","volume":"102","author":"Yue","year":"2016","journal-title":"Comput. Ind. Eng."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/j.knosys.2012.11.007","article-title":"Multi-criteria decision making method based on possibility degree of interval type-2 fuzzy number","volume":"43","author":"Hu","year":"2013","journal-title":"Knowl. Based Syst."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"256","DOI":"10.1016\/j.ins.2019.09.020","article-title":"A decision-making computational methodology for a class of type-2 fuzzy intervals: An interval-based approach","volume":"510","author":"Boukezzoula","year":"2020","journal-title":"Inf. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1016\/j.eswa.2018.06.008","article-title":"NEAT F-PROMETHEE\u2014A new fuzzy multiple criteria decision making method based on the adjustment of mapping trapezoidal fuzzy numbers","volume":"110","author":"Ziemba","year":"2018","journal-title":"Expert Syst. Appl."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/j.asoc.2019.03.034","article-title":"Word-level neutrosophic sentiment similarity","volume":"80","author":"Smarandache","year":"2019","journal-title":"Appl. Soft Comput."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"216","DOI":"10.1016\/j.cogsys.2018.10.023","article-title":"A hybrid neutrosophic multiple criteria group decision making approach for project selection","volume":"57","author":"Atef","year":"2019","journal-title":"Cogn. Syst. Res."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Fahmi, A., Amin, F., Smarandache, F., Khan, M., and Hassan, N. (2018). Triangular Cubic Hesitant Fuzzy Einstein Hybrid Weighted Averaging Operator and Its Application to Decision Making. Symmetry, 10.","DOI":"10.3390\/sym10110658"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Bolo\u0219, M.-I., Bradea, I.-A., and Delcea, C. (2019). Neutrosophic Portfolios of Financial Assets. Minimizing the Risk of Neutrosophic Portfolios. Mathematics, 7.","DOI":"10.3390\/math7111046"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"105882","DOI":"10.1016\/j.asoc.2019.105882","article-title":"A fully fuzzy best\u2013worst multi attribute decision making method with triangular fuzzy number: A case study of maintenance assessment in the hospitals","volume":"86","author":"Karimi","year":"2019","journal-title":"Appl. Soft Comput."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1016\/j.asoc.2018.01.020","article-title":"A group decision making model based on triangular fuzzy additive reciprocal matrices with additive approximation-consistency","volume":"65","author":"Liu","year":"2018","journal-title":"Appl. Soft Comput."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"286","DOI":"10.1016\/j.cie.2015.12.027","article-title":"Some new generalized aggregation operators for triangular intuitionistic fuzzy numbers and application to multi-attribute group decision making","volume":"93","author":"Wan","year":"2016","journal-title":"Comput. Ind. Eng."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1080\/01969722.2017.1412890","article-title":"Improved Accuracy Function for Interval-Valued Intuitionistic Fuzzy Sets and Its Application to Multi\u2013Attributes Group Decision Making","volume":"49","author":"Joshi","year":"2018","journal-title":"Cybern. Syst."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.knosys.2013.06.019","article-title":"The extended VIKOR method for multi-attribute group decision making with triangular intuitionistic fuzzy numbers","volume":"52","author":"Wan","year":"2013","journal-title":"Knowl. Based Syst."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"643","DOI":"10.1016\/j.cie.2019.06.031","article-title":"Approach for multi-attribute decision making based on novel intuitionistic fuzzy entropy and evidential reasoning","volume":"135","author":"Yuan","year":"2019","journal-title":"Comput. Ind. Eng."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"268","DOI":"10.1016\/j.scient.2011.03.022","article-title":"Dynamic multi-attribute decision making model based on triangular intuitionistic fuzzy numbers","volume":"18","author":"Chen","year":"2011","journal-title":"Sci. Iran."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/j.asoc.2017.01.041","article-title":"A TODIM-based multi-criteria group decision making with triangular intuitionistic fuzzy numbers","volume":"55","author":"Qin","year":"2017","journal-title":"Appl. Soft Comput."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1016\/j.ins.2016.04.008","article-title":"New method for solving Fuzzy transportation problems with LR flat fuzzy numbers","volume":"357","author":"Ebrahimnejad","year":"2016","journal-title":"Inf. Sci."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1016\/j.eswa.2017.03.040","article-title":"Constrained fuzzy arithmetic approach to fuzzy transportation problems with fuzzy decision variables","volume":"81","author":"Subulan","year":"2017","journal-title":"Expert Syst. Appl."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"4612","DOI":"10.1016\/j.apm.2015.10.053","article-title":"A simple parametric method to generate all optimal solutions of fuzzy solid transportation problem","volume":"40","author":"Kocken","year":"2016","journal-title":"Appl. Math. Model."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"4388","DOI":"10.1016\/j.apm.2014.02.024","article-title":"A novel method for solving linear programming problems with symmetric trapezoidal fuzzy numbers","volume":"38","author":"Ebrahimnejad","year":"2014","journal-title":"Appl. Math. Model."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"260","DOI":"10.1016\/j.procs.2019.11.283","article-title":"Solution approach to a special class of full fuzzy linear programming problems","volume":"162","author":"Dzitac","year":"2019","journal-title":"Procedia Comput. Sci."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1878","DOI":"10.1016\/j.mcm.2011.01.013","article-title":"Sensitivity analysis in fuzzy number linear programming problems","volume":"53","author":"Ebrahimnejad","year":"2011","journal-title":"Math. Comput. Model."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Zadeh, L.A., and Aliev, R.A. (2018). Fuzzy Logic Theory and Applications: Part I and Part II, World Scientific.","DOI":"10.1142\/10936"},{"key":"ref_47","unstructured":"Georgescu, I. (2015). Inteligen\u0163\u0103 Computa\u0163ional\u0103 (Engl. \u201cComputational Intelligence\u201d), Editura ASE."},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Zadeh, L.A., Klir, G.J., and Yuan, B. (1996). Fuzzy Sets, Fuzzy Logic, and Fuzzy Systems: Selected Papers, World Scientific.","DOI":"10.1142\/2895"},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Sangaiah, A.K., Gao, X.-Z., and Abraham, A. (2017). Handbook of Research on Fuzzy and Rough Set Theory in Organizational Decision Making, IGI Global.","DOI":"10.4018\/978-1-5225-1008-6"},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Bojadziev, G., and Bojadziev, M. (2007). Fuzzy Logic for Business, Finance, and Management, World Scientific.","DOI":"10.1142\/6451"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"613","DOI":"10.1080\/00207727808941724","article-title":"Operations on fuzzy numbers","volume":"9","author":"Dubois","year":"1978","journal-title":"Int. J. Syst. Sci."},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Vanderbei, R.J. (2013). Linear Programming: Foundations and Extensions, Springer.","DOI":"10.1007\/978-1-4614-7630-6"},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Thie, P.R., and Keough, G.E. (2008). An Introduction to Linear Programming and Game Theory, Wiley. [3rd ed.].","DOI":"10.1002\/9781118165447"},{"key":"ref_54","unstructured":"Greenwald, D.U. (2019). Linear Programming: An Explanation of the Simplex Algorithm, Forgotten Books."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/1\/121\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T14:05:24Z","timestamp":1760364324000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/1\/121"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1,19]]},"references-count":54,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,1]]}},"alternative-id":["e22010121"],"URL":"https:\/\/doi.org\/10.3390\/e22010121","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,1,19]]}}}