{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:33:33Z","timestamp":1760060013287,"version":"build-2065373602"},"reference-count":38,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,7,24]],"date-time":"2025-07-24T00:00:00Z","timestamp":1753315200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Modeling and valuation of financial instruments for climate and energy risk mitigation","award":["MUR PRIN 2022","2022FPLY97-CUP J53D23004530006"],"award-info":[{"award-number":["MUR PRIN 2022","2022FPLY97-CUP J53D23004530006"]}]},{"name":"European Union\u2014NextGenerationEU under the National Recovery and Resilience Plan (PNRR) M4C2","award":["MUR PRIN 2022","2022FPLY97-CUP J53D23004530006"],"award-info":[{"award-number":["MUR PRIN 2022","2022FPLY97-CUP J53D23004530006"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Interval Linear Programming (ILP) presents several compelling challenges when applied to real-world problems that cannot be easily captured by traditional robust uncertainty models. In this paper, we propose a novel solution method that employs a comparison index for interval ordering based on the generalized Hukuhara difference. This approach proves to be highly effective in comparing solutions within ILP frameworks. Additionally, we discuss the robustness of the proposed methodology and its implications for decision-making under uncertainty.<\/jats:p>","DOI":"10.3390\/axioms14080569","type":"journal-article","created":{"date-parts":[[2025,7,24]],"date-time":"2025-07-24T15:50:24Z","timestamp":1753372224000},"page":"569","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Comparison Index for Costs of Interval Linear Programming Models"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8944-2192","authenticated-orcid":false,"given":"Maria Letizia","family":"Guerra","sequence":"first","affiliation":[{"name":"Department of Statistical Sciences \u201cPaolo Fortunati\u201d, University of Bologna, 40126 Bologna, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0088-4005","authenticated-orcid":false,"given":"Laerte","family":"Sorini","sequence":"additional","affiliation":[{"name":"Department of Economics, Society and Politics, University of Urbino \u201cCarlo Bo\u201d, 61029 Urbino, Italy"}]},{"given":"Luciano","family":"Stefanini","sequence":"additional","affiliation":[{"name":"Department of Economics, Society and Politics, University of Urbino \u201cCarlo Bo\u201d, 61029 Urbino, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Dantzig, G.B. (1963). Linear Programming and Extensions, Princeton University Press.","DOI":"10.7249\/R366"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Moore, R.E. (1979). Methods and Applications of Interval Analysis, Philadelphia, Society for Industrial and Applied Mathematics.","DOI":"10.1137\/1.9781611970906"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Guerra, M.L., and Stefanini, L. (2011, January 11\u201315). A comparison index for interval ordering. Proceedings of the IEEE SSCI 2011, Symposium Series on Computational Intelligence (FOCI 2011), Paris, France.","DOI":"10.1109\/FOCI.2011.5949477"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1931","DOI":"10.1007\/s00500-012-0866-9","article-title":"A comparison index for interval ordering based on generalized Hukuhara difference","volume":"16","author":"Guerra","year":"2012","journal-title":"Soft Comput."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1016\/0377-2217(90)90375-L","article-title":"Multiobjective programming in optimization of the interval objective function","volume":"48","author":"Ishibuchi","year":"1990","journal-title":"Eur. J. Oper. Res."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1016\/j.cam.2015.07.034","article-title":"Inverse linear programming with interval coefficients","volume":"292","author":"Mostafaee","year":"2016","journal-title":"J. Comput. Appl. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1016\/j.fss.2014.07.013","article-title":"Interval and fuzzy average internal Rate of Return for investment appraisal","volume":"257","author":"Guerra","year":"2014","journal-title":"Fuzzy Sets Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1016\/0165-0114(94)90097-3","article-title":"Interval number, fuzzy number linear programming","volume":"66","author":"Tong","year":"1994","journal-title":"Fuzzy Sets Syst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1016\/S0377-2217(99)00319-7","article-title":"On comparing interval numbers","volume":"127","author":"Sengupta","year":"2000","journal-title":"Eur. J. Oper. Res."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Sengupta, A., and Pal, T.K. (2009). Fuzzy Preference Ordering of Interval Numbers in Decision Problems, Springer.","DOI":"10.1007\/978-3-540-89915-0"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1893","DOI":"10.1007\/s11590-012-0530-4","article-title":"The optimal solution set of the interval linear programming problems","volume":"7","author":"Allahdadi","year":"2013","journal-title":"Optim. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"224","DOI":"10.1016\/j.ins.2015.12.037","article-title":"Improving the modified interval linear programming method by new techniques","volume":"339","author":"Allahdadi","year":"2016","journal-title":"Inf. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1205","DOI":"10.1007\/s12351-018-0383-4","article-title":"Solving methods for interval linear programming problem: A review and an improved method","volume":"20","author":"Nehi","year":"2020","journal-title":"Oper. Res. Int. J."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/j.eswa.2017.10.020","article-title":"Solving the interval linear programming problem: A new algorithm for a general case","volume":"93","author":"Ashayerinasab","year":"2018","journal-title":"Expert Syst. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1434","DOI":"10.1016\/j.ejor.2005.12.042","article-title":"Multiple objective linear programming models with interval coefficients\u2014An illustrated overview","volume":"181","author":"Oliveira","year":"2007","journal-title":"Eur. J. Oper. Res."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ejor.2007.03.031","article-title":"A nonlinear interval number programming method for uncertain optimization problems","volume":"188","author":"Jiang","year":"2008","journal-title":"Eur. Oper. Res."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"873","DOI":"10.1007\/s40815-020-00800-5","article-title":"Obtaining Efficient Solutions of Interval Multi-objective Linear Programming Problems","volume":"22","author":"Bata","year":"2020","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1016\/j.laa.2015.06.029","article-title":"Necessary and sufficient conditions for unified optimality of interval linear program in the general form","volume":"484","author":"Li","year":"2015","journal-title":"Linear Algebr. Its Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"893","DOI":"10.1007\/s11590-011-0315-1","article-title":"Complexity of necessary efficiency in interval linear programming and multiobjective linear programming","volume":"6","year":"2012","journal-title":"Optim. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"705","DOI":"10.1016\/j.ejor.2016.04.032","article-title":"Robust optimal solutions in interval linear programming with forall-exists quantifiers","volume":"254","year":"2016","journal-title":"Eur. J. Oper. Res."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Alolyan, I. (2013, January 23\u201325). Algorithm for Interval Linear Programming Involving Interval Constraints. Proceedings of the WCECS 2013, San Francisco, CA, USA.","DOI":"10.1109\/IFSA-NAFIPS.2013.6608585"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1016\/j.ins.2018.01.006","article-title":"An ensemble framework for assessing solutions of interval programming problems","volume":"436\u2013437","author":"Sun","year":"2018","journal-title":"Inf. Sci."},{"key":"ref_23","first-page":"859","article-title":"A study of interval optimization problems","volume":"15","year":"2019","journal-title":"Optim. Lett."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1287\/mnsc.17.4.B141","article-title":"Decision-Making in a Fuzzy Environment","volume":"17","author":"Bellman","year":"1970","journal-title":"Manag. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1080\/01969727308545912","article-title":"On Fuzzy-Mathematical Programming","volume":"3","author":"Tanaka","year":"1974","journal-title":"J. Cybern."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/0165-0114(78)90031-3","article-title":"Fuzzy programming and linear programming with several objective functions","volume":"1","author":"Zimmermann","year":"1978","journal-title":"Fuzzy Sets Syst."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Pedrycz, W., Skowron, A., and Kreynovich, V. (2008). Fuzzy Linear Programming. Handbook of Granular Computing, Wiley & Sons.","DOI":"10.1002\/9780470724163"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1016\/j.fss.2015.08.023","article-title":"Progress on Fuzzy Mathematical Programming: A personal perspective","volume":"281","author":"Verdegay","year":"2015","journal-title":"Fuzzy Sets Syst."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1007\/s10898-019-00841-7","article-title":"On generating the set of nondominated solutions of a linear programming problem with parameterized fuzzy numbers","volume":"77","year":"2020","journal-title":"J. Glob. Optim."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"7421","DOI":"10.1002\/mma.4882","article-title":"Nondominated solutions in a fully fuzzy linear programming problem","volume":"41","year":"2018","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"274","DOI":"10.1007\/s40314-023-02408-5","article-title":"Developing solution algorithm for LR-type fully interval-valued intuitionistic fuzzy linear programming problems using lexicographic-ranking method","volume":"42","author":"Manisha","year":"2023","journal-title":"Comput. Appl. Math."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Supian, S., Megantara, T.R., and Bon, A.T. (2024). Ride-Hailing, Matching with Uncertain Travel Time: A Novel Interval-Valued Fuzzy Multi-Objective Linear Programming Approach. Mathematics, 12.","DOI":"10.3390\/math12091355"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1564","DOI":"10.1016\/j.fss.2009.06.009","article-title":"A generalization of Hukuhara difference and division for interval and fuzzy arithmetic","volume":"161","author":"Stefanini","year":"2010","journal-title":"Fuzzy Sets Syst."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Guerra, M.L., Sorini, L., and Stefanini, L. (2017, January 9\u201312). A new approach to linear programming with interval costs. Proceedings of the IEEE International Conference on Fuzzy Systems, Naples, Italy.","DOI":"10.1109\/FUZZ-IEEE.2017.8015661"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Stefanini, L., Guerra, M.L., and Amicizia, B. (2019). Interval Analysis and Calculus for Interval-Valued Functions of a Single Variable. Part I: Partial Orders, gH-Derivative, Monotonicity. Axioms, 8.","DOI":"10.3390\/axioms8040113"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Stefanini, L., Sorini, L., and Amicizia, B. (2019). Interval Analysis and Calculus for Interval-Valued Functions of a Single Variable\u2014Part II: Extremal Points, Convexity, Periodicity. Axioms, 8.","DOI":"10.3390\/axioms8040114"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"594","DOI":"10.1016\/0377-2217(95)00055-0","article-title":"Multiobjective programming in optimization of interval objective functions\u2014A generalized approach","volume":"94","author":"Chanas","year":"1996","journal-title":"Eur. J. Oper. Res."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1016\/S0377-2217(98)00044-7","article-title":"Multiobjective transportation problem with interval cost, source and destination parameters","volume":"117","author":"Das","year":"1999","journal-title":"Eur. J. Oper. Res."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/8\/569\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:15:42Z","timestamp":1760033742000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/8\/569"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,24]]},"references-count":38,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2025,8]]}},"alternative-id":["axioms14080569"],"URL":"https:\/\/doi.org\/10.3390\/axioms14080569","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,7,24]]}}}