{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,13]],"date-time":"2026-04-13T17:35:07Z","timestamp":1776101707504,"version":"3.50.1"},"reference-count":46,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2022,2,20]],"date-time":"2022-02-20T00:00:00Z","timestamp":1645315200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,2,20]],"date-time":"2022-02-20T00:00:00Z","timestamp":1645315200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Complex Intell. Syst."],"published-print":{"date-parts":[[2022,8]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This paper presents an optimization method to solve a multi-objective model of a bi-level linear programming problem with intuitionistic fuzzy coefficients. The idea is based on TOPSIS (technique for order preference by similarity to ideal solution) method. TOPSIS method is a multiple criteria method that identifies a satisfactory solution from a given set of alternatives based on the minimization of distance from an ideal point and maximization of distance from the nadir point simultaneously. A new model of multi-objective bi-level programming problem in an intuitionistic fuzzy environment has been considered. The problem is first reduced to a conventional multi-objective bi-level linear programming problem using accuracy function. Then the modified TOPSIS method is proposed to solve the problem at both the leader and the follower level where various linear\/non-linear membership functions are used to represent the flexibility in the approach of decision-makers (DMs). The problem is solved hierarchically, i.e., first the problem at the leader level is solved and then the feasible region is extended by relaxing the decision variables controlled by the leader. The feasible region is extended to obtain a satisfactory solution for the DMs at both levels. Finally, the application of the proposed approach in the production planning of a company has been presented. An illustrative numerical example is also given to explain the methodology defined in this paper.<\/jats:p>","DOI":"10.1007\/s40747-022-00662-4","type":"journal-article","created":{"date-parts":[[2022,2,20]],"date-time":"2022-02-20T06:08:26Z","timestamp":1645337306000},"page":"3263-3278","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["A novel multi-objective bi-level programming problem under intuitionistic fuzzy environment and its application in production planning problem"],"prefix":"10.1007","volume":"8","author":[{"given":"V. P.","family":"Singh","sequence":"first","affiliation":[]},{"given":"Kirti","family":"Sharma","sequence":"additional","affiliation":[]},{"given":"Debjani","family":"Chakraborty","sequence":"additional","affiliation":[]},{"given":"Ali","family":"Ebrahimnejad","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,2,20]]},"reference":[{"issue":"1","key":"662_CR1","doi-asserted-by":"publisher","first-page":"59","DOI":"10.1016\/0305-0548(82)90006-5","volume":"9","author":"W Candler","year":"1982","unstructured":"Candler W, Townsley R (1982) A linear two-level programming problem. Comp Oper Res 9(1):59\u201376","journal-title":"Comp Oper Res"},{"issue":"2","key":"662_CR2","first-page":"125","volume":"42","author":"UP Wen","year":"1991","unstructured":"Wen UP, Hsu ST (1991) Linear bi-level programming problems\u2019 a review. J Oper Res Soc 42(2):125\u2013133","journal-title":"J Oper Res Soc"},{"key":"662_CR3","doi-asserted-by":"crossref","unstructured":"Singh VP, Chakraborty D (2017) Solving bi-level programming problem with fuzzy random variable coefficients. J Intell Fuzzy Syst 32(1):521\u2013528","DOI":"10.3233\/JIFS-152354"},{"issue":"3","key":"662_CR4","first-page":"333","volume":"52","author":"S Dempe","year":"2003","unstructured":"Dempe S (2003) Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. J Math Program Oper Res 52(3):333\u2013359","journal-title":"J Math Program Oper Res"},{"key":"662_CR5","doi-asserted-by":"crossref","unstructured":"Singh VP, Chakraborty D (2019) Bi-level optimization based on fuzzy if-then rule. Croat Oper Res Rev 10:315\u2013328","DOI":"10.17535\/crorr.2019.0025"},{"key":"662_CR6","doi-asserted-by":"crossref","unstructured":"Moitra BN, Pal BB (2002) A fuzzy goal programming approach for solving bilevel programming problems. In: AFSS International Conference on Fuzzy Systems, pp 91\u201398. Springer","DOI":"10.1007\/3-540-45631-7_13"},{"issue":"2","key":"662_CR7","doi-asserted-by":"publisher","first-page":"368","DOI":"10.1016\/j.ejor.2007.12.019","volume":"194","author":"SR Arora","year":"2009","unstructured":"Arora SR, Gupta R (2009) Interactive fuzzy goal programming approach for bilevel programming problem. Eur J Oper Res 194(2):368\u2013376","journal-title":"Eur J Oper Res"},{"key":"662_CR8","doi-asserted-by":"crossref","unstructured":"Wan Z, Wang G, Hou K (2008) An interactive fuzzy decision making method for a class of bilevel programming. In: 2008 Fifth International Conference on Fuzzy Systems and Knowledge Discovery, volume\u00a01, pp 559\u2013564. IEEE","DOI":"10.1109\/FSKD.2008.537"},{"issue":"13","key":"662_CR9","doi-asserted-by":"publisher","first-page":"3136","DOI":"10.1016\/j.apm.2013.11.008","volume":"38","author":"Y Zheng","year":"2014","unstructured":"Zheng Y, Liu J, Wan Z (2014) Interactive fuzzy decision making method for solving bilevel programming problem. Appl Math Model 38(13):3136\u20133141","journal-title":"Appl Math Model"},{"issue":"2","key":"662_CR10","doi-asserted-by":"publisher","first-page":"1151","DOI":"10.1016\/j.ejor.2005.08.024","volume":"176","author":"S Pramanik","year":"2007","unstructured":"Pramanik S, Roy TK (2007) Fuzzy goal programming approach to multilevel programming problems. Eur J Oper Res 176(2):1151\u20131166","journal-title":"Eur J Oper Res"},{"issue":"2","key":"662_CR11","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1016\/S0165-0114(02)00362-7","volume":"136","author":"S Sinha","year":"2003","unstructured":"Sinha S (2003) Fuzzy programming approach to multi-level programming problems. Fuzzy Sets Syst 136(2):189\u2013202","journal-title":"Fuzzy Sets Syst"},{"issue":"1","key":"662_CR12","first-page":"1","volume":"9","author":"SK Maiti","year":"2019","unstructured":"Maiti SK, Roy SK (2019) Bi-level programming for stackelberg game with intuitionistic fuzzy number: a ranking approach. J Oper Res Soc China 9(1):1\u201319","journal-title":"J Oper Res Soc China"},{"issue":"1","key":"662_CR13","doi-asserted-by":"publisher","first-page":"141674","DOI":"10.1109\/ACCESS.2020.3013568","volume":"8","author":"R Said","year":"2020","unstructured":"Said R, Bechikh S, Louati A, Aldaej A, Said LB (2020) Solving combinatorial multi-objective bi-level optimization problems using multiple populations and migration schemes. IEEE Access 8(1):141674\u2013141695","journal-title":"IEEE Access"},{"issue":"3","key":"662_CR14","first-page":"563","volume":"27","author":"R Vidhya","year":"2017","unstructured":"Vidhya R, Irene\u00a0Hepzibah R (2017) A comparative study on interval arithmetic operations with intuitionistic fuzzy numbers for solving an intuitionistic fuzzy multi-objective linear programming problem. Int J Appl Math 27(3):563\u2013573","journal-title":"Int J Appl Math"},{"issue":"1","key":"662_CR15","first-page":"1","volume":"2016","author":"A Biswas","year":"2016","unstructured":"Biswas A, De AK (2016) An efficient ranking technique for intuitionistic fuzzy numbers with its application in chance constrained bilevel programming. Adv Fuzzy Syst 2016(1):1\u201312","journal-title":"Adv Fuzzy Syst"},{"key":"662_CR16","doi-asserted-by":"crossref","unstructured":"Nasseri SH, Goli M, Bavandi S (2018) An approach for solving linear programming problem with intuitionistic fuzzy objective coefficient. In: 2018 6th Iranian Joint Congress on Fuzzy and Intelligent Systems (CFIS), pp 105\u2013107. IEEE","DOI":"10.1109\/CFIS.2018.8336644"},{"issue":"4","key":"662_CR17","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s40747-021-00299-9","volume":"7","author":"F Ahmad","year":"2021","unstructured":"Ahmad F (2021) Robust neutrosophic programming approach for solving intuitionistic fuzzy multiobjective optimization problems. Complex Intell Syst 7(4):1\u201320","journal-title":"Complex Intell Syst"},{"issue":"2","key":"662_CR18","doi-asserted-by":"publisher","first-page":"1009","DOI":"10.1007\/s40747-020-00251-3","volume":"7","author":"S Ghosh","year":"2021","unstructured":"Ghosh S, Roy SK, Ebrahimnejad A, Verdegay JL (2021) Multi-objective fully intuitionistic fuzzy fixed-charge solid transportation problem. Complex Intell Syst 7(2):1009\u20131023","journal-title":"Complex Intell Syst"},{"issue":"4","key":"662_CR19","doi-asserted-by":"publisher","first-page":"447","DOI":"10.1007\/s10700-017-9280-1","volume":"17","author":"A Ebrahimnejad","year":"2018","unstructured":"Ebrahimnejad A, Verdegay JL (2018) A new approach for solving fully intuitionistic fuzzy transportation problems. Fuzzy Optim Decis Mak 17(4):447\u2013474","journal-title":"Fuzzy Optim Decis Mak"},{"issue":"4","key":"662_CR20","doi-asserted-by":"publisher","first-page":"2119","DOI":"10.1007\/s40747-021-00326-9","volume":"7","author":"S Gupta","year":"2021","unstructured":"Gupta S, Haq A, Ali I, Sarkar B (2021) Significance of multi-objective optimization in logistics problem for multi-product supply chain network under the intuitionistic fuzzy environment. Complex Intell Syst 7(4):2119\u20132139","journal-title":"Complex Intell Syst"},{"issue":"13","key":"662_CR21","doi-asserted-by":"publisher","first-page":"8635","DOI":"10.1007\/s00500-021-05791-5","volume":"25","author":"NA Alessa","year":"2021","unstructured":"Alessa NA (2021) Bi-level linear programming of intuitionistic fuzzy. Soft Comput 25(13):8635\u20138641","journal-title":"Soft Comput"},{"issue":"1","key":"662_CR22","doi-asserted-by":"publisher","first-page":"258","DOI":"10.1016\/j.eswa.2016.10.063","volume":"72","author":"X Zhao","year":"2017","unstructured":"Zhao X, Zheng Y, Wan Z (2017) Interactive intuitionistic fuzzy methods for multilevel programming problems. Expert Syst Appl 72(1):258\u2013268","journal-title":"Expert Syst Appl"},{"key":"662_CR23","first-page":"1","volume":"1","author":"Y Lv","year":"2020","unstructured":"Lv Y, Wang S, Gao Z, Cheng G, Huang G, He Z (2020) A sustainable road pricing oriented bilevel optimization approach under multiple environmental uncertainties. Int J Sustain Transp 1:1\u201328","journal-title":"Int J Sustain Transp"},{"issue":"2","key":"662_CR24","first-page":"45","volume":"17","author":"S Aggarwal","year":"2013","unstructured":"Aggarwal S, Gupta C (2013) Bi-level multi-objective linear programming under intuitionistic fuzzy environment. Int J Pure Appl Sci Technol 17(2):45\u201361","journal-title":"Int J Pure Appl Sci Technol"},{"issue":"Preprint","key":"662_CR25","first-page":"1","volume":"1","author":"M Malik","year":"2021","unstructured":"Malik M, Gupta S.K, Ahmad I (2021) A new approach to solve fully intuitionistic fuzzy linear programming problem with unrestricted decision variables. J Intell Fuzzy Syst 1(Preprint):1\u201314","journal-title":"J Intell Fuzzy Syst"},{"issue":"1","key":"662_CR26","first-page":"1","volume":"1","author":"M Abbassi","year":"2021","unstructured":"Abbassi M, Chaabani A, Said LB (2021) An efficient chemical reaction algorithm for multi-objective combinatorial bi-level optimization. Eng Optim 1(1):1\u201322","journal-title":"Eng Optim"},{"issue":"1","key":"662_CR27","doi-asserted-by":"publisher","first-page":"100492","DOI":"10.1016\/j.esr.2020.100492","volume":"30","author":"A Del Valle","year":"2020","unstructured":"Del Valle A, Wogrin S, Reneses J (2020) Multi-objective bi-level optimization model for the investment in gas infrastructures. Energy Strategy Rev 30(1):100492","journal-title":"Energy Strategy Rev"},{"issue":"1","key":"662_CR28","first-page":"1","volume":"235","author":"C Cao","year":"2021","unstructured":"Cao C, Liu Y, Tang O, Gao X (2021) A fuzzy bi-level optimization model for multi-period post-disaster relief distribution in sustainable humanitarian supply chains. Int J Prod Econ 235(1):1\u201314","journal-title":"Int J Prod Econ"},{"key":"662_CR29","doi-asserted-by":"crossref","unstructured":"Deb K, Sinha A, Malo P, Lu Z (2020) Approximate bilevel optimization with population-based evolutionary algorithms. In: Bilevel Optimization, pp 361\u2013402. Springer","DOI":"10.1007\/978-3-030-52119-6_13"},{"issue":"1","key":"662_CR30","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1016\/0165-0114(78)90031-3","volume":"1","author":"HJ Zimmermann","year":"1978","unstructured":"Zimmermann HJ (1978) Fuzzy programming and linear programming with several objective functions. Fuzzy Sets Syst 1(1):45\u201355","journal-title":"Fuzzy Sets Syst"},{"issue":"2","key":"662_CR31","doi-asserted-by":"publisher","first-page":"617","DOI":"10.1007\/s10479-017-2734-6","volume":"287","author":"TD Chuong","year":"2020","unstructured":"Chuong TD (2020) Optimality conditions for nonsmooth multiobjective bilevel optimization problems. Ann Oper Res 287(2):617\u2013642","journal-title":"Ann Oper Res"},{"issue":"9","key":"662_CR32","doi-asserted-by":"publisher","first-page":"2013","DOI":"10.1080\/02331934.2020.1768253","volume":"70","author":"N Gadhi","year":"2020","unstructured":"Gadhi N, Hamdaoui K, El Idrissi M (2020) Optimality conditions for a multiobjective bilevel optimization problem involving set valued constraints. Optimization 70(9):2013\u20132029","journal-title":"Optimization"},{"key":"662_CR33","doi-asserted-by":"publisher","DOI":"10.1201\/b11032","volume-title":"Multiple attribute decision making: methods and applications","author":"GH Tzeng","year":"2011","unstructured":"Tzeng GH, Huang JJ (2011) Multiple attribute decision making: methods and applications. CRC press, Boca Raton"},{"issue":"1","key":"662_CR34","first-page":"243","volume":"162","author":"MA Abo-Sinna","year":"2005","unstructured":"Abo-Sinna MA, Amer AH (2005) Extensions of topsis for multi-objective large-scale nonlinear programming problems. Appl Math Comput 162(1):243\u2013256","journal-title":"Appl Math Comput"},{"issue":"4","key":"662_CR35","first-page":"1","volume":"43","author":"MA Abo-Sinna","year":"2000","unstructured":"Abo-Sinna MA (2000) Extensions of the topsis for multi-objective dynamic programming problems under fuzziness. Adv Model Anal 43(4):1\u201324","journal-title":"Adv Model Anal"},{"issue":"1","key":"662_CR36","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/S0165-0114(97)00377-1","volume":"114","author":"CT Chen","year":"2000","unstructured":"Chen CT (2000) Extensions of the topsis for group decision-making under fuzzy environment. Fuzzy Sets Syst 114(1):1\u20139","journal-title":"Fuzzy Sets Syst"},{"issue":"3","key":"662_CR37","doi-asserted-by":"publisher","first-page":"1004","DOI":"10.1016\/j.apm.2012.03.002","volume":"37","author":"IA Baky","year":"2013","unstructured":"Baky IA, Abo-Sinna MA (2013) Topsis for bi-level modm problems. Appl Math Model 37(3):1004\u20131015","journal-title":"Appl Math Model"},{"key":"662_CR38","first-page":"1","volume":"20","author":"K Atanassov","year":"2016","unstructured":"Atanassov K (2016) Intuitionistic fuzzy sets. Int J Bioautom 20:1","journal-title":"Int J Bioautom"},{"issue":"4","key":"662_CR39","doi-asserted-by":"publisher","first-page":"B-141","DOI":"10.1287\/mnsc.17.4.B141","volume":"17","author":"RE Bellman","year":"1970","unstructured":"Bellman RE, Zadeh la (1970) Decision-making in a fuzzy environment. Manag Sci 17(4):B-141","journal-title":"Manag Sci"},{"issue":"12","key":"662_CR40","first-page":"1802","volume":"17","author":"R Parvathi","year":"2012","unstructured":"Parvathi R, Malathi C (2012) Intuitionistic fuzzy linear programming problems. World Appl Sci J 17(12):1802\u20131807","journal-title":"World Appl Sci J"},{"key":"662_CR41","unstructured":"Jana B, Roy TK (2007) Multi-objective intuitionistic fuzzy linear programming and its application in transportation model. Not Intuit Fuzzy Sets 13(1):34\u201351"},{"issue":"5","key":"662_CR42","doi-asserted-by":"publisher","first-page":"1430","DOI":"10.1007\/s40815-020-00878-x","volume":"22","author":"\u00d6 Atan","year":"2020","unstructured":"Atan \u00d6, Kutlu F, Castillo O (2020) Intuitionistic fuzzy sliding controller for uncertain hyperchaotic synchronization. Int J Fuzzy Syst 22(5):1430\u20131443","journal-title":"Int J Fuzzy Syst"},{"issue":"1","key":"662_CR43","doi-asserted-by":"publisher","first-page":"821","DOI":"10.3233\/JIFS-179451","volume":"38","author":"O Castillo","year":"2020","unstructured":"Castillo O, Kutlu F, Atan \u00d6 (2020) Intuitionistic fuzzy control of twin rotor multiple input multiple output systems. J Intell Fuzzy Syst 38(1):821\u2013833","journal-title":"J Intell Fuzzy Syst"},{"issue":"1","key":"662_CR44","doi-asserted-by":"publisher","first-page":"53","DOI":"10.1007\/s00500-019-04286-8","volume":"24","author":"F Kutlu","year":"2020","unstructured":"Kutlu F, Atan \u00d6, Silahtar O (2020) Intuitionistic fuzzy adaptive sliding mode control of nonlinear systems. Soft Comput 24(1):53\u201364","journal-title":"Soft Comput"},{"key":"662_CR45","first-page":"2018","volume":"1","author":"S Sotirov","year":"2018","unstructured":"Sotirov S, Sotirova E, Atanassova V, Atanassov K, Castillo O, Melin P, Petkov T, Surchev S (2018) A hybrid approach for modular neural network design using intercriteria analysis and intuitionistic fuzzy logic. Complexity 1:2018","journal-title":"Complexity"},{"issue":"1","key":"662_CR46","doi-asserted-by":"publisher","first-page":"693","DOI":"10.1007\/s10479-017-2551-y","volume":"269","author":"SK Singh","year":"2018","unstructured":"Singh SK, Yadav SP (2018) Intuitionistic fuzzy multi-objective linear programming problem with various membership functions. Ann. Oper. Res. 269(1):693\u2013707","journal-title":"Ann. Oper. Res."}],"container-title":["Complex &amp; Intelligent Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40747-022-00662-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40747-022-00662-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40747-022-00662-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,8,4]],"date-time":"2022-08-04T22:08:43Z","timestamp":1659650923000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40747-022-00662-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,20]]},"references-count":46,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2022,8]]}},"alternative-id":["662"],"URL":"https:\/\/doi.org\/10.1007\/s40747-022-00662-4","relation":{},"ISSN":["2199-4536","2198-6053"],"issn-type":[{"value":"2199-4536","type":"print"},{"value":"2198-6053","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,2,20]]},"assertion":[{"value":"7 June 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 January 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 February 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"On behalf of all authors, the corresponding author states that there is no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}