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When the game is cycle-free the H-average tree solution is the unique solution that satisfies component efficiency and component fairness. The interval average tree solution is introduced. Furthermore, we show the relationship between the H-average tree solution and the interval average tree solution. A practical application of the proposed model in outsource production is given.<\/jats:p>","DOI":"10.3233\/jifs-17502","type":"journal-article","created":{"date-parts":[[2018,11,2]],"date-time":"2018-11-02T12:08:18Z","timestamp":1541160498000},"page":"3851-3856","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":0,"title":["The graph games with fuzzy characteristic functions"],"prefix":"10.1177","volume":"36","author":[{"given":"Cuiping","family":"Nie","sequence":"first","affiliation":[{"name":"School of Science, Shijiazhuang College, Shijiazhuang, PR China"}]},{"given":"Tao","family":"Nie","sequence":"additional","affiliation":[{"name":"School of Science, Shijiazhuang College, Shijiazhuang, PR China"}]},{"given":"Hongxia","family":"Sun","sequence":"additional","affiliation":[{"name":"Business School, Beijing Technology and Business University, Beijing, PR China"}]}],"member":"179","published-online":{"date-parts":[[2018,10,31]]},"reference":[{"key":"e_1_3_2_2_2","doi-asserted-by":"publisher","DOI":"10.1287\/moor.2.3.225"},{"key":"e_1_3_2_3_2","doi-asserted-by":"publisher","DOI":"10.1086\/421171"},{"key":"e_1_3_2_4_2","doi-asserted-by":"publisher","DOI":"10.15807\/jorsj.51.203"},{"key":"e_1_3_2_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.geb.2007.03.007"},{"key":"e_1_3_2_6_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.geb.2009.10.002"},{"key":"e_1_3_2_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0165-0114(98)00006-2"},{"key":"e_1_3_2_8_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-7908-1820-8"},{"key":"e_1_3_2_9_2","doi-asserted-by":"publisher","DOI":"10.1142\/S021848850100082X"},{"key":"e_1_3_2_10_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4615-4429-6_11"},{"key":"e_1_3_2_11_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.fss.2009.08.001"},{"key":"e_1_3_2_12_2","first-page":"423","volume-title":"Cybernetics and Systems Research\u201992 [C]","author":"Sakawa M.","year":"1992","unstructured":"M.Sakawa and I.Nishizaki, A solution concept based on fuzzy decision in n-person cooperative game. 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