Fuzzy Course2

‫ﺩﺍﻧﺸﻜﺪﻩ ﺻﻨﻌﺖ ﺍﻟﻜﺘﺮﻭﻧﻴﻚ‬

‫ﻣﻨﻄﻖ ﻓﺎﺯﻱ ﻭ ﺳﻴﺴﺘﻢﻫﺎﻱ‬ ‫ﻓﺎﺯﻱ‬ ‫ﺍﻛﺒﺮ ﺭﻫﻴﺪﻩ‬ ‫‪١٣٨٢‬‬

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‫ﻣﻨﻄﻖ ﻓﺎﺯﻱ ﻭ ﺳﻴﺴﺘﻢﻫﺎﻱ ﻓﺎﺯﻱ‬ ‫ﻓﻬﺮﺳﺖ ﻣﻄﺎﻟﺐ‪:‬‬ ‫‪ .١‬ﻣﻘﺪﻣﻪ‬ ‫‪ .١,١‬ﻫﻮﺵ ﻣﺼﻨﻮﻋﻲ‬ ‫‪ .١,٢‬ﻣﻘﺪﻣﻪﺍﻱ ﺑﺮ ﻣﻨﻄﻖ ﻓﺎﺯﻱ‬ ‫‪ .١,٣‬ﺗﺎﺭﻳﺨﭽﻪ ﻣﻨﻄﻖ ﻓﺎﺯﻱ‬ ‫‪ .١,٤‬ﺳﺎﺧﺘﺎﺭ ﺳﻴﺴﺘﻢﻫﺎﻱ ﻓﺎﺯﻱ‬ ‫‪ .١,٥‬ﻣﺮﻭﺭﻱ ﻛﻠﻲ ﺑﺮ ﺗﺌﻮﺭﻱ ﻣﻨﻄﻖ ﻓﺎﺯﻱ‬ ‫‪ .١,٦‬ﺑﺮﺧﻲ ﻛﺎﺭﺑﺮﺩﻫﺎﻱ ﺳﻴﺴﺘﻢﻫﺎﻱ ﻓﺎﺯﻱ‬ ‫‪ .٢‬ﻣﺠﻤﻮﻋﻪﻫﺎﻱ ﻓﺎﺯﻱ‬ ‫‪ .٢,١‬ﻣﻘﺎﻳﺴﻪﺍﻱ ﺑﻴﻦ ﻣﺠﻤﻮﻋﻪﻫﺎﻱ ﻛﻼﺳﻴﻚ ﻭ ﻣﺠﻤﻮﻋﻪﻫﺎﻱ ﻓﺎﺯﻱ‬ ‫‪ .٢,٢‬ﺗﻌﺮﻳﻒ ﺗﺎﺑﻊ ﻋﻀﻮﻳﺖ‬ ‫‪ .٢,٣‬ﺍﻧﻮﺍﻉ ﺗﻮﺍﺑﻊ ﻋﻀﻮﻳﺖ‬ ‫‪ .٢,٤‬ﻣﻌﺮﻓﻲ ﻣﻔﺎﻫﻴﻢ ﺍﺳﺎﺳﻲ ﻣﺮﺗﺒﻂ ﺑﺎ ﻣﺠﻤﻮﻋﻪﻫﺎﻱ ﻓﺎﺯﻱ‬ ‫‪ .٣‬ﻋﻤﻠﻴﺎﺕ ﺑﺮ ﻣﺠﻤﻮﻋﻪﻫﺎﻱ ﻓﺎﺯﻱ‬ ‫‪ .٣,١‬ﻣﻌﺎﺩﻝ ﺑﻮﺩﻥ‬ ‫‪ .٣,٢‬ﺯﻳﺮ ﻣﺠﻤﻮﻋﻪ ﺑﻮﺩﻥ‬ ‫‪ .٣,٣‬ﻣﻜﻤﻞ‬ ‫‪ .٣,٤‬ﺍﺟﺘﻤﺎﻉ‬ ‫‪ .٣,٥‬ﺍﺷﺘﺮﺍﻙ‬ ‫‪ .٣,٦‬ﻣﻴﺎﻧﮕﻴﻦ‬ ‫‪ .٤‬ﺭﻭﺍﺑﻂ ﻓﺎﺯﻱ‬ ‫‪ .٤,١‬ﺗﺮﻛﻴﺐ ﺭﻭﺍﺑﻂ ﻓﺎﺯﻱ‬ ‫‪ .٥‬ﻣﺘﻐﻴﺮﻫﺎﻱ ﺯﺑﺎﻧﻲ ﻭ ﻗﻮﺍﻋﺪ ‪IF-THEN‬‬

‫‪ .٥,١‬ﻗﻴﻮﺩ ﺯﺑﺎﻧﻲ‬ ‫‪ .٦‬ﭘﺎﻳﮕﺎﻩ ﻗﻮﺍﻋﺪ ﻭ ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ ﻓﺎﺯﻱ‬ ‫‪ .٧‬ﻓﺎﺯﻱ ﺳﺎﺯﻫﺎ ﻭ ﻓﺎﺯﻱ ﺯﺩﺍﻫﺎ‬ ‫‪ .٨‬ﺑﻜﺎﺭ ﮔﻴﺮﻱ ﺟﻌﺒﻪ ﺍﺑﺰﺍﺭ ﻓﺎﺯﻱ ﺩﺭ ﻣﺤﻴﻂ ‪MATLAB‬‬

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‫ ﭼﻨﺪ ﻣﺜﺎﻝ ﺑﺎ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺟﻌﺒﻪ ﺍﺑﺰﺍﺭ ﻓﻮﻕ‬.٩ :‫ﻣﺮﺍﺟﻊ‬ A course in fuzzy system and control

by:Li-Xin Wang

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‫‪ -١‬ﻣﻘﺪﻣﻪ‬ ‫‪ -١-١‬ﻫﻮﺵ ﻣﺼﻨﻮﻋﻲ‬ ‫ﭘﻴﺸﺮﻓﺖﻫﺎﻱ ﺍﺧﻴﺮ ﺩﺭ ﺗﺌﻮﺭﻱ ﻛﻨﺘﺮﻝ ﺑﺎﻋﺚ ﺷﺪﻩ ﻛﻪ ﺭﻭﺷﻬﺎﻱ ﻣﺮﺳﻮﻡ ﺩﺭ ﻃﺮﺍﺣﻲ ﻛﻨﺘﺮﻟﺮﻫﺎ ﺟﺎﻱ ﺧﻮﺩ ﺭﺍ ﺑﻪ ﺗﻜﻨﻴﻚﻫﺎﻱ ﻣﺒﺘﻨﻲ‬ ‫ﺑﺮ ﻫﻮﺷﻬﺎﻱ ﻣﺼﻨﻮﻋﻲ )ﻋﺼﺒﻲ‪ ،‬ﻓﺎﺯﻱ‪ ،‬ﻓﺎﺯﻱ‪-‬ﻋﺼﺒﻲ ﻭ ﮊﻧﺘﻴﻚ( ﺑﺪﻫﻨﺪ‪ .‬ﺍﻳﻦ ﺭﻭﺷﻬﺎ ﺑﻮﺳﻴﻠﺔ ﺍﻃﻼﻋﺎﺕ ﻭ ﺩﺍﻧﺶﻫﺎﻱ ﻗﺒﻠﻲ ﺩﺭﺑﺎﺭﺓ‬ ‫ﺳﻴﺴﺘﻢ ﻭ ﻋﻤﻠﻜﺮﺩ ﺁﻥ ﺗﻮﺻﻴﻒ ﻣﻲﺷﻮﻧﺪ‪ .‬ﺑﺪﻟﻴﻞ ﻣﺰﺍﻳﺎﻱ ﺑﻴﺸﻤﺎﺭﻱ ﻛﻪ ﺍﻳﻦ ﺭﻭﺷﻬﺎ ﺩﺍﺭﻧﺪ ﺩﺭ ﺁﻳﻨﺪﻩ ﻧﺰﺩﻳﻚ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺍﻳﻦ‬ ‫ﺭﻭﺷﻬﺎﻱ ﻧﻮﻳﻦ ﺩﺭ ﺻﻨﻌﺖ ﺍﺟﺘﻨﺎﺏ ﻧﺎﭘﺬﻳﺮ ﺧﻮﺍﻫﺪ ﺑﻮﺩ‪ .‬ﺑﺮﺧﻲ ﺍﺯ ﻣﺰﺍﻳﺎﻱ ﺍﺻﻠﻲ ﻫﻮﺵﻫﺎﻱ ﻣﺼﻨﻮﻋﻲ ﻋﺒﺎﺭﺗﻨﺪ ﺍﺯ‪:‬‬ ‫• ﻃﺮﺍﺣـﻲ ﺍﻳـﻦ ﺳﻴﺴـﺘﻢﻫـﺎ ﻧـﻴﺎﺯﻱ ﺑﻪ ﻣﺪﻝ ﺭﻳﺎﺿﻲ ﭘﺮﻭﺳﻪ ﻧﺪﺍﺭﺩ‪) .‬ﺑﻌﺒﺎﺭﺕ ﺩﻳﮕﺮ ﻃﺮﺍﺣﻲ ﺑﺮ ﻣﺒﻨﺎﻱ ﺍﻃﻼﻋﺎﺕ‬ ‫ﺳﻴﺴﺘﻢ ﻫﺪﻑ ﻣﻲﺑﺎﺷﺪ(‬ ‫• ﺩﺭ ﻳـﻚ ﺳﻴﺴـﺘﻢ ﻓـﺎﺯﻱ ﻛـﻪ ﻳﻚ ﺳﻴﺴﺘﻢ ﺧﺒﺮﻩ ﻣﻲﺑﺎﺷﺪ‪ .‬ﻃﺮﺍﺣﻲ ﻣﻲﺗﻮﺍﻧﺪ ﻣﻨﺤﺼﺮﹰﺍ ﺑﺮﻣﺒﻨﺎﻱ ﺍﻃﻼﻋﺎﺕ ﺯﺑﺎﻧﻲ‬ ‫ﮔﺮﻓـﺘﻪ ﺷـﺪﻩ ﺍﺯ ﻛﺎﺭﺷﻨﺎﺳـﺎﻥ ﻭ ﻳﺎ )ﻭﻗﺘﻲ ﻛﻪ ﺍﻃﻼﻋﺎﺕ ﻛﺎﺭﺷﻨﺎﺳﻲ ﺩﺭ ﺩﺳﺘﺮﺱ ﻧﺒﺎﺷﺪ( ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺩﺳﺘﻪﺑﻨﺪﻱ‬ ‫ﺍﻃﻼﻋﺎﺕ ﺑﺎﺷﺪ‪.‬‬ ‫• ﺩﺭ ﻳـﻚ ﺳﻴﺴـﺘﻢ ﻓﺎﺯﻱ ـ ﻋﺼﺒﻲ ﻛﻪ ﻳﻚ ﺷﺒﻜﺔ ﻋﺼﺒﻲ ﻛﺎﺭﺷﻨﺎﺱ ﻣﻲﺑﺎﺷﺪ ﻃﺮﺍﺣﻲ ﻣﻲﺗﻮﺍﻧﺪ ﺑﺮﺍﺳﺎﺱ ﻗﻮﺍﻋﺪ‬ ‫ﺯﺑﺎﻧـﻲ ﺑﺪﺳـﺖ ﺁﻣـﺪﻩ ﺍﺯ ﻛﺎﺭﺷﻨﺎﺳـﺎﻥ ﻭ ﻳﺎ ﺩﺳﺘﻪﺑﻨﺪﻱ ﺍﻃﻼﻋﺎﺕ )ﻭﻗﺘﻲ ﻛﻪ ﺍﻃﻼﻋﺎﺕ ﻛﺎﺭﺷﻨﺎﺳﻲ ﺩﺭ ﺩﺳﺘﺮﺱ‬ ‫ﻧﺒﺎﺷـﺪ( ﺍﻧﺠـﺎﻡ ﮔﻴﺮﺩ‪ .‬ﺑﻨﺎﺑﺮﺍﻳﻦ ﻣﺪﻝ ﺷﺒﻜﻪ ﻣﻲﺗﻮﺍﻧﺪ ﻓﻘﻂ ﺑﺎ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺍﻃﻼﻋﺎﺕ ﺳﻴﺴﺘﻢ ﻫﺪﻑ ﺑﻨﺎ ﺷﻮﺩ ﻭ ﺩﺭ‬ ‫ﺍﺑـﺘﺪﺍﻱ ﺗﻌﻠـﻴﻢ ﻧـﻴﺎﺯﻱ ﺑـﻪ ﺍﻃﻼﻋـﺎﺕ ﻗﺒﻠـﻲ ﺍﺯ ﻗﻮﺍﻋـﺪ ﻓـﺎﺯﻱ ﻭ ﺗﻮﺍﺑـﻊ ﻋﻀﻮﻳﺖ ﻧﻤﻲﺑﺎﺷﺪ‪ .‬ﻫﺮ ﭼﻨﺪ ﺩﺍﻧﺶ‬ ‫ﻛﺎﺭﺷﻨﺎﺳـﺎﻧﻪ ﺍﺯ ﺳﻴﺴـﺘﻢ ﻫﺪﻑ ﺩﺭ ﺍﻧﺘﺨﺎﺏ ﺍﻭﻟﻴﻪ ﺳﺎﺧﺘﺎﺭ ﺷﺒﻜﻪ ﻛﻤﻚ ﻣﻲﻧﻤﺎﻳﺪ ﻭ ﺍﺯ ﺍﻳﻦ ﻃﺮﻳﻖ ﻣﻲﺗﻮﺍﻥ ﺧﻄﺎ‬ ‫ﻭ ﺯﻣﺎﻥ ﺗﻌﻠﻴﻢ ﺭﺍ ﻛﺎﻫﺶ ﺩﺍﺩ‪.‬‬ ‫• ﺩﺭ ﻳـﻚ ﺳﻴﺴـﺘﻢ ﺷﺒﻜﻪ ﻋﺼﺒﻲ )ﺑﺪﻭﻥ ﻓﺎﺯﻱ( ﺍﮔﺮ ﺷﺒﻜﻪ ﻋﺼﺒﻲ ﺑﺎ ﺭﻭﺵ ﺗﻌﻠﻴﻢ ﺗﺤﺖ ﻧﻈﺎﺭﺕ ﺁﻣﻮﺯﺵ ﺑﺒﻴﻨﺪ‪،‬‬ ‫ﻃﺮﺍﺣـﻲ ﻣﺒﻨـﻲ ﺑﺮ ﺍﻃﻼﻋﺎﺕ ﻣﻮﺟﻮﺩ ﺑﺮﺍﻱ ﺗﻌﻠﻴﻢ ﺑﻮﺩﻩ ﻭ ﺍﻳﻦ ﺍﻃﻼﻋﺎﺕ ﻣﻲﺗﻮﺍﻧﻨﺪ ﺍﺯ ﻣﻨﺎﺑﻊ ﻣﺘﻌﺪﺩﻱ ﺍﺳﺘﺨﺮﺍﺝ‬ ‫ﺷـﻮﻧﺪ )ﺑﻌـﻨﻮﺍﻥ ﻣـﺜﺎﻝ ﺍﺯ ﻃـﺮﻳﻖ ﺍﻧـﺪﺍﺯﻩ ﮔـﻴﺮﻱ(‪ .‬ﻫﺮﭼﻨﺪ ﻭﻗﺘﻲ ﺷﺒﻜﻪ ﻋﺼﺒﻲ ﺑﺎ ﺭﻭﺵ ﺗﻌﻠﻴﻢ ﺑﺪﻭﻥ ﻧﻈﺎﺭﺕ‬ ‫ﺁﻣـﻮﺯﺵ ﻣـﻲﺑﻴـﻨﺪ ﺑﻌﻨﻮﺍﻥ ﻣﺜﺎﻝ ﺷﺒﻜﻪ ﻋﺼﺒﻲ ﺧﻮﺩ ﺳﺎﺯﻣﺎﻧﺪﻩ‪ ،‬ﺷﺒﻜﻪ ﻋﺼﺒﻲ ﺍﻃﻼﻋﺎﺕ ﺭﺍ ﺑﺮ ﻃﺒﻖ ﺭﻭﺵ ﺑﻜﺎﺭ‬ ‫ﮔﺮﻓﺘﻪ‪ ،‬ﺩﺳﺘﻪﺑﻨﺪﻱ ﻣﻲﻧﻤﺎﻳﺪ‪.‬‬ ‫• ﺩﺭ ﺳﻴﺴﺘﻢﻫﺎﻱ ﻣﺒﺘﻨﻲ ﺑﺮ ﻫﻮﺵ ﻣﺼﻨﻮﻋﻲ ﺗﺎﺛﻴﺮﺍﺕ ﺗﻨﻈﻴﻢ ﻣﻲﺗﻮﺍﻧﺪ ﺑﺴﻴﺎﺭ ﻛﻤﺘﺮ ﺍﺯ ﺳﻴﺴﺘﻢﻫﺎﻱ ﻣﻌﻤﻮﻟﻲ ﺑﺎﺷﺪ‪.‬‬ ‫• ﺍﻳﻨﮕﻮﻧﻪ ﺳﻴﺴﺘﻢﻫﺎ ﺑﺨﻮﺑﻲ ﺧﻮﺩ ﺭﺍ ﺗﻌﻠﻴﻢ ﻭ ﻋﻤﻮﻣﻴﺖ ﻣﻲﺩﻫﻨﺪ )ﻳﻌﻨﻲ ﻣﻲﺗﻮﺍﻧﻨﺪ ﺗﺨﻤﻴﻦﻫﺎﻱ ﺧﻮﺑﻲ ﺭﺍ ﺩﺭ ﻗﺒﺎﻝ‬ ‫ﺍﻃﻼﻋﺎﺕ ﻭﺭﻭﺩﻱ ﻧﺎﺷﻨﺎﺧﺘﻪ ﺍﺯ ﺧﻮﺩ ﺍﺭﺍﺋﻪ ﺩﻫﻨﺪ( ﺑﻨﺎﺑﺮﺍﻳﻦ ﻣﺴﺘﻘﻞ ﺍﺯ ﺧﺼﻮﺻﻴﺎﺕ ﺩﺭﺍﻳﻮ ﻫﺴﺘﻨﺪ‪.‬‬ ‫• ﺍﻳﻦ ﺳﻴﺴﺘﻤﻬﺎ ﺧﺼﻮﺻﻴﺖ ﺣﺬﻑ ﻧﻮﻳﺰ ﺭﺍ ﺑﺨﻮﺑﻲ ﺍﺯ ﺧﻮﺩ ﻧﺸﺎﻥ ﻣﻲﺩﻫﻨﺪ‪.‬‬ ‫• ﺍﻳـﻦ ﺳﻴﺴـﺘﻢﻫﺎ ﺩﺭ ﻫﻨﮕﺎﻡ ﻭﻗﻮﻉ ﺧﻄﺎ ﺗﺤﻤﻞ ﻧﺴﺒﺘﹰﺎ ﺧﻮﺑﻲ ﺍﺯ ﺧﻮﺩ ﻧﺸﺎﻥ ﻣﻲﺩﻫﻨﺪ‪ .‬ﺑﻌﺒﺎﺭﺕ ﺩﻳﮕﺮ ﺍﮔﺮ ﺩﺭ ﻳﻚ‬ ‫ﺷـﺒﻜﻪ ﻋﺼﺒﻲ ﻳﻚ ﻋﺼﺐ ﺧﺮﺍﺏ ﻳﺎ ﺣﺬﻑ ﺷﻮﺩ ﻭ ﻳﺎ ﺍﻳﻨﻜﻪ ﺩﺭ ﻳﻚ ﺷﺒﻜﻪ ﻓﺎﺯﻱ ـ ﻋﺼﺒﻲ ﻳﻚ ﻗﺎﻋﺪﻩ ﺣﺬﻑ‬ ‫ﺷـﻮﺩ ﺳﻴﺴـﺘﻢ ﻣﺒﺘﻨﻲ ﺑﺮ ﻫﻮﺵ ﻣﺼﻨﻮﻋﻲ ﺑﺪﻟﻴﻞ ﺳﺎﺧﺘﺎﺭ ﻣﻮﺍﺯﻱ ﻣﻲﺗﻮﺍﻧﺪ ﺑﻜﺎﺭ ﺧﻮﺩ ﺍﺩﺍﻣﻪ ﺩﻫﺪ )ﺍﻟﺒﺘﻪ ﻋﻤﻠﻜﺮﺩ‬ ‫ﺳﻴﺴﺘﻢ ﻛﻤﻲ ﺩﭼﺎﺭ ﻣﺸﻜﻞ ﻣﻲﺷﻮﺩ(‪.‬‬

‫‪۴‬‬

‫• ﺳﻴﺴﺘﻢﻫﺎﻱ ﻣﺒﺘﻨﻲ ﺑﺮ ﻫﻮﺵ ﻣﺼﻨﻮﻋﻲ ﻣﻲﺗﻮﺍﻧﻨﺪ ﺑﻪ ﺁﺳﺎﻧﻲ ﺗﻌﻤﻴﻢ ﻭ ﺗﻮﺳﻌﻪ ﻭ ﻫﻤﭽﻨﻴﻦ ﺗﻌﺪﻳﻞ ﺷﻮﻧﺪ‪.‬‬ ‫• ﺍﻳﻦ ﺳﻴﺴﺘﻢﻫﺎ ﻫﻤﭽﻨﻴﻦ ﺩﺭ ﻣﻘﺎﺑﻞ ﺗﻐﻴﻴﺮ ﭘﺎﺭﺍﻣﺘﺮﻫﺎ ﺑﺴﻴﺎﺭ ﻣﻘﺎﻭﻡ ﻣﻲﺑﺎﺷﻨﺪ‪.‬‬ ‫‪ -٢-١‬ﻣﻘﺪﻣﻪﺍﻱ ﺑﺮ ﻣﻨﻄﻖ ﻓﺎﺯﻱ‬ ‫ﻛﻠﻤﻪ ﻓﺎﺯﻱ ﺩﺭ ﻓﺮﻫﻨﮓ ﻟﻐﺖ ﺑﺎ ﻣﻌﺎﻧﻲ ﻣﺒﻬﻢ‪ ،‬ﮔﻨﮓ‪ ،‬ﻧﺎﺩﻗﻴﻖ‪ ،‬ﮔﻴﺞ‪ ،‬ﻣﻐﺸﻮﺵ‪ ،‬ﺩﺭﻫﻢ ﻭ ﻧﺎﻣﺸﺨﺺ ﺁﻭﺭﺩﻩ ﺷﺪﻩ ﺍﺳﺖ‪ .‬ﺩﻗﺖ ﺷﻮﺩ‬ ‫ﺑﺎ ﻭﺟﻮﺩ ﺍﻳﻨﻜﻪ ﺳﻴﺴﺘﻢﻫﺎﻱ ﻓﺎﺯﻱ ﭘﺪﻳﺪﻩﻫﺎﻱ ﻏﻴﺮ ﻗﻄﻌﻲ ﻭ ﻧﺎﻣﺸﺨﺺ ﺭﺍ ﺗﻮﺻﻴﻒ ﻣﻲﻧﻤﺎﻳﻨﺪ ﻭﻟﻲ ﺧﻮﺩ ﺗﺌﻮﺭﻱ ﻓﺎﺯﻱ‪ ،‬ﺗﺌﻮﺭﻳﻲ‬ ‫ﻼ ﺩﻗﻴﻖ ﻣﻲﺑﺎﺷﺪ‪.‬‬ ‫ﻛﺎﻣ ﹰ‬ ‫ﺩﺭ ﺍﻳﻨﺠﺎ ﺩﻭ ﺗﻮﺟﻴﻪ ﺑﺮﺍﻱ ﺗﺌﻮﺭﻱ ﻓﺎﺯﻱ ﺑﻴﺎﻥ ﻣﻲﺷﻮﺩ‪:‬‬ ‫• ﭘﻴﭽﻴﺪﮔﻲ ﺳﻴﺴﺘﻢﻫﺎﻱ ﻭﺍﻗﻌﻲ ﻛﻪ ﺗﻮﺻﻴﻒ ﺩﻗﻴﻖ ﺑﺮﺍﻱ ﺁﻧﻬﺎ ﻣﻤﻜﻦ ﻧﻤﻲﺑﺎﺷﺪ‪.‬‬ ‫•‬

‫ﻓﺮﻣﻮﻟﻪ ﻛﺮﺩﻥ ﺩﺍﻧﺶ ﺑﺸﺮﻱ‬

‫‪ -٣-١‬ﺗﺎﺭﻳﺨﭽﻪ ﻣﻨﻄﻖ ﻓﺎﺯﻱ‬ ‫ﺗﺌﻮﺭﻱ ﻣﻨﻄﻖ ﻓﺎﺯﻱ ﺩﺭ ﺳﺎﻝ ‪ ١٩٦٥‬ﺗﻮﺳﻂ ﭘﺮﻓﺴﻮﺭ ﻟﻄﻔﻲ ﺯﺍﺩﻩ ﺩﺭ ﻣﻘﺎﻟﻪﺍﻱ ﺑﻨﺎﻡ " ﻣﺠﻤﻮﻋﻪﻫﺎﻱ ﻓﺎﺯﻱ" ﻣﻄﺮﺡ ﺷﺪ‪.‬‬ ‫‪Zadeh.L.A, [1965] , “Fuzzy Sets” , Information and Control, 8, pp. 338-353.‬‬

‫‪ -٤-١‬ﺳﺎﺧﺘﺎﺭ ﺳﻴﺴﺘﻢﻫﺎﻱ ﻓﺎﺯﻱ‬ ‫ﺑﻄﻮﺭ ﻛﻠﻲ ﺳﻪ ﻧﻮﻉ ﺳﺎﺧﺘﺎﺭ ﺑﺮﺍﻱ ﺳﻴﺴﺘﻢﻫﺎﻱ ﻓﺎﺯﻱ ﻭﺟﻮﺩ ﺩﺍﺭﺩ‪:‬‬ ‫•‬

‫ﺳﻴﺴﺘﻢﻫﺎﻱ ﻓﺎﺯﻱ ﺧﺎﻟﺺ‬

‫•‬

‫ﺳﻴﺴﺘﻢﻫﺎﻱ ﻓﺎﺯﻱ ﺗﺎﻛﺎﮔﻲ‪-‬ﺳﻮﮔﻨﻮ ﻭ ﻛﺎﻧﮓ )‪(TSK‬‬

‫•‬

‫ﺳﻴﺴﺘﻢﻫﺎﻱ ﻓﺎﺯﻱ ﺑﺎ ﻓﺎﺯﻱﺳﺎﺯ ﻭ ﻓﺎﺯﻱﺯﺩﺍ‬ ‫‪١‬‬

‫‪ .١-٤-١‬ﺳﻴﺴﺘﻢﻫﺎﻱ ﻓﺎﺯﻱ ﺧﺎﻟﺺ‬

‫ﺳﺎﺧﺘﺎﺭ ﺍﺻﻠﻲ ﺍﻳﻦ ﺳﻴﺴﺘﻢ ﺩﺭ ﺷﻜﻞ )‪ (١-١‬ﻧﺸﺎﻥ ﺩﺍﺩﻩ ﺷﺪﻩ ﺍﺳﺖ‪.‬‬

‫ﭘﺎﻳﮕﺎﻩ ﻗﻮﺍﻋﺪ ﻓﺎﺯﻱ‬

‫ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ‬

‫ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ ﻓﺎﺯﻱ‬

‫ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ‬

‫ﺷﻜﻞ )‪ :(١-١‬ﺳﺎﺧﺘﺎﺭ ﺍﺻﻠﻲ ﺳﻴﺴﺘﻢ ﻓﺎﺯﻱ ﺧﺎﻟﺺ‬ ‫ﻣﺸﻜﻞ ﺍﺻﻠﻲ ﺩﺭ ﺭﺍﺑﻄﻪ ﺑﺎ ﺳﻴﺴﺘﻢﻫﺎﻱ ﻓﺎﺯﻱ ﺧﺎﻟﺺ ﺍﻳﻦ ﺍﺳﺖ ﻛﻪ ﻭﺭﻭﺩﻱ ﻭ ﺧﺮﻭﺟﻲﻫﺎﻱ ﺁﻥ ﻣﺠﻤﻮﻋﻪﻫﺎﻱ ﻓﺎﺯﻱ ﻣﻲﺑﺎﺷﻨﺪ ﺣﺎﻝ ﺁﻧﻜﻪ‬ ‫‪1 - Pure Fuzzy system‬‬

‫‪۵‬‬

‫ﺩﺭ ﺳﻴﺴﺘﻢﻫﺎﻱ ﻣﻬﻨﺪﺳﻲ ﻭﺭﻭﺩﻱ ﻭ ﺧﺮﻭﺟﻲﻫﺎ ﻣﺘﻐﻴﺮﻫﺎﻳﻲ ﺑﺎ ﻣﻘﺎﺩﻳﺮ ﺣﻘﻴﻘﻲ ﻣﻲﺑﺎﺷﻨﺪ‪ .‬ﺑﺮﺍﻱ ﺣﻞ ﺍﻳﻦ ﻣﺸﻜﻞ ﺳﻴﺴﺘﻢﻫﺎﻱ ﻓﺎﺯﻱ ﺗﺎﻛﺎﮔﻲ‬ ‫ﺳﻮﮔﻨﻮ ﻭ ﻛﺎﻧﮓ ﻣﻌﺮﻓﻲ ﺷﺪ‪.‬‬

‫‪١‬‬

‫‪ .٢-٤-١‬ﺳﻴﺴﺘﻢﻫﺎﻱ ﻓﺎﺯﻱ ﺗﺎﻛﺎﮔﻲ‪-‬ﺳﻮﮔﻨﻮ ﻭ ﻛﺎﻧﮓ )‪(TSK‬‬

‫ﺳﺎﺧﺘﺎﺭ ﺍﺻﻠﻲ ﺳﻴﺴﺘﻢ ﻓﺎﺯﻱ ﺗﺎﻛﺎﮔﻲ‪-‬ﺳﻮﮔﻨﻮ ﻭ ﻛﺎﻧﮓ ﺩﺭ ﺷﻜﻞ )‪ (٢-١‬ﻧﺸﺎﻥ ﺩﺍﺩﻩ ﺷﺪﻩ ﺍﺳﺖ‪.‬‬

‫ﭘﺎﻳﮕﺎﻩ ﻗﻮﺍﻋﺪ ﻓﺎﺯﻱ‬

‫ﻣﻴﺎﻧﮕﻴﻦ ﻭﺯﻧﻲ‬

‫ﻣﺠﻤﻮﻋﻪ ﻏﻴﺮ ﻓﺎﺯﻱ‬

‫ﻣﺠﻤﻮﻋﻪ ﻏﻴﺮ ﻓﺎﺯﻱ‬

‫ﺷﻜﻞ )‪ :(٢-١‬ﺳﺎﺧﺘﺎﺭ ﺍﺻﻠﻲ ﺳﻴﺴﺘﻢ ﻓﺎﺯﻱ ﺗﺎﻛﺎﮔﻲ‪-‬ﺳﻮﮔﻨﻮ ﻭ ﻛﺎﻧﮓ‬ ‫ﺳﻴﺴﺘﻢ ﻓﺎﺯﻱ ﺗﺎﻛﺎﮔﻲ‪-‬ﺳﻮﮔﻨﻮ ﻭ ﻛﺎﻧﮓ ﻧﻴﺰ ﻣﺸﻜﻼﺗﻲ ﺩﺍﺭﺩ ﻛﻪ ﻋﺒﺎﺭﺗﻨﺪ ﺍﺯ‪:‬‬ ‫•‬

‫ﺑﺨﺶ ﺁﻧﮕﺎﻩ ﻗﺎﻋﺪﻩ ﻓﺎﺯﻱ ﻳﻚ ﻓﺮﻣﻮﻝ ﺭﻳﺎﺿﻲ ﻣﻲﺑﺎﺷﺪ ﻭ ﺑﻨﺎﺑﺮﺍﻳﻦ ﺑﺼﻮﺭﺕ ﺩﺍﻧﺶ ﺑﺸﺮﻱ ﺑﻴﺎﻥ ﻧﻤﻲﺷﻮﺩ‪.‬‬

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‫ﺍﻧﻌﻄﺎﻑ ﭘﺬﻳﺮﻱ ﺍﻳﻦ ﺳﻴﺴﺘﻢ ﻓﺎﺯﻱ ﺑﺪﻟﻴﻞ ﻋﺪﻡ ﺍﻣﻜﺎﻥ ﭘﻴﺎﺩﻩ ﺳﺎﺯﻱ ﺍﺻﻮﻝ ﻣﻨﻄﻖ ﻓﺎﺯﻱ ﻛﻢ ﻣﻲﺷﻮﺩ‪.‬‬ ‫‪٢‬‬

‫‪ .٣-٤-١‬ﺳﻴﺴﺘﻢﻫﺎﻱ ﻓﺎﺯﻱ ﺑﺎ ﻓﺎﺯﻱﺳﺎﺯ ﻭ ﻓﺎﺯﻱﺯﺩﺍ‬

‫ﺳﺎﺧﺘﺎﺭ ﺍﺻﻠﻲ ﺳﻴﺴﺘﻢ ﻓﺎﺯﻱ ﺑﺎ ﻓﺎﺯﻱﺳﺎﺯ ﻭ ﻓﺎﺯﻱﺯﺩﺍ ﺩﺭ ﺷﻜﻞ )‪ (٣-١‬ﻧﺸﺎﻥ ﺩﺍﺩﻩ ﺷﺪﻩ ﺍﺳﺖ‪.‬‬

‫ﭘﺎﻳﮕﺎﻩ ﻗﻮﺍﻋﺪ ﻓﺎﺯﻱ‬ ‫ﻣﺠﻤﻮﻋﻪ ﻏﻴﺮ ﻓﺎﺯﻱ‬

‫ﻓﺎﺯﻱﮔﺮ‬

‫ﻓﺎﺯﻱﺯﺩﺍ‬ ‫ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ‬

‫ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ ﻓﺎﺯﻱ‬

‫ﻣﺠﻤﻮﻋﻪ ﻏﻴﺮ ﻓﺎﺯﻱ‬

‫ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ‬

‫ﺷﻜﻞ )‪ :(٣-١‬ﺳﺎﺧﺘﺎﺭ ﺍﺻﻠﻲ ﺳﻴﺴﺘﻢ ﻓﺎﺯﻱ ﺑﺎ ﻓﺎﺯﻱﺳﺎﺯ ﻭ ﻓﺎﺯﻱﺯﺩﺍ‬ ‫ﺳﻴﺴﺘﻢ ﻓﺎﺯﻱ ﺑﺎ ﻓﺎﺯﻱﺳﺎﺯ ﻭ ﻓﺎﺯﻱﺯﺩﺍ ﻣﻌﺎﻳﺐ ﺩﻭ ﺳﻴﺴﺘﻢ ﻓﺎﺯﻱ ﻗﺒﻞ ﻳﻌﻨﻲ ﺳﻴﺴﺘﻢ ﻓﺎﺯﻱ ﺧﺎﻟﺺ ﻭ ﺳﻴﺴﺘﻢ ﻓﺎﺯﻱ ‪ TSK‬ﺭﺍ ﻧﺪﺍﺭﺩ‪ .‬ﺍﺯ ﺍﻳﻦ‬ ‫ﭘﺲ ﻣﻨﻈﻮﺭ ﺍﺯ ﺳﻴﺴﺘﻢ ﻓﺎﺯﻱ‪ ،‬ﺳﻴﺴﺘﻢ ﻓﺎﺯﻱ ﺑﺎ ﻓﺎﺯﻱﺳﺎﺯ ﻭ ﻓﺎﺯﻱﺯﺩﺍ ﻣﻲﺑﺎﺷﺪ ﻣﮕﺮ ﺧﻼﻑ ﺁﻥ ﻣﻄﺮﺡ ﺷﻮﺩ‪.‬‬ ‫‪1 -Takagi-Sugeno-Kang Fuzzy System‬‬ ‫‪2 -Fuzzy System with Fuzzifier & Defuzzifier‬‬

‫‪۶‬‬

‫‪ -٥-١‬ﻣﺮﻭﺭﻱ ﻛﻠﻲ ﺑﺮ ﺗﺌﻮﺭﻱ ﻣﻨﻄﻖ ﻓﺎﺯﻱ‬ ‫ﺍﺧـﻴﺮﹰﺍ ﻣـﻨﻄﻖ ﻓـﺎﺯﻱ ﺑﻌـﻨﻮﺍﻥ ﺯﻣﻴـﻨﻪﺍﻱ ﺟـﺬﺍﺏ ﺩﺭ ﺗﺤﻘﻴﻘﺎﺕ ﻛﻨﺘﺮﻟﻲ ﻇﻬﻮﺭ ﻳﺎﻓﺘﻪ ﺍﺳﺖ‪ .‬ﻣﻬﻤﺘﺮﻳﻦ ﺍﺻﻞ ﺩﺭ ﻣﻨﻄﻖ ﻓﺎﺯﻱ ﺳﺎﺧﺘﺎﺭ‬ ‫ﻛﻨـﺘﺮﻟﺮﻫﺎﻱ ﻓـﺎﺯﻱ ﻣﻲﺑﺎﺷﺪ ﻛﻪ ﺩﺍﻧﺶﻫﺎﻱ ﺯﺑﺎﻧﻲ ﺍﺷﺨﺎﺹ ﻛﺎﺭﺷﻨﺎﺱ ﺭﺍ ﺑﻜﺎﺭ ﻣﻲﮔﻴﺮﻧﺪ‪ .‬ﭼﻨﺪﻳﻦ ﻧﻤﻮﻧﻪ ﻛﻨﺘﺮﻟﺮ ﻓﺎﺯﻱ ﻭﺟﻮﺩ ﺩﺍﺭﺩ‬ ‫ﻛـﻪ ﺩﺭ ﺍﻳـﻨﺠﺎ ﻧـﻮﻉ ﻛﻨـﺘﺮﻟﺮ ﻓـﺎﺯﻱ ﻣﻤﺪﺍﻧﻲ‪ ١‬ﺗﺸﺮﻳﺢ ﻣﻲﺷﻮﺩ‪ .‬ﻫﻤﺎﻧﻄﻮﺭ ﻛﻪ ﺩﺭ ﺷﻜﻞ )‪ (٤-۱‬ﻣﺸﺎﻫﺪﻩ ﻣﻲﺷﻮﺩ‪ ،‬ﻳﻚ ﻛﻨﺘﺮﻟﺮ ﻓﺎﺯﻱ‬ ‫ﺷﺎﻣﻞ ﭼﻬﺎﺭ ﻗﺴﻤﺖ ﺍﺳﺖ ﻛﻪ ﺩﻭ ﻗﺴﻤﺖ ﺁﻥ ﻋﻤﻞ ﺗﺒﺪﻳﻞ ﺭﺍ ﺍﻧﺠﺎﻡ ﻣﻲﺩﻫﻨﺪ]‪:[۸‬‬ ‫•‬

‫ﻓﺎﺯﻱﮔﺮ ) ﺗﺒﺪﻳﻞ ‪( ۱‬‬

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‫ﭘﺎﻳﮕﺎﻩ ﺩﺍﺩﻩ‬

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‫ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ‬

‫•‬

‫ﻓﺎﺯﻱﺯﺩﺍ ) ﺗﺒﺪﻳﻞ ‪( ۲‬‬

‫ﻓﺎﺯﻱﮔﺮ ﻣﺘﻐﻴﺮﻫﺎﻱ ﻭﺭﻭﺩﻱ ) ﺳﻴﮕﻨﺎﻟﻬﺎﻱ ﻭﺍﻗﻌﻲ ( ﺭﺍ ﻓﺎﺯﻱ ﻣﻲﻧﻤﺎﻳﺪ‪ .‬ﺑﻨﺎﺑﺮﺍﻳﻦ ﺗﻤﺎﻣﻲ ﺳﻴﮕﻨﺎﻟﻬﺎﻱ ﻭﺭﻭﺩﻱ ﺑﻔﺮﻡ ﻓﺎﺯﻱ ﺩﺭ ﻣﻲﺁﻳﻨﺪ‪.‬‬ ‫ﺑﻌﺒﺎﺭﺕ ﺳﺎﺩﻩﺗﺮ ﻓﺎﺯﻱﮔﺮ ﻣﺘﻐﻴﺮﻫﺎﻱ ﻋﺪﺩﻱ ﺭﺍ ﺑﻪ ﻣﺘﻐﻴﺮﻫﺎﻱ ﻓﺎﺯﻱ ﻭ ﺑﻌﺒﺎﺭﺗﻲ ﻣﺘﻐﻴﺮﻫﺎﻱ ﺯﺑﺎﻧﻲ ﺗﺒﺪﻳﻞ ﻣﻲﻧﻤﺎﻳﺪ‪ .‬ﺍﻳﻦ ﺗﺒﺪﻳﻞ ﺗﻮﺳﻂ‬ ‫ﺗﻮﺍﺑﻊ ﻋﻀﻮﻳﺖ ﺍﻧﺠﺎﻡ ﻣﻲﮔﻴﺮﺩ‪.‬‬ ‫ﺑﻌـﻨﻮﺍﻥ ﻣـﺜﺎﻝ ﺍﮔـﺮ ﺳـﻴﮕﻨﺎﻝ ﻭﺭﻭﺩﻱ ﻛﻮﭼـﻚ ﻭﻟـﻲ ﻣﺜﺒﺖ ﺑﺎﺷﺪ ﺍﻳﻦ ﺳﻴﮕﻨﺎﻝ ﺑﻪ ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ ﻣﺜﺒﺖ ﻛﻮﭼﻚ ﺗﻌﻠﻖ ﺩﺍﺭﺩ ﻭﺍﮔﺮ‬ ‫ﻛﻮﭼـﻚ ﻭﻟـﻲ ﻣﻨﻔﻲ ﺑﺎﺷﺪ ﺑﻪ ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ ﻣﻨﻔﻲ ﻛﻮﭼﻚ ﻣﺘﻌﻠﻖ ﺍﺳﺖ‪ .‬ﺑﻪ ﻫﻤﻴﻦ ﺗﺮﺗﻴﺐ ﻣﺠﻤﻮﻋﻪﻫﺎﻱ ﻓﺎﺯﻱ ﺩﻳﮕﺮﻱ ﺑﺼﻮﺭﺕ‬ ‫ﻣﺜﺒـﺖ ﻣﺘﻮﺳـﻂ‪ ،‬ﻣﺜﺒـﺖ ﺑـﺰﺭﮒ ﻭ… ﻧـﻴﺰ ﻣﻲﺗﻮﺍﻧﺪ ﻭﺟﻮﺩ ﺩﺍﺷﺘﻪ ﺑﺎﺷﺪ‪ .‬ﺩﺭ ﻳﻚ ﻛﻨﺘﺮﻟﺮ ﻓﺎﺯﻱ ﻣﻌﻤﻮﻟﻲ‪ ،‬ﺗﻌﺪﺍﺩ ﺗﻮﺍﺑﻊ ﻋﻀﻮﻳﺖ ﻭ‬ ‫ﺷـﻜﻞ ﺁﻧﻬـﺎ ﺩﺭ ﺍﺑـﺘﺪﺍ ﺗﻮﺳـﻂ ﻛﺎﺭﺑﺮ ﺗﻌﻴﻴﻦ ﻣﻲﺷﻮﺩ‪ .‬ﺗﻮﺍﺑﻊ ﻋﻀﻮﻳﺖ ﻣﻘﺎﺩﻳﺮﻱ ﺑﻴﻦ ‪ ۰‬ﻭ ‪ ۱‬ﺩﺍﺭﻧﺪ ﻭ ﺩﺭﺟﻪ ﺗﻌﻠﻖ ﻳﻚ ﻛﻤﻴﺖ ﺭﺍ ﺑﻪ‬ ‫ﻣﺠﻤﻮﻋـﻪ ﻓﺎﺯﻱ ﻣﺸﺨﺺ ﻣﻲﻧﻤﺎﻳﻨﺪ‪ .‬ﺍﮔﺮ ﺗﻌﻠﻖ ﻳﻚ ﻛﻤﻴﺖ ﺑﻪ ﻳﻚ ﻣﺠﻤﻮﻋﺔ ﻓﺎﺯﻱ ﺑﻄﻮﺭ ﻣﻄﻠﻖ ﻣﻌﻴﻦ ﺑﺎﺷﺪ ﺁﻧﮕﺎﻩ ﺩﺭﺟﻪ ﺗﻌﻠﻖ ﺁﻥ‬ ‫ﺑﻪ ﻣﺠﻤﻮﻋﺔ ﻓﺎﺯﻱ ﻣﺰﺑﻮﺭ ﻳﻚ ﺑﺎﺷﺪ ) ﺑﻌﺒﺎﺭﺕ ﺩﻳﮕﺮ ﺍﻳﻦ ﻛﻤﻴﺖ ﺻﺪ ﺩﺭ ﺻﺪ ﺑﻪ ﻣﺠﻤﻮﻋﺔ ﻓﺎﺯﻱ ﻣﺰﺑﻮﺭ ﻣﺘﻌﻠﻖ ﺍﺳﺖ ( ﺍﻣﺎ ﺍﮔﺮ ﻳﻚ‬ ‫ﻛﻤﻴـﺖ ﺑـﻪ ﻫـﻴﭻ ﻋـﻨﻮﺍﻥ ﺑـﻪ ﻳﻚ ﻣﺠﻤﻮﻋﺔ ﻓﺎﺯﻱ ﻣﺘﻌﻠﻖ ﻧﺒﺎﺷﺪ ﺩﺭﺟﻪ ﺗﻌﻠﻖ ﺁﻥ ﺑﻪ ﻣﺠﻤﻮﻋﺔ ﻓﺎﺯﻱ ﻣﺬﻛﻮﺭ ﺻﻔﺮ ﺍﺳﺖ‪ .‬ﺑﻪ ﻫﻤﻴﻦ‬ ‫ﺗﺮﺗﻴـﺐ ﺍﮔـﺮ ﺑﻪ ﻋﻨﻮﺍﻥ ﻣﺜﺎﻝ ﺗﻌﻠﻖ ﻳﻚ ﻛﻤﻴﺖ ﺑﻪ ﻳﻚ ﻣﺠﻤﻮﻋﺔ ﻓﺎﺯﻱ ﺑﻪ ﺍﻧﺪﺍﺯﺓ ‪ ۵۰‬ﺩﺭﺻﺪ ﺑﺎﺷﺪ ﺁﻧﮕﺎﻩ ﺩﺭﺟﻪ ﺗﻌﻠﻖ ﺍﻳﻦ ﻛﻤﻴﺖ ﺑﻪ‬ ‫ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ ﻣﺬﻛﻮﺭ ‪ ۰/۵‬ﻣﻲﺑﺎﺷﺪ‪ .‬ﺗﻮﺍﺑﻊ ﻋﻀﻮﻳﺖ ﻣﻲﺗﻮﺍﻧﻨﺪ ﺍﺷﻜﺎﻝ ﻣﺘﻔﺎﻭﺗﻲ ﻣﺎﻧﻨﺪ ﻣﺜﻠﺜﻲ‪ ،‬ﮔﻮﺳﻲ‪ ،‬ﺫﻭﺯﻧﻘﻪﺍﻱ ﻭ ﺑﻄﻮﺭ ﻛﻠﻲﺗﺮ‬ ‫ﺷـﺒﻪ ﺫﻭﺯﻧﻘـﻪﺍﻱ ﺭﺍ ﺑﺨـﻮﺩ ﺑﮕـﻴﺮﺩ‪ .‬ﻓﺮﻡ ﺍﻭﻟﻴﻪ ﺗﻮﺍﺑﻊ ﻋﻀﻮﻳﺖ ﻣﻲﺗﻮﺍﻧﺪ ﺑﻮﺳﻴﻠﺔ ﺑﻜﺎﺭﮔﻴﺮﻱ ﻣﻼﺣﻈﺎﺕ ﻛﺎﺭﺷﻨﺎﺳﻲ ﻭ ﻳﺎ ﺩﺳﺘﻪﺑﻨﺪﻱ‬ ‫ﺍﻃﻼﻋﺎﺕ ﻭﺭﻭﺩﻱ ﺍﻧﺘﺨﺎﺏ ﮔﺮﺩﺩ‪.‬‬ ‫ﭘﺎﻳﮕـﺎﻩ ﺩﺍﺩﻩ ﺷـﺎﻣﻞ ﺍﻃﻼﻋـﺎﺕ ﻣﺒﻨﺎ ﻭ ﻗﻮﺍﻋﺪ ﺯﺑﺎﻧﻲ ﻣﻲﺑﺎﺷﺪ‪ .‬ﺩﺍﺩﻩﻫﺎﻱ ﻣﺒﻨﺎ ﺍﻃﻼﻋﺎﺗﻲ ﺭﺍ ﻛﻪ ﺩﺭ ﺗﻌﻴﻴﻦ ﻗﻮﺍﻋﺪ ﺯﺑﺎﻧﻲ ﻻﺯﻡ ﻣﻲﺑﺎﺷﺪ‬ ‫ﻓﺮﺍﻫﻢ ﻣﻲﺁﻭﺭﺩ‪ .‬ﭘﺎﻳﮕﺎﻩ ﺩﺍﺩﻩ ) ﻗﻮﺍﻋﺪ ﺧﺒﺮﻩ ( ﻫﺪﻑ ﺍﺻﻠﻲ ﻛﻨﺘﺮﻝ ﺭﺍ ﺗﻮﺳﻂ ﻣﺠﻤﻮﻋﻪﺍﻱ ﺍﺯ ﻗﻮﺍﻋﺪ ﻛﻨﺘﺮﻝ ﺯﺑﺎﻧﻲ ﺑﺮ ﺁﻭﺭﺩﻩ ﻣﻲﻛﻨﺪ‪.‬‬ ‫ﺑﻌـﺒﺎﺭﺕ ﺩﻳﮕـﺮ ﭘﺎﻳﮕـﺎﻩ ﺩﺍﺩﻩ ﺷﺎﻣﻞ ﻗﻮﺍﻋﺪﻱ ﺍﺳﺖ ﻛﻪ ﺗﻮﺳﻂ ﺍﻓﺮﺍﺩ ﻛﺎﺭﺷﻨﺎﺱ ﻓﺮﺍﻫﻢ ﺁﻣﺪﻩ ﺍﺳﺖ‪ .‬ﻛﻨﺘﺮﻟﺮ ﻣﻨﻄﻖ ﻓﺎﺯﻱ ﺳﻴﮕﻨﺎﻟﻬﺎﻱ‬ ‫ﻭﺭﻭﺩﻱ ﺭﺍ ﺗﻮﺳـﻂ ﻗﻮﺍﻋـﺪ ﺧﺒﺮﻩ ﺑﻪ ﺳﻴﮕﻨﺎﻟﻬﺎﻱ ﺧﺮﻭﺟﻲ ﻣﻨﺎﺳﺐ ﺗﺒﺪﻳﻞ ﻣﻲﻧﻤﺎﻳﺪ‪ .‬ﭘﺎﻳﮕﺎﻩ ﺩﺍﺩﻩ ﺷﺎﻣﻞ ﻣﺠﻤﻮﻋﻪﺍﻱ ﺍﺯ ﻗﻮﺍﻋﺪ –‪IF‬‬

‫‪ THEN‬ﻣﻲﺑﺎﺷﺪ‪ .‬ﺑﺮﺧﻲ ﺭﻭﺷﻬﺎﻱ ﺍﺻﻠﻲ ﺗﺸﻜﻴﻞ ﭘﺎﻳﮕﺎﻩ ﺩﺍﺩﻩ ﺑﻘﺮﺍﺭ ﺯﻳﺮﻧﺪ‪:‬‬ ‫•‬

‫ﺑﻜﺎﺭﮔﻴﺮﻱ ﺩﺍﻧﺶ ﻭ ﺗﺠﺮﺑﻴﺎﺕ ﻳﻚ ﻓﺮﺩ ﻛﺎﺭﺷﻨﺎﺱ ﺟﻬﺖ ﺑﺮﺁﻭﺭﺩﻩ ﻛﺮﺩﻥ ﺍﻫﺪﺍﻑ ﻛﻨﺘﺮﻝ‬ ‫‪1 - Mamdani‬‬

‫‪٧‬‬

‫• ﻣﺪﻟﺴﺎﺯﻱ ﻋﻤﻠﻜﺮﺩ ﻛﻨﺘﺮﻝ‬ ‫• ﻣﺪﻝ ﻛﺮﺩﻥ ﭘﺮﻭﺳﻪ‬ ‫• ﺑﻜﺎﺭﮔﻴﺮﻱ ﻳﻚ ﻛﻨﺘﺮﻟﺮ ﻓﺎﺯﻱ ﺧﻮﺩ ﺳﺎﺯﻣﺎﻧﺪﻩ‬ ‫• ﺑﻜﺎﺭﮔﻴﺮﻱ ﺷﺒﻜﻪﻫﺎﻱ ﻋﺼﺒﻲ ﻣﺼﻨﻮﻋﻲ‬ ‫ﭘﺎﻳﮕﺎﻩ ﺩﺍﺩﻩ‬ ‫ﻓﺎﺯﻱﮔﺮ‬

‫ﻓﺎﺯﻱﺯﺩﺍ‬ ‫ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ‬

‫ﺳﻴﺴﺘﻢ ﺗﺤﺖ ﻛﻨﺘﺮﻝ‬ ‫ﺟﻬﺖ ﺩﺍﺩﻩﻫﺎﻱ ﻓﺎﺯﻱ‬

‫ﺟﻬﺖ ﺩﺍﺩﻩﻫﺎﻱ‬ ‫ﺷﻜﻞ )‪ (٤-١‬ﺑﻠﻮﻙ ﺩﻳﺎﮔﺮﺍﻡ ﻳﻚ ﺳﻴﺴﺘﻢ ﻛﻨﺘﺮﻝ ﺷﺎﻣﻞ ﻛﻨﺘﺮﻟﺮ ﻓﺎﺯﻱ‬

‫ﻭﻗﺘـﻲ ﻛﻪ ﻗﻮﺍﻋﺪ ﺍﻭﻟﻴﻪ ﺑﻮﺳﻴﻠﺔ ﻣﻼﺣﻈﺎﺕ ﻛﺎﺭﺷﻨﺎﺳﻲ ﺑﺪﺳﺖ ﺁﻣﺪ ﺍﻳﻦ ﻗﻮﺍﻋﺪ ﻣﻲﺗﻮﺍﻧﻨﺪ ﺑﺎ ﺩﺭ ﻧﻈﺮ ﮔﺮﻓﺘﻦ ﺳﻪ ﻫﺪﻑ ﺍﺻﻠﻲ ﺑﺮﺍﻱ‬ ‫ﺍﺳﺘﻔﺎﺩﻩ ﺩﺭﻛﻨﺘﺮﻟﺮ ﻣﻨﻄﻖ ﻓﺎﺯﻱ ﻓﺮﻡ ﺩﺍﺩﻩ ﺷﻮﻧﺪ‪:‬‬ ‫• ﺣﺬﻑ ﻫﺮﮔﻮﻧﻪ ﺧﻄﺎﻱ ﻗﺎﺑﻞ ﻣﻼﺣﻈﻪ ﺩﺭ ﺧﺮﻭﺟﻲ ﭘﺮﻭﺳﻪ ﺑﻮﺳﻴﻠﻪ ﺗﻨﻈﻴﻢ ﻣﻨﺎﺳﺐ ﺧﺮﻭﺟﻲ ﻛﻨﺘﺮﻟﺮ‬ ‫• ﺗﺨﻤﻴﻦ ﻋﻤﻠﻜﺮﺩ ﻛﻨﺘﺮﻟﻲ ﻧﺰﺩﻳﻚ ﻣﻘﺪﺍﺭ ﻣﻄﻠﻮﺏ‬ ‫• ﺍﺟﺘﻨﺎﺏ ﺍﺯ ﺍﻳﻨﻜﻪ ﺧﺮﻭﺟﻲ ﭘﺮﻭﺳﻪ ﺍﺯ ﻣﻘﺎﺩﻳﺮ ﺗﻌﻴﻴﻦ ﺷﺪﻩ ﺗﻮﺳﻂ ﻛﺎﺭﺑﺮ ﺗﺠﺎﻭﺯ ﻧﻨﻤﺎﻳﺪ‪.‬‬ ‫ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ ﻣﻐﺰ ﻳﻚ ﻛﻨﺘﺮﻟﺮ ﻣﻨﻄﻖ ﻓﺎﺯﻱ ﻣﻲﺑﺎﺷﺪ ﻭ ﺗﻮﺍﻧﺎﻳﻲ ﺷﺒﻴﻪﺳﺎﺯﻱ ﺗﺼﻤﻴﻢﮔﻴﺮﻱ ﺑﺸﺮﻱ ﻣﺒﺘﻨﻲ ﺑﺮ ﺍﻳﺪﺓ ﻓﺎﺯﻱ ﻭ ﻫﻤﭽﻨﻴﻦ‬ ‫ﺗﻮﺍﻧﺎﻳـﻲ ﻧﺘـﻴﺠﻪﮔـﻴﺮﻱ ﻋﻤﻠﻜـﺮﺩ ﻛﻨﺘﺮﻝ ﻓﺎﺯﻱ ﺑﺎ ﺑﻜﺎﺭﮔﻴﺮﻱ ﻗﻮﺍﻋﺪ ﻣﻨﻄﻖ ﻓﺎﺯﻱ ﺭﺍ ﺩﺍﺭﺩ‪ .‬ﺑﻌﺒﺎﺭﺕ ﺩﻳﮕﺮ ﺗﻤﺎﻣﻲ ﻣﺘﻐﻴﺮﻫﺎﻱ ﻭﺭﻭﺩﻱ‬ ‫ﺗﻮﺳـﻂ ﻓـﺎﺯﻱﮔـﺮ ﺑـﻪ ﻣﺘﻐـﻴﺮﻫﺎﻱ ﺯﺑﺎﻧـﻲ ﻣﺮﺑﻮﻁ ﺑﻪ ﺧﻮﺩﺷﺎﻥ ﺗﺒﺪﻳﻞ ﺷﺪﻩ ﻭ ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ ﻣﺠﻤﻮﻋﻪﺍﻱ ﺍﺯ ﻗﻮﺍﻋﺪ ‪IF–THEN‬‬

‫ﻣﻮﺟـﻮﺩ ﺩﺭ ﭘﺎﻳﮕـﺎﻩ ﺩﺍﺩﻩ ﺭﺍ ﺍﺭﺯﻳﺎﺑـﻲ ﻧﻤـﻮﺩﻩ ﻭ ﺳـﭙﺲ ﻧﺘﻴﺠﺔ ﺑﺪﺳﺖ ﺁﻣﺪﻩ ﺍﺯ ﺍﻳﻦ ﺍﺭﺯﻳﺎﺑﻲ ﻛﻪ ﻳﻚ ﻣﻘﺪﺍﺭ ﺯﺑﺎﻧﻲ ﻣﻲﺑﺎﺷﺪ ﺗﻮﺳﻂ‬ ‫ﻓﺎﺯﻱﺯﺩﺍ ﺑﻪ ﺧﺮﻭﺟﻲ ﻭﺍﻗﻌﻲ ﺗﺒﺪﻳﻞ ﻣﻲﺷﻮﺩ‪.‬‬ ‫ﺗـﺒﺪﻳﻞ ﺩﻭﻡ ﻛـﻪ ﺗﻮﺳـﻂ ﻓـﺎﺯﻱﺯﺩﺍ ﺍﻧﺠﺎﻡ ﻣﻲﭘﺬﻳﺮﺩ ﻣﻘﺪﺍﺭ ﻓﺎﺯﻱ ﺩﺭ ﺧﺮﻭﺟﻲ ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ ﺭﺍ ﺗﻮﺳﻂ ﺗﻮﺍﺑﻊ ﻋﻀﻮﻳﺖ ﺑﻪ ﻣﻘﺪﺍﺭ‬ ‫ﻭﺍﻗﻌﻲ ﻭ ﻋﺪﺩﻱ ﺗﺒﺪﻳﻞ ﻣﻲﻧﻤﺎﻳﺪ‪ .‬ﭼﻨﺪﻳﻦ ﻧﻤﻮﻧﻪ ﺗﻜﻨﻴﻚ ﺑﺮﺍﻱ ﻓﺎﺯﻱ ﺯﺩﺍﻳﻲ ﻭﺟﻮﺩ ﺩﺍﺭﺩ ﺍﻣﺎ ﺑﺪﻟﻴﻞ ﺳﺎﺩﮔﻲ ﺑﻜﺎﺭﮔﻴﺮﻱ ﻭ ﺍﻟﮕﻮﺭﻳﺘﻢ‬ ‫ﺳﺎﺩﻩﺗﺮ ﺭﻭﺵ ﻣﻴﺎﻧﮕﻴﻦ ﻣﺮﺍﻛﺰ ﺑﻜﺎﺭ ﮔﺮﻓﺘﻪ ﻣﻲﺷﻮﺩ‪.‬‬ ‫‪ -٦-١‬ﺑﺮﺧﻲ ﻛﺎﺭﺑﺮﺩﻫﺎﻱ ﺳﻴﺴﺘﻢﻫﺎﻱ ﻓﺎﺯﻱ‬ ‫‪ .١-٦-١‬ﻣﺎﺷﻴﻦ ﺷﺴﺘﺸﻮﻱ ﻓﺎﺯﻱ‬ ‫ﺩﺭ ﺍﻳﻦ ﻛﺎﺭﺑﺮﺩ ﻫﺪﻑ ﺗﻌﻴﻴﻦ ﺗﻌﺪﺍﺩ ﺩﻭﺭ ﻣﻨﺎﺳﺐ ﺑﺮﺍﻱ ﻣﺎﺷﻴﻦ ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﻧﻮﻉ ﻛﺜﻴﻔﻲ‪ ،‬ﻣﻴﺰﺍﻥ ﻛﺜﻴﻔﻲ ﻭ ﺍﻧﺪﺍﺯﻩ ﺑﺎﺭ ﻣﻲﺑﺎﺷﺪ‪ .‬ﺑﻨﺎﺑﺮﺍﻳﻦ‬

‫‪٨‬‬

‫ﺳﻴﺴﺘﻢ ﻓﺎﺯﻱ ﺩﺍﺭﺍﻱ ﺳﻪ ﻭﺭﻭﺩﻱ ﻧﻮﻉ ﻛﺜﻴﻔﻲ‪ ،‬ﻣﻴﺰﺍﻥ ﻛﺜﻴﻔﻲ ﻭ ﺍﻧﺪﺍﺯﻩ ﺑﺎﺭ ﻭ ﻳﻚ ﺧﺮﻭﺟﻲ ﻳﻌﻨﻲ ﺗﻌﺪﺍﺩ ﺩﻭﺭ ﻣﻨﺎﺳﺐ ﻣﻲﺑﺎﺷﺪ‪.‬‬

‫‪٩‬‬

‫‪ -٢‬ﻣﺠﻤﻮﻋﻪﻫﺎﻱ ﻓﺎﺯﻱ‬ ‫‪١‬‬

‫‪ -١-٢‬ﻣﻘﺎﻳﺴﻪﺍﻱ ﺑﻴﻦ ﻣﺠﻤﻮﻋﻪﻫﺎﻱ ﻛﻼﺳﻴﻚ ﻭ ﻣﺠﻤﻮﻋﻪﻫﺎﻱ ﻓﺎﺯﻱ‬

‫ﺩﺭ ﺍﻳﻦ ﻗﺴﻤﺖ ﺑﺎ ﺫﻛﺮ ﻳﻚ ﻣﺜﺎﻝ ﻣﻘﺎﻳﺴﻪﺍﻱ ﺑﻴﻦ ﻣﺠﻤﻮﻋﻪﻫﺎﻱ ﻛﻼﺳﻴﻚ ﻭ ﻣﺠﻤﻮﻋﻪﻫﺎﻱ ﻓﺎﺯﻱ ﺍﻧﺠﺎﻡ ﻣﻲﮔﻴﺮﺩ‪:‬‬ ‫ﻣﺜﺎﻝ‪ :١‬ﻓﺮﺽ ﻛﻨﻴﺪ ‪ U‬ﻣﺠﻤﻮﻋﻪ ﻣﺮﺟﻊ‪ ٢‬ﺑﻔﺮﻡ ﺯﻳﺮ ﺑﺎﺷﺪ‪:‬‬

‫}‪U = {0 ,1, 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 ,10‬‬

‫ﺣـﺎﻝ ﺍﺑـﺘﺪﺍ ﻣﺠﻤﻮﻋـﻪ ﻛﻼﺳـﻴﻚ ‪ A‬ﺑﺼـﻮﺭﺕ ﻋﺪﺩ ﻣﻴﺎﻧﻲ ﻣﺠﻤﻮﻋﻪ ﻣﺮﺟﻊ ﻭ ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ '‪ A‬ﺑﺼﻮﺭﺕ ﺍﻋﺪﺍﺩ ﻧﺰﺩﻳﻚ ﺑﻪ ﻋﺪﺩ‬ ‫ﻣﻴﺎﻧـﻲ ﻣﺠﻤﻮﻋـﻪ ﻣـﺮﺟﻊ ﺭﺍ ﺑﺪﺳـﺖ ﺁﻭﺭﻳـﺪ ﻭ ﺳـﭙﺲ ﻣﺠﻤﻮﻋـﻪ ﻛﻼﺳﻴﻚ ‪ B‬ﺑﺼﻮﺭﺕ ﻛﻮﭼﻜﺘﺮﻳﻦ ﻋﺪﺩ ﺩﺭ ﻣﺠﻤﻮﻋﻪ ﻣﺮﺟﻊ ﻭ‬ ‫ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ'‪ B‬ﺑﺼﻮﺭﺕ ﺍﻋﺪﺍﺩ ﻛﻮﭼﻚ ﺩﺭ ﻣﺠﻤﻮﻋﻪ ﻣﺮﺟﻊ ﺭﺍ ﺑﺪﺳﺖ ﺁﻭﺭﻳﺪ‪.‬‬ ‫ﺣﻞ‪ :‬ﺍﺑﺘﺪﺍ ﺑﺮﺍﻱ ﻣﺠﻤﻮﻋﻪ ﻛﻼﺳﻴﻚ ‪ A‬ﻭ ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ '‪ A‬ﺩﺍﺭﻳﻢ‪:‬‬

‫ﻛﻪ ﺩﺭ ﻓﺮﻡ ﺩﻭﻡ ﻧﻤﺎﻳﺶ ﻣﺠﻤﻮﻋﻪ‬

‫‪0 0 0 0 0 1 0 0 0 0 0 ‬‬ ‫‪A= , , , , , , , , , ,‬‬ ‫‪‬‬ ‫‪ 0 1 2 3 4 5 6 7 8 9 10 ‬‬ ‫ﻛﻼﺳﻴﻚ ‪ A‬ﺍﻋﺪﺍﺩ ﻗﺮﺍﺭ ﮔﺮﻓﺘﻪ ﺩﺭ ﻣﺨﺮﺝ ﻛﺴﺮﻫﺎ ﻫﻤﺎﻥ ﺍﻋﺪﺍﺩ ﻣﺠﻤﻮﻋﻪ‬

‫}‪A = {5‬‬

‫‪or‬‬

‫ﻣﺮﺟﻊ ‪ U‬ﻫﺴﺘﻨﺪ ﻭ‬

‫ﺍﻋﺪﺍﺩ ﻗﺮﺍﺭ ﮔﺮﻓﺘﻪ ﺩﺭ ﺻﻮﺭﺕ ﻛﺴﺮﻫﺎ ﻣﻘﺪﺍﺭ ﻋﻀﻮﻳﺖ ﻋﺪﺩ ﻣﺨﺮﺝ ﺭﺍ ﺑﻪ ﻣﺠﻤﻮﻋﻪ ‪ A‬ﻧﺸﺎﻥ ﻣﻲﺩﻫﻨﺪ‪.‬‬ ‫‪ 0 0 .2 0 .4 0 .6 0 .8 1 0 .8 0 .6 0 .4 0 .2 0 ‬‬ ‫‪A′ =  ,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪, ,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪‬‬ ‫‪1‬‬ ‫‪2‬‬ ‫‪3‬‬ ‫‪4‬‬ ‫‪5‬‬ ‫‪6‬‬ ‫‪7‬‬ ‫‪8‬‬ ‫‪9 10 ‬‬ ‫‪0‬‬

‫ﻛﻪ ﻓﺮﻡ ﺩﻳﮕﺮ ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ '‪ A‬ﺑﺼﻮﺭﺕ ﺯﻳﺮ ﻣﻲﺑﺎﺷﺪ‪:‬‬ ‫‪0 0 .2 0 .4 0 .6 0 .8 1 0 .8 0 .6 0 .4 0 .2 0‬‬ ‫‪+‬‬ ‫‪+‬‬ ‫‪+‬‬ ‫‪+‬‬ ‫‪+ +‬‬ ‫‪+‬‬ ‫‪+‬‬ ‫‪+‬‬ ‫‪+‬‬ ‫‪0‬‬ ‫‪1‬‬ ‫‪2‬‬ ‫‪3‬‬ ‫‪4‬‬ ‫‪5‬‬ ‫‪6‬‬ ‫‪7‬‬ ‫‪8‬‬ ‫‪9 10‬‬

‫= ‪A′‬‬

‫ﺣﺎﻝ ﺑﺮﺍﻱ ﻣﺠﻤﻮﻋﻪ ﻛﻼﺳﻴﻚ ‪ B‬ﻭ ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ '‪ B‬ﺩﺍﺭﻳﻢ‪:‬‬ ‫‪1 0 0 0 0 0 0 0 0 0 0 ‬‬ ‫‪B= , , , , ,‬‬ ‫‪, , , , ,‬‬ ‫‪‬‬ ‫‪ 0 1 2 3 4 5 6 7 8 9 10 ‬‬ ‫‪ 1 0 .9 0 .8 0 .7 0 .6 0 .5 0 .4 0 .3 0 .2 0 .1 0 ‬‬ ‫‪B′ =  ,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪‬‬ ‫‪1‬‬ ‫‪2‬‬ ‫‪3‬‬ ‫‪4‬‬ ‫‪5‬‬ ‫‪6‬‬ ‫‪7‬‬ ‫‪8‬‬ ‫‪9 10 ‬‬ ‫‪0‬‬

‫}‪B = {0‬‬

‫‪or‬‬

‫‪٣‬‬

‫‪ -٢-٢‬ﺗﻌﺮﻳﻒ ﺗﺎﺑﻊ ﻋﻀﻮﻳﺖ‬

‫ﺗﺎﺑﻊ ﻋﻀﻮﻳﺖ ﻣﻴﺰﺍﻥ ﻭﺍﺑﺴﺘﮕﻲ ﻫﺮ ﻋﻀﻮ ﺭﺍ ﺑﻪ ﻣﺠﻤﻮﻋﻪ ﻣﻮﺭﺩ ﻧﻈﺮ ﺑﻔﺮﻡ ﻳﻚ ﺗﺎﺑﻊ ﺭﻳﺎﺿﻲ ﺑﻴﺎﻥ ﻣﻲﻛﻨﺪ‪ .‬ﺑﻌﻨﻮﺍﻥ ﻣﺜﺎﻝ ﺗﺎﺑﻊ‬ ‫ﻋﻀﻮﻳﺖ ﺑﺮﺍﻱ ﻣﺠﻤﻮﻋﻪ ﻛﻼﺳﻴﻚ ‪ A‬ﻭ ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ '‪ A‬ﻛﻪ ﺑﺎ ﻧﻤﺎﺩ ‪ µ‬ﻧﻤﺎﻳﺶ ﺩﺍﺩﻩ ﻣﻲﺷﻮﺩ‪ ،‬ﺑﺼﻮﺭﺕ ﺯﻳﺮ ﻣﻲﺑﺎﺷﺪ‪:‬‬ ‫)‪µA (x‬‬

‫‪x=5‬‬ ‫‪x≠5‬‬

‫‪1‬‬ ‫‪5‬‬

‫‪1‬‬ ‫‪µ A ( x) = ‬‬ ‫‪0‬‬

‫‪1 - Fuzzy Sets‬‬ ‫‪2 - Universal Set‬‬ ‫‪3 - Membership Function‬‬

‫‪١٠‬‬

µA’(x)

x < 0 or x > 10 0  0≤ x≤5 µ A′ ( x) =  x / 5 2 − x / 5 5 ≤ x ≤ 10 

1 5

‫ ﺍﻧﻮﺍﻉ ﺗﻮﺍﺑﻊ ﻋﻀﻮﻳﺖ‬-٣-٢ .‫( ﺁﻧﻬﺎ ﺭﺍ ﻧﺸﺎﻥ ﻣﻲﺩﻫﺪ‬١-٢) ‫ﺩﺭ ﺯﻳﺮ ﺍﻧﻮﺍﻉ ﺗﻮﺍﺑﻊ ﻋﻀﻮﻳﺖ ﺫﻛﺮ ﺷﺪﻩ ﺍﺳﺖ ﻛﻪ ﺷﻜﻞ‬ (trimf) ‫ﺗﺎﺑﻊ ﻋﻀﻮﻳﺖ ﻣﺜﻠﺜﻲ‬

•

(trapmf) 2‫ﺗﺎﺑﻊ ﻋﻀﻮﻳﺖ ﺫﻭﺯﻧﻘﻪﺍﻱ‬

•

(gaussmf) 3‫ﺗﺎﺑﻊ ﻋﻀﻮﻳﺖ ﮔﻮﺳﻲ‬

•

(gbellmf) 4‫ﺗﺎﺑﻊ ﻋﻀﻮﻳﺖ ﺯﻧﮕﻮﻟﻪﺍﻱ ﺗﻌﻤﻴﻢ ﻳﺎﻓﺘﻪ‬

•

(gauss2mf) 5‫ﺗﺎﺑﻊ ﻋﻀﻮﻳﺖ ﮔﻮﺳﻲ ﺩﻭ ﻃﺮﻓﻪ‬

•

(smf) 6‫ ﺷﻜﻞ‬S ‫ﺗﺎﺑﻊ ﻋﻀﻮﻳﺖ‬

•

(zmf) 7‫ ﺷﻜﻞ‬Z ‫ﺗﺎﺑﻊ ﻋﻀﻮﻳﺖ‬

•

(sigmf) 8‫ﺗﺎﺑﻊ ﻋﻀﻮﻳﺖ ﺳﻴﮕﻤﻮﻳﺪ‬

•

١

− ( x −c ) 2

f ( x; σ , c) = e

f ( x; a, b, c) =

f ( x; a, c) =

2σ 2

1 x−c 1+ a

2b

1 1 + e − a ( x −c )

1 - Triangular Membership Function 2 - Trapezoidal Membership Function 3 - Gaussian Curve Membership Function 4 - Generalized Bell Membership Function 5 - Two-sided Gaussian Curve Membership Function 6 - S-shaped Curve Membership Function 7 - Z-shaped Curve Membership Function 8 - Sigmoid Curve Membership Function

١١

‫•‬

‫ﺗﺎﺑﻊ ﻋﻀﻮﻳﺖ ﺣﺎﺻﻠﻀﺮﺏ ﺩﻭ ﺳﻴﮕﻤﻮﻳﺪ‪(psigmf) 1‬‬

‫•‬

‫ﺗﺎﺑﻊ ﻋﻀﻮﻳﺖ ﺗﻔﺎﺿﻞ ﺩﻭ ﺳﻴﮕﻤﻮﻳﺪ‪(dsigmf) 2‬‬

‫•‬

‫ﺗﺎﺑﻊ ﻋﻀﻮﻳﺖ ‪ π‬ﺷﻜﻞ‪(pimf) 3‬‬

‫ﺷﻜﻞ )‪ : (١-٢‬ﺍﻧﻮﺍﻉ ﺗﻮﺍﺑﻊ ﻋﻀﻮﻳﺖ‬

‫‪ -٤-٢‬ﻣﻌﺮﻓﻲ ﻣﻔﺎﻫﻴﻢ ﺍﺳﺎﺳﻲ ﻣﺮﺗﺐ ﺑﺎ ﻣﺠﻤﻮﻋﻪﻫﺎﻱ ﻓﺎﺯﻱ‬ ‫ﺩﺭ ﺍﻳﻦ ﻗﺴﻤﺖ ﻣﻔﺎﻫﻴﻢ ﺗﻜﻴﻪﮔﺎﻩ ﻓﺎﺯﻱ‪ ،‬ﻣﻨﻔﺮﺩ ﻓﺎﺯﻱ‪ ،‬ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ ﺗﻬﻲ‪ ،‬ﻣﺮﻛﺰ‪ ،‬ﻧﻘﻄﻪ ﺗﻘﺎﻃﻊ‪ ،‬ﺍﺭﺗﻔﺎﻉ ﻭ ﺑﺮﺵ ﺁﻟﻔﺎ ﺭﺍ ﺑﺮﺭﺳﻲ‬ ‫ﻣﻲﺷﻮﺩ‪:‬‬ ‫‪ -١‬ﺗﻜﻴﻪ ﮔﺎﻩ‪ :٤‬ﺗﻜﻴﻪﮔﺎﻩ ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ ‪ A‬ﺩﺭ ﻓﻀﺎﻱ ﻣﺠﻤﻮﻋﻪ ﻣﺮﺟﻊ ‪ U‬ﻳﻚ ﻣﺠﻤﻮﻋﻪ ﻏﻴﺮ ﻓﺎﺯﻱ )ﻛﻼﺳﻴﻚ( ﺍﺳﺖ ﻛﻪ ﺷﺎﻣﻞ‬ ‫ﺗﻤﺎﻣﻲ ﻋﻀﻮﻫﺎﻱ ﻏﻴﺮ ﺻﻔﺮ ‪ U‬ﻣﻲﺷﻮﺩ ﻳﻌﻨﻲ‬ ‫}‪Supp ( A) = {x ∈ U | µ A ( x) > 0‬‬

‫ﻣﺜﺎﻝ‪ :٢‬ﺩﺭ ﻣﺜﺎﻝ‪ ١‬ﺑﺮﺍﻱ ﻣﺠﻤﻮﻋﻪﻫﺎﻱ ﻓﺎﺯﻱ '‪ A‬ﻭ '‪ B‬ﺗﻜﻴﻪﮔﺎﻩ ﺭﺍ ﺑﺪﺳﺖ ﺁﻭﺭﻳﺪ‪.‬‬ ‫}‪Supp ( A′) = {1, 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9‬‬ ‫}‪Supp ( B ′) = {0 ,1, 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9‬‬

‫‪1 - Product of two Sigmoid Membership Function‬‬ ‫‪2 - Difference between two Sigmoid Membership Function‬‬ ‫‪3 - Pi-shaped Curve Membership Function‬‬ ‫‪4 - Support‬‬

‫‪١٢‬‬

‫‪ -٢‬ﻣﻨﻔﺮﺩ ﻓﺎﺯﻱ‪ :١‬ﻳﻚ ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ ﺍﺳﺖ ﻛﻪ ﺗﻜﻴﻪﮔﺎﻩ ﺁﻥ ﻳﻚ ﻧﻘﻄﻪ ﻭﺍﺣﺪ ﺩﺭ ‪ U‬ﺑﺎﺷﺪ‪.‬‬ ‫‪ -٣‬ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ ﺗﻬﻲ‪ :٢‬ﻳﻚ ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ ﺍﺳﺖ ﻛﻪ ﺗﻜﻴﻪﮔﺎﻩ ﺁﻥ ﺗﻬﻲ ﺑﺎﺷﺪ‪.‬‬ ‫‪ -٤‬ﻣﺮﻛﺰ‪ :٣‬ﻣﺮﻛﺰ ﻳﻚ ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ ﺑﻪ ﺻﻮﺭﺕ ﺯﻳﺮ ﺗﻌﺮﻳﻒ ﻣﻲﺷﻮﺩ‪:‬‬ ‫• ﺍﮔﺮ ﺣﺪﺍﻛﺜﺮ ﺗﺎﺑﻊ ﻋﻀﻮﻳﺖ ﻣﺘﻌﻠﻖ ﺑﻪ ﻧﻘﺎﻁ ﻣﺤﺪﻭﺩﻱ ﺑﺎﺷﺪ ﺁﻧﮕﺎﻩ ﻣﻴﺎﻧﮕﻴﻦ ﻧﻘﺎﻁ ﻣﺮﻛﺰ‪ ،‬ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ ﻣﻲﺑﺎﺷﺪ‪.‬‬

‫‪Center‬‬

‫‪Center‬‬

‫• ﺍﮔﺮ ﺣﺪﺍﻛﺜﺮ ﺗﺎﺑﻊ ﻋﻀﻮﻳﺖ ﻣﺘﻌﻠﻖ ﺑﻪ ﻧﻘﺎﻁ ﻧﺎﻣﺤﺪﻭﺩﻱ ﺑﺎﺷﺪ ﺁﻧﮕﺎﻩ ﻛﻮﭼﻜﺘﺮﻳﻦ ﻳﺎ ﺑﺰﺭﮔﺘﺮﻳﻦ ﻧﻘﻄﻪﺍﻱ ﻛﻪ ﺩﺭ ﺁﻥ ﻧﻘﻄﻪ‬ ‫ﺗﺎﺑﻊ ﺑﻪ ﺣﺪﺍﻛﺜﺮ ﻣﻘﺪﺍﺭ ﺧﻮﺩ ﻣﻲﺭﺳﺪ ﺑﻌﻨﻮﺍﻥ ﻣﺮﻛﺰ ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ ﺗﻌﺮﻳﻒ ﻣﻲﺷﻮﺩ‪.‬‬

‫‪Center‬‬

‫‪Center‬‬

‫‪ -٥‬ﻧﻘﻄﻪ ﺗﻘﺎﻃﻊ‪ :٤‬ﻧﻘﻄﻪ ﺗﻘﺎﻃﻊ ﻳﻚ ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ ﻧﻘﻄﻪﺍﻱ ﺩﺭ ‪ U‬ﺍﺳﺖ ﻛﻪ ﺩﺭ ﺁﻥ ﻣﻘﺪﺍﺭ ﺗﺎﺑﻊ ﻋﻀﻮﻳﺖ ﺑﺮﺍﺑﺮ ﺑﺎ ‪ ٠/٥‬ﻣﻲﺷﻮﺩ‪.‬‬

‫‪0.5‬‬ ‫‪Crossover‬‬

‫‪ -٦‬ﺍﺭﺗﻔﺎﻉ‪ :٥‬ﺍﺭﺗﻔﺎﻉ ﻳﻚ ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ ﺑﺰﺭﮔﺘﺮﻳﻦ ﻣﻘﺪﺍﺭ ﻳﻚ ﺗﺎﺑﻊ ﻋﻀﻮﻳﺖ ﺍﺳﺖ‪.‬‬ ‫ﺗﺬﻛﺮ‪ :‬ﺍﮔﺮ ﺍﺭﺗﻔﺎﻉ ﻳﻚ ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ ﺑﺮﺍﺑﺮ ﺑﺎ ﻳﻚ ﺑﺎﺷﺪ ﺩﺭ ﺁﻥﺻﻮﺭﺕ ﺁﻧﺮﺍ ﻳﻚ ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ ﻧﺮﻣﺎﻝ ﮔﻮﻳﻨﺪ‪.‬‬ ‫‪ -٧‬ﺑﺮﺵ ﺁﻟﻔﺎ‪ :٦‬ﺑﺮﺵ ﺁﻟﻔﺎﻱ ﻳﻚ ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ‪ ،‬ﻣﺠﻤﻮﻋﻪ ﻏﻴﺮ ﻓﺎﺯﻱ ‪ Aα‬ﺍﺳﺖ ﻛﻪ ﺷﺎﻣﻞ ﺗﻤﺎﻣﻲ ﻋﻀﻮﻫﺎﻱ ‪ U‬ﺍﺳﺖ ﻛﻪ ﻣﻘﺎﺩﻳﺮ‬ ‫‪1 - Fuzzy Singleton‬‬ ‫‪2 - Empty Fuzzy Set‬‬ ‫‪3 - Center‬‬ ‫‪4 - Crossover Point‬‬ ‫‪5 - Height‬‬ ‫‪6 - α-Cut‬‬

‫‪١٣‬‬

‫ﺑﺰﺭﮔﺘﺮ ﻳﺎ ﻣﺴﺎﻭﻱ ‪ α‬ﺩﺍﺭﻧﺪ‪:‬‬ ‫‪0 <α <1‬‬

‫} ‪Aα = {x ∈ U | µ A ( x) ≥ α‬‬

‫ﻣﺜﺎﻝ‪ :٣‬ﺑﺮﺍﻱ ﻣﺜﺎﻝ‪ ١‬ﺑﺮﺵ ‪ ٠/٨‬ﺑﺮﺍﻱ '‪ A‬ﻭ ﺑﺮﺵ ‪ ٠/٧‬ﺑﺮﺍﻱ '‪ B‬ﺭﺍ ﺑﺪﺳﺖ ﺁﻭﺭﻳﺪ‪.‬‬ ‫}‪A0′.8 = {4 , 5 , 6‬‬ ‫}‪B0′.7 = {0 ,1, 2 , 3‬‬

‫‪١۴‬‬

‫‪ -٣‬ﻋﻤﻠﻴﺎﺕ ﺑﺮ ﺭﻭﻱ ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ‬ ‫ﺩﺭ ﺍﻳﻦ ﻗﺴﻤﺖ ﻋﻤﻠﻴﺎﺕﻫﺎﻱ ﻣﻌﺎﺩﻝ ﺑﻮﺩﻥ‪ ،‬ﺯﻳﺮ ﻣﺠﻤﻮﻋﻪ ﺑﻮﺩﻥ‪ ،‬ﻣﻜﻤﻞ‪ ،‬ﺍﺟﺘﻤﺎﻉ ﻭ ﺍﺷﺘﺮﺍﻙ ﺭﺍ ﻣﻌﺮﻓﻲ ﻣﻲﻧﻤﺎﻳﻴﻢ‪:‬‬ ‫‪١‬‬

‫‪ -١‬ﻣﻌﺎﺩﻝ ﺑﻮﺩﻥ‬

‫ﺩﻭ ﻣﺠﻤﻮﻋﻪ ‪ A‬ﻭ ‪ B‬ﻣﻌﺎﺩﻝ ﻫﺴﺘﻨﺪ ﺍﮔﺮ ﻭ ﻓﻘﻂ ﺍﮔﺮ ﺑﺮﺍﻱ ﺗﻤﺎﻣﻲ ﻣﻘﺎﺩﻳﺮ ‪x ∈ U‬‬

‫)‪µ A ( x) = µ B ( x‬‬

‫‪٢‬‬

‫‪ -٢‬ﺯﻳﺮ ﻣﺠﻤﻮﻋﻪ ﺑﻮﺩﻥ‬

‫ﻣﺠﻤﻮﻋﻪ ‪ A‬ﺯﻳﺮ ﻣﺠﻤﻮﻋﻪ ‪ B‬ﺍﺳﺖ ﺍﮔﺮ ﻭ ﻓﻘﻂ ﺍﮔﺮ ﺑﺮﺍﻱ ﺗﻤﺎﻣﻲ ﻣﻘﺎﺩﻳﺮ ‪x ∈ U‬‬

‫)‪µ A ( x) ≤ µ B ( x‬‬ ‫)‪µB (x‬‬ ‫)‪µA (x‬‬

‫‪٣‬‬

‫‪ -٣‬ﻣﻜﻤﻞ‬

‫ﻣﻜﻤﻞ ‪ A‬ﺩﺭ ‪ A ،U‬ﺍﺳﺖ ﺍﮔﺮ ‪x ∈ U‬‬

‫)‪µ A ( x) = 1 − µ A ( x‬‬ ‫)‪1−µA (x‬‬

‫)‪µA (x‬‬

‫‪٤‬‬

‫‪ -٤‬ﺍﺟﺘﻤﺎﻉ‬

‫ﺍﺟﺘﻤﺎﻉ ‪ A‬ﻭ ‪ B‬ﻳﻚ ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ ﺍﺳﺖ ﻛﻪ ﺑﺎ‬

‫‪A U B‬‬

‫ﻧﺸﺎﻥ ﺩﺍﺩﻩ ﻣﻲﺷﻮﺩ ﻭ ﺩﺍﺭﺍﻱ ﺗﺎﺑﻊ ﻋﻀﻮﻳﺖ ﺯﻳﺮ ﺍﺳﺖ‪:‬‬

‫‪µ AU B‬‬

‫])‪µ AU B ( x) = max[ µ A ( x), µ B ( x‬‬ ‫)‪µB (x‬‬

‫)‪µA (x‬‬

‫‪٥‬‬

‫‪ -٥‬ﺍﺷﺘﺮﺍﻙ‬

‫ﺍﺷﺘﺮﺍﻙ ‪ A‬ﻭ ‪ B‬ﻳﻚ ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ ﺍﺳﺖ ﻛﻪ ﺑﺎ‬

‫‪A I B‬‬

‫ﻧﺸﺎﻥ ﺩﺍﺩﻩ ﻣﻲﺷﻮﺩ ﻭ ﺩﺍﺭﺍﻱ ﺗﺎﺑﻊ ﻋﻀﻮﻳﺖ ﺯﻳﺮ ﺍﺳﺖ‪:‬‬ ‫])‪µ AI B ( x) = min[ µ A ( x), µ B ( x‬‬

‫‪µ AI B‬‬

‫)‪µB (x‬‬

‫)‪µA (x‬‬

‫‪1 - Equal‬‬ ‫‪2 - Containment‬‬ ‫‪3 - Complement‬‬ ‫‪4 - Union‬‬ ‫‪5 - Intersection‬‬

‫‪١۵‬‬

‫ﻣﺜﺎﻝ‪ :٤‬ﺩﻭ ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ ‪ A‬ﻭ ‪ B‬ﺩﺍﺩﻩ ﺷﺪﻩ ﺍﺳﺖ‪ ،‬ﺍﺑﺘﺪﺍ ﻣﻜﻤﻞ ﻓﺎﺯﻱ ‪ A‬ﻭ‪ B‬ﺭﺍ ﺑﺪﺳﺖ ﺁﻭﺭﻳﺪ ﻭ ﺳﭙﺲ ﺍﺟﺘﻤﺎﻉ ﻭ ﺍﺷﺘﺮﺍﻙ ‪ A‬ﻭ‬ ‫‪ B‬ﺭﺍ ﻧﻴﺰ ﻣﺤﺎﺳﺒﻪ ﻧﻤﺎﻳﻴﺪ‪.‬‬ ‫‪1‬‬ ‫‪0 .8 0 .6 0 .4 0 .2 0 ‬‬ ‫‪ 0 0 .2 0 .4 0 .6 0 .8‬‬ ‫‪A= ,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪‬‬ ‫‪5‬‬ ‫‪6‬‬ ‫‪7‬‬ ‫‪8‬‬ ‫‪9 10 ‬‬ ‫‪1‬‬ ‫‪2‬‬ ‫‪3‬‬ ‫‪4‬‬ ‫‪0‬‬ ‫‪ 1 0 .9 0 .8 0 .7 0 .6 0 .5 0 .4 0 .3 0 .2 0 .1 0 ‬‬ ‫‪B= ,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪‬‬ ‫‪5‬‬ ‫‪6‬‬ ‫‪7‬‬ ‫‪8‬‬ ‫‪9 10 ‬‬ ‫‪1‬‬ ‫‪2‬‬ ‫‪3‬‬ ‫‪4‬‬ ‫‪0‬‬ ‫‪1‬‬ ‫‪‬‬ ‫‪10 ‬‬ ‫‪1‬‬ ‫‪,‬‬ ‫‪‬‬ ‫‪10 ‬‬ ‫‪0‬‬ ‫‪,‬‬ ‫‪‬‬ ‫‪10 ‬‬ ‫‪0‬‬ ‫‪,‬‬ ‫‪‬‬ ‫‪10 ‬‬ ‫‪,‬‬

‫‪0 .8‬‬ ‫‪9‬‬ ‫‪0 .9‬‬ ‫‪,‬‬ ‫‪9‬‬ ‫‪0 .2‬‬ ‫‪,‬‬ ‫‪9‬‬ ‫‪0 .1‬‬ ‫‪,‬‬ ‫‪9‬‬ ‫‪,‬‬

‫‪0 .6‬‬ ‫‪8‬‬ ‫‪0 .8‬‬ ‫‪,‬‬ ‫‪8‬‬ ‫‪0 .4‬‬ ‫‪,‬‬ ‫‪8‬‬ ‫‪0 .2‬‬ ‫‪,‬‬ ‫‪8‬‬ ‫‪,‬‬

‫‪0 .4‬‬ ‫‪7‬‬ ‫‪0 .7‬‬ ‫‪,‬‬ ‫‪7‬‬ ‫‪0 .6‬‬ ‫‪,‬‬ ‫‪7‬‬ ‫‪0 .3‬‬ ‫‪,‬‬ ‫‪7‬‬ ‫‪,‬‬

‫‪١۶‬‬

‫‪0 .2‬‬ ‫‪6‬‬ ‫‪0 .6‬‬ ‫‪,‬‬ ‫‪6‬‬ ‫‪0 .8‬‬ ‫‪,‬‬ ‫‪6‬‬ ‫‪0 .4‬‬ ‫‪,‬‬ ‫‪6‬‬ ‫‪,‬‬

‫‪0‬‬ ‫‪5‬‬ ‫‪0 .5‬‬ ‫‪,‬‬ ‫‪5‬‬ ‫‪1‬‬ ‫‪,‬‬ ‫‪5‬‬ ‫‪0 .5‬‬ ‫‪,‬‬ ‫‪5‬‬ ‫‪,‬‬

‫‪0 .2‬‬ ‫‪4‬‬ ‫‪0 .4‬‬ ‫‪,‬‬ ‫‪4‬‬ ‫‪0 .8‬‬ ‫‪,‬‬ ‫‪4‬‬ ‫‪0 .6‬‬ ‫‪,‬‬ ‫‪4‬‬ ‫‪,‬‬

‫‪0 .4‬‬ ‫‪3‬‬ ‫‪0 .3‬‬ ‫‪,‬‬ ‫‪3‬‬ ‫‪0 .7‬‬ ‫‪,‬‬ ‫‪3‬‬ ‫‪0 .6‬‬ ‫‪,‬‬ ‫‪3‬‬ ‫‪,‬‬

‫‪0 .6‬‬ ‫‪2‬‬ ‫‪0 .2‬‬ ‫‪,‬‬ ‫‪2‬‬ ‫‪0 .8‬‬ ‫‪,‬‬ ‫‪2‬‬ ‫‪0 .4‬‬ ‫‪,‬‬ ‫‪2‬‬ ‫‪,‬‬

‫‪0 .8‬‬ ‫‪1‬‬ ‫‪0 .1‬‬ ‫‪,‬‬ ‫‪1‬‬ ‫‪0 .9‬‬ ‫‪,‬‬ ‫‪1‬‬ ‫‪0 .2‬‬ ‫‪,‬‬ ‫‪1‬‬ ‫‪,‬‬

‫‪1‬‬ ‫‪A =‬‬ ‫‪0‬‬ ‫‪0‬‬ ‫‪B =‬‬ ‫‪0‬‬ ‫‪1‬‬ ‫‪AU B = ‬‬ ‫‪0‬‬ ‫‪0‬‬ ‫‪AI B = ‬‬ ‫‪0‬‬

‫‪ -٤‬ﻋﻤﻠﻴﺎﺕ ﺩﻳﮕﺮﻱ ﺑﺮ ﺭﻭﻱ ﻣﺠﻤﻮﻋﻪﻫﺎﻱ ﻓﺎﺯﻱ‬ ‫ﻋﻼﻭﻩ ﺑﺮ ﻣﻜﻤﻞ‪ ،‬ﺍﺟﺘﻤﺎﻉ ﻭ ﺍﺷﺘﺮﺍﻙ ﺗﻌﺮﻳﻒ ﺷﺪﻩ ﺑﺼﻮﺭﺕ ﻗﺒﻞ‪ ،‬ﺻﻮﺭﺕﻫﺎﻱ ﺩﻳﮕﺮﻱ ﻧﻴﺰ ﺍﺯ ﺗﻌﺮﻳﻒ ﻣﻜﻤﻞ‪ ،‬ﺍﺟﺘﻤﺎﻉ ﻭ ﺍﺷﺘﺮﺍﻙ‬ ‫ﻣﺠﻤﻮﻋﻪﻫﺎﻱ ﻓﺎﺯﻱ ﻭﺟﻮﺩ ﺩﺍﺭﻧﺪ ﻛﻪ ﺩﺭ ﺍﻳﻨﺠﺎ ﺑﻪ ﺑﺮﺭﺳﻲ ﺁﻧﻬﺎ ﻣﻲﭘﺮﺩﺍﺯﻳﻢ‪:‬‬ ‫‪١‬‬

‫‪ -١‬ﻣﻜﻤﻞ ﻓﺎﺯﻱ‬

‫• ﻣﻜﻤﻞ ﻓﺎﺯﻱ ﺍﺳﺎﺳﻲ‬

‫‪٢‬‬

‫)‪C [µ A ( x)] = 1 − µ A ( x‬‬

‫• ﻛﻼﺱ ﺳﻮﮔﻨﻮ‬

‫‪٣‬‬

‫)‪1 − µ A ( x‬‬ ‫)‪1 + λµ A ( x‬‬

‫) ∞ ‪λ ∈ ( −1 ,‬‬

‫= ])‪C λ [µ A ( x‬‬

‫• ﻛﻼﺱ ﻳﺎﮔﺮ‬

‫‪٤‬‬

‫) ∞ ‪w ∈ (0 ,‬‬

‫‪w‬‬

‫}‬

‫‪1‬‬

‫{‬

‫‪C w [µ A ( x)] = 1 − [ µ A ( x)] w‬‬

‫‪ -٢‬ﺍﺟﺘﻤﺎﻉ ﻓﺎﺯﻱ‪ -S) ٥‬ﻧﺮﻡ(‬ ‫• ﻛﻼﺱ ﻣﺎﻛﺰﻳﻤﻢ‬

‫‪٦‬‬

‫])‪S [µ A ( x) , µ B ( x)] = max[µ A ( x) , µ B ( x‬‬

‫• ﻛﻼﺱ ﺩﻭﻣﺒﻲ‬

‫‪٧‬‬

‫) ∞ ‪λ ∈ (0 ,‬‬

‫]‬

‫‪−1‬‬ ‫‪λ‬‬

‫‪1‬‬

‫[‬

‫‪1 + ( µ A1( x ) − 1) −λ + ( µ B1( x ) − 1) −λ‬‬

‫= ])‪S λ [µ A ( x) , µ B ( x‬‬

‫• ﻛﻼﺱ ﺩﺑﻮﻳﺲ‪-‬ﭘﺮﻳﺪ‬

‫‪٨‬‬

‫]‪α ∈ [0 ,1‬‬

‫] ‪µ A ( x) + µ B ( x) − µ A ( x) µ B ( x) − min[ µ A ( x), µ B ( x),1 − α‬‬ ‫] ‪max[1 − µ A ( x),1 − µ B ( x),α‬‬

‫= ])‪Sα [µ A ( x) , µ B ( x‬‬

‫• ﻛﻼﺱ ﻳﺎﮔﺮ‬

‫‪٩‬‬

‫) ∞ ‪w ∈ (0 ,‬‬

‫] ‪S w [µ A ( x) , µ B ( x)] = min[1, ([ µ A ( x)] w + [ µ B ( x)] w ) w‬‬ ‫‪1‬‬

‫‪1 - Fuzzy Complement‬‬ ‫‪2 - Basic Fuzzy Complement‬‬ ‫‪3 - Sugeno Class‬‬ ‫‪4 - Yager Class‬‬ ‫‪5 - Fuzzy Union‬‬ ‫‪6 - Maximum Class‬‬ ‫‪7 - Dombi Calss‬‬ ‫‪8 - Dubois-Prade Class‬‬ ‫‪9 - Yager Class‬‬

‫‪١٧‬‬

‫• ﺟﻤﻊ ﺩﺭﺍﺳﺘﻴﻚ‬

١

µ A ( x)  S ds [µ A ( x) , µ B ( x)] = µ B ( x) 1 

if if

µ B ( x) = 0 µ A ( x) = 0 otherwise

‫• ﺟﻤﻊ ﺍﻳﻨﺸﺘﻴﻦ‬

٢

S es [µ A ( x) , µ B ( x)] =

µ A ( x) + µ B ( x) 1 + µ A ( x) . µ B ( x)

‫• ﺟﻤﻊ ﺟﺒﺮﻱ‬

٣

S as [µ A ( x) , µ B ( x)] = µ A ( x) + µ B ( x) − µ A ( x) . µ B ( x)

(‫ ﻧﺮﻡ‬-T) ٤‫ ﺍﺷﺘﺮﺍﻙ ﻓﺎﺯﻱ‬-٣ ‫• ﻛﻼﺱ ﻣﻴﻨﻴﻤﻢ‬

t [µ A ( x) , µ B ( x)] = min[µ A ( x) , µ B ( x)]

‫• ﻛﻼﺱ ﺩﻭﻣﺒﻲ‬ t λ [µ A ( x) , µ B ( x)] =

[

1

1 + ( µ A1( x ) − 1) λ + ( µ B1( x ) − 1) λ

λ ∈ (0 , ∞ )

]

1 λ

‫ﭘﺮﻳﺪ‬-‫• ﻛﻼﺱ ﺩﺑﻮﻳﺲ‬ tα [µ A ( x) , µ B ( x)] =

µ A ( x) µ B ( x) max[µ A ( x), µ B ( x),α ]

α ∈ [0 ,1]

‫• ﻛﻼﺱ ﻳﺎﮔﺮ‬ t w [µ A ( x) , µ B ( x)] = 1 − min[1, {[1 − µ A ( x)] w + [1 − µ B ( x)] w } w ] 1

w ∈ (0 , ∞ )

‫• ﺿﺮﺏ ﺩﺭﺍﺳﺘﻴﻚ‬  µ A ( x)  t dp [µ A ( x) , µ B ( x)] =  µ B ( x) 0 

if if

µ B ( x) = 1 µ A ( x) = 1 otherwise

1 - Drastic Sum 2 - Einstein Sum 3 - Algebraic Sum 4 -Fuzzy Intersection

١٨

‫• ﺿﺮﺏ ﺍﻳﻨﺸﺘﻴﻦ‬ µ A ( x) .µ B ( x) 2 − [ µ A ( x) + µ B ( x) − µ A ( x) . µ B ( x)]

t ep [µ A ( x) , µ B ( x)] =

‫• ﺿﺮﺏ ﺟﺒﺮﻱ‬

t ap [µ A ( x) , µ B ( x)] = µ A ( x) . µ B ( x) Minimum Drastic Product Einstein Product Algebraic Product Dombi T-norm λ

Fuzzy AND

Fuzzy OR

Max-Min Averages λ

Yager T-norm ω

Maximum Drastic Sum Einstein Sum Algebraic Sum Dombi S-norm λ Yager S-norm ω

Generalized Means α

tdp (a,b)

min (a,b)

Intersection Operators

max (a,b) Averaging Operators

Sds (a,b) Union Operators

a = µ A ( x) , b = µ B ( x) ‫( ﻣﺤﺪﻭﺩﻩ ﻛﺎﻣﻞ ﻋﻤﻠﮕﺮﻫﺎﻱ ﻓﺎﺯﻱ‬١-٤) ‫ﺷﻜﻞ‬

‫( ﻣﻲﺗﻮﺍﻥ ﺩﻳﺪ ﻛﻪ ﻋﻤﻠﻜﺮﺩﻫﺎﻱ ﺍﺟﺘﻤﺎﻉ ﻭ ﺍﺷﺘﺮﺍﻙ ﻧﻤﻲﺗﻮﺍﻧﻨﺪ ﺗﻤﺎﻣﻲ ﻣﺤﺪﻭﺩﻩﻫﺎ ﺭﺍ ﭘﻮﺷﺶ ﺩﻫﻨﺪ‬١-٤) ‫ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﺷﻜﻞ‬ :‫ﺑﻨﺎﺑﺮﺍﻳﻦ ﻋﻤﻠﮕﺮ ﻣﻴﺎﻧﮕﻴﻦ ﻣﻌﺮﻓﻲ ﻣﻲﺷﻮﺩ‬ (‫ ﻧﺮﻡ‬-v) ١‫ ﻣﻴﺎﻧﮕﻴﻦ ﻓﺎﺯﻱ‬-٤ ٢

Vλ [µ A ( x) , µ B ( x)] = λ max[µ A ( x) , µ B ( x)] + (1 − λ ) min[µ A ( x) , µ B ( x)]

Max-Min ‫ﻣﻴﺎﻧﮕﻴﻦ‬

•

λ ∈ [0,1]

‫• ﻣﻴﺎﻧﮕﻴﻦ ﺗﻌﻤﻴﻢ ﻳﺎﻓﺘﻪ‬

٣

[ µ ( x)]α + [ µ B ( x)]α  Vα [µ A ( x) , µ B ( x)] =  A  2  

1 α

α ∈ R (α ≠ 0)

1 - Fuzzy Averaging 2 - Max-Min Averaging 3 - Generalized Means

١٩

V P [µ A ( x) , µ B ( x)] = P. min[µ A ( x) , µ B ( x)] +

Vγ [µ A ( x) , µ B ( x)] = γ . max[µ A ( x) , µ B ( x)] +

(1 − P ).[µ A ( x) + µ B ( x)] 2

(1 − γ ).[µ A ( x) + µ B ( x)] 2

٢٠

‫ ﻓﺎﺯﻱ‬AND • P ∈ [0,1]

‫ ﻓﺎﺯﻱ‬OR • γ ∈ [0,1]

‫‪ -٥‬ﺭﻭﺍﺑﻂ ﻓﺎﺯﻱ‬ ‫ﻓﺮﺽ ﻛﻨﻴﺪ ‪ A‬ﻭ ‪ B‬ﺩﻭ ﻣﺠﻤﻮﻋﻪ ﻛﻼﺳﻴﻚ ﺑﺎﺷﻨﺪ‪:‬‬ ‫}‪A = {1, 2 , 3‬‬

‫}‪B = {0 ,1‬‬

‫ﺁﻧﮕﺎﻩ ‪ A × B‬ﺧﻮﺩ ﻳﻚ ﻣﺠﻤﻮﻋﻪ ﻛﻼﺳﻴﻚ ﺷﺎﻣﻞ ﺯﻭﺝﻫﺎﻱ ﻣﺮﺗﺐ )‪ b‬ﻭ ‪ (a‬ﻣﻲﺑﺎﺷﻨﺪ‪:‬‬

‫})‪A × B = {(1, 0) , (1,1) , (2 , 0) , (2 ,1) , (3 , 0) , (3 ,1‬‬

‫ﺣﺎﻝ ﻓﺮﺽ ﻛﻨﻴﺪ ﻣﺠﻤﻮﻋﻪ ﻛﻼﺳﻴﻚ )‪ b‬ﻭ ‪ C (a‬ﻳﻚ ﺭﺍﺑﻄﻪ ﺑﺪﻳﻦ ﺻﻮﺭﺕ ﺍﺳﺖ ﻛﻪ " ﻋﻨﺼﺮ ﺍﻭﻝ ﻳﻚ ﻭﺍﺣﺪ ﺍﺯ ﻋﻨﺼﺮ ﺩﻭﻡ‬ ‫ﺑﺰﺭﮔﺘﺮ ﺑﺎﺷﺪ"‪:‬‬

‫})‪C ( a ,b ) = {(1, 0) , (2 ,1‬‬

‫ﺑﻨﺎﺑﺮﺍﻳﻦ ﺗﺎﺑﻊ ﻋﻀﻮﻳﺖ ﻣﺠﻤﻮﻋﻪ ‪ C‬ﺑﺼﻮﺭﺕ ﺯﻳﺮ ﻣﻲﺑﺎﺷﺪ‪:‬‬ ‫‪B‬‬ ‫‪0‬‬ ‫‪1‬‬ ‫‪0‬‬ ‫‪0‬‬

‫‪1‬‬ ‫‪0‬‬ ‫‪1‬‬ ‫‪0‬‬

‫‪µc‬‬ ‫‪1‬‬ ‫‪2‬‬ ‫‪3‬‬

‫‪A‬‬

‫‪١‬‬

‫‪ -١‬ﺭﺍﺑﻄﻪ ﻓﺎﺯﻱ‬

‫ﻓﺮﺽ ﻛﻨﻴﺪ '‪ A‬ﻭ '‪ B‬ﺩﻭ ﻣﺠﻤﻮﻋﻪ ﺑﺎﺷﻨﺪ‪:‬‬ ‫}‪A′ = {San Francisco , Hong Kong , Tokyo‬‬ ‫}‪B ′ = {Boston , Hong Kong‬‬

‫ﺣﺎﻝ ﻣﻲﺧﻮﺍﻫﻴﻢ ﺭﺍﺑﻄﻪ ﻓﺎﺯﻱ ﺧﻴﻠﻲ ﺩﻭﺭ ‪ Q‬ﺭﺍ ﺑﻴﻦ ﺍﻳﻦ ﺩﻭ ﻣﺠﻤﻮﻋﻪ ﺑﻨﻮﻳﺴﻴﻢ‪:‬‬ ‫'‪B‬‬

‫‪HK‬‬ ‫‪0.9‬‬ ‫‪0‬‬ ‫‪0.1‬‬

‫‪µQ‬‬

‫‪Boston‬‬ ‫‪0.3‬‬ ‫‪SF‬‬ ‫‪1‬‬ ‫‪HK‬‬ ‫‪Tokyo 0.95‬‬

‫'‪A‬‬

‫ﻛﻪ ﻣﻲﺗﻮﺍﻥ ﺭﺍﺑﻄﻪ ﻓﺎﺯﻱ ‪ Q‬ﺭﺍ ﺑﻔﺮﻡ ﺯﻳﺮ ﻧﻴﺰ ﻧﻤﺎﻳﺶ ﺩﺍﺩ‪:‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪0.3‬‬ ‫‪1‬‬ ‫‪0.95‬‬ ‫‪0.9‬‬ ‫‪0‬‬ ‫‪0.1‬‬ ‫‪Q=‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪‬‬ ‫‪ ( SF , Boston) ( HK , Boston) (Tokyo , Boston) ( SF , HK ) ( HK , HK ) (Tokyo , HK ) ‬‬

‫‪٢‬‬

‫‪ -٢‬ﺗﺼﺎﻭﻳﺮ‬

‫ﺗﺼﻮﻳﺮ ﺭﺍﺑﻄﻪ ﻓﺎﺯﻱ ‪ Q‬ﺑﺮ ﺭﻭﻱ '‪ A‬ﺑﺼﻮﺭﺕ ‪ Q1‬ﻧﺸﺎﻥ ﺩﺍﺩﻩ ﻣﻲﺷﻮﺩ‪:‬‬ ‫‪1 - Fuzzy Relation‬‬ ‫‪2 - Projections‬‬

‫‪٢١‬‬

‫)‪µ Q1 (a ′) = max µ Q (a ′, b ′‬‬ ‫‪b′∈B′‬‬

‫‪0.9‬‬ ‫‪1‬‬ ‫‪0.95‬‬ ‫‪+‬‬ ‫‪+‬‬ ‫‪SF HK Tokyo‬‬

‫= ‪Q1‬‬

‫ﺗﺼﻮﻳﺮ ﺍﻳﻦ ﺭﺍﺑﻄﻪ ﻓﺎﺯﻱ ﺑﺮ ﺭﻭﻱ '‪ B‬ﺑﺼﻮﺭﺕ ‪ Q2‬ﻧﺸﺎﻥ ﺩﺍﺩﻩ ﻣﻲﺷﻮﺩ‪:‬‬ ‫)‪µ Q2 (b ′) = max µ Q (a ′, b ′‬‬ ‫‪a′∈ A′‬‬

‫‪1‬‬ ‫‪0.9‬‬ ‫‪+‬‬ ‫‪Boston HK‬‬

‫= ‪Q2‬‬

‫‪١‬‬

‫‪ -٣‬ﺗﺮﻛﻴﺐ ﺭﻭﺍﺑﻂ ﻓﺎﺯﻱ‬

‫ﻓﺮﺽ ﻛﻨﻴﺪ )'‪ Q(A',B‬ﻭ )'‪ P(B',C‬ﺩﻭ ﺭﺍﺑﻄﻪ ﻓﺎﺯﻱ ﺑﺎﺷﻨﺪ ﻛﻪ ﻣﺠﻤﻮﻋﻪ '‪ B‬ﺩﺭ ﺁﻧﻬﺎ ﻣﺸﺘﺮﻙ ﻣﻲﺑﺎﺷﺪ‪:‬‬ ‫}‪A′ = {San Francisco , Hong Kong , Tokyo‬‬ ‫}‪B ′ = {Boston , Hong Kong‬‬ ‫}‪C ′ = {NYC , Beijing‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪0.3‬‬ ‫‪1‬‬ ‫‪0.95‬‬ ‫‪0.9‬‬ ‫‪0‬‬ ‫‪0.1‬‬ ‫‪Q( A′, B ′) = ‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪‬‬ ‫‪ ( SF , Boston) ( HK , Boston) (Tokyo , Boston) ( SF , HK ) ( HK , HK ) (Tokyo , HK ) ‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪0.95‬‬ ‫‪0.1‬‬ ‫‪0.1‬‬ ‫‪0.9‬‬ ‫‪P( B ′, C ′) = ‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪,‬‬ ‫‪‬‬ ‫‪( HK , NYC ) ( HK , Beijing ) ‬‬ ‫) ‪ ( Boston , NYC ) ( Boston , Beijing‬‬

‫ﻛﻪ )'‪ Q(A',B‬ﺭﺍﺑﻄﻪ ﻓﺎﺯﻱ ﺧﻴﻠﻲ ﺩﻭﺭ ﺑﻴﻦ ﻣﺠﻤﻮﻋﻪﻫﺎﻱ '‪ A‬ﻭ '‪ B‬ﻭ )'‪ P(B',C‬ﺭﺍﺑﻄﻪ ﻓﺎﺯﻱ ﺧﻴﻠﻲ ﻧﺰﺩﻳﻚ ﺑﻴﻦ ﻣﺠﻤﻮﻋﻪﻫﺎﻱ '‪B‬‬

‫ﻭ '‪ C‬ﻣﻲﺑﺎﺷﺪ‪.‬‬ ‫‪ P o Q‬ﺗﺮﻛﻴﺐ )'‪ Q(A',B‬ﻭ )'‪ P(B',C‬ﺍﺳﺖ ﺍﮔﺮ ﻭ ﻓﻘﻂ ﺍﮔﺮ‪:‬‬ ‫])‪µ PoQ (a ′, c ′) = max t [ µ P (b′, c ′) , µ Q (a ′ , b ′‬‬ ‫‪b′∈B ′‬‬

‫ﺩﺭ ﺍﻳﻨﺠﺎ ﺩﻭ ﻧﻮﻉ ﺗﺮﻛﻴﺐ ﺭﺍ ﻣﻌﺮﻓﻲ ﻣﻲﻛﻨﻴﻢ‪:‬‬ ‫‪٢‬‬

‫‪ -١‬ﺗﺮﻛﻴﺐ ﻣﺎﻛﺰﻳﻤﻢ‪-‬ﻣﻴﻨﻴﻤﻢ‬

‫])‪µ PoQ (a ′, c ′) = max min [ µ P (b ′, c ′) , µ Q (a ′ , b ′‬‬ ‫‪b′∈B ′‬‬

‫‪٣‬‬

‫‪ -٢‬ﺗﺮﻛﻴﺐ ﻣﺎﻛﺰﻳﻤﻢ‪-‬ﺣﺎﺻﻠﻀﺮﺏ‬

‫])‪µ PoQ (a ′, c ′) = max [ µ P (b ′, c ′) . µ Q (a ′ , b′‬‬ ‫‪b′∈B′‬‬

‫ﻣﺜﺎﻝ‪ :‬ﺍﮔﺮ )'‪ P(B',C‬ﺭﺍﺑﻄﻪ ﻓﺎﺯﻱ ﺧﻴﻠﻲ ﻧﺰﺩﻳﻚ ﺑﻴﻦ ﻣﺠﻤﻮﻋﻪﻫﺎﻱ '‪ B‬ﻭ '‪ C‬ﻭ )'‪ Q(A',B‬ﺭﺍﺑﻄﻪ ﻓﺎﺯﻱ ﺧﻴﻠﻲ ﺩﻭﺭ ﺑﻴﻦ‬ ‫ﻣﺠﻤﻮﻋﻪﻫﺎﻱ '‪ A‬ﻭ '‪ B‬ﺑﺎﺷﺪ ﺗﺮﻛﻴﺐ ﻣﺎﻛﺰﻳﻤﻢ‪-‬ﻣﻴﻨﻴﻤﻢ ﻭ ﺗﺮﻛﻴﺐ ﻣﺎﻛﺰﻳﻤﻢ‪-‬ﺣﺎﺻﻠﻀﺮﺏ ﺑﻴﻦ ‪ P‬ﻭ ‪ Q‬ﺭﺍ ﺑﺪﺳﺖ ﺁﻭﺭﻳﺪ‪:‬‬

‫‪1 - Composition of Fuzzy Relation‬‬ ‫‪2 - Max-Min Composition‬‬ ‫‪3 - Max-Product Composition‬‬

‫‪٢٢‬‬

  0.3 1 0.95 0.9 0 0.1 Q( A′, B ′) =  , , , , ,   ( SF , Boston) ( HK , Boston) (Tokyo , Boston) ( SF , HK ) ( HK , HK ) (Tokyo , HK )    0.95 0.1 0.1 0.9 P( B ′, C ′) =  , , ,  ( HK , NYC ) ( HK , Beijing )   ( Boston , NYC ) ( Boston , Beijing )

A′ × C ′ = {( SF , NYC ) , ( SF , Beijing ) , ( HK , NYC ) , ( HK , Beijing ) , (Tokyo , NYC ) , ( Tokyo , Beijing )}

:‫ ﻣﻲﺑﺎﺷﺪ‬A′ × C ′ ‫ﺩﺭ ﺍﻳﻦ ﻣﺮﺣﻠﻪ ﻫﺪﻑ ﺗﻌﻴﻴﻦ ﻣﻴﺰﺍﻥ ﻋﻀﻮﻳﺖ ﻫﺮ ﻋﻀﻮ ﻣﺠﻤﻮﻋﻪ‬ ‫ ﺑﺮﺍﻱ ﻣﺤﺎﺳﺒﻪ ﺗﺮﻛﻴﺐ‬Max-Min ‫ ﻋﻤﻠﮕﺮ‬:‫ﺍﻟﻒ‬ 1 : µ P oQ ( SF , NYC ) = max{min[µ P ( SF , Boston) , µ Q ( Boston, NYC )] , min[ µ P ( SF , HK ) , µ Q ( HK , NYC )]} ⇒ µ P oQ ( SF , NYC ) = max{min[0.3,0.95] , min[0.9,0.1]} = 0.3 2 : µ P oQ ( SF , Beijing ) = max{min[µ P ( SF , Boston) , µ Q ( Boston, Beijing )] , min[ µ P ( SF , HK ) , µ Q ( HK , Beijing )]} ⇒ µ P oQ ( SF , Beijing ) = max{min[0.3,0.1] , min[0.9,0.9]} = 0.9

M

:‫ ﺑﺼﻮﺭﺕ ﺯﻳﺮ ﻣﻲﺑﺎﺷﺪ‬min-max ‫ﺑﻨﺎﺑﺮﺍﻳﻦ ﻧﺘﻴﺠﻪ ﺗﺮﻛﻴﺐ‬   0.3 0.9 0.95 0.1 0.95 0.1 PoQ =  , , , , ,   ( SF , NYC ) ( SF , Beijing ) ( HK , NYC ) ( HK , Beijing ) (Tokyo , NYC ) ( Tokyo , Beijing ) 

‫ ﺑﺮﺍﻱ ﻣﺤﺎﺳﺒﻪ ﺗﺮﻛﻴﺐ‬Max-Product ‫ ﻋﻤﻠﮕﺮ‬:‫ﺏ‬ 1 : µ P oQ ( SF , NYC ) = max{[µ P ( SF , Boston).µ Q ( Boston, NYC )] , [ µ P ( SF , HK ). µ Q ( HK , NYC )]} ⇒ µ P oQ ( SF , NYC ) = max{[0.3 × 0.95] , [0.9 × 0.1]} = 0.285 2 : µ P oQ ( SF , Beijing ) = max{[µ P ( SF , Boston).µ Q ( Boston, Beijing )] , [ µ P ( SF , HK ) . µ Q ( HK , Beijing )]} ⇒ µ P oQ ( SF , Beijing ) = max{[0.3 × 0.1] , [0.9 × 0.9]} = 0.81

M

:‫ ﺑﺼﻮﺭﺕ ﺯﻳﺮ ﻣﻲﺑﺎﺷﺪ‬Max-Product ‫ﺑﻨﺎﺑﺮﺍﻳﻦ ﻧﺘﻴﺠﻪ ﺗﺮﻛﻴﺐ‬  0.285  0.81 0.95 0.1 0.9025 0.095 PoQ= , , , , ,   ( SF , NYC ) ( SF , Beijing ) ( HK , NYC ) ( HK , Beijing ) (Tokyo , NYC ) ( Tokyo , Beijing ) 

‫ ﺍﺑﺘﺪﺍ ﺩﻭ ﻣﺠﻤﻮﻋﻪﺍﻱ ﺭﺍ ﻛﻪ ﻣﻲﺧﻮﺍﻫﻴﻢ ﺗﺮﻛﻴﺐ ﻛﻨﻴﻢ ﺑﺼﻮﺭﺕ ﻣﺎﺗﺮﻳﺲ ﻧﻮﺷﺘﻪ ﻭ ﻣﺜﻞ ﺿﺮﺏ‬Max-Min ‫ ﺑﺮﺍﻱ ﺗﺮﻛﻴﺐ‬:١ ‫ﻧﺘﻴﺠﻪ‬ :‫ ﺍﺳﺘﻔﺎﺩﻩ ﻣﻲﻛﻨﻴﻢ‬min ‫ ﻭ ﺑﻪ ﺟﺎﻱ ﺿﺮﺏ ﺍﺯ‬Max ‫ﺩﻭ ﻣﺎﺗﺮﻳﺲ ﻋﻤﻠﻴﺎﺕ ﺭﺍ ﺍﻧﺠﺎﻡ ﻣﻲﺩﻫﻴﻢ ﺑﺎ ﺍﻳﻦ ﺗﻔﺎﻭﺕ ﻛﻪ ﺑﻪ ﺟﺎﻱ ﺟﻤﻊ ﺍﺯ‬  0.3 0.9  0.3 0.9 0 . 95 0 . 1    1   0  o   = 0.95 0.1  0 . 1 0 . 9  0.95 0.1 0.95 0.1   

٢٣

‫ﻧﺘﻴﺠﻪ ‪ :٢‬ﺑﺮﺍﻱ ﺗﺮﻛﻴﺐ ‪ Max-Product‬ﺩﻭ ﻣﺠﻤﻮﻋﻪﺍﻱ ﺭﺍ ﻛﻪ ﻣﻲﺧﻮﺍﻫﻴﻢ ﺗﺮﻛﻴﺐ ﻛﻨﻴﻢ ﺑﺼﻮﺭﺕ ﻣﺎﺗﺮﻳﺲ ﻧﻮﺷﺘﻪ ﻭ ﻣﺜﻞ ﺿﺮﺏ‬ ‫ﺩﻭ ﻣﺎﺗﺮﻳﺲ ﻋﻤﻠﻴﺎﺕ ﺭﺍ ﺍﻧﺠﺎﻡ ﻣﻲﺩﻫﻴﻢ ﺑﺎ ﺍﻳﻦ ﺗﻔﺎﻭﺕ ﻛﻪ ﺑﻪ ﺟﺎﻱ ﺟﻤﻊ ﺍﺯ ‪ Max‬ﺍﺳﺘﻔﺎﺩﻩ ﻣﻲﻛﻨﻴﻢ‪:‬‬ ‫‪ 0.3 0.9‬‬ ‫‪ 0.285 0.81 ‬‬ ‫‪0.95 0.1 ‬‬ ‫‪ 1‬‬ ‫‪‬‬ ‫‪0 o‬‬ ‫‪0.1 ‬‬ ‫‪ =  0.95‬‬ ‫‪‬‬ ‫‪0‬‬ ‫‪.‬‬ ‫‪1‬‬ ‫‪0‬‬ ‫‪.‬‬ ‫‪9‬‬ ‫‪‬‬ ‫‪ 0.9025 0.095‬‬ ‫‪0.95 0.1‬‬ ‫‪‬‬ ‫‪‬‬

‫ﺗﻤﺮﻳﻦ‪ :‬ﺳﻪ ﺭﺍﺑﻄﻪ ﻓﺎﺯﻱ ﺗﻮﺳﻂ ﻣﺎﺗﺮﻳﺲﻫﺎﻱ ﺭﺍﺑﻄﻪﺍﻱ ﺯﻳﺮ ﺗﻌﺮﻳﻒ ﻣﻲﺷﻮﻧﺪ‪ .‬ﺗﺮﻛﻴﺐﻫﺎﻱ ﻣﺎﻛﺰﻳﻤﻢ‪-‬ﻣﻴﻨﻴﻤﻢ ﻭ ﻣﺎﻛﺰﻳﻤﻢ‪-‬‬ ‫ﺣﺎﺻﻠﻀﺮﺏ ‪Q1 o Q2‬‬

‫‪,‬‬

‫‪Q1 o Q3‬‬

‫‪ 1 0 0.7 ‬‬ ‫‪Q3 =  0 1 0 ‬‬ ‫‪0.7 0 1 ‬‬

‫‪ Q1 o Q2 o Q3‬ﺭﺍ ﻣﺤﺎﺳﺒﻪ ﻧﻤﺎﻳﻴﺪ‪.‬‬

‫‪,‬‬ ‫‪,‬‬

‫‪0.6 0.6 0 ‬‬ ‫‪Q2 =  0 0.6 0.1‬‬ ‫‪ 0 0.1 0 ‬‬

‫‪٢۴‬‬

‫‪,‬‬

‫‪0 0.7 ‬‬ ‫‪1‬‬ ‫‪‬‬ ‫‪Q1 = 0.3 0.2 0 ‬‬ ‫‪ 0 0.5 1 ‬‬

‫‪ -٦‬ﻣﺘﻐﻴﺮﻫﺎﻱ ﺯﺑﺎﻧﻲ ﻭ ﻗﻮﺍﻋﺪ ﻓﺎﺯﻱ‬

‫‪١‬‬

‫‪IF THEN‬‬

‫ﻣﺜﺎﻝ‪ :‬ﻓﺮﺽ ﻛﻨﻴﺪ ﺳﺮﻋﺖ ﻳﻚ ﺧﻮﺩﺭﻭ ﺩﺭ ﻣﺤﺪﻭﺩﻩ ‪ [0,110]mph‬ﺑﺎﺷﺪ ﻣﻲﺧﻮﺍﻫﻴﻢ ﺳﺮﻋﺖ ﺧﻮﺩﺭﻭ ﺭﺍ ﺑﺼﻮﺭﺕ ﻣﺘﻐﻴﺮ ﺯﺑﺎﻧﻲ ﻛﻪ‬ ‫ﺩﺍﺭﺍﻱ ﺳﻪ ﺗﺎﺑﻊ ﻋﻀﻮﻳﺖ ‪ Medium ،Fast‬ﻭ ‪ Slow‬ﻣﻲﺑﺎﺷﺪ ﺩﺭﺁﻭﺭﻳﻢ‪:‬‬ ‫‪Fast‬‬

‫)‪Speed(mph‬‬

‫‪110‬‬

‫‪Medium‬‬

‫‪75‬‬

‫‪Slow‬‬

‫‪35‬‬

‫‪55‬‬

‫‪0‬‬

‫ﺗﻌﺮﻳﻒ‪ :‬ﻳﻚ ﻣﺘﻐﻴﺮ ﺯﺑﺎﻧﻲ ﺑﻮﺳﻴﻠﻪ ﭼﻬﺎﺭ ﭘﺎﺭﺍﻣﺘﺮ )‪ (X, T, U, M‬ﻣﺸﺨﺺ ﻣﻲﮔﺮﺩﺩ‪:‬‬ ‫‪ X‬ﻧﺎﻡ ﻣﺘﻐﻴﺮ ﺯﺑﺎﻧﻲ ﺍﺳﺖ‪ .‬ﺩﺭ ﺍﻳﻦ ﻣﺜﺎﻝ ﺳﺮﻋﺖ ﺧﻮﺩﺭﻭ‬ ‫‪ T‬ﻣﺠﻤﻮﻋﻪ ﻣﻘﺎﺩﻳﺮ ﺯﺑﺎﻧﻲ ﻣﺮﺑﻮﻃﻪ ﺍﺳﺖ‪ .‬ﺩﺭ ﺍﻳﻦ ﻣﺜﺎﻝ } ﺁﻫﺴﺘﻪ‪ ،‬ﻣﺘﻮﺳﻂ‪ ،‬ﺳﺮﻳﻊ{‬ ‫‪ U‬ﺩﺍﻣﻨﻪ ﻭﺍﻗﻌﻲ ﺍﺳﺖ‪ .‬ﺩﺭ ﺍﻳﻦ ﻣﺜﺎﻝ ‪[0,110]mph‬‬

‫‪ M‬ﻳﻚ ﻗﺎﻋﺪﻩ ﻟﻐﻮﻱ ﺍﺳﺖ ﻛﻪ ﻫﺮ ﻣﻘﺪﺍﺭ ﺯﺑﺎﻧﻲ ‪ T‬ﺭﺍ ﺑﻪ ﻳﻚ ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ ﺩﺭ ‪ U‬ﻣﺮﺗﺒﻂ ﻣﻲﻛﻨﺪ‪.‬‬ ‫‪٢‬‬

‫ﻗﻴﻮﺩ ﺯﺑﺎﻧﻲ‬

‫• ﺍﺻﻄﻼﺣﺎﺕ ﭘﺎﻳﻪ‪ :‬ﻣﺎﻧﻨﺪ‬

‫‪Slow, Medium, Fast‬‬

‫• ﻣﻜﻤﻞ ﻛﻨﻨﺪﻩ‪ :‬ﻣﺎﻧﻨﺪ‬

‫‪Not‬‬

‫• ﻣﺘﺼﻞ ﻛﻨﻨﺪﻩ‪ :‬ﻣﺎﻧﻨﺪ‬

‫‪And, Or‬‬

‫• ﻗﻴﻮﺩ‪ :‬ﻣﺎﻧﻨﺪ‬

‫‪Very, Slightly, More or less‬‬

‫• ﻋﻼﻣﺖ‪ :‬ﻣﺎﻧﻨﺪ‬

‫‪Positive, Negative‬‬ ‫‪1‬‬

‫‪µ More or less A ( x) = [ µ A ( x)] 2‬‬

‫‪µVery A ( x) = [ µ A ( x)]2‬‬

‫ﻣﺜﺎﻝ‪ :‬ﺍﮔﺮ ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ ‪ A‬ﺑﺼﻮﺭﺕ ﺯﻳﺮ ﺗﻌﺮﻳﻒ ﺷﻮﺩ ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ ‪ Very very A ، Very A‬ﻭ ‪ More or less A‬ﺭﺍ‬ ‫ﺑﺪﺳﺖ ﺁﻭﺭﻳﺪ‪:‬‬ ‫‪1 0.8 0.6 0.4 0.2‬‬ ‫‪+‬‬ ‫‪+‬‬ ‫‪+‬‬ ‫‪A= +‬‬ ‫‪1 2‬‬ ‫‪3‬‬ ‫‪4‬‬ ‫‪5‬‬ ‫‪1 0.64 0.36 0.16 0.04‬‬ ‫‪+‬‬ ‫‪+‬‬ ‫‪+‬‬ ‫‪Very A = +‬‬ ‫‪1‬‬ ‫‪2‬‬ ‫‪3‬‬ ‫‪4‬‬ ‫‪5‬‬ ‫‪1 0.4096 0.1296 0.0256 0.0016‬‬ ‫‪+‬‬ ‫‪+‬‬ ‫‪+‬‬ ‫‪Very very A = +‬‬ ‫‪1‬‬ ‫‪2‬‬ ‫‪3‬‬ ‫‪4‬‬ ‫‪5‬‬

‫‪1 - Linguistic Variables and Fuzzy IF-THEN Rules‬‬ ‫‪2 - Linguistic Hedge‬‬

‫‪٢۵‬‬

‫ﻗﻮﺍﻋﺪ ﻓﺎﺯﻱ ‪IF-THEN‬‬

‫ﻓﺮﻡ ﻛﻠﻲ ﻳﻚ ﻗﺎﻋﺪﻩ ﻓﺎﺯﻱ ﺑﺼﻮﺭﺕ ﺯﻳﺮ ﻣﻲﺑﺎﺷﺪ‪:‬‬ ‫> ﻋﺒﺎﺭﺕ ﻓﺎﺯﻱ <‬

‫‪ > THEN‬ﻋﺒﺎﺭﺕ ﻓﺎﺯﻱ < ‪IF‬‬

‫ﺣﺎﻝ ﺑﻪ ﺑﺮﺭﺳﻲ ﻋﺒﺎﺭﺕ ﻓﺎﺯﻱ ﻣﻲﭘﺮﺩﺍﺯﻳﻢ‪:‬‬ ‫‪١‬‬

‫ﻋﺒﺎﺭﺕ ﻓﺎﺯﻱ‬

‫ﺩﻭ ﻧﻮﻉ ﻋﺒﺎﺭﺕ ﻓﺎﺯﻱ ﻭﺟﻮﺩ ﺩﺍﺭﺩ‪ :‬ﺳﺎﺩﻩ ﻭ ﻣﺮﻛﺐ‪.‬‬

‫ﻣﺜﺎﻝ‪ :‬ﺳﻪ ﻋﺒﺎﺭﺕ ﻓﺎﺯﻱ ﺳﺎﺩﻩ ﻭ ﺳﻪ ﻋﺒﺎﺭﺕ ﻓﺎﺯﻱ ﻣﺮﻛﺐ‪:‬‬

‫‪M‬‬

‫‪X is‬‬

‫‪is S‬‬ ‫‪is M‬‬ ‫‪is F‬‬ ‫‪is S or X is not M‬‬ ‫‪is not S or X is not F‬‬ ‫‪is S and X is not F) or‬‬

‫‪X‬‬ ‫‪X‬‬ ‫‪X‬‬ ‫‪X‬‬ ‫‪X‬‬ ‫‪(X‬‬

‫ﻛﻪ ‪ S‬ﻭ ‪ M‬ﻭ ‪ F‬ﻣﺠﻤﻮﻋﻪﻫﺎﻱ ﻓﺎﺯﻱ ‪ Medium ، Slow‬ﻭ ‪ Fast‬ﻣﻲﺑﺎﺷﻨﺪ ﻭ ‪ X‬ﻳﻚ ﻣﺘﻐﻴﺮ ﺯﺑﺎﻧﻲ ﻣﻲﺑﺎﺷﺪ‪.‬‬ ‫ﻣﺜﺎﻝ‪ :‬ﺳﻴﺴﺘﻤﻲ ﺭﺍ ﺑﺎ ﺳﻪ ﻭﺭﻭﺩﻱ ‪ X3 ، X2 ، X1‬ﻭ ﻳﻚ ﺧﺮﻭﺟﻲ ‪ Y‬ﺩﺭ ﻧﻈﺮ ﺑﮕﻴﺮﻳﺪ‪:‬‬

‫ﭘﺎﻳﮕﺎﻩ ﻗﻮﺍﻋﺪ ﻓﺎﺯﻱ‬

‫‪Y‬‬

‫ﻓﺎﺯﻱﺯﺩﺍ‬

‫ﻓﺎﺯﻱﮔﺮ‬

‫ﻣﺠﻤﻮﻋﻪ ﻏﻴﺮ ﻓﺎﺯﻱ‬ ‫ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ‬

‫ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ ﻓﺎﺯﻱ‬

‫‪X1‬‬ ‫‪X2‬‬ ‫‪X3‬‬ ‫ﻣﺠﻤﻮﻋﻪ ﻏﻴﺮ ﻓﺎﺯﻱ‬

‫ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ‬

‫ﻼ ﺫﻛﺮﺷﺪ ﺩﺭ ﺍﻳﻦ ﺳﻴﺴﺘﻢ ﺍﺑﺘﺪﺍ ﺳﻪ ﻭﺭﻭﺩﻱ ﻏﻴﺮ ﻓﺎﺯﻱ ‪ X3 ، X2 ، X1‬ﺗﻮﺳﻂ ﻓﺎﺯﻱﮔﺮ ﺗﺒﺪﻳﻞ ﺑﻪ ﻣﺘﻐﻴﺮﻫﺎﻱ ﻓﺎﺯﻱ‬ ‫ﻫﻤﺎﻧﻄﻮﺭ ﻛﻪ ﻗﺒ ﹰ‬ ‫ﻣﻲﺷﻮﻧﺪ‪.‬‬ ‫ﻓﺮﺽ ﻛﻨﻴﺪ ‪ X1‬ﺩﺍﺭﺍﻱ ‪ ٧‬ﺗﺎﺑﻊ ﻋﻀﻮﻳﺖ ﺑﺼﻮﺭﺕ ﻣﺠﻤﻮﻋﻪﻫﺎﻱ ﻓﺎﺯﻱ ‪ NL ،NM ،NS ،Z ،PS ،PM ،PL‬ﻣﻲﺑﺎﺷﺪ‪ X2 ،‬ﺩﺍﺭﺍﻱ ‪٥‬‬ ‫ﺗﺎﺑﻊ ﻋﻀﻮﻳﺖ ﺑﺼﻮﺭﺕ ﻣﺠﻤﻮﻋﻪﻫﺎﻱ ﻓﺎﺯﻱ ‪ NL ،NS ،Z ،PS ،PL‬ﻣﻲﺑﺎﺷﺪ ﻭ ‪ X3‬ﺩﺍﺭﺍﻱ ‪ ٣‬ﺗﺎﺑﻊ ﻋﻀﻮﻳﺖ ﺑﺼﻮﺭﺕ‬ ‫ﻣﺠﻤﻮﻋﻪﻫﺎﻱ ﻓﺎﺯﻱ ‪ N ،Z ،P‬ﻣﻲﺑﺎﺷﺪ‪ .‬ﺍﺯ ﻃﺮﻓﻲ ﺧﺮﻭﺟﻲ ‪ Y‬ﻧﻴﺰ ﺩﺍﺭﺍﻱ ‪ ٥‬ﺗﺎﺑﻊ ﻋﻀﻮﻳﺖ ﺍﺳﺖ ﻛﻪ ﺑﺼﻮﺭﺕ ﻣﺠﻤﻮﻋﻪﻫﺎﻱ ﻓﺎﺯﻱ‬ ‫‪1 - Fuzzy Proposition‬‬

‫‪٢۶‬‬

‫‪ NL ،NS ،Z ،PS ،PL‬ﻫﺴﺘﻨﺪ‪.‬‬ ‫ﻧﻜﺘﻪ‪ :‬ﻫﺮ ﻛﺪﺍﻡ ﺍﺯ ﻣﺘﻐﻴﺮﻫﺎﻱ ﻏﻴﺮ ﻓﺎﺯﻱ ﻳﻚ ﺩﺍﻣﻨﻪ ﺗﻐﻴﻴﺮﺍﺕ ﺩﺍﺭﻧﺪ ﻛﻪ ﺍﻳﻦ ﺗﻮﺍﺑﻊ ﻋﻀﻮﻳﺖ ﺩﺭ ﺍﻳﻦ ﻣﺤﺪﻭﺩﻩ ﺗﻌﺮﻳﻒ ﻣﻲﺷﻮﻧﺪ‪.‬‬ ‫‪.‬‬ ‫ﺗﺬﻛﺮ‪ :‬ﺣﺪﺍﻛﺜﺮ ﺗﻌﺪﺍﺩ ﻗﻮﺍﻋﺪ ﺯﺑﺎﻧﻲ ﺑﺮﺍﻱ ﺳﻴﺴﺘﻢ ﺑﺎ ‪ N‬ﻭﺭﻭﺩﻱ ﺍﺯ ﺭﺍﺑﻄﻪ ﺯﻳﺮ ﺑﺪﺳﺖ ﻣﻲﺁﻳﺪ‪:‬‬ ‫)ﺗﻌﺪﺍﺩ ﺗﻮﺍﺑﻊ ﻋﻀﻮﻳﺖ ﻣﺘﻐﻴﺮ ‪ N‬ﺍﻡ ( * … * )ﺗﻌﺪﺍﺩ ﺗﻮﺍﺑﻊ ﻋﻀﻮﻳﺖ ﻣﺘﻐﻴﺮ ﺍﻭﻝ(= )ﺣﺪﺍﻛﺜﺮ ﺗﻌﺪﺍﺩ ﻗﻮﺍﻋﺪ ﻓﺎﺯﻱ ‪(IF-THEN‬‬ ‫ﺑﻨﺎﺑﺮﺍﻳﻦ ﺣﺪﺍﻛﺜﺮ ﺗﻌﺪﺍﺩ ﻗﻮﺍﻋﺪ ﺯﺑﺎﻧﻲ ﺑﺮﺍﻱ ﺳﻴﺴﺘﻢ ﻣﺬﺑﻮﺭ ‪ ١٠٥‬ﻣﻲﺑﺎﺷﺪ‪ .‬ﻓﺮﺽ ﻛﻨﻴﺪ ﻳﻜﻲ ﺍﺯ ﻗﻮﺍﻋﺪ ﻓﺎﺯﻱ ﺩﺭ ﭘﺎﻳﮕﺎﻩ ﻗﻮﺍﻋﺪ‬ ‫ﻓﺎﺯﻱ ﺑﻪ ﻓﺮﻡ ﺯﻳﺮ ﺑﺎﺷﺪ‪:‬‬ ‫‪Y is NS‬‬

‫‪THEN‬‬

‫}‪is Z‬‬

‫‪X3‬‬

‫‪PS) and‬‬

‫‪X2 is‬‬

‫‪or‬‬

‫‪IF {(X1 is NL‬‬

‫ﺍﮔﺮ ﻣﺠﻤﻮﻋﻪﻫﺎﻱ ﺑﻜﺎﺭ ﺭﻓﺘﻪ ﺩﺭ ﺍﻳﻦ ﺳﻴﺴﺘﻢ ﻣﺠﻤﻮﻋﻪﻫﺎﻱ ﻛﻼﺳﻴﻚ )ﻏﻴﺮ ﻓﺎﺯﻱ( ﺑﻮﺩﻧﺪ‪ ،‬ﻣﺸﻜﻠﻲ ﻧﺒﻮﺩ ﻭﻟﻲ ﭼﻮﻥ ﺍﻳﻦ‬ ‫ﻣﺠﻤﻮﻋﻪﻫﺎ ﻓﺎﺯﻱ ﻣﻲﺑﺎﺷﻨﺪ ﺑﻨﺎﺑﺮﺍﻳﻦ‪:‬‬ ‫ﺑﻪ ﺟﺎﻱ ‪ not‬ﺍﺯ ﻣﻜﻤﻞﻫﺎﻱ ﻓﺎﺯﻱ‬ ‫ﺑﻪ ﺟﺎﻱ ‪ or‬ﺍﺯ ﺍﺟﺘﻤﺎﻉ ﻓﺎﺯﻱ‬ ‫ﺑﻪ ﺟﺎﻱ ‪ and‬ﺍﺯ ﺍﺷﺘﺮﺍﻙ ﻓﺎﺯﻱ ﺍﺳﺘﻔﺎﺩﻩ ﻣﻲﺷﻮﺩ‪.‬‬

‫‪٢٧‬‬

‫‪ -٧‬ﺗﻔﺴﻴﺮ ﻗﻮﺍﻋﺪ ﻓﺎﺯﻱ‬

‫‪١‬‬

‫‪IF-THEN‬‬

‫ﺩﺭ ﺗﻔﺴﻴﺮ ﻗﻮﺍﻋﺪ ﻓﺎﺯﻱ ‪ IF-THEN‬ﺍﮔﺮ ﻋﺒﺎﺭﺍﺕ ﻛﻼﺳﻴﻚ ﺑﺎﺷﻨﺪ ﺩﺍﺭﻳﻢ‪:‬‬ ‫)‪q‬‬ ‫ﺟﺪﻭﻝ ﺻﺤﺖ ‪q‬‬

‫‪p‬‬

‫‪q‬‬ ‫‪T‬‬ ‫‪F‬‬ ‫‪T‬‬ ‫‪T‬‬ ‫ﺑﻨﺎﺑﺮﺍﻳﻦ ﻣﻌﺎﺩﻝ )‪q‬‬

‫‪IF p THEN q‬‬

‫‪(p‬‬

‫‪p‬‬

‫‪q‬‬

‫‪p‬‬

‫‪T‬‬ ‫‪F‬‬ ‫‪T‬‬ ‫‪F‬‬

‫‪T‬‬ ‫‪T‬‬ ‫‪F‬‬ ‫‪F‬‬

‫‪ (p‬ﻣﻲﺗﻮﺍﻧﺪ ﻳﻜﻲ ﺍﺯ ﺟﻤﻼﺕ ﺯﻳﺮ ﺑﺎﺷﺪ‪:‬‬ ‫‪p → q ≡ p∨ q‬‬ ‫‪p → q ≡ ( p ∧ q) ∨ p‬‬

‫ﻛﻪ ‪ ∧ , ∨ ,−‬ﺑﺘﺮﺗﻴﺐ ‪ or ، not‬ﻭ ‪ and‬ﻣﻲﺑﺎﺷﻨﺪ‪ .‬ﺍﺯ ﺍﻳﻦ ﻣﻌﺎﺩﻝﻫﺎ ﺩﺭ ﻗﻮﺍﻋﺪ ﻓﺎﺯﻱ ‪ IF-THEN‬ﻧﻴﺰ ﺍﺳﺘﻔﺎﺩﻩ ﻣﻲﻛﻨﻴﻢ ﺑﺎ ﺍﻳﻦ‬ ‫ﺗﻔﺎﻭﺕ ﻛﻪ ﺑﻪ ﺟﺎﻱ ‪ or ،not‬ﻭ ‪ and‬ﺍﺯ ﻣﻜﻤﻞ ﻓﺎﺯﻱ‪ ،‬ﺍﺟﺘﻤﺎﻉ ﻓﺎﺯﻱ ﻭ ﺍﺷﺘﺮﺍﻙ ﻓﺎﺯﻱ ﺍﺳﺘﻔﺎﺩﻩ ﻣﻲﺷﻮﺩ‪.‬‬ ‫ﺑﺮﺍﻱ ﺗﻌﻴﻴﻦ ﻧﻮﻉ ﻣﻜﻤﻞ ﻓﺎﺯﻱ‪ ،‬ﺍﺟﺘﻤﺎﻉ ﻓﺎﺯﻱ ﻭ ﺍﺷﺘﺮﺍﻙ ﻓﺎﺯﻱ ﺍﺯ ﺍﺳﺘﻠﺰﺍﻡ ﻓﺎﺯﻱ ﺍﺳﺘﻔﺎﺩﻩ ﻣﻲﺷﻮﺩ‪.‬‬

‫ﺍﺳﺘﻠﺰﺍﻡ ﻓﺎﺯﻱ‬

‫‪٢‬‬ ‫‪٣‬‬

‫‪ -١‬ﺍﺳﺘﻠﺰﺍﻡ ﺩﻧﻴﺲ‪-‬ﺭﺷﺮ‬

‫ﺩﺭ ﺍﻳﻦ ﺍﺳﺘﻠﺰﺍﻡ ﺍﺯ ﻣﻌﺎﺩﻝ ‪ p → q ≡ p ∨ q‬ﺍﺳﺘﻔﺎﺩﻩ ﻣﻲﺷﻮﺩ ﻭ ﺑﻪ ﺟﺎﻱ ﻋﻤﻠﮕﺮ ‪ not‬ﺍﺯ ﻣﻜﻤﻞ ﻓﺎﺯﻱ ﺍﺳﺎﺳﻲ ﻭ ﺑﻪ ﺟﺎﻱ ‪ or‬ﺍﺯ‬ ‫ﺍﺟﺘﻤﺎﻉ ﻓﺎﺯﻱ ﻣﺎﻛﺰﻳﻤﻢ ﺍﺳﺘﻔﺎﺩﻩ ﻣﻲﮔﺮﺩﺩ‪:‬‬ ‫〉 ‪IF 〈 FP1〉 THEN 〈 FP 2‬‬

‫ﺑﻨﺎﺑﺮﺍﻳﻦ ﻗﺎﻋﺪﻩ ﻓﺎﺯﻱ ﻓﻮﻕ ﺑﺼﻮﺭﺕ ﻳﻚ ﺭﺍﺑﻄﻪ ﻓﺎﺯﻱ ‪ QD‬ﺑﺎ ﺗﺎﺑﻊ ﻋﻀﻮﻳﺖ ﺯﻳﺮ ﺗﺒﺪﻳﻞ ﻣﻲﺷﻮﺩ‪:‬‬ ‫]) ‪µ QD ( x, y ) = max [1 − µ FP1 ( x) , µ FP 2 ( y‬‬

‫ﻛﻪ ‪ x‬ﻭ ‪ y‬ﻣﺘﻐﻴﺮﻫﺎﻱ ﻓﺎﺯﻱ ‪ FP1‬ﻭ ‪ FP2‬ﻣﻲﺑﺎﺷﻨﺪ‪.‬‬ ‫‪٤‬‬

‫‪ -٢‬ﺍﺳﺘﻠﺰﺍﻡ ﻟﻮﻛﺎﺯﻭﻳﭻ‬

‫ﺩﺭ ﺍﻳﻦ ﺍﺳﺘﻠﺰﺍﻡ ﺍﺯ ﻣﻌﺎﺩﻝ ‪ p → q ≡ p ∨ q‬ﺍﺳﺘﻔﺎﺩﻩ ﻣﻲﺷﻮﺩ ﻭ ﺑﻪ ﺟﺎﻱ ﻋﻤﻠﮕﺮ ‪ not‬ﺍﺯ ﻣﻜﻤﻞ ﻓﺎﺯﻱ ﺍﺳﺎﺳﻲ ﻭ ﺑﻪ ﺟﺎﻱ ‪ or‬ﺍﺯ‬ ‫‪1 - Interpretation of Fuzzy IF-THEN Rules‬‬ ‫‪2 - Fuzzy Implication‬‬ ‫‪3 - Dienes Rescher Implication‬‬ ‫‪4 - Lukasiewicz Implication‬‬

‫‪٢٨‬‬

‫‪ S-norm‬ﻳﺎﮔﺮ ﺑﺎ ‪ w=1‬ﺍﺳﺘﻔﺎﺩﻩ ﻣﻴﮕﺮﺩﺩ‪:‬‬ ‫〉 ‪IF 〈 FP1〉 THEN 〈 FP 2‬‬

‫ﺑﻨﺎﺑﺮﺍﻳﻦ ﻗﺎﻋﺪﻩ ﻓﺎﺯﻱ ﻓﻮﻕ ﺑﺼﻮﺭﺕ ﻳﻚ ﺭﺍﺑﻄﻪ ﻓﺎﺯﻱ ‪ QL‬ﺑﺎ ﺗﺎﺑﻊ ﻋﻀﻮﻳﺖ ﺯﻳﺮ ﺗﺒﺪﻳﻞ ﻣﻲﺷﻮﺩ‪:‬‬ ‫]) ‪µ QL ( x, y ) = min [1 , 1 − µ FP1 ( x) + µ FP 2 ( y‬‬

‫‪١‬‬

‫‪ -٣‬ﺍﺳﺘﻠﺰﺍﻡ ﺯﺍﺩﻩ‬

‫ﺩﺭ ﺍﻳﻦ ﺍﺳﺘﻠﺰﺍﻡ ﺍﺯ ﻣﻌﺎﺩﻝ ‪ p → q ≡ ( p ∧ q) ∨ p‬ﺍﺳﺘﻔﺎﺩﻩ ﻣﻲﺷﻮﺩ ﻭ ﺑﻪ ﺟﺎﻱ ﻋﻤﻠﮕﺮ ‪ not‬ﺍﺯ ﻣﻜﻤﻞ ﻓﺎﺯﻱ ﺍﺳﺎﺳﻲ ﻭ ﺑﻪ ﺟﺎﻱ‬ ‫‪ or‬ﺍﺯ ﺍﺟﺘﻤﺎﻉ ﻓﺎﺯﻱ ﻣﺎﻛﺰﻳﻤﻢ ﻭ ﺑﻪ ﺟﺎﻱ ‪ and‬ﺍﺯ ﺍﺷﺘﺮﺍﻙ ﻓﺎﺯﻱ ﻣﻴﻨﻴﻤﻢ ﺍﺳﺘﻔﺎﺩﻩ ﻣﻲﺷﻮﺩ‪:‬‬ ‫〉 ‪IF 〈 FP1〉 THEN 〈 FP 2‬‬

‫ﺑﻨﺎﺑﺮﺍﻳﻦ ﻗﺎﻋﺪﻩ ﻓﺎﺯﻱ ﻓﻮﻕ ﺑﺼﻮﺭﺕ ﻳﻚ ﺭﺍﺑﻄﻪ ﻓﺎﺯﻱ ‪ QZ‬ﺑﺎ ﺗﺎﺑﻊ ﻋﻀﻮﻳﺖ ﺯﻳﺮ ﺗﺒﺪﻳﻞ ﻣﻲﺷﻮﺩ‪:‬‬ ‫} )‪µ QZ ( x, y ) = max{min [ µ FP1 ( x) , µ FP 2 ( y )] , 1 − µ FP1 ( x‬‬ ‫‪٢‬‬

‫‪ -٤‬ﺍﺳﺘﻠﺰﺍﻡ ﮔﻮﺩﻝ‬

‫〉 ‪IF 〈 FP1〉 THEN 〈 FP 2‬‬

‫ﺑﻨﺎﺑﺮﺍﻳﻦ ﻗﺎﻋﺪﻩ ﻓﺎﺯﻱ ﻓﻮﻕ ﺑﺼﻮﺭﺕ ﻳﻚ ﺭﺍﺑﻄﻪ ﻓﺎﺯﻱ ‪ QG‬ﺑﺎ ﺗﺎﺑﻊ ﻋﻀﻮﻳﺖ ﺯﻳﺮ ﺗﺒﺪﻳﻞ ﻣﻲﺷﻮﺩ‪:‬‬ ‫) ‪µ FP1 ( x) ≤ µ FP 2 ( y‬‬ ‫‪otherwise‬‬

‫‪if‬‬ ‫‪ 1‬‬ ‫‪µ QG ( x, y ) = ‬‬ ‫) ‪ µ FP 2 ( y‬‬

‫‪٣‬‬

‫‪ -٥‬ﺍﺳﺘﻠﺰﺍﻡ ﻣﻤﺪﺍﻧﻲ‬

‫ﺩﺭ ﻣﻨﻄﻖ ﻓﺎﺯﻱ ﻣﻲﺗﻮﺍﻥ ﺍﺯ ﻣﻌﺎﺩﻝ ) ‪ p → q ≡ ( p ∧ q‬ﺍﺳﺘﻔﺎﺩﻩ ﻛﺮﺩ ﻭ ﺑﻪ ﺟﺎﻱ ﻋﻤﻠﮕﺮ ‪ and‬ﺍﺯ ‪ min‬ﻳﺎ ﺿﺮﺏ ﺟﺒﺮﻱ ﺍﺳﺘﻔﺎﺩﻩ‬ ‫ﻣﻲﺷﻮﺩ‪:‬‬ ‫]) ‪µ QMM ( x, y ) = min [ µ FP1 ( x) , µ FP 2 ( y‬‬ ‫) ‪µ QMP ( x, y ) = µ FP1 ( x) . µ FP 2 ( y‬‬

‫ﻣﺜﺎﻝ‪ :‬ﻓﺮﺽ ﻛﻨﻴﺪ ‪ x1‬ﺳﺮﻋﺖ ﻳﻚ ﻣﺎﺷﻴﻦ‪ x2 ،‬ﺷﺘﺎﺏ ﻭ ‪ y‬ﻧﻴﺰ ﻧﻴﺮﻭﻱ ﺍﻋﻤﺎﻟﻲ ﺑﻪ ﭘﺪﺍﻝ ﮔﺎﺯ ﺑﺎﺷﺪ‪ .‬ﻗﺎﻋﺪﻩ ﺯﻳﺮ ﺭﺍ ﺩﺭ ﻧﻈﺮ ﺑﮕﻴﺮﻳﺪ‪:‬‬ ‫〉 ‪IF 〈 x1 is slow and x 2 is small 〉 THEN 〈 y is l arg e‬‬

‫ﻛﻪ ﻣﺠﻤﻮﻋﻪﻫﺎﻱ ﻓﺎﺯﻱ ‪ small ، slow‬ﻭ ‪ large‬ﺑﺼﻮﺭﺕ ﺯﻳﺮ ﺗﻌﺮﻳﻒ ﻣﻲﺷﻮﻧﺪ‪:‬‬ ‫)‪µ(x1‬‬

‫‪Slow‬‬

‫‪1‬‬ ‫‪x1‬‬

‫‪100‬‬

‫‪75‬‬

‫‪55‬‬

‫‪35‬‬

‫‪x1 ≤ 35‬‬

‫‪if 35 < x1 ≤ 55‬‬ ‫‪x1 > 55‬‬

‫‪0‬‬

‫‪if‬‬ ‫‪if‬‬

‫‪ 1‬‬ ‫‪ 55 − x1‬‬ ‫‪µ slow ( x1 ) = ‬‬ ‫‪ 20‬‬ ‫‪ 0‬‬

‫‪1 - Zadeh Implication‬‬ ‫‪2 - Godel Implication‬‬ ‫‪3 - Mamdani Implication‬‬

‫‪٢٩‬‬

µ(x2)

10 − x 2  µ small ( x 2 ) =  10  0

if if

x 2 > 10

0

10

20

x2

30

µ(y)

if y ≤ 1 if 1 < y ≤ 2 if y > 2

0  µ l arg e ( y ) =  y − 1 1 

Small

1

0 ≤ x 2 ≤ 10

Large

1

0

1

2

y

3

.‫ ﺑﺎﺷﺪ‬V = [0,3] , U 2 = [0,30] , U 1 = [0,100] ‫ ﺑﺘﺮﺗﻴﺐ‬y ‫ ﻭ‬x2 ، x1 ‫ﻓﺮﺽ ﻛﻨﻴﺪ ﺩﺍﻣﻨﻪ ﻣﺘﻐﻴﺮﻫﺎﻱ ﺯﺑﺎﻧﻲ‬ .‫ ﻋﻼﻭﻩ ﺑﺮ ﻣﺠﻤﻮﻋﻪﻫﺎﻱ ﻓﻮﻕ ﺩﺍﺭﺍﻱ ﻣﺠﻤﻮﻋﻪﻫﺎﻱ ﻓﺎﺯﻱ ﺩﻳﮕﺮﻱ ﻧﻴﺰ ﻣﻲﺑﺎﺷﻨﺪ‬y ‫ ﻭ‬x2 ، x1 ‫ ﺩﻗﺖ ﺷﻮﺩ ﻛﻪ ﻣﺘﻐﻴﺮﻫﺎﻱ‬:‫ﺗﺬﻛﺮ‬ .‫ ﺭﺍ ﺑﺪﺳﺖ ﺁﻭﺭﻳﺪ‬µ Q ( x1 , x 2 , y ) ‫ﺭﺷﺮ‬-‫ ﻭ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺍﺳﺘﻠﺰﺍﻡ ﺩﻧﻴﺲ‬and ‫ﺑﺎ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺣﺎﺻﻠﻀﺮﺏ ﺟﺒﺮﻱ ﺑﻪ ﺟﺎﻱ‬ D

FP1 = x1 is slow and x 2 is small  0  10 − x 2 µ FP1 ( x1 , x 2 ) = µ slow ( x1 ) µ small ( x 2 ) =  10   (55 − x1 )(10 − x 2 )  200

if

x1 > 55

if

x1 ≤ 35

if

35 < x1 ≤ 55 and x 2 ≤ 10

or

x 2 > 10

and x 2 ≤ 10

x2

10

Small µ(x2) 1

0

µ(x1)

Slow

1

0

35

55

x1

:‫ﺭﺷﺮ‬-‫ﺑﺎ ﺍﺳﻠﺰﺍﻡ ﺩﻧﻴﺲ‬ µ QD ( x1 , x 2 , y ) = max [1 − µ FP1 ( x1 , x 2 ) , µ l arg e ( y )]  1  x 1 − µ FP1 ( x1 , x 2 ) =  2  10 1 − (55 − x1 )(10 − x 2 ) 200 

if

x1 > 55

or

if

x1 ≤ 35

and x 2 ≤ 10

if

35 < x1 ≤ 55 and x 2 ≤ 10

٣٠

x 2 > 10

if x1 > 55 or x2 > 10 or y > 2  1  x2 if x1 ≤ 35 and x2 ≤ 10 and y ≤ 1   10  1− (55− x1 )(10− x2 ) if 35 < x1 ≤ 55 and x2 ≤ 10 and y ≤ 1 µQD (x1, x2 , y) =  200  x2 if x1 ≤ 35 and x2 ≤ 10 and 1 < y < 2 max[y −1, ] 10  max[y −1,1− (55− x1 )(10− x2 ) ] if 35 < x1 ≤ 55 and x2 ≤ 10 and 1 < y < 2  200 y>2 1< y ≤ 2 y ≤1 x1 > 55 or x 2 > 10

x1 < 35 and 35 < x1 ≤ 55 x 2 < 10 and x 2 < 10

.‫ ﻣﺜﺎﻝ ﻓﻮﻕ ﺭﺍ ﺑﺎ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺍﺳﺘﻠﺰﺍﻡ ﻟﻮﻛﺎﺯﻭﻳﭻ ﻭ ﻣﻤﺪﺍﻧﻲ ﺍﻧﺠﺎﻡ ﺩﻫﻴﺪ‬:‫ﺗﻤﺮﻳﻦ‬

٣١

١

‫ ﻣﻨﻄﻖ ﻓﺎﺯﻱ ﻭ ﺍﺳﺘﺪﻻﻝ ﺗﻘﺮﻳﺒﻲ‬-٨ :‫ﺳﻪ ﻧﻤﻮﻧﻪ ﺍﺯ ﻗﻮﺍﻋﺪ ﺍﺳﺘﻨﺘﺎﺝ ﺑﺼﻮﺭﺕ ﺯﻳﺮﻧﺪ‬ ٢

‫ ﻣﻮﺩﺱ ﭘﻮﻧﻨﺲ‬-١

:‫ ﻧﺘﻴﺠﻪ ﮔﺮﻓﺘﻪ ﻣﻲﺷﻮﺩ‬q ‫ ﺩﺭﺳﺘﻲ ﻋﺒﺎﺭﺕ‬p → q , p ‫ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﺩﻭ ﻋﺒﺎﺭﺕ‬ ( p ∧ ( p → q)) → q Premise 1: x is A Premise 2: IF x is A THEN Conclusion: y is B

y is B

٣

‫ ﻣﻮﺩﺱ ﺗﻮﻟﻨﺲ‬-٢

:‫ ﻧﺘﻴﺠﻪ ﮔﺮﻓﺘﻪ ﻣﻲﺷﻮﺩ‬p ‫ ﺩﺭﺳﺘﻲ ﻋﺒﺎﺭﺕ‬p → q , q ‫ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﺩﻭ ﻋﺒﺎﺭﺕ‬ (q ∧ ( p → q)) → p Premise 1: y is not B Premise 2: IF x is A THEN Conclusion: x is not A

y is B

٤

‫ ﻗﻴﺎﺱ ﻓﺮﺿﻲ‬-٣

:‫ ﻧﺘﻴﺠﻪ ﮔﺮﻓﺘﻪ ﻣﻲﺷﻮﺩ‬p → r ‫ ﺩﺭﺳﺘﻲ ﻋﺒﺎﺭﺕ‬q → r , p → q ‫ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﺩﻭ ﻋﺒﺎﺭﺕ‬ (( p → q) ∧ (q → r ) → ( p → r ) Premise 1: IF x is A THEN y is B Premise 2: IF y is B THEN z is C Conclusion: IF x is A THEN z is C

:‫ﺣﺎﻝ ﺍﻳﻦ ﻗﻮﺍﻋﺪ ﺍﺳﺘﻨﺘﺎﺝ ﺭﺍ ﺑﺮﺍﻱ ﻋﺒﺎﺭﺍﺕ ﻓﺎﺯﻱ ﺗﻌﻤﻴﻢ ﻣﻲﺩﻫﻴﻢ‬ ٥

‫ ﻣﻮﺩﺱ ﭘﻮﻧﻨﺲ ﺗﻌﻤﻴﻢ ﻳﺎﻓﺘﻪ‬-١

Premise 1: x is A' Premise 2: IF x is A THEN Conclusion: y is B'

y is B

1 - Fuzzy Logic and Approximate Reasoning 2 - Modus Ponens 3 - Modus Tollens 4 - Hypothetical Syllogism 5 - Generalized Modus Ponens

٣٢

‫ﻫﺮ ﭼﻘﺪﺭ '‪ A‬ﺑﻪ ‪ A‬ﻧﺰﺩﻳﻜﺘﺮ ﺑﺎﺷﺪ '‪ B‬ﻧﻴﺰ ﺑﻪ ‪ B‬ﻧﺰﺩﻳﻜﺘﺮ ﺧﻮﺍﻫﺪ ﺑﻮﺩ‪.‬‬ ‫‪١‬‬

‫‪ -٢‬ﻣﻮﺩﺱ ﺗﻮﻟﻨﺲ ﺗﻌﻤﻴﻢ ﻳﺎﻓﺘﻪ‬

‫‪y is B‬‬

‫‪Premise 1:‬‬ ‫'‪y is B‬‬ ‫‪Premise 2: IF x is A THEN‬‬ ‫'‪Conclusion: x is A‬‬

‫ﻫﺮ ﭼﻘﺪﺭ ﺍﺧﺘﻼﻑ'‪ B‬ﻭ ‪ B‬ﺯﻳﺎﺩﺗﺮ ﺑﺎﺷﺪ ﺍﺧﺘﻼﻑ'‪ A‬ﻭ ‪ A‬ﻧﻴﺰ ﺑﻴﺸﺘﺮ ﺧﻮﺍﻫﺪ ﺑﻮﺩ‪.‬‬ ‫‪٢‬‬

‫‪ -٣‬ﻗﻴﺎﺱ ﻓﺮﺿﻲ ﺗﻌﻤﻴﻢ ﻳﺎﻓﺘﻪ‬

‫‪Premise 1:‬‬ ‫‪IF x is A THEN y is B‬‬ ‫‪Premise 2:‬‬ ‫‪IF y is B' THEN z is C‬‬ ‫'‪Conclusion: IF x is A THEN z is C‬‬

‫ﻫﺮ ﭼﻘﺪﺭ ‪ B‬ﺑﻪ '‪ B‬ﻧﺰﺩﻳﻜﺘﺮ ﺑﺎﺷﺪ ‪ C‬ﻧﻴﺰ ﺑﻪ '‪ C‬ﻧﺰﺩﻳﻜﺘﺮ ﻣﻲﺑﺎﺷﺪ‪.‬‬

‫‪٣‬‬

‫ﻗﻮﺍﻋﺪ ﺗﺮﻛﻴﺒﻲ ﺍﺳﺘﻨﺘﺎﺝ‬

‫ﺣﺎﻝ ﻣﻲ ﺧﻮﺍﻫﻴﻢ ﺑﺎ ﺩﺍﺷﺘﻦ ﺗﻮﺍﺑﻊ ﻋﻀﻮﻳﺖ ﻗﺴﻤﺖ ﻣﻘﺪﻣﻪ ﺗﻮﺍﺑﻊ ﻋﻀﻮﻳﺖ ﻗﺴﻤﺖ ﻧﺘﻴﺠﻪ ﺭﺍ ﺑﺪﺳﺖ ﺁﻭﺭﻳﻢ‪:‬‬ ‫‪ -١‬ﻣﻮﺩﺱ ﭘﻮﻧﻨﺲ ﺗﻌﻤﻴﻢ ﻳﺎﻓﺘﻪ‬ ‫]) ‪µ B′ ( y ) = SUP t [ µ A′ ( x), µ A→ B ( x, y‬‬ ‫‪x∈U‬‬

‫‪ -٢‬ﻣﻮﺩﺱ ﺗﻮﻟﻨﺲ ﺗﻌﻤﻴﻢ ﻳﺎﻓﺘﻪ‬ ‫]) ‪µ A′ ( x) = SUP t [ µ B′ ( y ), µ A→ B ( x, y‬‬ ‫‪y∈V‬‬

‫‪ -٣‬ﻗﻴﺎﺱ ﻓﺮﺿﻲ ﺗﻌﻤﻴﻢ ﻳﺎﻓﺘﻪ‬ ‫]) ‪µ A→C ′ ( x, z ) = SUP t [ µ A→ B ( x, y ), µ B′→C ( y, z‬‬ ‫‪y∈V‬‬

‫‪1 - Generalizes Modus Tollens‬‬ ‫‪2 - Generalized Hypothetical Syllogism‬‬ ‫‪3 - The Compositional Rule of Inference‬‬

‫‪٣٣‬‬

‫‪ -٩‬ﭘﺎﻳﮕﺎﻩ ﻗﻮﺍﻋﺪ ﻓﺎﺯﻱ ﻭ ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ ﻓﺎﺯﻱ‬

‫‪١‬‬

‫‪ -١‬ﭘﺎﻳﮕﺎﻩ ﻗﻮﺍﻋﺪ ﻓﺎﺯﻱ‬ ‫ﭘﺎﻳﮕﺎﻩ ﻗﻮﺍﻋﺪ ﻓﺎﺯﻱ ﺑﺎﻳﺪ ﺗﻤﺎﻣﻲ ﺩﺍﺩﻩﻫﺎﻱ ﻭﺭﻭﺩﻱ ﺭﺍ ﭘﻮﺷﺶ ﺩﻫﺪ‪:‬‬ ‫ﻗﻮﺍﻋﺪ ﻓﺎﺯﻱ ﻛﺎﻣﻞ ‪ :‬ﻳﻚ ﻣﺠﻤﻮﻋﻪ ﺍﺯ ﻗﻮﺍﻋﺪ ‪ IF-THEN‬ﻓﺎﺯﻱ ﺭﺍ ﻛﺎﻣﻞ ﮔﻮﻳﻨﺪ ﺍﮔﺮ ﺑﺮﺍﻱ ﻫﺮ ‪ x ∈ U‬ﺣﺪﺍﻗﻞ ﻳﻚ ﻗﺎﻋﺪﻩ ﺩﺭ‬ ‫ﭘﺎﻳﮕﺎﻩ ﻗﻮﺍﻋﺪ ﻓﺎﺯﻱ ﻭﺟﻮﺩ ﺩﺍﺷﺘﻪ ﺑﺎﺷﺪ‪.‬‬ ‫ﻗﻮﺍﻋﺪ ﻓﺎﺯﻱ ﺳﺎﺯﮔﺎﺭ‪ :‬ﻳﻚ ﻣﺠﻤﻮﻋﻪ ﺍﺯ ﻗﻮﺍﻋﺪ ‪ IF-THEN‬ﻓﺎﺯﻱ ﺭﺍ ﺳﺎﺯﮔﺎﺭ ﮔﻮﻳﻨﺪ ﺍﮔﺮ ﻗﻮﺍﻋﺪﻱ ﻳﺎﻓﺖ ﻧﺸﻮﻧﺪ ﻛﻪ ﺑﺨﺶﻫﺎﻱ‬ ‫ﺍﮔﺮ ﻳﻜﺴﺎﻥ ﻭ ﺑﺨﺶﻫﺎﻱ ﺁﻧﮕﺎﻩ ﻣﺘﻔﺎﻭﺕ ﺩﺍﺷﺘﻪ ﺑﺎﺷﻨﺪ‪.‬‬ ‫ﻗﻮﺍﻋﺪ ﻓﺎﺯﻱ ﭘﻴﻮﺳﺘﻪ‪ :‬ﻳﻚ ﻣﺠﻤﻮﻋﻪ ﺍﺯ ﻗﻮﺍﻋﺪ ‪ IF-THEN‬ﻓﺎﺯﻱ ﺭﺍ ﭘﻴﻮﺳﺘﻪ ﮔﻮﻳﻨﺪ ﺍﮔﺮ ﻗﻮﺍﻋﺪ ﻫﻤﺴﺎﻳﻪﺍﻱ ﻭﺟﻮﺩ ﻧﺪﺍﺷﺘﻪ ﺑﺎﺷﻨﺪ‬ ‫ﻛﻪ ﺍﺷﺘﺮﺍﻙ ﻣﺠﻤﻮﻋﻪﻫﺎﻱ ﻓﺎﺯﻱ ﻗﺴﻤﺖ ‪ THEN‬ﺁﻧﻬﺎ ﺗﻬﻲ ﺑﺎﺷﺪ‪.‬‬ ‫ﻣﺜﺎﻝ‪ :‬ﻓﺮﺽ ﻛﻨﻴﺪ ﻳﻚ ﺳﻴﺴﺘﻢ ﻓﺎﺯﻱ ﺷﺎﻣﻞ ﺩﻭ ﻭﺭﻭﺩﻱ ﻭ ﻳﻚ ﺧﺮﻭﺟﻲ ﺑﺎﺷﺪ ﻛﻪ ﻭﺭﻭﺩﻱ ﺍﻭﻝ ‪ x1‬ﺷﺎﻣﻞ ﺳﻪ ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ ‪S1‬‬

‫‪ M1 ،‬ﻭ ‪ L1‬ﻭ ﻭﺭﻭﺩﻱ ﺩﻭﻡ ‪ x2‬ﺷﺎﻣﻞ ﺩﻭ ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ ‪ L2 ، S2‬ﺑﺎﺷﺪ ﻣﻲﺗﻮﺍﻥ ﭘﺎﻳﮕﺎﻩ ﻗﻮﺍﻋﺪ ﻓﺎﺯﻱ ﺭﺍ ﺑﻪ ﻓﺮﻡ ﺯﻳﺮ ﻧﻮﺷﺖ‪:‬‬ ‫‪B1‬‬ ‫‪B2‬‬ ‫‪B3‬‬ ‫‪B4‬‬ ‫‪B5‬‬ ‫‪B6‬‬

‫‪is‬‬ ‫‪is‬‬ ‫‪is‬‬ ‫‪is‬‬ ‫‪is‬‬ ‫‪is‬‬

‫‪y‬‬ ‫‪y‬‬ ‫‪y‬‬ ‫‪y‬‬ ‫‪y‬‬ ‫‪y‬‬

‫‪THEN‬‬ ‫‪THEN‬‬ ‫‪THEN‬‬ ‫‪THEN‬‬ ‫‪THEN‬‬ ‫‪THEN‬‬

‫‪S2‬‬ ‫‪L2‬‬ ‫‪S2‬‬ ‫‪L2‬‬ ‫‪S2‬‬ ‫‪L2‬‬

‫‪is‬‬ ‫‪is‬‬ ‫‪is‬‬ ‫‪is‬‬ ‫‪is‬‬ ‫‪is‬‬

‫‪x2‬‬ ‫‪x2‬‬ ‫‪x2‬‬ ‫‪x2‬‬ ‫‪x2‬‬ ‫‪x2‬‬

‫‪and‬‬ ‫‪and‬‬ ‫‪and‬‬ ‫‪and‬‬ ‫‪and‬‬ ‫‪and‬‬

‫‪S1‬‬ ‫‪S1‬‬ ‫‪M1‬‬ ‫‪M1‬‬ ‫‪L1‬‬ ‫‪L1‬‬

‫‪is‬‬ ‫‪is‬‬ ‫‪is‬‬ ‫‪is‬‬ ‫‪is‬‬ ‫‪is‬‬

‫‪IF‬‬ ‫‪IF‬‬ ‫‪IF‬‬ ‫‪IF‬‬ ‫‪IF‬‬ ‫‪IF‬‬

‫‪x1‬‬ ‫‪x1‬‬ ‫‪x1‬‬ ‫‪x1‬‬ ‫‪x1‬‬ ‫‪x1‬‬

‫‪ -٢‬ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ ﻓﺎﺯﻱ‬ ‫ﺑﺮﺭﺳﻲ ﭼﮕﻮﻧﮕﻲ ﻧﺘﻴﺠﻪﮔﻴﺮﻱ ﺍﺯ ﺭﻭﻱ ﻳﻚ ﻣﺠﻤﻮﻋﻪ ﺍﺯ ﻗﻮﺍﻋﺪ‪:‬‬ ‫‪٢‬‬

‫‪ -١‬ﺍﺳﺘﻨﺘﺎﺝ ﻣﺒﺘﻨﻲ ﺑﺮ ﺗﺮﻛﻴﺐ ﻗﻮﺍﻋﺪ‬

‫ﺗﻤﺎﻣﻲ ﻗﻮﺍﻋﺪ ﻣﻮﺟﻮﺩ ﺩﺭ ﭘﺎﻳﮕﺎﻩ ﻗﻮﺍﻋﺪ ﻓﺎﺯﻱ ﺩﺭ ﻳﻚ ﺭﺍﺑﻄﻪ ﻓﺎﺯﻱ ﺗﺮﻛﻴﺐ ﺷﺪﻩ ﻭ ﺁﻧﮕﺎﻩ ﺑﺪﻳﺪﻩ ﻳﻚ ﻗﺎﻋﺪﻩ ‪ IF-THEN‬ﻓﺎﺯﻱ‬ ‫ﺗﻨﻬﺎ ﻧﮕﺮﻳﺴﺘﻪ ﻣﻲﺷﻮﺩ‪.‬‬ ‫ﻣﺮﺍﺣﻞ ﻣﺤﺎﺳﺒﺎﺕ ﺍﺳﺘﻨﺘﺎﺝ ﻣﺒﺘﻨﻲ ﺑﺮ ﺗﺮﻛﻴﺐ ﻗﻮﺍﻋﺪ‪:‬‬ ‫ﻣﺮﺣﻠﻪ ﺍﻭﻝ‪ :‬ﺑﺮﺍﻱ ‪ M‬ﻗﺎﻋﺪﻩ ﻓﺎﺯﻱ ﺗﻮﺍﺑﻊ ﻋﻀﻮﻳﺖ ﺭﺍ ﻣﺤﺎﺳﺒﻪ ﻣﻲﻛﻨﻴﻢ‬ ‫‪l = 1, 2 , K , M‬‬

‫) ‪µ Al ×K× Al ( x1 ,L, xn ) = µ Al ( x1 ) ∗ L ∗ µ Al ( xn‬‬ ‫‪n‬‬

‫‪1‬‬

‫‪n‬‬

‫‪1‬‬

‫ﻛﻪ ﻋﻼﻣﺖ * ﻧﻤﺎﻳﺎﻧﮕﺮ ‪-t‬ﻧﺮﻡ ﻭ ‪ n‬ﺗﻌﺪﺍﺩ ﻭﺭﻭﺩﻱ ﻣﻲﺑﺎﺷﺪ‪.‬‬ ‫‪1 - Fuzzy Rule Base & Fuzzy Inference Engine‬‬ ‫‪2 - Composition Based Inference‬‬

‫‪٣۴‬‬

‫ﻣﺮﺣﻠﻪ ﺩﻭﻡ‪:‬‬

‫‪l‬‬

‫‪× K × An‬‬

‫‪ A1l‬ﺭﺍ ﺑﻪ ﻋﻨﻮﺍﻥ ﻣﻘﺪﻣﻪ )‪ (FP1‬ﻭ ‪ Bl‬ﺭﺍ ﺑﻪ ﻋﻨﻮﺍﻥ ﻧﺘﻴﺠﻪ )‪ (FP2‬ﺩﺭ ﺍﺳﺘﻠﺰﺍﻡﻫﺎﻱ ﮔﻔﺘﻪ ﺷﺪﻩ ﺩﺭ ﻧﻈﺮ‬

‫ﻣﻲﮔﻴﺮﻳﻢ ﻭ ﺩﺍﺭﻳﻢ‪:‬‬ ‫) ‪µ Ru ( l ) ( x1 , L , x n , y ) = µ Al ×K× Al → B ( x1 , L , x n , y‬‬

‫‪l = 1, 2 , K , M‬‬

‫‪n‬‬

‫‪1‬‬

‫ﻣﺮﺣﻠﻪ ﺳﻮﻡ‪ :‬ﻣﺤﺎﺳﺒﻪ ) ‪ µ Q ( x, y‬ﻳﺎ ) ‪) µ Q ( x, y‬ﺑﺘﺮﺗﻴﺐ ﺗﺮﻛﻴﺐ ﻣﻤﺪﺍﻧﻲ ﻳﺎ ﮔﻮﺩﻝ(‪:‬‬ ‫‪G‬‬

‫‪M‬‬

‫•‬

‫•‬

‫) ‪µ QM ( x, y ) = µ Ru (1) ( x, y ) + L + µ Ru ( M ) ( x, y‬‬

‫⇒‬

‫) ‪µ QM ( x, y ) = µ Ru (1) ( x, y ) ∗ L ∗ µ Ru ( M ) ( x, y‬‬

‫⇒‬

‫‪M‬‬

‫) ‪QM = U Ru ( l‬‬ ‫‪l =1‬‬

‫‪M‬‬

‫) ‪QG = I Ru (l‬‬ ‫‪l =1‬‬

‫•‬

‫ﻛﻪ ﻋﻼﻣﺖ * ﻧﻤﺎﻳﺎﻧﮕﺮ ‪-t‬ﻧﺮﻡ ﻭ ‪ +‬ﻧﻤﺎﻳﺎﻧﮕﺮ ‪-S‬ﻧﺮﻡ ﺍﺳﺖ‪.‬‬ ‫ﻣﺮﺣﻠﻪ ﭼﻬﺎﺭﻡ‪ :‬ﺑﺮﺍﻱ ﻳﻚ ﻭﺭﻭﺩﻱ ﺩﺍﺩﻩ ﺷﺪﻩ '‪ A‬ﺧﺮﻭﺟﻲ ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ ﺭﺍ ﻛﻪ ﻫﻤﺎﻥ '‪ B‬ﺍﺳﺖ ﺍﺯ ﺭﻭﺍﺑﻂ ﺯﻳﺮ ﻣﺤﺎﺳﺒﻪ ﻣﻲﻛﻨﻴﻢ‪:‬‬ ‫ﺑﺮﺍﻱ ﺗﺮﻛﻴﺐ ﻣﻤﺪﺍﻧﻲ‬

‫]) ‪µ B′ ( y ) = SUP t [ µ A′ ( x), µ QM ( x, y‬‬

‫ﺑﺮﺍﻱ ﺗﺮﻛﻴﺐ ﮔﻮﺩﻝ‬

‫]) ‪µ B′ ( y ) = SUP t [ µ A′ ( x), µ QG ( x, y‬‬

‫‪x∈U‬‬

‫‪x∈U‬‬

‫‪١‬‬

‫‪ -٢‬ﺍﺳﺘﻨﺘﺎﺝ ﻣﺒﺘﻨﻲ ﺑﺮ ﻗﻮﺍﻋﺪ ﺟﺪﺍﮔﺎﻧﻪ‬

‫ﻫﺮ ﻗﺎﻋﺪﻩ ﺩﺭ ﭘﺎﻳﮕﺎﻩ ﻗﻮﺍﻋﺪ ﻓﺎﺯﻱ ﻳﻚ ﺧﺮﻭﺟﻲ ﻓﺎﺯﻱ ﺭﺍ ﻣﻌﻴﻦ ﻛﺮﺩﻩ ﻭ ﺧﺮﻭﺟﻲ ﻧﻬﺎﻳﻲ ﺗﺮﻛﻴﺐ ‪ M‬ﺧﺮﻭﺟﻲ ﺟﺪﺍﮔﺎﻧﻪ‬ ‫ﻣﺠﻤﻮﻋﻪﻫﺎﻱ ﻓﺎﺯﻱ ﺧﻮﺍﻫﺪ ﺑﻮﺩ‪.‬‬ ‫ﻣﺮﺍﺣﻞ ﻣﺤﺎﺳﺒﺎﺕ ﺍﺳﺘﻨﺘﺎﺝ ﻣﺒﺘﻨﻲ ﺑﺮ ﻗﻮﺍﻋﺪ ﺟﺪﺍﮔﺎﻧﻪ‪:‬‬ ‫ﻣﺮﺣﻠﻪ ﺍﻭﻝ ﻭ ﺩﻭﻡ‪ :‬ﻣﺸﺎﺑﻪ ﺍﺳﺘﻨﺘﺎﺝ ﻣﺒﺘﻨﻲ ﺑﺮ ﺗﺮﻛﻴﺐ ﻗﻮﺍﻋﺪ‬ ‫ﻣﺮﺣﻠﻪ ﺳﻮﻡ‪ :‬ﺑﺮﺍﻱ ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ ﺩﺍﺩﻩ ﺷﺪﻩ '‪ A‬ﺩﺭ ‪ ، U‬ﺧﺮﻭﺟﻲ ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ '‪ Bl‬ﺩﺭ ‪ V‬ﺭﺍ ﺑﺮﺍﻱ ﻫﺮ ﻗﺎﻋﺪﻩ ﺟﺪﺍﮔﺎﻧﻪ )‪Ru(l‬‬

‫ﻣﻄﺎﺑﻖ ﺑﺎ ﻣﻮﺩﺱ ﭘﻮﻧﻨﺲ ﺗﻌﻤﻴﻢ ﻳﺎﻓﺘﻪ ﻣﺤﺎﺳﺒﻪ ﻣﻲﻛﻨﻴﻢ‪:‬‬ ‫‪l = 1, 2 ,K, M‬‬

‫]) ‪µ Bl′ ( y ) = SUP t [ µ A′ ( x), µ Ru ( l ) ( x, y‬‬ ‫‪x∈U‬‬

‫ﻣﺮﺣﻠﻪ ﭼﻬﺎﺭﻡ‪ :‬ﺧﺮﻭﺟﻲ ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ ﻓﺎﺯﻱ‪ ،‬ﺗﺮﻛﻴﺐ ﺧﺮﻭﺟﻲ ﻓﺎﺯﻱ }'‪ {B1' , B2' , ..., BM‬ﺧﻮﺍﻫﺪ ﺑﻮﺩ‪:‬‬ ‫•‬

‫•‬

‫ﺑﺼﻮﺭﺕ ﺍﺟﺘﻤﺎﻉ‪:‬‬

‫) ‪µ B′ ( y ) = µ B ′ ( y ) + L + µ B ′ ( y‬‬

‫ﺑﺼﻮﺭﺕ ﺍﺷﺘﺮﺍﻙ‪:‬‬

‫) ‪µ B′ ( y ) = µ B ′ ( y ) ∗ L ∗ µ B ′ ( y‬‬

‫‪M‬‬

‫‪M‬‬

‫‪1‬‬

‫‪1‬‬

‫‪1 - Individual Rule Based Inference‬‬

‫‪٣۵‬‬

‫ﺟﺰﺋﻴﺎﺕ ﭼﻨﺪ ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ‬ ‫‪ -١‬ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ ﺣﺎﺻﻠﻀﺮﺏ ﻣﻤﺪﺍﻧﻲ‬ ‫ﺩﺭ ﺍﻳﻦ ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ ﺍﺯ‪:‬‬ ‫•‬

‫ﺍﺳﺘﻨﺘﺎﺝ ﻣﺒﺘﻨﻲ ﺑﺮ ﻗﻮﺍﻋﺪ ﺟﺪﺍﮔﺎﻧﻪ ﺑﺎ ﺗﺮﻛﻴﺐ ﺍﺟﺘﻤﺎﻉ‬

‫•‬

‫ﺍﺳﺘﻠﺰﺍﻡ ﺣﺎﺻﻠﻀﺮﺏ ﻣﻤﺪﺍﻧﻲ‬

‫•‬

‫ﺿﺮﺏ ﺟﺒﺮﻱ ﺑﺮﺍﻱ ‪ t‬ﻧﺮﻡﻫﺎ ﻭ ‪ max‬ﺑﺮﺍﻱ ‪ s‬ﻧﺮﻡﻫﺎ‬

‫ﺍﺳﺘﻔﺎﺩﻩ ﻣﻲﺷﻮﺩ‪.‬‬ ‫‪n‬‬ ‫‪M ‬‬ ‫‪‬‬ ‫‪µ B′ ( y ) = max  SUP( µ A′ ( x)∏ ( µ Al ( xi )) µ B l ( y ))‬‬ ‫‪i‬‬ ‫‪l =1‬‬ ‫‪i =1‬‬ ‫‪ x∈U‬‬ ‫‪‬‬

‫‪ -٢‬ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ ﻣﻴﻨﻴﻤﻢ ﻣﻤﺪﺍﻧﻲ‬ ‫ﺩﺭ ﺍﻳﻦ ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ ﺍﺯ‪:‬‬ ‫•‬

‫ﺍﺳﺘﻨﺘﺎﺝ ﻣﺒﺘﻨﻲ ﺑﺮ ﻗﻮﺍﻋﺪ ﺟﺪﺍﮔﺎﻧﻪ ﺑﺎ ﺗﺮﻛﻴﺐ ﺍﺟﺘﻤﺎﻉ‬

‫•‬

‫ﺍﺳﺘﻠﺰﺍﻡ ﻣﻴﻨﻴﻤﻢ ﻣﻤﺪﺍﻧﻲ‬

‫•‬

‫‪ min‬ﺑﺮﺍﻱ ‪ t‬ﻧﺮﻡﻫﺎ ﻭ ‪ max‬ﺑﺮﺍﻱ ‪ s‬ﻧﺮﻡﻫﺎ‬

‫ﺍﺳﺘﻔﺎﺩﻩ ﻣﻲﺷﻮﺩ‪.‬‬

‫]‬

‫[‬

‫‪M‬‬

‫)) ‪µ B′ ( y ) = max SUP min(µ A′ ( x), µ Al ( x1 ),..., µ Al ( xn ), µ B l ( y‬‬ ‫‪n‬‬

‫‪1‬‬

‫‪x∈U‬‬

‫‪l =1‬‬

‫‪ -٣‬ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ ﻟﻮﻛﺎﺯﻭﻳﺞ‬ ‫ﺩﺭ ﺍﻳﻦ ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ ﺍﺯ‪:‬‬ ‫•‬

‫ﺍﺳﺘﻨﺘﺎﺝ ﻣﺒﺘﻨﻲ ﺑﺮ ﻗﻮﺍﻋﺪ ﺟﺪﺍﮔﺎﻧﻪ ﺑﺎ ﺗﺮﻛﻴﺐ ﺍﺷﺘﺮﺍﻙ‬

‫• ﺍﺳﺘﻠﺰﺍﻡ ﻟﻮﻛﺎﺯﻭﻳﺞ‬ ‫•‬

‫‪ min‬ﺑﺮﺍﻱ ‪ t‬ﻧﺮﻡﻫﺎ‬

‫ﺍﺳﺘﻔﺎﺩﻩ ﻣﻲﺷﻮﺩ‪.‬‬

‫)‬

‫(‬

‫‪n‬‬ ‫‪M‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪µ B′ ( y ) = min SUP min  µ A′ ( x),1 − min µ Al ( xi ) + µ B l ( y ) ‬‬ ‫‪i‬‬ ‫=‬ ‫‪i‬‬ ‫‪l =1‬‬ ‫‪1‬‬ ‫‪x∈U‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬

‫‪ -٤‬ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ ﺯﺍﺩﻩ‬ ‫ﺩﺭ ﺍﻳﻦ ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ ﺍﺯ‪:‬‬ ‫•‬

‫ﺍﺳﺘﻨﺘﺎﺝ ﻣﺒﺘﻨﻲ ﺑﺮ ﻗﻮﺍﻋﺪ ﺟﺪﺍﮔﺎﻧﻪ ﺑﺎ ﺗﺮﻛﻴﺐ ﺍﺷﺘﺮﺍﻙ‬

‫• ﺍﺳﺘﻠﺰﺍﻡ ﺯﺍﺩﻩ‬ ‫•‬

‫‪ min‬ﺑﺮﺍﻱ ‪ t‬ﻧﺮﻡﻫﺎ‬

‫ﺍﺳﺘﻔﺎﺩﻩ ﻣﻲﺷﻮﺩ‪.‬‬

‫‪٣۶‬‬

‫‪) ‬‬ ‫‪‬‬

‫(‬

‫‪M ‬‬ ‫‪n‬‬ ‫‪‬‬ ‫‪‬‬ ‫) ‪µ B′ ( y ) = min SUP min  µ A′ ( x), max min( µ Al ( x1 ),..., µ Al ( xn ), µ B l ( y )),1 − min µ Al ( xi‬‬ ‫‪n‬‬ ‫‪i‬‬ ‫‪1‬‬ ‫‪l =1‬‬ ‫‪i =1‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ x∈U‬‬

‫‪ -٥‬ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ ﺩﻧﻴﺲ‪ -‬ﺭﺷﺮ‬ ‫ﺩﺭ ﺍﻳﻦ ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ ﺍﺯ‪:‬‬ ‫•‬

‫ﺍﺳﺘﻨﺘﺎﺝ ﻣﺒﺘﻨﻲ ﺑﺮ ﻗﻮﺍﻋﺪ ﺟﺪﺍﮔﺎﻧﻪ ﺑﺎ ﺗﺮﻛﻴﺐ ﺍﺷﺘﺮﺍﻙ‬

‫• ﺍﺳﺘﻠﺰﺍﻡ ﺩﻧﻴﺲ‪ -‬ﺭﺷﺮ‬ ‫•‬

‫‪ min‬ﺑﺮﺍﻱ ‪ t‬ﻧﺮﻡﻫﺎ‬

‫ﺍﺳﺘﻔﺎﺩﻩ ﻣﻲﺷﻮﺩ‪.‬‬

‫)‬

‫(‬

‫‪M ‬‬ ‫‪n‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ ‬‬ ‫‪µ B′ ( y ) = min SUP min  µ A′ ( x), max1 − min µ Al ( xi ) , µ B l ( y )  ‬‬ ‫‪i‬‬ ‫‪l =1‬‬ ‫‪i =1‬‬ ‫‪‬‬ ‫‪ ‬‬ ‫‪‬‬ ‫‪ x∈U‬‬

‫ﻛﻪ ﺩﺭ ﺭﻭﺍﺑﻂ ﻓﻮﻕ ‪ M‬ﺗﻌﺪﺍﺩ ﻗﻮﺍﻋﺪ ﻣﻮﺟﻮﺩ ﺩﺭ ﭘﺎﻳﮕﺎﻩ ﻗﻮﺍﻋﺪ ﻣﻲﺑﺎﺷﺪ ﻭ ‪ n‬ﺗﻌﺪﺍﺩ ﻛﻞ ﻭﺭﻭﺩﻳﻬﺎ ﻣﻲﺑﺎﺷﺪ‪.‬‬ ‫ﺩﺭ ﺍﻳﻦ ﺭﻭﺍﺑﻂ )‪ µ A′ (x‬ﺑﻪ ﻳﻜﻲ ﺍﺯ ﺳﻪ ﻓﺮﻡ ﺯﻳﺮ ﻣﻌﺮﻓﻲ ﻣﻲﮔﺮﺩﺩ‪:‬‬ ‫•‬

‫ﻣﻨﻔﺮﺩ ﻓﺎﺯﻱ‬ ‫*‪X = X‬‬ ‫‪otherwise‬‬

‫‪1‬‬ ‫‪µ A′ ( x) = ‬‬ ‫‪0‬‬

‫‪if‬‬

‫ﻛﻪ ] ‪ X = [ x1 x 2 L x n‬ﻭ ] *‪ X * = [ x1* x 2* L x n‬ﻣﻲﺑﺎﺷﺪ‪.‬‬ ‫•‬

‫ﮔﻮﺳﻴﻦ‬ ‫‪2‬‬

‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬

‫*‪ x − x‬‬ ‫‪− n n‬‬ ‫‪ an‬‬ ‫‪‬‬

‫‪2‬‬

‫‪*L* e‬‬ ‫ﻛﻪ ‪ai‬ﻫﺎ ﭘﺎﺭﺍﻣﺘﺮﻫﺎﻱ ﻣﺜﺒﺖ ﻭ * ﻧﺸﺎﻥ ﺩﻫﻨﺪﻩ ‪ t‬ﻧﺮﻡ ﻣﻲﺑﺎﺷﺪ ﻭ ﻣﻌﻤﻮ ﹰﻻ ﺍﺯ ﻧﻮﻉ ﺿﺮﺏ ﻳﺎ ‪ min‬ﺍﻧﺘﺨﺎﺏ ﻣﻲﮔﺮﺩﺩ‪.‬‬ ‫•‬

‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬

‫*‪ x − x‬‬ ‫‪− 1 1‬‬ ‫‪ a‬‬ ‫‪1‬‬ ‫‪‬‬

‫‪µ A′ ( x ) = e‬‬

‫ﻣﺜﻠﺜﻲ‬

‫‪‬‬ ‫‪‬‬ ‫‪x1 − x1* ‬‬ ‫‪x n − x n* ‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ 1−‬‬ ‫‪*L * 1 −‬‬ ‫* ‪if X = X‬‬ ‫‪‬‬ ‫‪µ A′ ( x) = ‬‬ ‫‪b1 ‬‬ ‫‪bn ‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪0‬‬ ‫‪otherwise‬‬ ‫‪‬‬ ‫ﻛﻪ ‪bi‬ﻫﺎ ﭘﺎﺭﺍﻣﺘﺮﻫﺎﻱ ﻣﺜﺒﺖ ﻭ * ﻧﺸﺎﻥ ﺩﻫﻨﺪﻩ ‪ t‬ﻧﺮﻡ ﻣﻲﺑﺎﺷﺪ ﻭ ﻣﻌﻤﻮ ﹰﻻ ﺍﺯ ﻧﻮﻉ ﺿﺮﺏ ﻳﺎ ‪ min‬ﺍﻧﺘﺨﺎﺏ ﻣﻲﮔﺮﺩﺩ‪.‬‬

‫ﺍﺯ ﻣﻴﺎﻥ ﺍﻧﺘﺨﺎﺑﻬﺎﻱ ﻓﻮﻕ ﺑﺮﺍﻱ )‪ µ A′ (x‬ﻧﻮﻉ ﻣﻨﻔﺮﺩ ﻓﺎﺯﻱ ﻣﺤﺎﺳﺒﺎﺕ ﺭﺍ ﺑﺴﻴﺎﺭ ﺳﺎﺩﻩ ﻛﺮﺩﻩ ﻭ ﺑﻨﺎﺑﺮﺍﻳﻦ ﺑﺴﻴﺎﺭ ﻣﻮﺭﺩ ﺍﺳﺘﻔﺎﺩﻩ ﻗﺮﺍﺭ ﻣﻲﮔﻴﺮﺩ‪.‬‬ ‫ﻭﻟﻲ ﺍﺯ ﻃﺮﻓﻲ ﺗﻮﺍﻧﺎﻳﻲ ﺣﺬﻑ ﻧﻮﻳﺰ ﺭﺍ ﻧﺪﺍﺭﺩ‪.‬‬

‫‪٣٧‬‬

:‫ ﻣﻮﺗﻮﺭﻫﺎﻱ ﺍﺳﺘﻨﺘﺎﺝ ﺫﻛﺮ ﺷﺪﻩ ﺑﺼﻮﺭﺕ ﺯﻳﺮ ﺳﺎﺩﻩ ﻣﻲﺷﻮﻧﺪ‬µ A′ (x) ‫ﺑﺎ ﺍﻧﺘﺨﺎﺏ ﻣﻨﻔﺮﺩ ﻓﺎﺯﻱ‬

‫ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ ﺣﺎﺻﻠﻀﺮﺏ ﻣﻤﺪﺍﻧﻲ‬

(

)

M  n  µ B′ ( y ) = max ∏ µ Al ( xi* ) µ B l ( y ) i l =1  i =1 

M

[

µ B′ ( y ) = max min(µ Al ( x1* ),..., µ Al ( x n* ), µ Bl ( y )) l =1

n

1

‫ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ ﻣﻴﻨﻴﻤﻢ ﻣﻤﺪﺍﻧﻲ‬

]

‫ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ ﻟﻮﻛﺎﺯﻭﻳﺞ‬

(

)

M n   µ B ′ ( y ) = min 1,1 − min µ Al ( xi* ) + µ B l ( y ) i l =1  i =1 

‫ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ ﺯﺍﺩﻩ‬

(

)

M n    µ B ′ ( y ) = min max min( µ Al ( x1* ),..., µ Al ( xn* ), µ B l ( y )),1 − min µ Al ( xi* )  1 n i 1 = l =1 i   

‫ ﺭﺷﺮ‬-‫ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ ﺩﻧﻴﺲ‬

(

)

M n    µ B ′ ( y ) = min max1 − min µ Al ( xi* ) , µ B l ( y )  i l =1 i =1   

٣٨

‫ﻓﺎﺯﻱ ﺳﺎﺯﻫﺎ ﻭ ﻓﺎﺯﻱ ﺯﺩﻫﺎ‬

‫‪-١٠‬‬

‫‪١‬‬

‫‪ -١‬ﻓﺎﺯﻱﺳﺎﺯﻫﺎ‬ ‫ﺩﺭ ﻗﺴﻤﺖ ﺗﻮﺍﺑﻊ ﻋﻀﻮﻳﺖ ﺍﻧﻮﺍﻉ ﺁﻧﻬﺎ ﻣﻄﺮﺡ ﺷﺪ ﺍﺯ ﻫﺮ ﻛﺪﺍﻡ ﺍﺯ ﺍﻳﻦ ﻧﻤﻮﻧﻪﻫﺎ ﻣﻲﺗﻮﺍﻥ ﺑﻌﻨﻮﺍﻥ ﻓﺎﺯﻱ ﺳﺎﺯ ﺍﺳﺘﻔﺎﺩﻩ ﻛﺮﺩ‪.‬‬

‫‪ -٢‬ﻓﺎﺯﻱ ﺯﺩﺍﻫﺎ‬ ‫‪٢‬‬

‫‪ -١‬ﻓﺎﺯﻱ ﺯﺩﺍﻱ ﻣﺮﻛﺰ ﺛﻘﻞ‬

‫‪( y ) dy‬‬

‫‪B′‬‬

‫‪∫ y.µ‬‬

‫‪V‬‬

‫‪( y ) dy‬‬ ‫‪V‬‬

‫‪B′‬‬

‫‪∫µ‬‬

‫∗‬

‫= ‪y‬‬

‫‪V‬‬

‫*‪y‬‬ ‫‪٣‬‬

‫‪ -٢‬ﻓﺎﺯﻱ ﺯﺩﺍﻱ ﻣﻴﺎﻧﮕﻴﻦ ﻣﺮﺍﻛﺰ‬

‫‪W1‬‬

‫‪M‬‬

‫‪∑ yˆ l wl‬‬

‫‪W2‬‬

‫‪l =1‬‬ ‫‪M‬‬

‫‪∑w‬‬

‫‪l‬‬

‫‪yˆ1‬‬

‫‪yˆ 2‬‬

‫= ∗‪y‬‬

‫‪l =1‬‬

‫ﻛﻪ ‪ yˆ l‬ﻣﺮﻛﺰ ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ ‪ l‬ﺍﻡ ‪ wl ،‬ﺩﺭﺟﻪ ﺍﺭﺗﻔﺎﻉ ﺁﻥ ﻭ ‪ M‬ﺗﻌﺪﺍﺩ ﻛﻞ ﻣﺠﻤﻮﻋﻪﻫﺎﻱ ﻓﺎﺯﻱ ﻣﻲﺑﺎﺷﺪ‪.‬‬ ‫‪٤‬‬

‫‪ -٣‬ﻓﺎﺯﻱ ﺯﺩﺍﻱ ﻣﺎﻛﺰﻳﻤﻢ‬

‫‪1 - Fuzzifiers & Defuzzifiers‬‬ ‫‪2 - Center of Gravity Defuzzifier‬‬ ‫‪3 - Center Average Defuzzifier‬‬ ‫‪4 - Maximum Defuzzifier‬‬

‫‪٣٩‬‬

‫}‬

‫{‬

‫) ‪y ∗ = hgt ( B′) = y ∈ V | µ B ′ ( y ) = SUP µ B′ ( y‬‬

‫ﺍﮔﺮ )'‪ hgt(B‬ﻳﻚ ﻧﻘﻄﻪ ﺑﺎﺷﺪ‬

‫‪y∈V‬‬

‫ﺩﺭ ﻏﻴﺮ ﺍﻳﻦ ﺻﻮﺭﺕ ﺍﺯ ﻳﻜﻲ ﺍﺯ ﻣﻮﺍﺭﺩ ﺯﻳﺮ ﺍﺳﺘﻔﺎﺩﻩ ﻣﻲﻛﻨﻴﻢ‪:‬‬ ‫ﻓﺎﺯﻱ ﺯﺩﺍﻱ ﻛﻮﭼﻜﺘﺮﻳﻦ ﻣﺎﻛﺰﻳﻤﺎ‬

‫})‪y ∗ = inf {y ∈ hgt ( B ′‬‬

‫ﻓﺎﺯﻱ ﺯﺩﺍﻱ ﺑﺰﺭﮔﺘﺮﻳﻦ ﻣﺎﻛﺰﻳﻤﺎ‬

‫})‪y ∗ = SUP{y ∈ hgt (B ′‬‬

‫∫‬ ‫∫‬

‫‪y dy‬‬

‫ﻓﺎﺯﻱ ﺯﺩﺍﻱ ﻣﻴﺎﻧﮕﻴﻦ ﻣﺎﻛﺰﻳﻤﺎ‬

‫‪dy‬‬

‫) ‪hgt ( B′‬‬

‫= ∗‪y‬‬

‫) ‪hgt ( B ′‬‬

‫ﻣﺜﺎﻝ‪ :‬ﻓﺮﺽ ﻛﻨﻴﺪ ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ '‪ B‬ﺍﺟﺘﻤﺎﻉ ﺩﻭ ﻣﺠﻤﻮﻋﻪ ﻓﺎﺯﻱ ﺯﻳﺮ ﺑﺎﺷﺪ‪ .‬ﺣﺎﻝ ﻓﺎﺯﻱ ﺯﺩﺍﻱ ﻣﻴﺎﻧﮕﻴﻦ ﻣﺮﺍﻛﺰ‪ ،‬ﻣﺮﻛﺰ ﺛﻘﻞ ﻭ‬ ‫)‪µB'(y‬‬

‫ﻣﺎﻛﺰﻳﻤﻢ ﺭﺍ ﻣﺤﺎﺳﺒﻪ ﻛﻨﻴﺪ‪(w1>w2) :‬‬

‫‪W1‬‬ ‫‪W2‬‬

‫‪y‬‬

‫‪1‬‬

‫‪2‬‬

‫‪0‬‬

‫‪0.8‬‬

‫‪-0.8‬‬

‫‪ -١‬ﻓﺎﺯﻱ ﺯﺩﺍﻱ ﻣﻴﺎﻧﮕﻴﻦ ﻣﺮﺍﻛﺰ‬ ‫‪yˆ 1 w1 + yˆ 2 w2‬‬ ‫‪w2‬‬ ‫=‬ ‫‪w1 + w2‬‬ ‫‪w1 + w2‬‬

‫=‬

‫‪wl‬‬

‫‪l‬‬

‫‪2‬‬

‫ˆ‪∑ y‬‬ ‫‪l =1‬‬ ‫‪2‬‬

‫‪∑w‬‬

‫‪l‬‬

‫∗‬

‫‪yˆ = 1‬‬

‫= ‪y‬‬

‫‪2‬‬

‫‪yˆ = 0‬‬ ‫‪1‬‬

‫‪l =1‬‬

‫‪ -٢‬ﻓﺎﺯﻱ ﺯﺩﺍﻱ ﻣﺮﻛﺰ ﺛﻘﻞ‬ ‫‪yw2 (2 − y) dy‬‬

‫‪2‬‬

‫∫‪y(w2 y) dy +‬‬

‫‪1‬‬

‫‪w1‬‬

‫‪1‬‬ ‫‪w1‬‬

‫∫‪y( 0w.18 y + w1 ) dy +∫ w2 + 0w.18 y(w1 − 0w.18 y) dy +‬‬ ‫‪0‬‬

‫‪w‬‬ ‫‪w2 + 0.18‬‬

‫‪0.8 w1 w2‬‬ ‫‪w2 + 0w.18‬‬

‫‪0‬‬

‫∫‬

‫‪−0.8‬‬

‫=‬

‫‪∫ y.µ B′ ( y) dy‬‬ ‫‪∫ µ B′ ( y) dy‬‬

‫‪0.8w1 + w2 − 12‬‬

‫‪y∗ = V‬‬

‫‪V‬‬

‫‪ -٣‬ﻓﺎﺯﻱ ﺯﺩﺍﻱ ﻣﺎﻛﺰﻳﻤﻢ‬ ‫‪y ∗ = hgt ( B ′) = 0‬‬

‫ﻣﺜﺎﻝ‪ :‬ﺩﺭ ﺷﻜﻞﻫﺎﻱ ﺯﻳﺮ ﻓﺎﺯﻱ ﺯﺩﺍﻱ ﻣﺎﻛﺰﻳﻤﻢ ﺭﺍ ﻣﺸﺨﺺ ﻧﻤﻮﺩﻩ ﻭ ﺍﮔﺮ ﺑﻴﺶ ﺍﺯ ﻳﻚ ﻧﻘﻄﻪ ﻣﻲﺑﺎﺷﺪ ﺑﺎ ﺗﻌﻴﻴﻦ ﻧﻮﻉ ﺁﻥ ﻫﺮ ﺳﻪ‬ ‫ﻧﻮﻉ ﺁﻧﺮﺍ ﻣﺸﺨﺺ ﻧﻤﺎﻳﻴﺪ‪:‬‬

‫‪V‬‬

‫‪V‬‬

‫*‪y‬‬

‫‪V‬‬

‫*‪y‬‬ ‫*‪y‬‬ ‫‪y* Largest of Maxima‬‬ ‫‪Mean of Maxima‬‬

‫‪۴٠‬‬

‫*‪y‬‬

‫‪Smallest of Maxima‬‬

‫ﻣﺜﺎﻝ‪ :‬ﻳﻚ ﺳﻴﺴﺘﻢ ﻓﺎﺯﻱ ﺑﺎ ﺩﻭ ﻭﺭﻭﺩﻱ ‪ X1‬ﻭ ‪ X2‬ﻭ ﻳﻚ ﺧﺮﻭﺟﻲ ‪ Y‬ﺑﺎ ﺗﻮﺍﺑﻊ ﻋﻀﻮﻳﺖ ﺯﻳﺮ ﻣﻔﺮﻭﺽ ﺍﺳﺖ‪ .‬ﻗﻮﺍﻋﺪ ﻓﺎﺯﻱ ﺁﻥ ﻧﻴﺰ ﺩﺭ‬ ‫ﺟﺪﻭﻝ ﺯﻳﺮ ﺩﺍﺩﻩ ﺷﺪﻩ ﺍﺳﺖ‪ .‬ﺑﻪ ﺍﺯﺍﺀ ﻣﻘﺎﺩﻳﺮ ‪ X1=20‬ﻭ ‪ X2=8.5‬ﻣﻘﺪﺍﺭ ﺧﺮﻭﺟﻲ ‪ Y‬ﺭﺍ ﺑﺎ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ ﺣﺎﺻﻠﻀﺮﺏ‬ ‫ﻣﻤﺪﺍﻧﻲ‪ ،‬ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ ﻣﻴﻨﻴﻤﻢ ﻣﻤﺪﺍﻧﻲ‪ ،‬ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ ﻟﻮﻛﺎﺯﻭﻳﺞ‪ ،‬ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ ﺯﺍﺩﻩ ﻭ ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ ﺩﻧﻴﺲ‪ -‬ﺭﺷﺮ ﺑﺎ ﻓﺮﺽ‬ ‫ﻓﺎﺯﻱ ﮔﺮ ﻣﻨﻔﺮﺩ ﺑﺪﺳﺖ ﺁﻭﺭﻳﺪ‪.‬‬

‫ﺗﻮﺍﺑﻊ ﻋﻀﻮﻳﺖ ﻭﺭﻭﺩﻱ ﺍﻭﻝ‬

‫ﺗﻮﺍﺑﻊ ﻋﻀﻮﻳﺖ ﻭﺭﻭﺩﻱ ﺩﻭﻡ‬

‫ﺗﻮﺍﺑﻊ ﻋﻀﻮﻳﺖ ﺧﺮﻭﺟﻲ‬ ‫ﺟﺪﻭﻝ ﻗﻮﺍﻋﺪ ﻓﺎﺯﻱ‬ ‫‪L2‬‬ ‫‪My‬‬ ‫‪Ly‬‬ ‫‪Ly‬‬

‫‪X2‬‬ ‫‪S2‬‬ ‫‪Sy‬‬ ‫‪Sy‬‬ ‫‪My‬‬

‫‪S1‬‬ ‫‪M1‬‬ ‫‪L1‬‬

‫‪۴١‬‬

‫‪X1‬‬

µ S 1 ( X 1 = 20) = 0.6 µ M 1 ( X 1 = 20) = 0.4 µ L1 ( X 1 = 20) = 0

µ S 2 ( X 2 = 8.5) = 0.15 µ L 2 ( X 2 = 8.5) = 0.85

‫ﻣﻮﺗﻮﺭ ﺍﺳﺘﻨﺘﺎﺝ ﻣﻴﻨﻴﻤﻢ ﻣﻤﺪﺍﻧﻲ‬ µ S 1× S 2 ( X 1* , X 2* ) = min[ µ S 1 ( X 1* ), µ S 2 ( X 2* )] = min [0.6,0.15] = 0.15 µ S 1× L 2 ( X 1* , X 2* ) = min[ µ S 1 ( X 1* ), µ L 2 ( X 2* )] = min [0.6,0.85] = 0.6 µ M 1× S 2 ( X 1* , X 2* ) = min[ µ M 1 ( X 1* ), µ S 2 ( X 2* )] = min [0.4,0.15] = 0.15 µ M 1× L 2 ( X 1* , X 2* ) = min[ µ M 1 ( X 1* ), µ L 2 ( X 2* )] = min [0.4,0.85] = 0.4 µ L1× S 2 ( X 1* , X 2* ) = min[ µ L1 ( X 1* ), µ S 2 ( X 2* )] = min [0,0.15] = 0 µ L1× L 2 ( X 1* , X 2* ) = min[ µ L1 ( X 1* ), µ L 2 ( X 2* )] = min [0,0.85] = 0

۴٢

۴٣

۴۴

Fuzzy Logic and Systems

Contents: 1. Introduction to fuzzy logic 1.1. The history of fuzzy logic 1.2. Configuration of fuzzy systems 1.3. some applications of fuzzy systems

2. Fuzzy sets 2.1. A comparison between classic and fuzzy sets 2.2. Introduction to basic concepts associated with fuzzy sets 3. Operation on fuzzy sets 3.1. Equal 3.2. Containment 3.3. Complement 3.4. Union 3.5. Intersection 3.6. Average 4. Fuzzy relations and the extension principle 4.1. Composition of fuzzy relations 5. Linguistic variable and fuzzy IF-THEN rules 5.1. Linguistic hedges 6. Fuzzy rule base & fuzzy inference engine 7. Fuzzifiers & Defuzzifiers 8. Using fuzzy toolbox in MATLAB 9. Some examples with fuzzy toolbox References: A course in fuzzy system and control

by:Li-Xin Wang

۴۵