X is - Tech7

‫‪ ‬ﻣﻨﻄﻖ‪ ‬ﻓﺎﺯﯼ‬
‫‪ ‬ﺍﻋﺪﺍﺩﻓﺎﺯﯼ‬
‫‪ ‬ﺗﻮﺍﺑﻊ‪ ‬ﻋﻀﻮﻳﺖ‬
‫ﺩﮐﺘﺮ‪ ‬ﻣﺤﻤﺪ‪ ‬ﻋﻠﯽ‪ ‬ﺍﻓﺸﺎﺭ‪ ‬ﮐﺎﻇﻤﯽ‬
‫‪ ۱ ¢ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﻓﺎﺯﯼ‬
‫‪ ۲ ¢ ‬ﺗﺎﺑﻊ‪ ‬ﻭ‪ ‬ﺩﺭﺟﻪ‪ ‬ﻋﻀﻮﻳﺖ‬
‫‪ ۳ ¢ ‬ﻋﻀﻮ‪ ‬ﭘﺸﺘﻴﺒﺎﻥ‬
‫‪ ۴ ¢ ‬ﺑﺮﺵ‪ ‬ﺁﻟﻔﺎ‬
‫‪ ۵ ¢ ‬ﮐﺎﻧﻮﻥ‬
‫‪ ۶ ¢ ‬ﺑﻠﻨﺪﯼ‬
‫‪ ۷ ¢ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﻣﺴﺎﻭﯼ‪ ‬ﻳﺎ‪ ‬ﺗﺮﺍﺯ‬
‫‪ ۸ ¢ ‬ﺯﻳﺮﻣﺠﻤﻮﻋﻪ‬
‫‪ ۹ ¢ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﺗﻬﯽ‪ ‬ﻓﺎﺯﯼ‬
‫‪ ۱۰ ¢ ‬ﺍﻋﻤﺎﻝ‪ ‬ﺍﺳﺎﺳﯽ‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﻫﺎ‬
‫‪ ۱۱ ¢ ‬ﺧﻮﺍﺹ‪ ‬ﺍﻋﻤﺎﻝ‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﺍﯼ‬
‫‪ ۱۲ ¢ ‬ﺗﻔﺎﻭﺕ‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﮐﻼﺳﻴﮏ‪ ‬ﻭ‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﻓﺎﺯﯼ‬
‫‪ ۱۳ ¢ ‬ﻣﺜﺎﻝ‪ ‬ﻫﺎ‬
‫‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﻓﺎﺯﯼ‬
‫‪ ¢ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﻓﺎﺯﯼ‪ ‬ﺑﺮﺍﺳﺎﺱ‪ ‬ﺗﺎﺑﻊ‪ ‬ﻋﻀﻮﻳﺖ‪ ‬ﺗﻌﺮﻳﻒ‪ ‬ﻣﯽ‪ ‬ﺷﻮﺩ‪ ‬ﮐﻪ‪ ‬ﺗﺼﻮﻳﺮ‬
‫‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﻓﺮﺍﮔﻴﺮ‪ ‬ﺩﺭ‪ ‬ﺑﺎﺯﻩ‪ ] ‬ﺻﻔﺮ‪ ‬ﻭ‪ ‬ﻳﮏ‪ [ ‬ﺍﺳﺖ‪. ‬‬
‫‪ ¢ ‬ﻫﺮ‪ ‬ﻳﮏ‪ ‬ﺍﺯ‪ ‬ﺍﻋﻀﺎ‪ ‬ﺩﺭﺟﻪ‪ ‬ﻋﻀﻮﻳﺖ‪ ‬ﺩﺍﺭﻧﺪ‪ . ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﻓﺎﺯﯼ‪ ‬ﺍﺯ‪ ‬ﺗﻌﻤﻴﻢ‪ ‬ﻭ‬
‫‪ ‬ﻋﻤﻮﻣﻴﺖ‪ ‬ﺩﺍﺩﻥ‪ ‬ﺗﺌﻮﺭﯼ‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﻫﺎﯼ‪ ‬ﮐﻼﺳﻴﮏ‪ ‬ﺍﻳﺠﺎﺩ‪ ‬ﺷﺪ‪ . ‬ﺩﺭ‪ ‬ﺗﺌﻮﺭﯼ‬
‫‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﻫﺎﯼ‪ ‬ﮐﻼﺳﻴﮏ‪ ٬‬ﻋﻀﻮﻳﺖ‪ ‬ﺍﻋﻀﺎ‪ ‬ﺩﺭ‪ ‬ﻳﮏ‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﺑﻪ‪ ‬ﺻﻮﺭﺕ‬
‫‪ ‬ﺟﻤﻼﺕ‪ ‬ﺑﺎﻳﻨﺮﯼ‪ ‬ﺑﺮ‪ ‬ﺍﺳﺎﺱ‪ ‬ﺷﺮﻁ‪ ‬ﺩﻭﺩﻭﺋﯽ‪ ‬ﺗﻌﻴﻴﻦ‪ ‬ﻣﯽ‪ ‬ﺷﻮﻧﺪ‪ ‬ﮐﻪ‪ ‬ﻳﮏ‪ ‬ﻋﻀﻮ‪ ‬ﻳﺎ‪ ‬ﺑﻪ‬
‫‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﺗﻌﻠﻖ‪ ‬ﺩﺍﺭﺩ‪ ‬ﻳﺎ‪ ‬ﻧﺪﺍﺭﺩ‪ . ‬ﺩﺭ‪ ‬ﺣﺎﻟﯽ‪ ‬ﮐﻪ‪ ‬ﺩﺭ‪ ‬ﺗﺌﻮﺭﯼ‪ ‬ﻓﺎﺯﯼ‪ ‬ﺩﺭﺟﺎﺕ‪ ‬ﻧﺴﺒﯽ‬
‫‪ ‬ﻋﻀﻮﻳﺖ‪ ‬ﺍﻋﻀﺎ‪ ‬ﺩﺭ‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﻣﺠﺎﺯ‪ ‬ﺍﺳﺖ‪.‬‬
‫‪ ‬ﺗﺎﺑﻊ‪ ‬ﻭ‪ ‬ﺩﺭﺟﻪ‪ ‬ﻋﻀﻮﻳﺖ‬
‫‪ ¢ ‬ﺗﺎﺑﻊ‪ ‬ﻋﻀﻮﻳﺖ‪ ‬ﺗﺎﺑﻌﯽ‪ ‬ﺍﺳﺖ‪ ‬ﺍﺯ‪ ‬ﺗﺼﻮﻳﺮﻣﺠﻤﻮﻋﻪ‪ ‬ﮐﻠﯽ‪ ‬ﺑﻪ‪ Ù ‬ﻧﺴﺒﺖ‪ ‬ﺑﻪ‪ ‬ﺑﺎﺯﻩ‪ ‬ﺑﺴﺘﻪ‬
‫‪ .[ 0٬1 ] ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﻓﺎﺯﯼ‪ A ‬ﺑﺎ‪ ‬ﺗﺎﺑﻊ‪ ‬ﻋﻀﻮﻳﺖ‪ μ A ‬ﺩﺭ‪ U ‬ﺗﻌﺮﻳﻒ‪ ‬ﺷﺪﻩ‪ ‬ﺍﺳﺖ‪. ‬‬
‫‪ ¢ ‬ﻋﺪﺩﯼ‪ ‬ﮐﻪ‪ ‬ﺗﺎﺑﻊ‪ ‬ﺑﻪ‪ ‬ﻫﺮ‪ ‬ﻋﻀﻮ‪ ‬ﺍﺭﺯﺵ‪ ‬ﺩﻫﯽ‪ ‬ﻣﯽ‪ ‬ﻧﻤﺎﻳﺪ‪ ‬ﺩﺭﺟﻪ‪ ‬ﻋﻀﻮﻳﺖ‪ ‬ﺁﻥ‪ ‬ﻋﻀﻮ‪ ‬ﺩﺭ‪ ‬ﺁﻥ‬
‫‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﺭﺍ‪ ‬ﻣﺸﺨﺺ‪ ‬ﻣﯽ‪ ‬ﺳﺎﺯﺩ‪ . ‬ﺍﮔﺮ‪ ‬ﺩﺭﺟﻪ‪ ‬ﻋﻀﻮﻳﺖ‪ ‬ﻳﮏ‪ ‬ﻋﻨﺼﺮ‪ ‬ﺍﺯ‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﺑﺮﺍﺑﺮ‬
‫‪ ‬ﺑﺎ‪ ‬ﺻﻔﺮ‪ ‬ﺑﺎﺷﺪ‪ ‬ﺁﻥ‪ ‬ﻋﻀﻮ‪ ‬ﮐﺎﻣﻼ‪ ‬ﺍﺯ‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﺧﺎﺭﺝ‪ ‬ﺍﺳﺖ‪ ‬ﻭﺍﮔﺮ‪ ‬ﺩﺭﺟﻪ‪ ‬ﻋﻀﻮﻳﺖ‪ ‬ﻳﮏ‬
‫‪ ‬ﻋﻀﻮ‪ ‬ﺑﺮﺍﺑﺮ‪ ‬ﺑﺎ‪ ‬ﻳﮏ‪ ‬ﺑﺎﺷﺪﺁﻥ‪ ‬ﻋﻀﻮ‪ ‬ﮐﺎﻣﻼ‪ ‬ﺩﺭ‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﻗﺮﺍﺭ‪ ‬ﺩﺍﺭﺩ‪ ‬ﻣﯽ‪ ‬ﺗﻮﺍﻥ‪ ‬ﻧﺘﻴﺠﻪ‬
‫‪ ‬ﮔﺮﻓﺖ‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﮐﻼﺳﻴﮏ‪ ‬ﻳﮏ‪ ‬ﺣﺎﻟﺖ‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﻓﺎﺯﯼ‪ ‬ﻳﻌﻨﯽ‪ ‬ﺯﻳﺮﻣﺠﻤﻮﻋﻪ‪ ‬ﻣﺠﻤﻮﻋﻪ‬
‫‪ ‬ﻓﺎﺯﯼ‪ ‬ﺍﺳﺖ‪ . ‬ﻭ‪ ‬ﺣﺎﻝ‪ ‬ﺍﮔﺮ‪ ‬ﺩﺭﺟﻪ‪ ‬ﻋﻀﻮﻳﺖ‪ ‬ﻳﮏ‪ ‬ﻋﻀﻮ‪ ‬ﻣﺎﺑﻴﻦ‪ ‬ﺻﻔﺮ‪ ‬ﻭﻳﮏ‪ ‬ﺑﺎﺷﺪ‪ ‬ﺍﻳﻦ‬
‫‪ ‬ﻋﺪﺩ‪ ‬ﺑﻴﺎﻧﮕﺮ‪ ‬ﺩﺭﺟﻪ‪ ‬ﻋﻀﻮﻳﺖ‪ ‬ﺗﺪﺭﻳﺠﯽ‪ ‬ﻣﯽ‪ ‬ﺑﺎﺷﺪ‪. ‬‬
‫‪ ¢ ‬ﺍﺯﻟﺤﺎﻅ‪ ‬ﻣﻔﻬﻮﻣﯽ‪ ‬ﺩﺭ‪ ‬ﺿﻤﻦ‪ ‬ﻣﯽ‪ ‬ﺗﻮﺍﻧﺪ‪ ‬ﻫﺮ‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﺑﺼﻮﺭﺕ‪ ‬ﺗﺪﺍﺧﻠﯽ‪ ‬ﺑﺎ‪ ‬ﺩﺭﺟﻪ‪ ‬ﺍﯼ‪ ‬ﺩﺭ‬
‫‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﺩﻳﮕﺮ‪ ‬ﻗﺮﺍﺭ‪ ‬ﮔﻴﺮﺩ‪ . ‬ﻣﺜﻼ‪ ‬ﺩﺭ‪ ‬ﻣﺘﻐﻴﺮ‪ ‬ﺯﺑﺎﻧﯽ‪ ‬ﺳﻦ‪ ‬ﺻﻔﺖ‪ ‬ﺟﻮﺍﻧﯽ‪ ‬ﺭﺍ‪ ‬ﻣﺪ‪ ‬ﻧﻈﺮ‬
‫‪ ‬ﺑﮕﻴﺮﻳﻢ‪ ‬ﺣﺎﻝ‪ ‬ﺑﺎ‪ ‬ﺗﻮﺟﻪ‪ ‬ﺑﻪ‪ ‬ﺍﻧﺘﺨﺎﺏ‪ ‬ﺗﺎﺑﻊ‪ ‬ﻋﻀﻮﻳﺖ‪ ‬ﻣﺎﻧﻨﺪ‪ ‬ﮔﺎﻭﺳﻴﺎﻥ‪ ‬ﺻﻔﺖ‪ ‬ﻣﻴﺎﻥ‪ ‬ﺳﺎﻟﯽ‪ ‬ﺑﺎ‬
‫‪ ‬ﺩﺭﺟﻪ‪ ‬ﻋﻀﻮﻳﺖ‪ ‬ﮐﻢ‪ ‬ﻣﯽ‪ ‬ﺗﻮﺍﻧﺪ‪ ‬ﺩﺭ‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﺻﻔﺖ‪ ‬ﺟﻮﺍﻧﯽ‪ ‬ﻗﺮﺍﺭ‪ ‬ﮔﻴﺮﺩ‪ ‬ﻭ‪ ‬ﺻﻔﺖ‪ ‬ﭘﻴﺮﯼ‬
‫‪ ‬ﻧﻴﺰ‪ ‬ﺑﺎ‪ ‬ﺩﺭﺟﻪ‪ ‬ﻋﻀﻮﻳﺖ‪ ‬ﮐﻤﺘﺮﯼ‪ ‬ﺩﺭﻣﺠﻤﻮﻋﻪ‪ ‬ﺻﻔﺖ‪ ‬ﺟﻮﺍﻧﯽ‪ ‬ﻇﺎﻫﺮ‪ ‬ﻣﯽ‪ ‬ﺷﻮﺩ‪.‬‬
MEMBERSHIP FUNCTIONS
¢A
‫ﻋﻀﻮﻳﺖ‬ ‫ﺗﺎﺑﻊ‬ membership function (MF) is a curve
that defines how each point in the input
space is mapped to a membership value
(or degree of membership) between 0
and 1. The input space is sometimes
referred to as the universe of discourse.
¢
One of the most commonly used examples of a fuzzy
set is the set of tall people. In this case, the universe
of discourse is all potential heights, say from 3 feet to
9 feet, and the word tall would correspond to a curve
that defines the degree to which any person is tall. If
the set of tall people is given the welldefined (crisp)
boundary of a classical set, you might say all people
taller than 6 feet are officially considered tall.
‫ﻧﻤﺎﺩﯼ‪ ‬ﺍﺯ‪ ‬ﻓﺎﺯﯼ‪ ‬ﻭ‪ ‬ﮐﻼﺳﻴﮏ‬
‫ﻣﻘﺎﺩﻳﺮ‪ ‬ﻓﺎﺯﯼ‬
‫ﻣﻘﺎﺩﻳﺮ‪ ‬ﻓﺎﺯﯼ‬
‫‪ ‬ﻋﻀﻮ‪ ‬ﭘﺸﺘﻴﺒﺎﻥ‬
‫‪ ‬ﺍﻋﻀﺎﯼ‪ ‬ﺍﺯﻣﺠﻤﻮﻋﻪ‪ ‬ﺍﺻﻠﯽ‪ ‬ﺍﻧﺪ‪ ‬ﺑﺮﺍﯼ‪ ‬ﺁﻧﻬﺎ‪ ‬ﺩﺭﺟﻪ‪ ‬ﻋﻀﻮﻳﺖ‪ ‬ﻏﻴﺮ‪ ‬ﺻﻔﺮ‪ ‬ﺑﺮﺍﺳﺎﺱ‬
‫‪ ‬ﺗﺎﺑﻊ‪ ‬ﻋﻀﻮﻳﺖ‪ ‬ﺗﻌﻴﻴﻦ‪ ‬ﻣﯽ‪ ‬ﮔﺮﺩﺩ‪ ‬ﺩﺭﻭﺍﻗﻊ‪ ‬ﺣﺎﻣﯽ‪ ‬ﻭﭘﺸﺘﻴﺒﺎﻥ‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﻓﺎﺯﯼ‪ ‬ﺍﻧﺪ‪.‬‬
‫‪ ‬ﺑﺮﺵ‪ ‬ﺁﻟﻔﺎ‪: ‬‬
‫‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﺍﯼ‪ ‬ﺍﺯ‪ ‬ﺗﻤﺎﻡ‪ ‬ﻋﻨﺎﺻﺮ‪ ‬ﻣﺮﺑﻮﻁ‪ ‬ﺑﻪ‪ ‬ﺩﺍﻣﻨﻪ‪ ‬ﺍﯼ‪ ‬ﺍﺯ‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﺍﺻﻠﯽ‪ ‬ﺑﺎ‪ ‬ﺩﺭﺟﻪٴ‬
‫ﻋﻀﻮﻳﺖ‪ ‬ﺑﻴﺸﺘﺮ‪ ‬ﻳﺎ‪ ‬ﻣﺴﺎﻭﯼ‪ ‬ﺁﻟﻔﺎ‬
‫‪CORE ‬‬
‫‪ ‬ﮐﺎﻧﻮﻥ‬
‫‪ ‬ﺍﻋﻀﺎﯼ‪ ‬ﮐﺎﻧﻮﻥ‪ ‬ﺍﻋﻀﺎﻳﯽ‪ ‬ﺍﺯ‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﺍﺻﻠﯽ‪ ‬ﺍﻧﺪ‪ ‬ﮐﻪ‪ ‬ﺑﺮﺍﯼ‪ ‬ﺁﻥ‪ ‬ﻫﺎ‪ ‬ﺩﺭﺟﻪ‬
‫‪ ‬ﻋﻀﻮﻳﺖ‪ ٬‬ﺑﺮﺍﺳﺎﺱ‪ ‬ﺗﺎﺑﻊ‪ ‬ﻋﻀﻮﻳﺖ‪ ‬ﺑﺮﺍﺑﺮ‪ » ‬ﻳﮏ‪ « ‬ﺍﺭﺯﺵ‪ ‬ﺩﻫﯽ‪ ‬ﻣﯽ‪ ‬ﺷﻮﺩ‪.‬‬
‫‪ ‬ﺍﺭﺗﻔﺎﻉ‬
‫‪HEIGHT‬‬
‫‪ ‬ﺩﺍﻣﻨﻪ‪ ‬ﻓﻮﻗﺎﻧﯽ‪ ‬ﺩﺭﺟﺎﺕ‪ ‬ﻋﻀﻮﻳﺖ‪ ‬ﺭﺍ‪ ‬ﮔﻮﻳﻨﺪ‪ ‬ﺩﺭﺣﺎﻟﺖ‪ ‬ﺍﺳﺘﺎﻧﺪﺍﺭﺩ‪ ‬ﺑﺮﺍﺑﺮ‪ " ‬ﻳﮏ‪" ‬‬
‫‪ ‬ﺍﺳﺖ‪.‬‬
‫‪¢ ‬‬
TRIMF
TRIANGULARSHAPED MEMBERSHIP FUNCTION
¢
The triangular curve is a function of a vector, x, and depends on
three scalar parameters a, b, and c, as given by
¢
Example
x=0:0.1:10;
y=trimf(x,[3 6 8]);
plot(x,y)
xlabel('trimf, P=[3 6 8]')
TRAPMF
TRAPEZOIDALSHAPED MEMBERSHIP FUNCTION
¢
Description
The trapezoidal curve is a function of a vector, x, and depends on four
scalar parameters a, b, c, and d, as given by
y = trapmf(x,[a b c d])
¢
Examples
x=0:0.1:10;
y=trapmf(x,[1 5 7 8]);
plot(x,y)
xlabel('trapmf, P=[1 5 7 8]')
GAUSSMF
GAUSSIAN CURVE MEMBERSHIP FUNCTION
¢
Description
The symmetric Gaussian function depends on two parameters
σ and c as given by
y = gaussmf(x,[sig c])
Examples
x=0:0.1:10;
¢
y=gaussmf(x,[2 5]);
plot(x,y)
xlabel('gaussmf, P=[2 5]')
GAUSS2MF
GAUSSIAN COMBINATION MEMBERSHIP FUNCTION
¢
Description
The Gaussian function depends on two parameters sig and c as given by
The function gauss2mf is a combination of two of these two
parameters. The first function, specified by sig1 and c1,
determines the shape of the leftmost curve. The second function
specified by sig2 and c2 determines the shape of the rightmost
curve. Whenever c1 < c2, the gauss2mf function reaches a
maximum value of 1. Otherwise, the maximum value is less than
one. The parameters are listed in the order:
[sig1, c1, sig2, c2]
GAUSS2MF
GAUSSIAN COMBINATION MEMBERSHIP FUNCTION
¢
Example
x = (0:0.1:10)';
y1 = gauss2mf(x, [2 4 1 8]);
y2 = gauss2mf(x, [2 5 1 7]);
y3 = gauss2mf(x, [2 6 1 6]);
y4 = gauss2mf(x, [2 7 1 5]);
y5 = gauss2mf(x, [2 8 1 4]);
plot(x, [y1 y2 y3 y4 y5]);
set(gcf, 'name', 'gauss2mf', 'numbertitle', 'off');
GBELLMF
GENERALIZED BELLSHAPED MEMBERSHIP FUNCTION
¢
Description
The generalized bell function depends on three parameters a, b, and c
as given by
x=0:0.1:10;
y=gbellmf(x,[2 4 6]);
plot(x,y) xlabel('gbellmf,
P=[2 4 6]')
where the parameter b is usually positive. The parameter c locates
the center of the curve. Enter the parameter vector params, the
second argument for gbellmf, as the vector whose entries are a,
b, and c, respectively
SIGMF
SIGMOID SHAPED MEMBERSHIP FUNCTION
¢
Syntax
¢
y = sigmf(x,[a c])
¢
Description
The sigmoidal function, sigmf(x,[a c]), as given in the following
equation by f(x,a,c) is a mapping on a vector x, and depends on
two parameters a and c.
Examples
x=0:0.1:10;
y=sigmf(x,[-2 4]);
plot(x,y)
xlabel('sigmf, P=[2 4]')
Depending on the sign of the parameter a, the sigmoidal membership
function is inherently open to the right or to the left, and thus is appropriate
for representing concepts such as "very large" or "very negative." PSIGMF
BUILTIN MEMBERSHIP FUNCTION COMPOSED OF PRODUCT OF
TWO SIGMOIDALLY SHAPED MEMBERSHIP FUNCTIONS
Syntax
y = psigmf(x,[a1 c1 a2 c2])
Description
The sigmoid curve plotted for the vector x depends on two parameters
a and c as given by
psigmf is simply the product of two such curves plotted for the values
of the vector x
f1(x; a1, c1) × f2(x; a2, c2)
The parameters are listed in the order [a1 c1 a2 c2].
¢
Examples
¢
x=0:0.1:10;
¢
y=psigmf(x,[2 3 5 8]);
¢
plot(x,y) xlabel('psigmf, P=[2 3 5 8]')
DSIGMF
BUILTIN MEMBERSHIP FUNCTION COMPOSED OF DIFFERENCE
BETWEEN TWO SIGMOIDAL MEMBERSHIP FUNCTIONS
Syntax
y = dsigmf(x,[a1 c1 a2 c2])
Description
The sigmoidal membership function used depends on the two parameters a
and c and is given by
¢
¢
The membership function dsigmf depends on four parameters, a1, c1, a2,
and c2, and is the difference between two of these sigmoidal functions.
f1(x; a1, c1) f2(x; a2, c2)
¢
The parameters are listed in the order: [a1 c1 a2 c2].
¢
Examples
¢
x=0:0.1:10;
¢
y=dsigmf(x,[5 2 5 7]);
¢
plot(x,y)
¢
xlabel('dsigmf, P=[5 2 5 7]')
PIMF
ΠSHAPED BUILTIN MEMBERSHIP FUNCTION
Syntax
y = pimf(x,[a b c d])
Description
The membership function is evaluated at the points determined by
the vector x. The parameters a and d locate the "feet" of the curve,
while b and c locate its "shoulders." The membership function is a
product of smf and zmf membership functions, and is given by:
PIMF
ΠSHAPED BUILTIN MEMBERSHIP FUNCTION
¢
Examples
¢
x=0:0.1:10;
¢
y=pimf(x,[1 4 5 10]);
¢
plot(x,y) xlabel('pimf, P=[1 4 5 10]')
SMF
SSHAPED BUILTIN MEMBERSHIP FUNCTION
Syntax
y = smf(x,[a b])
Description
This splinebased curve is a mapping on the vector x, and is named
because of its Sshape. The parameters a and b locate the
extremes of the sloped portion of the curve, as given by:
SMF
SSHAPED BUILTIN MEMBERSHIP FUNCTION
¢
Examples
¢
x=0:0.1:10;
¢
y=smf(x,[1 8]);
¢
plot(x,y)
¢
xlabel('smf, P=[1 8]')
ZMF
ZSHAPED BUILTIN MEMBERSHIP FUNCTION
Syntax
y = zmf(x,[a b])
Description
This splinebased function of x is so named because of its Z
shape. The parameters a and b locate the extremes of the
sloped portion of the curve as given by.
ZMF
ZSHAPED BUILTIN MEMBERSHIP FUNCTION
Examples
x=0:0.1:10;
y=zmf(x,[3 7]);
plot(x,y)
xlabel('zmf, P=[3 7]')
Logical Operations
‫‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﻣﺴﺎﻭﯼ‪ ‬ﻳﺎ‪ ‬ﺗﺮﺍﺯ‬
‫‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﺍﯼ‪ ‬ﮐﻪ‪ ‬ﺩﺭﺟﺎﺕ‪ ‬ﻋﻀﻮﻳﺖ‪ ‬ﺁﻥ‪ ‬ﺑﺎ‪ ‬ﺩﺭﺟﺎﺕ‪ ‬ﻋﻀﻮﻳﺖ‬
‫‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﻣﻮﺭﺩ‪ ‬ﻧﻈﺮ ‪ ǚ ‬ﺍﺳﺖ‪. ‬‬
‫‪ ‬ﺯﻳﺮﻣﺠﻤﻮﻋﻪ‪: ‬‬
‫‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﺍﯼ‪ ‬ﮐﻪ‪ ‬ﺗﻤﺎﻣﯽ‪ ‬ﺩﺭﺟﺎﺕ‪ ‬ﻋﻀﻮﻳﺖ‪ ‬ﺁﻥ‪ ‬ﺍﺯﺩﺭﺟﺎﺕ‪ ‬ﻋﻀﻮﻳﺖ‪ ‬ﻣﺠﻤﻮﻋﻪ‬
‫‪ ‬ﻣﻮﺭﺩﻧﻈﺮ‪ ‬ﮐﻤﺘﺮﺍﺳﺖ‪. ‬‬
‫‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﺗﻬﯽ‪ ‬ﻓﺎﺯﯼ‪: ‬‬
‫‪ ‬ﺍﺳﺖ‪ ‬ﮐﻪ‪ ‬ﺑﺮﺍﯼ‪ ‬ﺗﻤﺎﻣﯽ‪ ‬ﻋﻨﺎﺻﺮ‪ ‬ﺁﻥ‪ ٬‬ﺍﺭﺯﺵ‪ ‬ﺗﺎﺑﻊ‪Φ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﻓﺎﺯﯼ‬
‫ﻋﻀﻮﻳﺖ‪ ‬ﺻﻔﺮ‪ ‬ﺑﺎﺷﺪ‬
‫‪ ‬ﺍﻋﻤﺎﻝ‪ ‬ﺍﺳﺎﺳﯽ‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﻫﺎ‬
‫‪ ( 1 ‬ﺍﺟﺘﻤﺎﻉ‬
‫‪ ( 2 ‬ﺍﺷﺘﺮﺍﮎ‬
‫‪( 3 ‬ﻣﺘﻤﻢ‬
‫‪ ‬ﺍﺟﺘﻤﺎﻉ‪:‬‬
‫ﺍﺷﺘﺮﺍﮎ‬
‫ﻣﺘﻤﻢ‪ ‬ﻳﺎ‪ ‬ﻣﮑﻤﻞ‬
‫‪ ‬ﺗﻔﺎﻭﺕ‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﮐﻼﺳﻴﮏ‪ ‬ﻭ‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﻓﺎﺯﯼ‬
‫‪ ¢ ‬ﺩﻟﻴﻞ‪ ‬ﺍﺻﻠﯽ‪ ‬ﺗﻘﺴﻴﻢ‪ ‬ﺑﻨﺪﯼ‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﮐﻼﺳﻴﮏ‪ ‬ﻭ‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﻓﺎﺯﯼ‪ ‬ﺑﺎ‪ ‬ﻭﺟﻮﺩ‬
‫‪ ‬ﺗﺸﺎﺑﻬﺎﺕ‪ ‬ﺧﺎﺹ‪ ٬‬ﻋﺪﻡ‪ ‬ﺗﺒﻌﻴﺖ‪ ‬ﺑﻌﻀﯽ‪ ‬ﺍﺯ‪ ‬ﻗﻮﺍﻧﻴﻦ‪ ‬ﺍﺳﺖ‪: ‬‬
‫‪ ( 1 ‬ﺩﺭ‪ ‬ﺗﺌﻮﺭﯼ‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﻓﺎﺯﯼ‪ ‬ﺗﻮﺍﺑﻊ‪ ‬ﻋﻀﻮﻳﺖ‪ ‬ﺑﮑﺎﺭ‪ ‬ﻣﯽ‪ ‬ﺭﻭﺩ‪. ‬‬
‫‪. Җ Ắ Ǜһ Қ Ǜ Ұ ấ ǚ Қ ǚ ‬‬
‫‪( 2 ‬‬
‫‪ ( 3 ‬ﺍﺟﺘﻤﺎﻉ‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﺑﺎ‪ ‬ﻣﺘﻤﻤﺶ‪ ‬ﺑﺮﺍﺑﺮﺑﺎﻳﮏ‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﮐﻞ‪ ‬ﻧﻴﺴﺖ‪.‬‬
‫ﺧﻮﺍﺹ‪ ‬ﺍﻋﻤﺎﻝ‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﺍﯼ‬
‫ﺗﻮﺍﺑﻊ‪ ‬ﻋﻀﻮﻳﺖ‪ ‬ﺯﻧﮕﻮﻟﻪ‪ ‬ﺍﯼ‪ ‬ﺷﮑﻞ‬
‫‪ ‬ﻓﺎﺯﯼ‪ ‬ﺳﺎﺯﯼ‪ ‬ﻣﺘﻐﻴﺮﻫﺎ‬
‫‪FUZZYFICATION‬‬
‫‪ ¢ ‬ﺷﻨﺎﺳﺎﻳﯽ‪ ‬ﺻﻔﺖ‪ ‬ﻣﻮﺭﺩ‪ ‬ﻧﻈﺮ‪ ‬ﺑﺮﺍﯼ‪ ‬ﺍﻓﺮﺍﺯ‪ ‬ﻣﺠﻤﻮﻋﻪ‬
‫‪ ¢ ‬ﻣﺸﺨﺺ‪ ‬ﮐﺮﺩﻥ ‪ MIN ‬ﻭ ‪ MAX ‬ﺻﻔﺖ‬
‫‪ ¢ ‬ﺗﻘﺴﻴﻢ‪ ‬ﺑﻨﺪﯼ‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﻣﺮﺟﻊ‪ ‬ﺑﻪ‪ ‬ﺯﻳﺮﻣﺠﻤﻮﻋﻪ‪ ‬ﻫﺎﯼ‪ ‬ﻣﻨﺎﺳﺐ‪ ) ‬ﺧﻄﯽ‪ – ‬ﻏﻴﺮ‪ ‬ﺧﻄﯽ‪( ‬‬
‫‪ ¢ ‬ﺍﻧﺘﺨﺎﺏ‪ ‬ﻳﮏ‪ ‬ﻧﺎﻡ‪ ‬ﻣﻨﺎﺳﺐ‪ ‬ﺑﺮﺍﯼ‪ ‬ﻫﺮ‪ ‬ﻣﺠﻤﻮﻋﻪ‬
‫‪ ¢ ‬ﺗﺨﺼﻴﺺ‪ ‬ﺗﺎﺑﻊ‪ ‬ﻋﻀﻮﻳﺖ‪ ‬ﻣﻨﺎﺳﺐ‬
‫‪ ‬ﻣﺜﺎﻝ‪ ‬ﻫﺎ‬
‫‪ ‬ﺧﻮﺩ‪ ‬ﻟﻄﻔﯽ‪ ‬ﺯﺍﺩﻩ‪ ‬ﻣﺜﺎﻝ‪ ‬ﺧﻮﺑﯽ‪ ‬ﺍﺯ‪ ‬ﺗﻌﺮﻳﻒ‪ ‬ﺗﺎﺑﻊ‪ ‬ﻋﻀﻮﻳﺖ‪ ‬ﺩﺭ‪ ‬ﻣﺠﻤﻮﻋﻪ‪ ‬ﻓﺎﺯﯼ‬
‫‪ ‬ﺍﺳﺖ‪ . ‬ﺗﻌﻴﻴﻦ‪ ‬ﻗﻮﻣﻴﺖ‪ ‬ﻟﻄﻔﯽ‪ ‬ﺯﺍﺩﻩ‪ ‬ﺗﺎ‪ ‬ﺣﺪﯼ‪ ‬ﺳﺨﺖ‪ ‬ﺍﺳﺖ‪ . ‬ﭘﺪﺭ‪ ‬ﺍﻭ‪ ‬ﻳﮏ‪ ‬ﺗﺮﮎ‬
‫‪ ‬ﺍﻳﺮﺍﻧﯽ‪ ) ‬ﺁﺫﺭﺑﺎﻳﺠﺎﻧﯽ‪ ( ‬ﻭ‪ ‬ﻣﺎﺩﺭﺵ‪ ‬ﺭﻭﺳﯽ‪ ‬ﻳﻬﻮﺩﯼ‪ ‬ﺑﻮﺩ‪ . ‬ﭘﺪﺭ‪ ‬ﺍﻭ‪ ‬ﻳﮏ‬
‫‪ ‬ﺭﻭﺯﻧﺎﻣﻪ‪ ‬ﻧﮕﺎﺭ‪ ‬ﻣﺸﻐﻮﻝ‪ ‬ﺑﻪ‪ ‬ﮐﺎﺭ‪ ‬ﺩﺭ‪ ‬ﺑﺎﮐﻮ‪ ٬‬ﺟﻤﻬﻮﺭﯼ‪ ‬ﺁﺫﺭﺑﺎﻳﺠﺎﻥ‪ ‬ﺩﺭ‪ ‬ﺍﺗﺤﺎﺩ‬
‫‪ ‬ﺟﻤﺎﻫﻴﺮ‪ ‬ﺷﻮﺭﻭﯼ‪ ‬ﺳﺎﺑﻖ‪ ‬ﺑﻮﺩ‪ . ‬ﺍﻭ‪ ‬ﺑﻪ‪ ‬ﻋﻨﻮﺍﻥ‪ ‬ﻳﮏ‪ ‬ﺧﺒﺮﻧﮕﺎﺭ‪ ‬ﺑﺮﺍﯼ‪ ‬ﺭﻭﺯﻧﺎﻣﻪ‪ ‬ﻫﺎﯼ‬
‫‪ ‬ﺍﻳﺮﺍﻥ‪ ‬ﺧﺪﻣﺖ‪ ‬ﮐﺮﺩﻩ‪ ‬ﺍﺳﺖ‪ ‬ﺩﺭ‪ ‬ﺣﺎﻟﯽ‪ ‬ﮐﻪ‪ ‬ﺧﺮﻳﺪ‪ ‬ﻭ‪ ‬ﻓﺮﻭﺵ‪ ‬ﺗﺠﺎﺭﺕ‪ ‬ﺻﺎﺩﺭﺍﺕ‪ ‬ﻭ‬
‫‪ ‬ﻭﺍﺭﺩﺍﺕ‪ ‬ﻧﻴﺰﻣﯽ‪ ‬ﮐﺮﺩ‪ . ‬ﻣﺎﺩﺭ‪ ‬ﺍﻭ‪ ‬ﭘﺰﺷﮏ‪ ‬ﻣﺘﺨﺼﺺ‪ ‬ﺍﻃﻔﺎﻝ‪ ‬ﺑﻮﺩ‪ . ‬ﻟﻄﻔﯽ‪ ‬ﺯﺍﺩﻩ‪ ‬ﺩﺭ‬
‫‪ ‬ﺑﺎﮐﻮ‪ ‬ﺩﺭ‪ ‬ﺳﺎﻝ‪ ۱۹۲۱ ‬ﻣﺘﻮﻟﺪ‪ ‬ﺷﺪ‪ ‬ﻭ‪ ‬ﺩﺭ‪ ‬ﺁﻧﺠﺎ‪ ‬ﺯﻧﺪﮔﯽ‪ ‬ﻣﯽ‪ ‬ﮐﺮﺩﻧﺪ‪ ‬ﺗﺎ‪ ‬ﺧﺎﻧﻮﺍﺩﻩ‪ ‬ﺩﺭ‬
‫‪ ‬ﺳﺎﻝ‪ ۱۹۳۱ ‬ﺑﻪ‪ ‬ﺗﻬﺮﺍﻥ‪ ‬ﻣﻨﺘﻘﻞ‪ ‬ﺷﺪ‪ ‬ﺩﺑﻴﺮﺳﺘﺎﻥ‪ ‬ﻭﺩﺍﻧﺸﮕﺎﻩ‪ ‬ﺩﺭ‪ ‬ﺍﻳﺮﺍﻥ‪ ‬ﺗﻤﺎﻡ‪ ‬ﮐﺮﺩ‬
‫‪ ‬ﻭﻟﯽ‪ ‬ﻓﻮﻕ‪ ‬ﻟﻴﺴﺎﻧﺲ‪ ‬ﻭ‪ ‬ﺩﮐﺘﺮﯼ‪ ‬ﺭﺍ‪ ‬ﺩﺭ‪ ‬ﺍﻳﺎﻻﺕ‪ ‬ﻣﺘﺤﺪﻩ‪ ‬ﺧﻮﺍﻧﺪ‪.‬‬
‫‪RELATION ‬‬
‫‪ ‬ﺭﺍﺑﻄﻪ‬
‫‪FUZZY CRISP ‬‬
‫ﺩﮐﺘﺮ‪ ‬ﺍﻓﺸﺎﺭ‪ ‬ﮐﺎﻇﻤﯽ‬
‫‪ ‬ﺍﻧﻮﺍﻉ‪ ‬ﺭﺍﺑﻄﻪ‬
‫‪ ( 1 ‬ﮐﻼﺳﻴﮏ‬
‫‪ ( 2 ‬ﻓﺎﺯﯼ‬
CLASSICAL RELATIONS
Cartesian Product
‫ﮐﻼﺳﻴﮏ‬ ‫ﺭﻭﺍﺑﻂ‬ CLASSICAL RELATIONS
‫ﮐﻼﺳﻴﮏ‬ ‫ﺭﻭﺍﺑﻂ‬ ‫ﮐﻼﺳﻴﮏ‬ ‫ﺭﺍﺑﻄﻪ‬ X={x1,x2,x3,x4}
Y={y1,y2,y3,y4,y5}
A={(x1,0),(x2,1),(x3,1),(x4,0)}
B={(y1,1),(y2,0),(y3,1),(y4,1),(y5,0)} ‫ﻣﺎﺗﺮﻳﺲ‬ ‫ﺭﺍﺑﻄﻪ‬ Y1
Y2
Y3
Y4
Y5
X1
0
0
0
0
0
X2
1
0
1
1
0
X3
1
0
1
1
0
X4
0
0
0
0
0
MIN{0,1}=0
MIN{1,1}=1
NULLRELATION
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 ‫ﮐﺎﻣﻞ‬ ‫ﺭﺍﺑﻄﻪ‬ COMPLEMENT RELATION 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
OPERATIONS ON CRISP RELATION
R1 ‫ﻣﺘﻤﻢ‬ R1
Y1
Y2
Y3
Y4
Y5
X1
0
1
1
0
1
X2
0
0
0
1
1
X3
1
1
0
0
0
R2
Y1
Y2
Y3
Y4
Y5
X1
1
1
0
0
1
X2
1
0
0
1
0
X3
1
0
0
0
0 ‫ﻣﺘﻤﻢ‬ R1
Y1
Y2
Y3
Y4
Y5
X1
1
0
0
1
0
X2
1
1
1
0
0
X3
0
0
1
1
1
R 1 U R 2
x 1 y1
1
y2
1
y3
1
y4
0
y 5 1 x 2 1
0
0
1
1 x 3 1
1
0
0
0 R1 I R 2
y1
y2
y3
y4
y 5 x 1 x 2 0
0
1
0
0
0
0
1
1 0 x 3 1
0
0
0
0 COMPOSITION
‫ﺗﺮﮐﻴﺐ‬ XRY
&
YSZ
‫ﺭﻭﺍﺑﻂ‬ ‫ﺗﺮﮐﻴﺐ‬ R
Y1
Y2
Y3
Y4
X1
1
0
1
0
X2
1
0
1
1
X3
1
0
1
1
S
Z1
Z2
Y1
0
0
Y2
1
0
Y3
0
1
Y4
1
0
T=ROS
MAXMIN
Z1
Z2
X1
0
1
X2
1
1
X3
1
0
MAXPRODUCT
Z1
Z2
X1
0
1
X2
1
1
X3
1
1
FUZZY RELATIONS
ì 0.4 0.7 0.1 ü
°
A = í
,
, ý
î x1 x 2 x 3 þ
ì 0.5 0.8 ü
° B = í
, ý
î y1 y 2 þ
‫ﻓﺎﺯﯼ‬ ‫ﺭﻭﺍﺑﻂ‬ ‫ﻓﺎﺯﯼ‬ ‫ﺭﻭﺍﺑﻂ‬ X={X1,X2,X3,X4}
Y={Y1,Y2,Y3,Y4,Y5}
A={(X1,0.2),(X2,0.8),(X3,1),(X4,0.4)}
B={(Y1,0),(Y2,0.3),(Y3,0.6),(Y4,0.8),(Y5,0.5)}
Y1
Y2
Y3
Y4
Y5
X1
0
0.2
0.2
0.2
0.2
X2
0
0.3
0.6
0.8
0.5
X3
0
0.3
0.6
0.8
0.5
Min(0.2,0.6)=0.
2
Min(0.2,0.8)=0.2
X4
0
0.3
0.4
0.4
0.4
Min(0.2,0.5)=0.2
Min(0.2,0)=
0
Min(0.2,0.3)=0.2
‫‪ ‬ﻋﻤﻠﻴﺎﺕ‪ ‬ﺑﺮ‪ ‬ﺭﻭﯼ‪ ‬ﺭﺍﺑﻄﻪ‪ ‬ﻫﺎﯼ‪ ‬ﻓﺎﺯﯼ‬
‫‪ ‬ﺍﺟﺘﻤﺎﻉ‪: ‬‬
‫‪ ‬ﺍﺷﺘﺮﺍﮎ‪: ‬‬
‫‪ ‬ﻣﺘﻤﻢ‪:‬‬
‫‪OPERATIONS ON FUZZY RELATIONS ‬‬
COMPOSITION OF FUZZY RELATION
‫‪ ‬ﺗﺮﮐﻴﺐ‪ ‬ﻓﺎﺯﯼ‬
‫‪ ‬ﻓﺎﺯﯼ‪R‬‬
‫‪Y4‬‬
‫‪Y3‬‬
‫‪Y2‬‬
‫‪Y1‬‬
‫‪0‬‬
‫‪0.8‬‬
‫‪0‬‬
‫‪0.4‬‬
‫‪X1‬‬
‫‪0.8‬‬
‫‪0.4‬‬
‫‪0‬‬
‫‪0.5‬‬
‫‪X2‬‬
‫‪0.5‬‬
‫‪0‬‬
‫‪0.2‬‬
‫‪0.3‬‬
‫‪X3‬‬
‫‪Z2‬‬
‫‪Z1‬‬
‫‪ ‬ﻓﺎﺯﯼ ‪S‬‬
‫‪0‬‬
‫‪0‬‬
‫‪Y1‬‬
‫‪0‬‬
‫‪0.3‬‬
‫‪Y2‬‬
‫‪0.5‬‬
‫‪0‬‬
‫‪Y3‬‬
‫‪0‬‬
‫‪0.7‬‬
‫‪Y4‬‬
T=ROS=
T = ROS=
MAXMIN
Z1
Z2
X1
0
0.5
X2
0.7
0.4
X3
0.5
0
MAX
PRODUCT
Z1
Z2
X1
0
0.4
X2
0.56
0.2
X3
0.35
0
‫ﺩﮐﺘﺮ‪ ‬ﻣﺤﻤﺪ‪ ‬ﻋﻠﯽ‪ ‬ﺍﻓﺸﺎﺭ‪ ‬ﮐﺎﻇﻤﯽ‬
‫‪ ‬ﺍﺳﺘﻨﺘﺎﺝ‪ ‬ﻓﺎﺯﯼ‬
FIS
FUZZY INFERENCE SYSTEM
Knowledge Base
Data base
Non
fuzzy
FUZZIFY
Rule base
Inference
Engine
FIS
DEFUZZY
Non
fuzzy
– a rule base containing a number of fuzzy IF–
THEN rules;
– a database which defines the membership
functions of the fuzzy sets used in the fuzzy rules;
– a decisionmaking unit which performs the
inference operations on the rules;
– a fuzzification interface which transforms the
crisp inputs into degrees of match with linguistic
values;
– a defuzzification interface which transforms the
fuzzy results of the inference into a crisp output.
The steps of fuzzy reasoning (inference operations
upon fuzzy IF–THEN rules) performed by FISs are:
1. Compare the input variables with the membership
functions on the antecedent part to obtain the
membership values of each linguistic label. (this step is
often called fuzzification.)
2. Combine (through a specific tnorm operator, usually
multiplication or min) the membership values on the
premise part to get firing strength (weight) of each rule.
3. Generate the qualified consequents (either fuzzy or
crisp) or each rule depending on the firing strength.
4. Aggregate the qualified consequents to produce a crisp
output. (This stepis called defuzzification.)
FORMATION OF RULES
(1) Assignment statements
¢ (2) Conditional statements
¢ (3) Unconditional statements
¢
ASSIGNMENT STATEMENTS
These statements are those in which the variable is
assignment with the value.
The variable and the value assigned are combined by
the assignment operator “=.” The assignment
statements are necessary in forming fuzzy rules. The
value to be assigned may be a linguistic term.
¢ y = low,
¢ Sky color = blue,
¢ Climate = hot
¢a=5
¢p=q+r
¢ Temperature = high
CONDITIONAL STATEMENTS
In this statements, some specific conditions are
mentioned, if the conditions are satisfied then it enters
the following statements, called as restrictions.
If x = y Then both are equal,
¢ If Mark > 50 Then pass,
¢ If Speed > 1, 500 Then stop.
¢
These statements can be said as fuzzy conditional
statements, such as If condition C Then restriction F
UNCONDITIONAL STATEMENTS
There is no specific condition that has to be satisfied in this form of
statements. Some of the unconditional statements are:
¢
¢
¢
Go to F/o
Push the value
Stop
The control may be transferred without any appropriate conditions. The
unconditional restrictions in the fuzzy form can be:
¢
¢
¢
¢
¢
R1 : Output is B1
AND
R2 : Output is B2
AND
. . . , etc.
DECOMPOSITION OF RULES
IF x = y THEN both are equal
ELSE
¢ IF x = y
THEN
¢ IF x > y THEN X is highest
ELSE
¢ IF y > x THEN Y is highest
ELSE
¢ IF x and y are equal to zero THEN no output is
obtained
¢
MULTIPLE CONJUNCTION ANTECEDENTS
¢ There
are various methods for decomposition
of rules. They are : Since it involves linguistic
“AND” connective
¢ IF x is P1 AND P2 · · · AND Pn THEN y is Qr,
where
¢ Pr = P1 AND P2 · · · OR Pn .
The membership for this can be
¢ μPr(x) = min [μP1(x) , μP2(x) , . . . , μPn(x)]
Hence the rule can be
¢ IF x is Pr THEN Qr
MULTIPLE DISJUNCTIVE ANTECEDENTS
This uses fuzzy union operations. It involves linguistic
“OR” connections
¢ IF x is P1 OR P2 · · · OR Pn THEN y is Qr,
where
¢ Pr = P1 OR P2· · · OR Pn
The membership for this can be
¢ μPr (x) = max[μP1(x) , μP2(x) , . . . , μPn(x)].
Hence the rule can be
¢ IF x is Pr THEN y is Qr
¢
CONDITIONAL STATEMENTS WITH ELSE
(a) IF P1 THEN (Q1 ELSE Q2).
Considering this as one compound statement, splitting
this into two canonical form rules, we get
IF P1 THEN Q1 OR IF NOT P1 THEN Q2.
¢
(b) IF P1 THEN (Q1 ELSE P2 THEN (Q2)).
The decomposition for this can be of the form
¢
IF P1 THEN Q1 OR
IF NOT P1 AND P2 THEN Q2.
NESTED IF–THEN RULES
IF P1 THEN (IF P2 THEN (Q2).
This can be decomposed into
¢ IF P1 AND P2 THEN Q1.
¢
AGGREGATION OF FUZZY RULES
IF–THEN RULES
“IF X is A THEN Y is B,”
where A and B are fuzzy sets
If A1 then B1
¢ If A2 then B2
¢ If A3 then B3
¢.
¢.
¢ If An then Bn
¢
ZADHE MAX – MIN
¢
: IF X is A THEN Y is B R = ( A% * B% ) U ( A% * Y ) m R ( x , y ) = M AX [m in{ m A ( x ), m B ( y )}, m in{1 - m A ( x ),1}] m R ( x , y ) = MAX [min{ m A ( x ), m B ( y )},{1 - m A ( x )}] X_: UNIVERSE SET OF PRICE
Y_: UNIVERSE SET OF DEMAND X = {100, 200, 300, 400) Y = {1, 2 , 3 , 4 , 5 , 6} 1
1
1
1 X = {
,
,
,
) 100 200 300 400 1 1 1 1 1 1 Y = { , , , , , } 1 2 3 4 5 6 Very cheap=A1
¢ cheap=A2
¢ Medium price=A3
¢ expensive =A4
¢ Very expensive=A5
¢
Very low demand=B1
¢ Low demand=B2
¢ Medium demand=B3
¢ High demand=B4
¢ Very high demand=B5
¢
X = {100, 200, 300, 400) 1
1
1
1 X = {
,
,
,
) 100 200 300 400 Y = {1, 2 , 3 , 4 , 5 , 6} 1 1 1 1 1 1 Y = { , , , , , } 1 2 3 4 5 6 IF X is Medium price of CPU THEN
Y is Medium demand of CPU In market
R3: IF A3 then B3 R 3 = ( A% 3 * B% 3 ) U ( A% 3 * Y ) m R 3 ( x, y ) = MAX [min{m A ( x), m B ( y )},{1 - m A 3 ( x)}] A* B = min{mA ( x), mB ( y)} A3=MEDIUM PRICE OF CPU ={(1,0),(2,0.6),(3,1),(4,0.2)}
B3=MEDIUM DEMAND IN MARKET={(2,0.4),(3,1),(4,0.8),(5,0.3)}
Or
A3 ={(1 ,0),(2,0.6),(3,1),(4,0.2)}
B3 ={(1,0),(2,0.4),(3,1),(4,0.8),(5,0.3),(6,0)}
A3*B3
1
2
3
4
5
6
100
0
0
0
0
0
0
200
0
0.4
0.6
0.6
0.3
0
300
0
0.4
1
0.8
0.3
0
400
0
0.2
0.2
0.2
0.2
0 A*Y=min{1- mA (x),1} Y={(1,1),(2,1),(3,1),(4,1),(5,1),(6,1)}
A3={(1,0),(2,0.6),(3,1),(4,0.2)}
A 3 ={(1,1),(2,0.4),(3,0),(4,0.8)} A 3 *Y 1
2
3
4
5
6
100
1
1
1
1
1
1
200
0.4
0.4
0.4
0.4
0.4
0.4
300
0
0
0
0
0
0
400
0.8
0.8
0.8
0.8
0.8
0.8 R = ( A% * B% ) U ( A% * Y ) m R 3 ( x, y ) = MAX [min{m A3 ( x), m B 3 ( y )},{1 - m A 3 ( x)}] A*B
1
2
3
4
5
6
A 3 *Y 1
2
3
4
5
6
1
1
1
1
1
1
1
2
0.4
0.4
0.4
0.4
0.4
0.4
100
0
0
0
0
0
0
200
0
0.4
0.6
0.6
0.3
0
300
0
0.4
1
0.8
0.3
0
3
0
0
0
0
0
0
400
0
0.2
0.2
0.2
0.2
0
4
0.8
0.8
0.8
0.8
0.8
0.8 Rule
1
2
3
4
5
6
1
1
1
1
1
1
1
2
0.4
0.4
0.6
0.6
0.4
0.4
3
0
0.4
1
0.8
0.3
0
4
0.8
0.8
0.8
0.8
0.8
0.8
INFERENCE
IF X IS A THEN Y IS ? ¢
IF A THEN B :RULEà expert
¢ IF A` THEN ?: Artificial Intelligence (Inference Engine)
? = MAX [min{ m A ' ( x ), m RULE ( x , y )} IF X is cheapprice of CPU THEN Y is ?
R: IF A2 then ?
¢ A2={(1,0.5),(2,1),(3,0.3),(4,0)}
A`
Rule 1
2
3
4
5
6
0.5
1
1
1
1
1
1
1
1
2
0.4
0.4
0.6
0.6
0.4
0.4
0.3
3
0
0.4
1
0.8
0.3
0
0
4
0.8
0.8
0.8
0.8
0.8
0.8
MAX{MIN[(0.5,1),(1,0.4),(0.3,0),(0,0.8)}
¢ MAX{0.5,0.4,0,0}=0.5
¢ B?={(1,0.5),(2,0.5),(3,0.6),(4,0.6),(5,0.5),(6,0.5)}
¢
MAMDANI
TAKAGI–SUGENO FUZZY METHOD (TKS METHOD)
¢
A typical fuzzy rule in a Sugeno fuzzy model has the format
IF x is A and y is B THEN z = f(x, y),
where A&B are fuzzy sets in the antecedent; Z = f(x, y) is a crisp function in
the consequent
The main difference between Mamdani and Sugeno is that the sugeno
output membership functions are either linear or constant.
IF Input 1 = x AND Input 2 = y, THEN Output is z = ax + by + c.
¢
The output level zi of each rule is weighted by the firing strength wi of
the rule. for an AND rule with Input x and Input y, thefiring strength is:
wi = AndMethod(F1(x), F2(y))
¢
where F1,2(·) are the membership functions for Inputs. The final output
of the system is the weighted average of all rule outputs
TAKAGI–SUGENO FUZZY METHOD (TKS METHOD)
TAKAGI–SUGENO FUZZY METHOD (TKS METHOD)
DEFUZZY
¢
The output of an entire fuzzy process can be union of two or
more fuzzy membership functions. To explain this in detail,
consider a fuzzy output, which is formed by two parts, one part
being triangular shape (Fig. 5.1a) and other part being
trapezoidal (Fig. 5.1b). The union of these two forms (Fig. 5.1c)
the outer envelop of the two shapes.
DEFUZZY
there are various defuzzification methods employed
to convert the fuzzy quantities into crisp quantities.
¢ There are seven methods used for defuzzifying the
fuzzy output functions.
¢ (1) Maxmembership principle,
¢ (2) Centroid method,
¢ (3) Weighted average method,
¢ (4) Mean–max membership,
¢ (5) Centre of sums,
¢ (6) Centre of largest area, and
¢ (7) First of maxima or last of maxima
¢
MAXMEMBERSHIPPRINCIPLE
This method is given by the expression,
μ (z∗) ≥ μ (z)
for all z ∈ z.
¢ This method is also referred as height method.
¢
DEFUZZY
CENTER OF GRAVITY : m ( z ). zdz Z = ò
m ( z ) dz
* ‫ﺛﻘﻞ‬ ‫ﻣﺮﮐﺰ‬ INFERENCE
If A1 then B1 R1=A1àB1
¢ If A2 then B2
R2=A2àB2
¢ If A3 then B3 A1 then R3=A3àB3
¢.
¢.
¢ If An then Bn
R4=AnàBn
¢
RULE BASE
·
If
A1 & A2 & A3 & A3 &…. At
then Bs
As
¢
¢
¢
¢
¢
If A1 then
If A2 then
If A3 then
.
.
.
.
¢
If At then Bs
¢
If
As:
A1 & A2 & A3 & A3 &…. At= As
è If As then Bs
‫‪ ‬ﻧﮑﺘﻪ‪: ‬‬
‫‪ ‬ﺍﮔﺮ‪ ‬ﺑﻴﻦ‪ ‬ﻗﻮﺍﻋﺪ‪ ) ‬ﺭﺍﺑﻄﻪ‪ ‬ﻫﺎ‪ ( ‬ﻋﺒﺎﺭﺕ‪ “ ‬ﻳﺎ‪ ” ‬ﺭﺍ‪ ‬ﺑﻪ‪ ‬ﮐﺎﺭ‪ ‬ﺑﺒﺮﻳﻢ‪ ‬ﺩﺭ‪ ‬ﻓﺮﻣﻮﻝ‪ ‬ﻓﻮﻕ‪ ‬ﺑﻪ‬
‫‪ ‬ﺟﺎﯼ‪ ‬ﻣﺎﮐﺰﻳﻤﻢ‪ ‬ﺍﺯ‪ ‬ﻣﻴﻨﻴﻤﻢ‪ ‬ﺍﺳﺘﻔﺎﺩﻩ‪ ‬ﻣﯽ‪ ‬ﮐﻨﻴﻢ‪.‬‬
‫ﺩﻳﮕﺮ‬ ‫ﺷﺮﻃﯽ‬ ‫ﺟﻤﻠﻪ‬ ‫ﭼﻨﺪ‬ 1)
2)
If A1 then (B1 else B2)
If not A1 then B . ‫ﻫﺴﺘﻨﺪ‬ ‫ﻣﻌﺎﺩﻝ‬ 2 ‫ﻭ‬ 1 ‫ﺟﻤﻼﺕ‬ 1)
2)
If A1 then B1 unless A2
If A1 then B1
If A2 then not B1 .‫ﻫﺴﺘﻨﺪ‬ ‫ﻣﻌﺎﺩﻝ‬ 2 ‫ﻭ‬ 1 ‫ﺟﻤﻼﺕ‬

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