$$ \chi : X \rightarrow \big\{0,1\big\} $$
$$ \mu_{a} : X \rightarrow [0,1] $$
Funkcija pripadanja \( \mu_{a}\) određuje stepen pripadanja elemanta \(x\) fazi skupa \(A\). $$ ^{ \alpha }A= \big\{x \in X | \mu_{A}\big(x\big) \geq \alpha \big\} $$
\( A\) - normalan tj. $$ h\big(a\big)= \sup_{x\in X} \mu _{a} \big(x\big)=1 $$
\(^{ \alpha }A\) je zatvoren interval
Nosač \(^{ \alpha }A= \big\{x\in X | \mu _{a} \big(x\big) > 0\big\} \) je ograničen.
$$ ^{ \alpha }A= [A^{*}_{ \alpha},A^{**}_{ \alpha}]=[ \alpha \cdot m + \big(1 - \alpha \big) \cdot l, \alpha \cdot m + \big(1 - \alpha \big) \cdot r ] $$
Operacije sa fazi brojevima se defenišu preko \( { \alpha }\) -preseka tj. zatvorenih intervala, npr: $$ [a,b]+[c,d]=[a+c,b+d] $$ $$ [a,b] \cdot [c,d]=[a \cdot c,b \cdot d] $$ $$ k \cdot [a,b] =[k \cdot a,k \cdot b] $$ $$ [a,b]^{n}=[a^{n},b^{n}] $$
$$ E= \big(\sum_{i=1}^{k} w_{i} \cdot e^{r}_{i} \big)^{ \frac{1}{r} }\; , \;0 \leq w_{i} \leq 1 \;, \; \sum_{i=1}^{k} w_{i} = 1 $$ $$ e \in [0,1] \; , \; E \in [0,1] \; , \; k \geq 2 $$
$$ E= \big(\sum_{i=1}^{k} w_{i} \cdot e^{r}_{i} \big)^{ \frac{1}{r} }\; , \;0 \leq w_{i} \leq 1 \;, \; \sum_{i=1}^{k} w_{i} = 1 $$ $$ e \in [0,1] \; , \; E \in [0,1] \; , \; k \geq 2 $$ $$ \Downarrow $$ $$ \widehat{E} = \big(\widehat{w}_{1} \cdot \big(\widehat{e}_{1}\big) ^{r}+ ... + \widehat{w}_{n} \cdot \big(\widehat{e}_{n}\big) ^{r} \big) ^{ \frac{1}{r} } $$ $$ \Downarrow $$ $$ ^{\alpha}\widehat{E} =[ E^{*}, E^{**} ]= [ \big(\sum_{i=1}^{k} w^{*}_{i} \cdot e^{r}_{i} \big)^{ \frac{1}{r} }\; ,\big(\sum_{i=1}^{k} w^{**}_{i} \cdot e^{r}_{i} \big)^{ \frac{1}{r} }] $$ $$ \Downarrow $$ $$ UCL = [ 1.1 \cdot \frac{1}{n} \sum_{i=1}^{n} E^{*}_{i},1.1 \cdot \frac{1}{n} \sum_{i=1}^{n} E^{**}_{i}] \; \; LCL = [ 0.9 \cdot \frac{1}{n} \sum_{i=1}^{n} E^{*}_{i},0.9 \cdot \frac{1}{n} \sum_{i=1}^{n} E^{**}_{i}] \Longleftarrow 10\% $$ $$ LCL = [ 1.05 \cdot \frac{1}{n} \sum_{i=1}^{n} E^{*}_{i},1.05 \cdot \frac{1}{n} \sum_{i=1}^{n} E^{**}_{i}] \; \; LCL = [ 0.95 \cdot \frac{1}{n} \sum_{i=1}^{n} E^{*}_{i},0.95 \cdot \frac{1}{n} \sum_{i=1}^{n} E^{**}_{i}] \Longleftarrow 5\% $$
$$ LCL = [ \color{red} 0\color{red}.\color{red}6\color{red}9\color{red}3 \; \; 0.772] \; \; UCL = [ 0.765 \; \; \color{red}0\color{red}.\color{red}8\color{red}1\color{red}1 ] $$ $$ \Downarrow $$
$$ \Downarrow $$ $$ SCM14 \rightleftharpoons SCM5 $$