4. (7 points) The numbers $a, b, c, d$ belong to the interval $[-5.5,5.5]$. Find the maximum value of the expression $a+2 b+c+2 d-a b-b c-c d-d a$.
Answer: 132

Solution: Note that $a+2 b+c+2 d-a b-b c-c d-d a=(a+c)+2(b+d)-(a+$ $c)(b+d)$. Let $x=a+c, y=b+d$, then we will find the maximum value of the expression $x+2 y-x y=(y-1)(2-x)+2$, where $-12 \leqslant y-1 \leqslant 10$ and $-9 \leqslant 2-x \leqslant 13$. Therefore, $(y-1)(2-x)+2 \leqslant 10 \cdot 13+2=132$. The maximum value of the expression is achieved when $a=c=-5.5, b=d=5.5$.