Write each of the following as compactly as possible using summation and “for all” indexed notation.

(a) min 3y3,1 + 3y3, 2 + 4y4,1 + 4y4, 2

(b) max 1y1, 3 + 2y2, 3 + 3y3, 3 + 4y4,3

(c) max a1 y1, 4 + a2 y2, 4 + c + apyp, 4

(d) min d1 y1 + d2 y2 + c + dt yt

(e) y1, 1 + y1, 2 + y1, 3 + y1, 4 = s1 y2, 1 + y2, 2 + y2, 3 + y2, 4 = s2 y3, 1 + y3, 2 + y3, 3 + y3, 4 = s3

(f) a1, 1y1 + a2, 1y2 + a3, 1y3 + a4, 1y4 = c1 a1, 2y1 + a2, 2y2 + a3, 2y3 + a4,2y4 = c2 a1, 3y1 + a2, 3y2 + a3, 3y3 + a4, 3y4 = c3