2-11 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 + 232,3 + 3y3,3 + 4y4,3 (c) max (1)1,4 + + + (d) min 81Y1 + 82/22 ... + 8,31 (c) Y1,1 Y1,4 = 51 Y2,1 + Y2,2 + Y2,3 + Y2,4 = 52 Y3,1 + Y3,2 + Y3,3 + Y3,4 = 53 (f) + + a3,1133 + 14,174 = C1 + + + + 02,382 + 03,343 + = C3 2-14 Suppose that the decision variables of a mathematical programming model are Kigh st. acres of land plot i allocated to crop j in year t where i = 1 47:j= 1 9: t = 1 10. Use summation and "for all" indexed notation to write expressions for each of the following systems of constraints in terms of these decision variables, and determine how many constraints belong to each system. (a) The total acres allocated in each year to each plot i cannot exceed the available acres there (call it P1). (b) At least 25% of the total acreage allocated in each of the first 5 years years should be de- voted to corn (crop j = 4). (c) More acres should be devoted to beans (crop j = 1) in each year and each plot than to any other crop. 2-17 Taking the by as variables, and all other sym- bols as given constants. determine whether each of the following is a linear or a nonlinear con- straint, and briefly explain why. & (a) 3x1 + 2x2 - X17 = 9 = 4x6 + 9x (c) a/xg + 10x 13 s 100 (d) x4/a + Bx 13 Z 29 2-20 The Forest Service can build a firewatch tower on any of 8 mountains. Using decision variables X1 = 1 if a tower is built on mountain j and = 0 otherwise, write constraint(s) enforc- ing each of the following requirements. (a) In all 3 sites will be selected. (b) Atleast 2 of mountains 1,2,4, and 5 must be selected (c) A tower should not be built on both mountains 3 and 8. (d) A tower can be built on mountain 1 only if one is built on mountain 4.