Ronnie Runner distilleries blends i = 1,c, m scotch whiskeys to create its j = 1,c, n products with properties k = 1,c, p. Unblended whiskey i measures ai, k on scale k. Express each of the following as linear constraint(s) in these parameters and the nonnegative decision variables xi, j ! barrels of whiskey i used in product j. Assume that the properties combine in proportion to volume and that the total production of any blend is free to vary with the optimization.

(a) Property k = 11 should fall between 45 and 48 in all products.

(b) Product j = 14 must have all properties k = 5,c, 9 between 90 and 95.

(c) Product j = 26 must measure at least 116 on property k = 15.

(d) No product should measure more than 87 on property k = 8.

(e) Products 6 through 11 should combine input whiskey 1 with others in at most the ratio 3: 7.

(f) Input whiskeys 4 and 7 should be in ratio 2:3 for all blends.

(g) At least one-third of all output must come from inputs i = 3,c, 6.

(h) No more than 5% of all output can come from input i = 13.