Let x, y and z be the quantities of I, II and III used respectively, and x, y and z are positive integers.

Vit A: 400x + 1200y + 800z = 8800
Vit C: 110x + 570y + 340z = 3380
Calcium: 90x + 30y + 60z = 1020

(x, y, z) = (6, 0, 8) + λ(1, 1, -2)

Given x, y and z are positive integers -> λ =1, 2, 3

(x, y, z) = (7, 1, 6), (8, 2, 4) and (9, 3, 2)

Let x, y and z be the quantities of Walkers, Runners and Trainers produced respectively, and x, y and z are non-negative integers.

x + y + z = 20
30x + 40y + 50z = 980

(x, y, z) = (0, 2, 18) + λ(1, -2, 1)

Given x, y and z are non-negative integers -> λ = 0, 1

(x, y, z) = (0, 2, 18) and (1, 0, 19)

Revenue for (0, 2, 18) = 130 × 2 + 150 × 18 = 2960
Revenue for (1, 0, 19) = 80 + 150 × 19 = 2930

Since cost of goods are identical, the company should manufacture 0 Walkers, 2 Runners and 18 Trainers to maximise profit.