| x1 | x2 | x3 --+----+----+---- M1| 4 | 2 | 3 M2| 3 | 6 | 5The first row in the table says that it takes 4 days for M1 to produce 1000 units of x1, 2 days for M1 to produce 1000 units of x2, and 3 days for M1 to produce 1000 units of x3. The second row reads similarly, but then for M2.
The problem is to find an optimal assignment of the machines, given that we need 1000 units of products, in any combination, e.g., 200 units of x1, 500 units of x2, and 300 units of x3. Assume that we have one machine per product.
If we let one machine work on one product, then it is clear that it does not make sense to produce anything from x3, because M1 is faster at x2, and M2 is faster at x1. So we assign M1 to produce x2 and M2 to produce x1.
Denote by t1 the number of days M1 spends on x2 and by t2 the number of days M2 spends on x1. Our goal is to minimize t1+t2, under the constraint that x1 + x2 = 1,000. Since it takes M2 3 days to make 1,000 units of x1, we replace x1 by t2/3. Similarly we substitute x2 by t1/2. Scaling the units from 1,000 to 1, we have the constraint t2/3 + t1/2 = 1, which allows us to eliminate t1: t1 = 2 - 2*t2/3. So we minimize 2-5*t2/3, which becomes zero when t2 = 6/5. So we let both machines work for 6/5 days and produce 600 units of x2 and 400 units of x1.