A transportation model fills one-half of its cells under the non-degenerate optimal solution. Which of the following most closely describes the number of sources compared to the number of destinations?
A) 1 to 1
B) 2 to 2
C) 3 to 3
D) 4 to 3
E) 4 to 4

Question

Quizplus · Accepted Answer

If the transportation model fills one-half of its cells under the non-degenerate optimal solution, that means the number of non-zero variables is equal to m+n-1, where m is the number of sources and n is the number of destinations. Since half of the cells are filled, that means the number of non-zero variables is (m*n)/2.

So we have:

(m*n)/2 = m+n-1

Simplifying this equation, we get:

m*n - 2m - 2n + 4 = 0

Using the quadratic formula, we can solve for m and n. The discriminant must be a perfect square for m and n to be integers:

b^2 - 4ac = 4m^2 - 4n^2 + 8m - 16 = k^2

Simplifying this equation, we get:

(m + 1)^2 - n^2 = (k/2)^2

This is a difference of squares, so we can factor it:

(m + n + 1)(m - n + 1) = (k/2)^2

The factors on the left-hand side must both be even or both be odd for m and n to be integers. But since m and n are positive, they cannot both be odd. Therefore, one factor is odd and the other factor is even. The difference between the factors is 2n, which is even, so n is an integer. This means that m is also an integer, since (m*n)/2 is an integer.

The only answer choice that results in integers for both m and n is D, 4 to 3.

A Transportation Model Fills One-Half of Its Cells Under the Non-Degenerate