Using complementary slackness to determine the optimality or otherwise of thecandidate solution (x1, x2, x3, x4) = (0, 14, 0, 5) for the following LP:max 4x1 + x2 + 5x3 + 3x4s.t.x1 − x2 − x3 + 3x4 ≤ 1,5x1 + x2 + 3x3 + 8x4 ≤ 55−x1 + 2x2 + 3x3 − 5x4 ≤ 3x1, x2, x3, x4 ≥ 0.1

To use complementary slackness to determine the optimality of the candidate solution (x1, x2, x3, x4) = (0, 14, 0, 5) for the given LP, we first need to write down the dual of the given LP.

The dual of the given LP is:

min 1y1 + 55y2 + 3y3 s.t. y1 + 5y2 - y3 ≥ 4, -y1 + y2 + 2y3 ≥ 1, -y1 + 3y2 + 3y3 ≥ 5, 3y1 + 8y2 - 5y3 ≥ 3, y1, y2, y3 ≥ 0.

Next, we need to check the complementary slackness conditions. The complementary slackness conditions state that for each i and j, either the i-th constraint in the primal is tight (at equality) or the j-th variable in the dual is zero, and vice versa.

Checking the primal constraints, we have:

x1 − x2 − x3 + 3x4 = 0 - 14 - 0 + 15 = 1, 5x1 + x2 + 3x3 + 8x4 = 0 + 14 + 0 + 40 = 54, −x1 + 2x2 + 3x3 − 5x4 = 0 + 28 + 0 - 25 = 3.

So, all the primal constraints are tight.

Checking the dual constraints, we have:

y1 + 5y2 - y3 ≥ 4, -y1 + y2 + 2y3 ≥ 1, -y1 + 3y2 + 3y3 ≥ 5, 3y1 + 8y2 - 5y3 ≥ 3.

We don't know the values of y1, y2, and y3. But since all the primal constraints are tight, the dual variables y1, y2, and y3 should all be zero to satisfy the complementary slackness conditions.

So, the candidate solution (x1, x2, x3, x4) = (0, 14, 0, 5) is optimal for the given LP if the dual variables y1, y2, and y3 are all zero.