A biologist has 400 kg of nutrient A, 600 kg of nutrient B, and 300 kg of nutrie

ID: 3216143 • Letter: A

Question

A biologist has 400 kg of nutrient A, 600 kg of nutrient B, and 300 kg of nutrient C. These nutrients will be used to make 4 types of food, whose contents (in percent per kilogram of food) and whose 'growth values' are shown on the right Set up the initial simplex tableau that would be used to determine how many kilograms of each food should be produced in order to maximize total growth 0-1 12.5 | 37.5 A B 0 50 625 62 5 C100 50 25 0 Growth Value Let x, be the number of klograms of food type P Let X2 be the number of kilograms of Q Let xg be the number of kilograms of R, and let xq be the number of kilograms of s. What is the objective function?

Explanation / Answer

Solution

Formulation

Objective Function: Maximise Z = 110x1 + 90x2 + 80x3 + 60x4, subject to

Constraints:

12.5x3 + 37.5x4 ? 400 [nutrient A]

50x2 + 62.5x3 + 62.5x4 ? 600 [nutrient B]

100x1 + 50x2 + 25x3 ? 300 [nutrient C]

x1, x2, x3, x4 ? 0 [non-negativity condition]

Standard Format

To convert the above constraints to standard format, let s1, s2 and s3 be the slack variables connected to nutrients A, B and C respectively. Then, the above problem turns into:

Maximise Z = 110x1 + 90x2 + 80x3 + 60x4 + 0.s1 + 0.s2 + 0.s3, subject to

Constraints:

12.5x3 + 37.5x4 + s1= 400 [nutrient A]

50x2 + 62.5x3 + 62.5x4 + s2= 600 [nutrient B]

100x1 + 50x2 + 25x3 + s3 = 300 [nutrient C]

x1, x2, x3, x4 ? 0 [non-negativity condition]

Initial Simplex Tableau

110

400

12.5

37.5

600

62.5

300

100

30000

45000

50000

52500

400

600

300

zj - ci

29890

44910

49020

52440

400

600

300

DONE

110

400