Answer to Question #174032 in Operations Research for Alicia

Question

QUESTION 1

Food For U Sdn Bhd., a catering company is to make breakfast for a business meeting. It will serve chicken sandwiches, egg sandwiches, and vegetarian sandwiches. A chicken sandwich has 1 serving of vegetables, 4 slices of chicken, 2 slices of cheese, and 2 slices of bread. An egg sandwich has 2 serving of vegetables, 2 servings of egg, 2 slices of cheese and 2 slices of bread. A vegetarian sandwich has 3 servings of vegetables, 2 slices of cheese, and 2 slices of bread. A total of 5 bags of chicken are available, each of which has 40 slices; 19 loaves of bread are available, each with 14 slices; 200 servings of vegetables and eggs are available, and 15 bags of cheese, each with 60 slices, are available.

a) Formulate linear programming model based on the above problem by using Microsoft Excel. Show the answer and sensitivity report. (9 marks)

Answer 1

Solution:

Let the number of chicken, egg and vegetable sandwiches produced be $x,y,z$ respectively.

(a) : Objective function, maximise $Z=x+y+z$

subject to the constraints:

$x+2y+3z\le200 \\ 4x\le5\times40 \Rightarrow x\le50 \\ 2x+2y+2z\le15\times60 \Rightarrow x+y+z\le450 \\ 2x+2y+2z\le19\times 14 \Rightarrow x+y+z\le133 \\ 2y\le 200 \Rightarrow y\le100 \\ x,y\ge0$

(b): Consider above 5 inequations to be equations.

$x+2y+3z=200 ...(i) \\ x=50...(ii) \\ x+y+z=450...(iii) \\ x+y+z=133...(iv) \\ y=100...(v)$

Solving (i), (ii), (v), we get $A(50,100,-\frac{50}3)$

Solving (ii), (iii), (v), we get $B(50,100,300)$

Solving (ii), (iv), (v), we get $C(50,100,-17)$

Clearly, out of these points,

maximum value of $Z=50+100+300=450$ at $B(50,100,300)$.

Thus, the number of chicken, egg and vegetable sandwiches produced are $50,100, 300$ respectively.

Sensivity report: Reducing or adding a certain number of serving of vegetables and loaves of bread cause no change in the optimal number of sandwiches.

Example: Reducing serving of vegetables by 2, we get inequation: $x+2y+3z\le198$

which causes no change in the optimal number of sandwiches.

Similarly, by adding 5 more loaves of bread, we get inequation:

$2x+2y+2z\le24\times14\Rightarrow x+y+z\le168$

which causes no change in the optimal number of sandwiches.