Question

2.         From Unit 11:            

Ray’s Gun Shop produces 2 models Macho and Rambo which yield $ 2,700 and $ 3,600 as profits respectively. The three resources to produce these models are wood (W), steel (S) and labor (L).   The table below indicates the input-output relationship.

           

		Units of input Required
INPUTS	Total Available	Macho	Rambo
Wood (W)	9600	600	1200
Steel (S)	3000	300	300
Labor (L)	6300	900	0

Set up the LP problem and derive the optimal output mix. Put your answers in the space below:

Optimal units of Macho =

Optimal units of Rambo =

Optimal profits =

3.         Dogs require protein, carbohydrate and fat to lead healthy and productive lives. The local SPCA buys two types of feed, S and G that sell for $ 8.00 and $ 5.00 per unit. Dogs require a minimum of 300 gms of protein and 200 gms of carbohydrate and 200 gms of fat. The S-feed contains 20 gms of protein, 12 gms of carbohydrate and 20 gms of fat per unit. The G-feed contains 15 gms of protein, 15 gms of carbohydrate and 8 gms of fat per unit. Find the optimal (the least expensive) S-G combination for dogs.

           

Type the solutions in the space below:

Optimal units of S =

Optimal units of G

Optimal cost =

4.         UnaB produces bombs A and B, which retail for $ 150 and $ 100 respectively. Inputs are Man (M), Women (W) and Child (C) with total capacity of 1500, 1000 and 500 hours. A requires 2 hours of W and M, while B requires 1 hr of W, 5 hrs of M and 2 hours of C. Solve for the optimal mix of outputs.

Optimal units of A =

Optimal units of B =

Optimal Revenue =

4.         The Waco Company in Texas has two furnaces (A and B) used for producing 3 types of fertilizers: Hectic, Moderate and Loud. Currently, the Company has agreed to supply 24 units of Hectic, 16 units of Moderate and 48 units of Loud to its Militia friends in PA. It costs the Company $ 10,000 per-day to operate A and $ 5,000 per-day to operate B. Per-day, A can produce 6 units of Hectic, 2 units of Moderate and 4 units of Loud. Per-day B produces of 2 units of Hectic and Moderate and 12 units of Loud. Set-up and solve the LP problem.

Optimal units of A =

Optimal units of B =

Optimal cost =

Answer 1

2)

Optimal units of Macho = 4

Optimal units of Rambo = 6

Optimal profits = 32400

The problem set-up

Maximise Z = 2700 * x₁ + 3600 * x₂

x₁ -number of macho , x₂ - number of rambo

Subject to,

600 * x₁ + 1200 * x₂ <= 9600 (Wood)

300 * x₁ + 300 * x₂ <= 3000 (Steel)

900 * x₁ <= 6300 (Labour)

x₁ >= 0 (Non-negativity)

x₂ >= 0 (Non-negativity)

Solved using excel solver. Screenshot below,

3)

Optimal units of S = 4

Optimal units of G = 15

Optimal cost = 107

(Please note that the values will change slightly if integer contraint is not applied. But, integer constraint is applied assuming that the food cannot be brought as a fraction of unit)

The problem set-up

Minimise Z = 8 * S+ 5 * G

Subject to,

20 * S+ 15 * G >= 300 (Protein)

12 * S+ 15 * G >= 200 (Carbohydrate)

20 * S+ 8 * G >= 200 (Fat)

_S >= 0 (Non-negativity)

_G >= 0 (Non-negativity)

4)

Optimal units of A = 438

Optimal units of B = 124

Optimal Revenue = 78100

(Please note that the values will change slightly if integer contraint is not applied. But, integer constraint is applied assuming that the bomb cannot be madet as a fraction of unit)

The problem set-up

Maximise Z = 150 * A+ 100 * B

Subject to,

2 * A+ 5 * B <= 1500 (Man)

2 * A+ 1 * B <= 1000 (Woman)

2 * B <= 500 (Child)

_A >= 0 (Non-negativity)

_B >= 0 (Non-negativity)

4)

Optimal units of A = 2

Optimal units of B = 6

Optimal cost = 50000

The problem set-up

Minimize Z = 10000 * A+ 5000 * B

Subject to,

6 * A+ 2 * B >= 24 (Hectic)

2 * A+ 2 * B >= 16 (Moderate)

4 * A + 12 * B >= 48 (Loud)

_A >= 0 (Non-negativity)

_B >= 0 (Non-negativity)