Answer to Question #169966 in Operations Research for mari

Question

A bank has two types of branches. A satellite branch employs 3 people, requires P2,500,000 to construct and open, and generates an average daily revenue of P4,000,000. A full-service branch employs 6 people, requires P5,500,000 to construct and open, and generates an average daily revenue of P7,000,000. The bank has up to P80,000,000 available to open new branches, and has decided to limit the new branches to a maximum of 20 and to hire at most 120 employees. How many branches of each type should the bank open in order to maximize the average daily revenue?

REQUIREMENTS:

1. Formulate the LP Model;

2. Identify the decision variables used in the model; and

3. Determine the optimal solution.

Answer 1

Answer:

1) Let the 2 branches be represented by x₁ and x₂.

Max z= 4000000 x₁ + 7000000 x₂

Subject to

2500000 x₁ + 5500000x₂ $≤$ 80,000,000

x₁ +x₂ $≤$ 20

3x₁ + 6x₂ $≤$ 120

2)

The decision variables are given as;

x₁ $\rarr$

x₂ $\rarr$full-service branch

3)

2500000 x₁ + 5500000x₂ $≤$ 80,000,000

Divide the both sides by 100,000

25 x₁ + 55x₂ $≤$ 800

 x₁ = 0 , x₂= 14.6

x₁ = 32 ,  x₂= 0

x₁ +x₂= 20

 x₁ = 0 ,  x₂= 20

x₁ = 20 ,  x₂= 0

3x₁ + 6x₂= 120

 x₁ = 0 ,  x₂= 20

x₁ = 40 ,  x₂= 0

The feasible region is OABC.

Z(O)=0

Z(A)= 7000000(20)= 14,000,000

Z(C)= 4000000(20)= 8,000,000

B$\rarr$25 x₁ + 55x₂ = 800

25 x₁ + 25x₂ = 500

 25x₂ = 300

x₁ = -80 ,  x₂= 100

Z(B)=  4000000(-80)+ 7000000(100)

= 380,000,000

Therefore, the optimal solution is 380,000,000.