Operations Research Answers

The decision variables represent the amounts of ingredients 1, 2, and 3 to put into a blend. The objective function represents profit. The first three constraints measure the usage and availability of resources A, B, and C. The fourth constraint is a minimum requirement for ingredient 3.

LP Problem

Maximize 4x₁ + 6x₂ + 7x₃

s.t.

3x₁+2x₂+5x₃ ≤ 120

  x₁+3x₂+3x₃ ≤  80

5x₁+5x₂+8x₃ ≤ 160

    x₃ ≥  10

         x₁, x₂ , x₃ ≥ 0

Answer the following questions.

a. How much of ingredient 1 will be put into the blend?

b. How much of ingredient 2 will be put into the blend?

c. How much of ingredient 3 will be put into the blend?

d. How much resource A is used?

e. How much resource B will be left unused?

f. What will the profit be?

g. What will happen to the solution if the profit from ingredient 2 drops to 4?

h. What will happen to the solution if the profit from ingredient 3 increases by 1?

i. What will happen to the solution if the amount of resource C increases by 2?

j. What will happen to the solution if the minimum requirement for ingredient 3 increases to 15?

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.

To use the dual simplex method to solve the following LPP :

Minimise z=x₁+2x₂+3x₃

subject to

x₁-x₂+x₃>= 4

x₁+x₂+2x₃<= 8

x₁-x₃ >= 2

x₁,x₂, x₃>= 0

A company has three warehouses F₁, F₂ and F₃ which supply goods to four warehouses W₁, W₂, W₃ and W₄. The daily factory capacities of F₁, F₂ and F₃ are respectively, six units, one unit and ten units. The demand of the warehouses W₁, W₂, W₃ and W₄ are respectively, seven, five, three and two units. Unit transportation costs are as follows :

W₁ W₂ W₃ W₄

F₁ 2 3 11 7

F₂ 1 0 6 1

F₃ 5 8 15 9

To find an initial base feasible solution by the Vogel's approximation method.

10. A can of cat food, guaranteed by the manufacturer to contain at least 10 units of protein, 20 units of mineral matter, and 6 units of fat, consists of a mixture of four different ingredients. Ingredient A contains 10 units of protein, 2 units of mineral matter, and 1 2 unit of fat per 100g. Ingredient B contains 1 unit of protein, 40 units of mineral matter, and 3 units of fat per 100g. Ingredient C contains 1 unit of protein, 1 unit of mineral matter, and 6 units of fat per 100g. Ingredient D contains 5 units of protein, 10 units of mineral matter, and 3 units of fat per 100g. The cost of each ingredient is Birr 3, Birr 2, Birr 1, and Birr 4 per 100g, respectively. How many grams of each should be used to minimize the cost of the cat food, while still meeting the guaranteed composition? (Hint: Solve through simplex model)

To find the sequence of jobs that maximises the total elapsed time required to complete the following task on two machines -

Task A B C D E F G

1 2 5 4 9 8 5 4

2 6 8 7 4 9 8 11

Also to find the optimal elapsed time.

A firm makes two products A and B. It has a total production capacity of 9 tonnes per day, with A and B utilising the same production facilities. The firm has a permanent contract to supply at least 2 tonnes of A per day ti another company. Each tonne of A requires 20 machine hours of production time abd each tonne of B requires 50 machine hours of production time. The daily maximum possible number of machine hours is 360. All the firm's output can be sold and the profit made is $80 per tonne of A and $120 per tonne of B. To formulate the problem of maximising the profit as an LPP and to solve it graphically.

9. A company manufactures two products P1 and P2. Profit per unit for P1 is $200 and for P2 is $300. Three raw materials M1, M2 and M3 are required. One unit of P1 needs 5 units of M1 and 10 units of M2. One unit of P2 needs 18 units of M2 and 10 units of M3. Availability is 50 units of M1, 90 units of M2 and 50 units of M3.

A. Formulate as LPP

B. Find the optimal by using Simplex method

Use the graphical method to solve the following LPP:

Minimize, Z = 3x + 5y

Subject to the constraints:

−3x + 4y ≤ 12

2x − y ≥ −2

2x + 3y ≥ 12

x ≤ 4; y ≥ 2 and x, y ≥ 0

Which of the following is not an assumption underlying linear programming?

a. the objective function can be expressed in terms of a linear equation.

b. the constraints can be expressed in terms of linear equations or inequalities

c. the usage of resources is known with certainty

d. the decision variables can take only integral values

e. the total usage of a resource is the sum of the resources used by each decision variable.