Operations Research Answers

Solve the following LPP by the two-phase simplex method.

 Max Z = x₁ + x₂ − x₃

 Subject to

4x₁ + x₂ + x₃ = 4

3x₁ + 2x₂ - x₄ = 6

x₁,x₂,x₃ ≥ 0

Solve the (4x3) game with pay off matrix.

8 5 8

[A] = 8 6 5

7 4 5

6 5 6

 At each stage, clearly explain the steps involved.

Which statements about linear programming are true?

There is more than one correct answer. Select all correct answers.

• Linear programming involves values represented by quadratic equations.
• A computer or computational tool is required to solve linear programming problems.
• Constraints are a component of linear programming problems.
• Linear programming is a theoretical method and cannot be shown graphically.
• Linear programming is a method to achieve the best outcome of a situation.
• Linear programming is a special type of optimization.

(more than one answer)

Solve using dual simplex method

Minimize z 2x₁ + 2x₂ + 4x3

2x+3x2 + 5x3 2 2

Subject to 3x1 + x₂ + 7x3 <3

x1 + 4x₂ + 6x3 ≤ 5

Answer the questions related to the model below:

max. 3 x1 + 2 x2

st 2 x1 + 2 x2 ≤ 5

2 x1 + x2 ≤ 4

x1 + 2 x2 ≤ 4

x1, x2 ≥ 0

a. Use the graphical solution technique to find the optimal solution to the model.

b. Use the simplex algorithm to find the optimal solution to the model.

c. For which objective function coefficient value ranges of x1 and x2 does the solution remain optimal?

d. Find the dual of the model.

The mean arrival rate to a service centre is 3 per hour. The mean service time is found

to be 10 minutes foe service. Assuming Poisson arrival and exponential service time,

find

(i) the utilisation factor for this service facility,

(ii) the probability of two units in the system,

(iii) the expected number of units in the system, and

(iv) the expected time in hours that a customer has to spend in the system.

A company that produces two kinds of office tables, T1 and T2. It takes 2 hours to produce the parts of one unit of T1, 1 hour to assemble and 2 hours to polish. It takes 4 hours to produce the parts of one unit of T2, 2.5 hour to assemble and 1.5 hours to polish. Per month, 7000 hours are available for producing the parts, 4000 hours for assembling the parts and 5500 hours for polishing the tables. The profit per unit of T1 is $90 and per unit of T2 is $110. cOMPUTE THE DUAL PRICE.

The following payoff matrix describes the increase in market share for L.G Company and decrease in market share for Samsung Company: 

Company 

L.G

Samsung 

Strategy 

Low advt. 

High advt.

Low advt. 

4 

-6

High advt. 

-12 

-10

By using Games theory, answer the following points: ∙ Game strategy is considered as: 

∙ Saddle point will be: 

∙ Game value will be: 

∙ Game result is for: 

A furniture company manufactures dining room tables and chairs. The company has 150 hours of assembly time available per week and workers must spend at least 100 hours on finishing per week. A table requires 540 minutes for assembly and 180 minutes for finishing. A chair requires 150 minutes for assembly and 60 minutes for finishing. Each table is sold for R4 000 and each chair for R1 500. If x is the number of tables and y the number of chairs produced per week, the constraints and the objective function of the company are [1] 2,5x + y ≥ 1 500; 9x + 3y ≤ 4 000; x, y ≥ 0; π = 150x + 100y. [2] 9x + 2,5y ≥ 150; 3x + y ≤ 100; x, y ≥ 0; T C = 4 000x + 1 500y. [3] 9x + 2,5y ≤ 150; 3x + y ≥ 100; x, y ≥ 0; T R = 4 000x + 1 500y. [4] 540x + 150y ≥ 150; 180x + 60y ≤ 100; π = 4x + 1,5y.