Operations Research Answers

To use the simplex method to solve the following LLP :

Maximise z = 4x₁ + 3x₂

Subject to :

2x₁ + x₂ <= 1000

x₁ + x₂ <= 800

x₁ <= 400

x₂ <= 700

x₁ , x₂ >= 0

To write the dual of the following LLP :

Minimise Z = 16x₁ + 9x₂ + 21x₃

Subject to the constraints :

x₁ + x₂ + x₃ = 16

2x₁ + x₂ + x₃ >= 12

x₁ , x₂ >= 0

x₃ - unrestricted.

To solve the ILLP given below by the graphical method :

Maximum Z = 95x₁ + 100x₂

Subject to the constraints

5x₁ + 2x₂ <= 20

x₁ >= 3

x₂ <= 5

x₁ , x₂ are non-negative integers.

A factory has decided to diversify its activities. The data collected for the sales and production departments are summarized below: Potential demand exists for two products A and B. Market can absorb any quantity of A, whereas the share of B for this organization is expected to be not more than 400 units per month. For every three units of B produced there is one unit of a by - product which sells at K3 per unit and only 100 units of this by - product can be sold per month. Contribution per unit of products A and B is expected to be K6 and K8 respectively. These products require three different processes and the time required per unit of product is given in the table below: Process Product Product B Available hours 900 600 1200 III Find the product mix to optimize the contribution by using Simplex Method.

If the availability and requirements of a balanced transportation problem are integers , the optimal solution to the problem will have integer value . Justifiy the statement are true and false ? Give a proofs or a counter example

1.  The Eastern Iron and steel company makes nails, bolts, and washers from steel and coats them with zinc. The company has 24 tons of steel and 30 tons of zinc. The following table gives detailed information on the objective of the company and items production. (6 Points)

Products

Steel used in ton per unit

 Zinc used in tons/unit

Profit per unit

Nails (X1)

4

2

6

Bolts (X2)

1

6

2

Washer (X3)

3

3

12

Total Resource available

24

30

a)     Formulate the model for LP

b)     Solve the LPM using the simplex algorithm.

c)     If the company plans to make only nails and bolts with the existed steel and zinc, Solve the LPM using a graphical method.

d)    Interpret the optimal solution. 

1. Consider the weighted voting system [q : 8, 3, 3, 2], where q is an integer and 9 ≤ q ≤ 16.

(a) For what values of q is there a dummy?

(b) For what values of q do all the voters have the same power?

(c) If a voter is a dummy for a given quota, must the voter be a dummy for all larger quotas? Explain

A business executive has the option to invest money in two plans: Plan A guarantees that each dollar invested will earn $0.70 a year later, and plane B guarantees that each dollar invested will earn $2 after 2 years. In plan A, investments can be made annually, and in plan B investments are allowed for periods that are multiples of 2 years only. How should the executive invest $100,000 to maximize the earnings at the end of 3 years?

Solve the following problem using the simplex method:

Maximise: z = −x1 + 2x2 + x3

subject to

3x2 + x3 ≤ 120,

x1 −x2 −4x3 ≤80,

−3x1 +x2 +2x3 ≤100

(no non-negativity constraints). You should follow the following steps.

(a) First reformulate the problem so that all variables have non-negativity constraints.

(b) Then work through the simplex method step by step to solve the problem.

(c) State the values of the decision variables x1, x2, x3 as well as the objective function z in an optimal solution. 

The demand for a certain product is 2000 units per year and the items are withdrawn at a constant rate. The ordering cost incurred each time an order is placed to replenish inventory is £50. The unit cost of purchasing the product is £470 per item, and the holding cost is £4.10 per item per year.

Apply a basic inventory model to determine the optimal size of each order and how often an order should be placed. You should follow the following steps:

(a) Formulate the mathematical problem.

(b) Determine the optimal size of each order.

(c) Determine how often an order should be placed.