A company has three operational departments (weaving, processing and packaging) with capacity to produce three different types of clothes namely suiting, shirting and woollens yielding profit of Rs 20, Rs 40 and Rs 30 per metre, respectively. One metre suiting requires 3 minutes in weaving, 2 minutes in processing and 1 minute in packing. One metre of shirting requires 4 minutes in weaving, 1 minute in processing and 3 minutes in packing while one metre of woollen requires 3 minutes in each department. In a week, total run time of each department is 60, 40 and 80 hours for weaving, processing and packaging departments, respectively. Formulate as Linear programming problem to maximize the profit.

A company is manufacturing two different types of products A and B. Each product has to be processed in three different departments casting, machining, and finally quality inspection. The capacity of the departments is limited to 35 hours, 32 hours and 24 hours, per week, respectively. Product A requires 7 hours in casting department, 8 hours in machining shop and 4 hours in inspection, whereas product B requires 5 hours, 4 hours, and 6 hours respectively in each shop. The profit contribution for a unit product of A and B is Rs 40 and Rs 30 respectively.

Find the optimal quantities of products A and B.

What is the total profit contribution?

2. Solve the game whose pay-off matrix is

<img src='./qimages/13375-2a.jpg'>

Also calculate the game value.

Differentiate between constrained and unconstrained problems with the help of an example.

Use dynamic programming to find the shortest path from city 1 to city 7 of the following route network. (Distance between the cities are given in kilometres)

<img src='./qimages/13375-3a.jpg'>

Use the Kuhn-Tucker conditions to solve the following problem:

Maximize z =2xl X2

subject to:

-x1+X2 0

x1^2

0

Use the Newton-Raphson method to find the roots of the equation

x^3 =0.

Given the values:

x 5 7 11 13 17
150 392 1452 2366 5202

Evaluate using Newton's divided difference formula.

Apply Runge-Kutta fourth order method to find an approximate value of y when x given that

y and y=1 when O.

Evaluate <img src='./qimages/13375-5b.jpg'> by using Trapezoidal rule.

Solve the following transportation problem to minimize the total transportation cost

<img src='./qimages/13375-6a.jpg'>

(b)Find the maximum value of

z 2xl 3x2

subject to:

xl x2 30

x2 3

x2 12

xl-x2 0

20

What is dynamic programming? What sort of problems can be solved by it Explain.

What is simulation Describe its advantages in solving the problems. Give its main limitations with suitable examples.