Operations Research Answers

Maximize Z = 2a + b – 3c + 5d

Subjected to;

a + 7b + 3c + 7d ≤ 46

3a – b + c +2d ≤ 8

2a + 3b – c + d ≤ 10

Where,

a, b, c, and d all are ≥ 0

State which of the following statements are true and which are false. Give reasons for

your answers with a short proof or a counter-example. (10)

i) The set of all convex combinations of a finite number of points 1 2 Xn X is not ,X ,...,

a convex set.

ii) If the pay-off matrix of a game is transformed, saddle point of the game if it exists,

changes.

iii) If a negative value appears in the solution values ) (XB

column of the simplex

method, then the basic solution is optimum.

iv) In an assignment problem, if a constant is added to each element of the matrix, the

optimal assignment does not change.

v) In an LPP, every feasible solution is optimal.

Player A and B play a game in which each has three coins, a 5p, 10p and a 20p.

Each selects a coin without the knowledge of the other’s choice. If the sum of the

coins is an odd amount, then A wins B’s coin. But, if the sum is even, then B wins

A’s coin. Find the best strategy for each player and the values of the game.

An advertising agency whishes to reach two types of audiences: customers with

annual income greater than one lakh rupees (target audience A) and customers

with annual income of less than one lakh rupees (target audience B). The total

advertising budget is ` 2,00,000. One programme of TV advertising costs ` 50,000;

one programme of radio advertising costs ` 20,000. For contract reasons, at least

three programmes should to be on TV and the number of radio programmes must be

limited to 5. Surveys indicate that a single TV programme reaches 4,50,000

propective customers in target audience A and 50,000 in target audience B. One

radio programme reaches 20,000 prospective customers in target audience A and

80,000 in target audience B. Formulate it as a LPP and solve it graphically to

maximize the total reach for the programmes.

THE MEANS IN WHICH PEOPLE FIND THE RIGHT WAY TO ACHIEVE THEIR OBJECTIVES IS KNOWN AS

Solve the following LP problem using the two-phase simplex method.


Manimize Z=X1-X2-3X3


Subjects to constraints

-2X1+2X2+3X3=2



2X1+3X2+4X3=1

And

X1, X2, X3≥0

Player A and B play a game in which each has three coins, a 5p, 10p and a 20p. Each selects a coin without the knowledge of the other’s choice. If the sum of the coins is an odd amount, then A wins B’s coin. But, if the sum is even, then B wins A’s coin. Find the best strategy for each player and the values of the game.

V = xy+ λ(2,000-20x-10y)

where λ is the Lagrange multiplier.

Now, the first-order conditions for constrained output maximisation are

how i slove it

Suppose PIA is offering a new route: from Lahore to New York. The aircraft on this flight has a total of 280 seats. These seats can be converted into two categories: Business Class or Economy class, before flight schedule, depending on passengers buying any particular class of ticket. For the Lahore-New York flight to be profitable, PIA must sell a minimum of 80 Business class tickets and a minimum of 100 Economy class tickets. However, PIA does not want to have more than 150 seats in economy class to promote business class travelling. The airline earns a profit of $150 for each Business Class ticket and $100 for each Economy class ticket. How many of each category of ticket should be sold in order to MAXIMIZE total profit from of a flight. Use linear programming in following steps:

3. Using linear modelling method that you have learnt in the class, find out the solution.
4. Countercheck the answer with coordinates on the graph as well.
5. Find out the MINIMUM TOTAL PROFIT as well.