Operations Research Answers

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:
1. Prepare a mathematical model for this problem
2. Plot the problem conditions on a GRAPH PAPER accurately.

Use KKT conditions to find the optimal solution to this problem: maximize x1 - x2, such that x1^2 + x2^2 <= 1

Maximize Z = -x1 + 2x2 + x3
Subject To
3x2 + x3 =< 120
x1 - x2 - 4x3 =< 80
-3x1 + x2 + 2x3 =< 100
(no nongeative constraints)
A) Reformulate this problem so tha all variables have nonnegative constraints
B) Work through the simplex method step by step to solve the problem

1.4 Given the constraints (10)
A+B + C <= 24, B +C >=8 and A >= 0, B >= 0, C>=  0.
Maximize 24-A-B - C
A: amount of time spent on school work
B: amount of time spent on fun
C: amount of time spent on pay work

Maximize z=3a+b+2c
Subject to: a + b+ 3c <=30, a>=0, b>=0, c>=0.

Maximize z = 3a + b + 2c
Subject to: 1. a + b + 3c <= 30
2. 2a + 2b + 5c<=  24
3. 4a + b + 2c<=  36
4. a,b,c>=  0
NB : a= Computers b= Network devices c= IP cameras
Z= Performance
-Numbers are costs.
The problem above consist of maximizing the performance of our computer network by reducing
the total cost.

The Make-It-Good manufacturing company produces three products, Widgets, Mingets, and Tringles.

During a given year they plan to produce a total of 13,000 units of these products.

The per unit production costs for Widgets, Mingets, and Tringles are $4, $5, and $7 respectively.

The per unit profit for the Widgets, Mingets, and Tringles is $1, $2, and $3 respectively.

If the production costs are to be $70,000 and the desired profit is $27,000, how many of each product should the company produce?

Set up the solution: define variables and determine the equations. Then, solve the system of equations using any valid method, and answer the question asked in the problem.