Operations Research Answers

Take an ordinary job that you can parcel out pieces of it to another person. The job needs at least 10 tasks. Pick something you do regularly and know the steps necessary, such as baking a particular item, cooking a certain meal, cleaning house, any mult-task chore will work nicely.. You need to be specific in the tasks that make up the job. . Write down each task, symbol, and precedence relations in a chart (see table 8-1). . Then translate that chart into a project digraph (see figure 8-8 for an example . Now find an optimal solution using the critical-path algorithm and two processors (you and your helper on this job). Remember each task must be done by one person. See the list for doing the critical-path algorithm at the beginning of section 8.6.

A company produces three products P1,P2 and P3 from two raw materials A and B, and lobar L. one unit of product P1 requires one unit of A, three unit of B, and two units of L. one unit of product P2 requires 2 unit of A and B each, three unit of L, while one unit of P3 needs two unit of A six unit of B and four unit of L. the company has a daily available of 8 unit of A, 12 unit of B and 12 unit of L. it is further known that the unit contribution margin for the product is birr 3,2 and 5 respectively for P1, P2 and P3.
a. Formulate the above problem as a linear programing model
b. Obtain optimal solution to the problem by using the simplex method. Which of the three products shall not by produced by the firm? Why?
c. Calculate the unused capacity if any

A firm uses three machines in the manufacture of three products. Each unit of product A requires 3 hours on machine I, two hours on machine II, and one hour on machine III. While each unit of product B requires four hours on machine I, one hour on machine II, and three hours on machine III. While each unit of product C requires two hours on each of the three machines. The contribution margin of the three products is birr 30, birr 40 and birr 35 per unit respectively. The machine hours available on the three machines are 90, 54, and 93 respectively.
a. Formulate the above problem as a linear programing model
b. Obtain optimal solution to the problem by using the simplex method. Which of the three products shall not by produced by the firm? Why?
c. Calculate the un used capacity if any
d. What are the shadow prices of the machine hours?

Assuming that KKT applies to the following problem, use it to find the optimal solution: minimize (x1-1)^2 + x2^2 such that x1 - x2^2 <=0

Assuming that KKT applies to the following problem, use it to find the optimal solution: maximize e^(-x1) + e^(-2*x2) such that x1+x2<=1, x1>=0, x2>=0

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.

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

The final table of the LP relaxation of an integer linear programming problem is as follows:
Cost Basic 2 1 1 0 0 Solution
variables x₁ x₂ x₃ x₄ x₅
0 x₄ 0 -1/2 1/2 1 -1/2 3/2
2 x₁ 1 3/2 1/2 0 1/2 7/2
0 -2 0 0 -1 7
To use the branch and bound method to find an optimal solution to the original integer linear programming problem.