Operations Research Answers

1)    Complete the regularization of the following primal problem

            Min Z = 15x₁  +  15x₂

               s.t       3x₁ + 2x₂ > 2

                          7x₁ + 2x₂ = 6

                          5x₁ + 7x₂ < 4

                           x₁ ,  x₂ > 0

For model:

3x1+2x2+7x3+5x4+2x5>= 13000

2x2+x4+2x5+3x6>=20000

F(c)=0,2x1+0,1x2+0,2x3+0,3x4+0,4x5+0x6

Build a dual model, solve using the graphical method

Solve the following LPP using dual simplex method: Max Z= -3x1- x2,

Subject to: x1+x2 ≥ 1, 2x1+3x2 ≥ 2, x1,x2 ≥ 0.

Placid company makes 3 production components A, B, C using 3 machines Cutting, Polishing and Packaging whose allocated and available hours are not more than 96hrs, 40hrs and 60hrs respectively. Product A spends 6 hours on cutting machine, 2 hours on polishing and 5 hours on packaging machine. Product B goes through 8 hours of cutting, 1 hour of polishing and 3 hours of packaging. Product C takes 4 hours on cutting machine, 4 hours of polishing and 2 hours of packaging. Contribution margins for each component product are N2, N5 and N8.

Use the Simplex Algorithm to determine how many of each component the Placid Company will make to maximize contribution

A manufacturer makes two products, doors and windows. Each must processed through two work areas. Work area #1 has 60 hours of available production time. Work area #2 has 48 hours of available production time. Manufacturing of a door requires 4 hours in work area #1 and 2 hours in work area #2. Manufacturing of a window requires 2 hours in work area #1 and 4 hours in work area #2. Profit is $8 per door and $6 per window

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 unused capacity if any.