  Linear Equations - Maximizing Time Used

Linear Equation problem-solving model

Golden Jewel wishes to produce three types of bracelets: Amethyst, Pearly and Emerald.

• Manufacturing Time Required

- To manufacture an Amethyst bracelet requires 2 min on machine I and II and 1 min on machine III.

- Manufacturing a Pearly bracelet requires 2 min on machine I, 9 min on machine II and 4 min on machine III.

- Manufacturing an Emerald bracelet requires 4 min on machine I, 7 min on machine II and 1 min on machine III.

• Manufacturing Time Available

- 48 min available on machine I

- 105 min available on machine II

- 37 min available on machine III

How many bracelets of each type will be manufactured in order to use all the available time?

• Denote x, y and z as the number of bracelet type Amethyst, Pearly and Emerald manufactured.

• Represent the above information as a system of linear equations.

• Use the Gaussian Elimination method.

Comments for Linear Equations - Maximizing Time Used

 Sep 03, 2012 Solving Linear Equations Using Gaussian Elimination by: Staff Answer: Part I Golden Jewel wishes to produce three types of bracelet: Amethyst, Pearly and Emerald. x = number of Amethyst bracelets manufactured y = number of Pearly bracelets manufactured z = number of Emerald bracelets manufactured m₁ = total time used on Machine I m₂ = total time used on Machine II m₃ = total time used on Machine III P = total machine time used = m₁ + m₂ + m₃ Constraints          - Amethyst bracelet requires:               2 min on machine I               2 min on machine II               1 min on machine III          - Pearly bracelet requires:               2 min on machine I               9 min on machine II               4 min on machine III          - Emerald bracelet requires:               4 min on machine I               7 min on machine II               1 min on machine III      • Machine Time:          - maximum machine time available               48 min available on machine I        (2 min) * (number of Amethyst bracelets) + (2 min) * (number of Pearly bracelets) + (4 min) * (number of Emerald bracelets) ≤ 48 min        2x + 2y + 4z ≤ 48 ---------------------------------------------------

 Sep 03, 2012 Solving Linear Equations Using Gaussian Elimination by: Staff --------------------------------------------------- Part II               105 min available on machine II        (2 min) * (number of Amethyst bracelets) + (9 min) * (number of Pearly bracelets) + (7 min) * (number of Emerald bracelets) ≤ 105 min        2x + 9y + 7z ≤ 105               37 min available on machine III        (1 min) * (number of Amethyst bracelets) + (4 min) * (number of Pearly bracelets) + (1 min) * (number of Emerald bracelets) ≤ 37 min        1x + 4y + 1z ≤ 37 The objective function:      • maximize machine time utilized:          - Machine I:               m₁ = 2x + 2y + 4z          - Machine II:               m₂ = 2x + 9y + 7z          - Machine III:               m₃ = 1x + 4y + 1z        P = total machine time used = m₁ + m₂ + m₃        P = (2x + 2y + 4z) + (2x + 9y + 7z) + (1x + 4y + 1z)        P = 5x + 15y + 12z Represent the above information as a system of linear equations.:      • Solve for the values of x, y, and z which maximize the following objective function :          P = 5x + 15y + 12z ---------------------------------------------------

 Sep 03, 2012 Solving Linear Equations Using Gaussian Elimination by: Staff --------------------------------------------------- Part III      • Subject to the following restrictions:          2x + 2y + 4z ≤ 48          2x + 9y + 7z ≤ 105          1x + 4y + 1z ≤ 37          x ≥ 0          y ≥ 0          z ≥ 0 By using the Gaussian Elimination method, how many bracelets of each type will be manufactured in order to use all the available time? Use slack variables r, s, and t to convert the inequalities into equations: 2x + 2y + 4z + 1r + 0 + 0 + 0 = 48 2x + 9y + 7z + 0 + 1s + 0 + 0 = 105 1x + 4y + 1z + 0 + 0 + 1t + 0 = 37 -5x - 15y - 12z + 0 + 0 + 0 + 1P = 0 Convert the four equations are converted into the augmented matrix (Initial simplex tableau) shown below: Initial simplex tableau x y z r s t p 2 2 4 1 0 0 0 48 2 9 7 0 1 0 0 105 1 4 1 0 0 1 0 37 -5 -15 -12 0 0 0 1 0 ---------------------------------------------------

 Sep 03, 2012 Solving Linear Equations Using Gaussian Elimination by: Staff --------------------------------------------------- Part IV Use the Gaussian Elimination method to determine the maximum value for P Tableau #2 x y z r s t p 3/2 0 7/2 1 0 -1/2 0 59/2 -1/4 0 19/4 0 1 -9/4 0 87/4 1/4 1 1/4 0 0 1/4 0 37/4 -5/4 0 -33/4 0 0 15/4 1 555/4 Tableau #3 x y z r s t p 32/19 0 0 1 -14/19 22/19 0 256/19 -1/19 0 1 0 4/19 -9/19 0 87/19 5/19 1 0 0 -1/19 7/19 0 154/19 -32/19 0 0 0 33/19 -3/19 1 3354/19 Tableau #4 x y z r s t p 1 0 0 19/32 -7/16 11/16 0 8 0 0 1 1/32 3/16 -7/16 0 5 0 1 0 -5/32 1/16 3/16 0 6 0 0 0 1 1 1 1 190 x = 8 Amethyst bracelets manufactured y = 6 Pearly bracelets manufactured z = 5 Emerald bracelets manufactured P = 190 minutes Final Answer 8 Amethyst bracelets manufactured 6 Pearly bracelets manufactured 5 Emerald bracelets manufactured Total Machine Time Used = 190 minutes ---------------------------------------------------------------- check the solution: Maximize the objective function P = 5x + 15y + 12z subject to the following constraints 2x + 2y + 4z ≤ 48 2x + 9y + 7z ≤ 105 1x + 4y + 1z ≤ 37 x ≥ 0 y ≥ 0 z ≥ 0 P = 5x + 15y + 12z P = 5*8 + 15*6 + 12*5 = 190, OK 2x + 2y + 4z ≤ 48 2*8 + 2*6 + 4*5 ≤ 48, OK 2x + 9y + 7z ≤ 105 2*8 + 9*6 + 7*5 ≤ 105, OK 1x + 4y + 1z ≤ 37 1*8 + 4*6 + 1*5 ≤ 37, OK Thanks for writing. Staff www.solving-math-problems.com