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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

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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

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Sep 03, 2012
Solving Linear Equations Using Gaussian Elimination
by: Staff


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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

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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



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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



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