Problem 726 Presentation Piraeus, Greece Olive Yield: Pruning vs' nonPruning Methods

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Problem 7-26 PresentationPiraeus, Greece - Olive
Yield Pruning vs. non-Pruning Methods
Pruning Yield More trees per acre Increased
output, smaller olives One barrel 5 hours labor
on 1 acre One barrel sells for 20
Non-pruning Yield One barrel 2 hours labor on 2
acres One barrel sells for 30
Grower will produce no more than 40 barrels of
pruned olives. Grower has 250 hours of labor
available. Grower has 150 acres available for
growing.
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Use graphical Linear Programming to find

• The maximum possible profit
• The best combination of barrels of pruned and
  regular olives
• The number of acres that the olive grower should
  devote to each growing process

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Equations Labor Hours and Acres Available for
Use
Labor Hours Pruned Olives Hours 5P
Unpruned Olives Hours 2U 5P 2U lt 250 Acres
Available Pruned Olives Acres 1P
Unpruned Olives Acres 2U 1P 2U lt 150 Other
Constraints Plt 40 barrels
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Equations and Solutions
Labor When U 0 5P 2(0) 250, so 5P
250, then P 50 When P 0 5(0) 2U 250,
so 2U 250, then U 125 Acres When U 0
1P 2(0) 150, then P 150 When P 0
1(0) 2U 150, so 2U 150, then U 75 Use
the points derived from the above equations to
graph the lines for labor hours and acres and to
display the Feasible Region
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Graph of the Equations and Feasible Region
Pruned Olives P axis
To find the coordinates of the point of maximum
profit, the constraint equations must be solved
simultaneously. The equations are 5P 2U lt
250, and 1P 2U lt 150
(150, 0)
100
Corner Point, the possible point of maximum
profit within the Feasible Region as
outlined by the constraint equations
(50,0)
Feasible Region
(0,125)
Unpruned Olives (Not to Scale) U
axis
10
100
(0, 75)
10
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Solving to find the solutions for maximum profit,
barrels of olives to be produced and amount of
acreage to be allocated for growing methods
To isolate one variable (P) for solution,
multiply the second equation by -1, so -1(1P 2U
150), which yields -P - 2U -150. This
equation is then added to the first. -P - 2U
-150 5P 2U 250 which solves as 4P
100, so P 25. This solution for P is
now substituted back into the first equation to
solve for U. 5(25) 2U 250, so 125 2U
250, 2U 125. U 62.5 These solutions give
you the coordinates for maximum profit, (25,
62.5) which is the number of barrels of each
type of olive to be grown (25 pruned olives and
62.5 unpruned). Maximum profit (the Objective)
can be solved (z 20P 30U) by substituting in
the number of barrels of each, so 20(25)
30(62.5) z. So, z 500 1875 2375 The
amount of acreage to be used for each growth
method can be solved as 1(25) 2(62.5) lt 150,
so 25 125 lt 150. This is true, so 25 acres
will be used for pruned olives, and 125 acres for
unpruned olives.