Question

The seasonal yield of olives in a Piraeus, Greece, vineyard is greatly influenced by a process of branch pruning. If olive trees are pruned every two weeks, output is increased. The pruning process, however, requires considerably more labor than permitting the olives to grow on their own and results in a smaller size olive. It also permits olive trees to be spaced closer together. The yield of 1 barrel of olives by pruning requires 5 hours of labor and 1 acre of land. The production of a barrel of olives by the normal process requires only 2 labor hours but takes 2 acres of land. An olive grower has 240 hours of labor available and a total of 150 acres for growing. Because of the olive size difference, a barrel of olives produced on pruned trees sells for $20, whereas a barrel of regular olives has a market price of $35. The grower has determined that because of uncertain demand, no more than 40 barrels of pruned olives should be produced.

Use LP to find:

a. the maximum possible profit.

b. the best combination of barrels of pruned and regular olives.

c. the number of acres that the olive grower should devote to each growing process.

Please show excel formulas

Answer 1

a) Calculation of maximum possible profit :

Using X1 which denote the number of barrels of olives by pruning and X₂ which denotes the number of barrels of olives by normal process.

Inserting the values then Lp problem is given by

Maximum profit = 20 X₁+35 X₂

Subject to 5 X₁+2 X₂ <= 240

X₁+ 2 X₂  <= 150

  X₁ <= 40

  X1, X₂ >= 0

Using Isoprofit line graphical method, first we deprive the graph with constraints and feasible region.

The blue line denotes the Isoprofit line, the intersection points D, E results by

D : 5X1 + 2 X₂ = 240

5(40) + 2X₂ = 240

X₂ = 20

X1 = 40

E : 5X1+ 2X₂ = 240

X1 + 2X₂ = 150

4X1 = 90

X1 = 22.5

X₂= 63.75

As denoted in the graph we have five corner points. They are (0.0) , (0.75) , (40,0) ,(40,22.5) , (22.5 , 63.75).

According to the isocost line , the slope of that is -2/3 which is less than -1/2 nut larger than -5/2 then we find that the maximum profit happens when X₁  = 22.5 and X₂ = 63.75.

The maximum profit (22.5 , 63.75) = 20 * 22.5 + 35 * 63.75

= $ 2681.25

b)

Hence the best combination is to produce 22.5 barrels of pruned plives and  63.75 barrels of normal olives.

c)

The number of acres that the olives grower should be used during the process is

X₁+ 2 X₂ = 22.5 + 2 * 63.75

= $ 150