Question

2. 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.

Answer 1

Hi

The answer of the following question is given below as follows:

Ans.A) let's assume X_{​​​​​​1} to indicate the number of barrels of olives per pruning, and X₂ to denote the number of barrels of olives per normal process.
Inserting the values, then the LP problem is given as follows :

Max. Profit= $20X₁ + $30X₂

5x₁ + 2x₂ <= 250.

X_{​​​​​1} +2X₂ <=150

X_{​​​​​​1} <=40.

So X_{​​​​​​1} , X_{​​​​​​2 >=0.}

Using the isoprofit line graph method, we first derive the graph with constraints and feasible region. The blue line denotes the isoprofit line, the  results of the intersection point by D,E.

D=5X₁ =250

5X₁+ 2X₂ = 250

2X₂ =50

ie X_{​​​​​​1} = 40 & X_{​​​​​​2} =25

Now E

X_{​​​​​​1}+2X₂ = 150

5X₁ +2X₂ = 250

By solving the above equation we get the value of X_{​​​​​​1} As 25 and X_{​​​​​​2} as 62.5

As indicated in the graph, below we have five vertex points, they are (0,0), (0,75), (40,0), (40,25) and (25,62.5).

According to the isocost line, the slope of that is -2/3, which is less than -1/2 but greater than then -5/2, so we find that the maximum gain occurs when The maximum profit is when X_{​​​​​​1} as 25 and X_{​​​​​​2 as} ​​​​ 62.5.

Max Profit (25,62.5)=$20*25+$30*62.5

$2375.will be max profit.

Ans.B) Therefore, the best combination is to produce 25 barrels of pruned olives and 62.5 barrels of normal olives.

Ans.C) The number of acres that the olive grower must use during the process is given below as follows

No. Of acres: X_{​​​​​​1} +2X₂

25+2*62.5

150.

I hope I have served the purpose well.

Thanks.