You own 50 acres of land where you can plant any amount of corn, soybeans and le

Question

You own 50 acres of land where you can plant any amount of corn, soybeans and
lettuce. The following table shows the relevant information regarding the production, the
net gain and water requirements of each crop.

Cultive(No).

Given that 100,000 liters of water are available and that the company has committed to sell
At least 5,120 kg of corn, the following PL model was developed, where Xi = Acres to sow
each type of crop i = 1, 2, 3:

Maximize 640X1 +400X2 + 240X ($)

Sujeto a:

X1 + X2 + X3 ? 50 acres of land

5600X1 +2500X2 + 900X3 ? 100000 liters of water

By solving this model, you get the following optimal solution and sensitivity analysis (Only
show the variables greater than or equal to zero):

coefficient of

objective Function

Minimum coefficient in

f. objective

Value of the objective function= $16,940

Value right

side

(RHS)

Maximum

(RHS)

Answer the following questions and clearly support your answer:

a) With the same amount of money, what would be better, acquire 10 acres of additional land or
Buy 13,000 additional liters of water? (You can only choose one of the two alternatives).

b) Maintaining the same amount of land and corn demand condition, you want to obtain
a profit of $ 21,000. How many additional liters of water would you need to achieve it?

c) You know that the inverse of the optimal base is the following matrix:

i) Rebuild the Simplex final board. Note that a variable must have been added
artificial A in the demand restriction of corn (in addition to the variable of excess S3), and, therefore,
the third column of the previous inverse matrix corresponds to the column of said artificial variable
and the first two to the slack variables S1 and S2 added in the first two constraints of
Earth and water.

ii) Check the range of sensitivity given by the program of the coefficient of the variable X2 in
the objective function.

iii) Check the sensitivity range given by the ground resource program in the function
objective.

d) Explain why the opportunity cost of the corn demand restriction is negative.
According to the above, if you must satisfy a minimum demand of 10 acres of corn, what would be the
new value of the objective function? If there is no restriction of minimum corn demand,
then it would be profitable to produce corn? What would be the value of your total net profit?

e) You have the option to grow cotton whose production is 300 Kg / acre, consume 4
liters / Kg and produces a net profit of $ 0.70 / Kg and has no demand restrictions. It is profitable
plant any extension of land with cotton?

f) Formulate the dual problem of the primal model formulated in this point and find its solution
optimal without solving it from the beginning.

Cultive(No).

Type of cultive Production(Kg/acre) Net income($/Kg) Water required (Lt/Kg) 1 Crop 640 1,00 8,75 2 Soy 500 0,80 5,00 3 Lettuce 400 0,60 2,25

Explanation / Answer

a) With the same amount of money, what would be better, acquire 10 acres of additional land or

Buy 13,000 additional liters of water?

With 10 acres of land the profit will increase by 150*10 = $1500

With 13000 liters of water the profit will increase by 13000*0.10 = 1300

So 10 acres of land will be better as it is giving higher increase in profit.

b) Maintaining the same amount of land and corn demand condition, you want to obtain a profit of $ 21,000. How many additional liters of water would you need to achieve it?

(Desired profit – actual profit)/shadow price of water

(21000-16940)/0.10 = 40,600 liters of water

d) Explain why the opportunity cost of the corn demand restriction is negative. According to the above, if you must satisfy a minimum demand of 10 acres of corn, what would be the new value of the objective function? If there is no restriction of minimum corn demand, then it would be profitable to produce corn? What would be the value of your total net profit?

The opportunity cost of the corn demand is negative because a unit increase in RHS of the constraint will result in reduction in the objective value of the problem by an amount equal to shadow price.

If we satisfy minimum demand of 10, the objective value will decrease by 70*(10-8) = $140.

If there are not restrictions there will be no production because of the two things, one the shadow price is negative, and second the Minimum (RHS) is zero.   

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