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mavroulis · Feb 16, 2010, 01:12 PM

The following part is the beginning of an assignment. It needs to be solved by linear programming and the thing is that although I have tried solving it, it seems that I am doing something wrong. I am unsure which should be the decision variables(products, or ingredients) and what are the equation, inequalities. Could you help me please?

The problem
A company blends number 1 and number 2 oranges to produce canned oranges and orange juice. The proportions of each number orange in each product, together with the selling price and limit on demand
for each product are given in the following table.

Product.. Proportion of n.of orange in each product..
... N1... N2.. …
Canned oranges... 0.75.. 0.25
Orange juice... 0.40.. 0.60
Selling price for Canned Oranges is 240 pounds/tonne and for Orange Juices is 190pounds/tonne. Also, the Demand for Canned oranges is 360 tonnes and for orange juice is 900 tonnes.

The Company has an agreement with a merchant that it may buy up to a given limit of each number of orange at the price given in the table below:
... Number1... Number2
Price (£/tonne)... 240... 120
Limit (tonnes)... 600... 640

(I) How many tonnes of canned oranges and orange juice should be manufactured in order to max-
imize Company’s total income from selling its products less the total cost of buying the oranges?

Model this problem as an LP problem and solve it using the simplex method.

galactus · Feb 18, 2010, 06:45 AM

Originally Posted by mavroulis

The following part is the beginning of an assignment. It needs to be solved by linear programming and the thing is that although I have tried solving it, it seems that I am doing something wrong. I am unsure which should be the decision variables(products, or ingredients) and what are the equation, inequalities. Could you help me please?

The problem
A company blends number 1 and number 2 oranges to produce canned oranges and orange juice. The proportions of each number orange in each product, together with the selling price and limit on demand
for each product are given in the following table.

Product..Proportion of n.of orange in each product..
............................N1....N2..…
Canned oranges....0.75..0.25
Orange juice.........0.40..0.60
Selling price for Canned Oranges is 240 pounds/tonne and for Orange Juices is 190pounds/tonne. Also, the Demand for Canned oranges is 360 tonnes and for orange juice is 900 tonnes.

The Company has an agreement with a merchant that it may buy up to a given limit of each number of orange at the price given in the table below:
....................Number1....Number2
Price (£/tonne)....240.......120
Limit (tonnes).....600........640

(i) How many tonnes of canned oranges and orange juice should be manufactured in order to max-
imize Company’s total income from selling its products less the total cost of buying the oranges?

Model this problem as an LP problem and solve it using the simplex method.

We have to maximize profit. That is Revenue - Cost.

Let $x_{1}=\text{tons of canned oranges produced}$

Let $x_{2}=\text{tons of orange juice produced}$

The revenue is $R=240x_{1}+190x_{2}$

The cost is $C=(.75x_{1})N_{1}+(.25x_{1})N_{2}+(.40x_{2})N_{1}+ (.60x_{2})N_{2}$

Given the constraints we have a total cost of :

$240\left(.75x_{1}+.40x_{2})+120(.25x_{1}+.60x_{2}) =210x_{1}+168x_{2}$

Subtract cost from revenue to find the objective function, which is profit.

$z=30x_{1}+22x_{2}$

The constraints on the oranges are:

$N_{1}: \;\ .75x_{1}+.40x_{2}\leq 600$

$N_{2}:.25x_{1}+.60x_{2}\leq 640$

Demand constraints of canned oranges and OJ:

$x_{1}\leq 360, \;\ x_{2}\leq 900$

There is the objective function and the constraints.

Now, use Excel Solver or some other software to find the solution you are aiming for. I will leave you set up the Simplex Tableau.