[Math] Why were the ratios disregarded when forming the model.

linear programmingoperations research

I came across this question in a textbook and I dont fully understand why they omitted one of the conditons when they were forming the model for the LPP.

A manufacturer produces three models I, II and III of a certain product. He uses two
types of raw materials (A and B) of which 5000 and 8000 units respectively are available. Raw material of type A requires 3, 4 and 6 units of each model. Whereas type B requires 6, 4 and 8 of model I, II and III respectively.

The labour time of each unit of model I is twice that of model
II and three times of model III. The entire labour force of the factory can produce equivalent of 3000 units of model I. A market survey indicates that the minimum demand of three models is 600, 400 and 350 units respectively.

However, the ratios of number of units produced must be
equal to 3 : 2 : 5. Assume that the profit per unit of models I, II and III are Rs 80, 50, and 120 respectively. Formulate this problem as linear programming model to determine the number of units of each product which will maximize the profit.

Solution given as 

x1 <= 600
x2 <= 400
x3 <= 350
3x1 + 4x2 + 6x3 <= 5000
6x1 + 4x2 + 8x3 <= 8000
x1 + 0.5x2 +.333x3 <= 3000
 x1,x2,x3 >= 0

from the question , i thought we would need another constraint such as 

3/10x1 + 2/10x2 +5/10x3 = 1

to cater for the ratio of units that must be produced.

Why was there no mention of the ratio in the solution ?prog

Best Answer

I agree that the ratio constraints are missing. Additionally the relation signs of the minimum demand constraints are wrong:

... that the minimum demand of three models is $600, 400$ and $350$ units respectively.

$x_1 \geq 600$

$x_2 \geq 400$

$x_3 \geq 350$

However, the ratios of number of units produced must be equal to $3 : 2 : 5$

Here you need two separate constraints such as

$2x_1=3x_2$

$5x_2=2x_3$

A third constraint is not necessary since from this two constraints it can be deduced that $\frac{10}{3}x_1=2x_3$.

You can check it by plugging in some values, for instance $x_1=600$, $x_2=400$, $x_3=1000$

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