How do I formulate this linear programming problem? (especially the second restriction)

Question

A factory creates different types of oils and mixes them together. There exists two types of vegetarian oils (veg1,veg2) and three types of non-vegetarian oils (oil1,oil2,oil3), the price of each oil (per ton) is:

veg1=$115$, veg2=$128$, oil1=$132$, oil2=$109$, oil3=$114$.

The final product is a mix of these oils and is sold for $180$ dollars for each ton. The goal is to maximize the profit of the factory, under the following restrictions:

1. We can't refine more than $210$ tons of vegetarian oil, and more than $260$ tons of non vegetarian oil.
   
2. There exists a restrictment on the viscosities (from google translate, hope it makes sense) of the final product, it can't be lower than $3.5$ and not higher than $6.2$. We assume that the oils are mixed in a linear manner and the viscosity of each oil is:
   veg1=$8.8$, veg2=$6.2$, oil1=$1.9$, oil2=$4.3$, oil3=$5.1$

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Note: The main part of my question is understanding the second restriction. Here's my work until I reached that:

Decision Variables:
let $x_1,x_2$ be the amount in tons that we bought from veg1 and veg2 oils.
$y_1,y_2,y_3$ the amount in tons that we bought from oil1,oil2 and oil3.

Our goal function should be the profit of the factory, and we need to maximize that.
First, we need to know our product sales money, and that is $180$ for each ton of the mix, I got confused of how to express it, but I think the most intuitive way is $180*(x_1+x_2+y_1+y_2+y_3)$, since the mix is made from all the oils (and they didn't mention them needing to have the same proportions).
Second, the cost of the oils which is $115x_1+128x_2+132y_1+109y_2+114y_3$.

So Our function is: $180*(x_1+x_2+y_1+y_2+y_3)-115x_1+128x_2+132y_1+109y_2+114y_3$.

From the first restriction, we get these inequalities: $x_1+x_2\le 210$ and $y_1+y_2+y_3\le 260$.

But here I got stuck, because I don't understand what that second restriction is, I'm not sure what they meant by linear manner here.

Answer 1

I suppose it means that the resulting viscosity will be a weighed average of the individual viscosities: $$ Visc = \dfrac{8.8 x_1 + 6.2 x_2 + 1.9 y_1 + 4.3 y_2 + 5.1 y_3}{x_1+x_2+y_1+y_2+y_3} $$

So, for instance, the restriction $Visc \leq 6.2$ would be equivalent to $$ \dfrac{8.8 x_1 + 6.2 x_2 + 1.9 y_1 + 4.3 y_2 + 5.1 y_3}{x_1+x_2+y_1+y_2+y_3}\leq 6.2\Leftrightarrow $$

$$8.8 x_1 + 6.2 x_2 + 1.9 y_1 + 4.3 y_2 + 5.1 y_3 \leq 6.2(x_1+x_2+y_1+y_2+y_3) \Leftrightarrow $$

$$ 2.6 x_1-4.3 y_1-1.9y_2-1.1y_3 \leq 0. $$