What kind of Linear Programming is this?

Question

I started study Operations Research (OR) recently and I faced a problem which I struggled to modeling it to traditional OR problems. I'm not sure if such problem is a linear one nor if it can be solved using LP methods.

Problem Description I need to find the minimum purchase amount (T) to buy a batch of products, observing a restriction that each product must be purchased at least ax% of total purchase amount.

Table describing products and restrictions

Data	Corn(x1)	Soy(x2)	Wheat(x3)	Rice (x4)
Minimum Bag Weight (kg)	100	100	100	100
Unitary Price Bag ($)	4,000	3,200	4,500	1,600
Perc(% of T) of each Product	32%	27%	15%	26%

Conditions The amount of bags of each product (x1,x2,x3,x4) must be integer, ie, no fractional bag is allowed.

The objective is to determine the minimum purchase amount to buy all of the products observing the percentage of total purchase value for each one.

If you find a solution for this, please describe which technique/method will be used and make clear what kind of OR problem is this?

Thanks.

Answer 1

It is a linear programming problem, but it looks like there is something wrong about the question. You can't really buy, let's say 'at least 27% corn', at the moment. Percentages of each product are given with a total of 100%. So in this case they all have to be in that exact percentage or increasing one will decrease the other which is infeasible. Mathematical model for strict percentages would be:

Minimize T = 4000x1 +3200x2 +4500x3 +1600x4

4000x1 = T*(32/100)

3200x2 = T*(27/100)

4500x3 = T*(15/100)

1600x4 = T*(26/100)

x1,x2,x3,x4 >= 100

x1,x2,x3,x4 = integer