I started studying Operations Research (OR) recently and I faced a problem in which I struggled to model it using the traditional LP models. After some attempts, I'm not sure if such a 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 x% of the total purchase amount.
Table describing products and restrictions
|Minimum Bag Weight (kg)||100||100||100||100|
|Unitary Bag Price ($)||4,000||3,200||4,500||1,600|
|Perc(% of T) of each Product||32%||27%||15%||26%|
Critical Condition/Requirement The amount of bags of each product (x1,x2,x3,x4) must be integer, i.e., no fractional bag is allowed.
This seemed to me a very unusual condition, once the total is the target value to be minimized and at the same time is still the basis value of a condition (the amount in which percentage of each product will be based on).
The objective is to determine the minimum purchase amount to buy all of the products observing the percentage of the total purchase value for each one.
What kind/category of OR problem is this and which technique/method should be used to solve it?
Additional Details (an attempt to clarify the problem)
1-Minimum Bag weight 100kg ==> it means that amounts to be purchased must be integer (bags of 100kg) and not fractions (portions of the bag or kg).
2-Unitary bag Price ==> is the price of one bag and it contains 100kg
3-Perc(% of T) of each Product ==> it is the percentage of the total amount in value US$, that must be addressed to each product.
For example, suppose I have US$ 10,00 to buy those products. Then this value must be distribute according the table :
|Possible to buy?||No||No||No||Yes|
however, the total amount of US$ 10,000 can only buy one product (Rice) because the others have bag price higher than the amount addressed to each product.
Now, let's suppose we have the amount to buy at least one bag of each product, this amount would be US$ 13,300. However observe that with such amount the restriction Percent% of T (T=13,300) for each product is not attended !
|Unitary Bag Price ($)||4,000||3,200||4,500||1,600||--|
|Perc% of Total||30%||24%||34%||12%||100%|
|Min Perc% required||32%||27%||15%||26%||100%|
|Attended Min Perc%||No||No||No||No||--|
This Min Perc% of Total , must be exactly the required number, not less, not more.