A shop has 6 products and two types of packaging. The shop administrator wishes to determine the composition in which they can get the best income. You are given the following information.

The max weight of package 1 is 5lbs and the max weight of package 2 is 3lbs.

The prices and weights of each artice go as following.

Article 1: 0.1 lbs, 5$

Article 2: 0.4 lbs, 0.75$

Article 3: 1.2 lbs, 3.50$

Article 4: 1.3 lbs, 12$

Article 5: 0.8 lbs, 2$

Article 6: 2lbs, 15$




To satisfy the wished demand the package 1 must have at least 3 articles and the package 2 must have at lest 2 articles.




Give the mathematic model to maximize benefits.

Extra: if the same product can't be in both packages and every product must end up in one package, give a mathematic model that can consider these aspects.




As of right now I have the following.




Variables: x1, x2 . . . x6 each one representing one product.




Objetive function:

Maximize Z = 5x1 + 0.75x2 + 3.5x3 + 12x4 + 2x5 +15x6




Constraints


  1. 0.1x1 + 0.4x2 + 1.2x3 + 1.3 x4 + 08x5 + 2x6 <= 5 //Package 1
  2. 0.1x1 + 0.4x2 + 1.2x3 + 1.3 x4 + 08x5 + 2x6 <= 3 //Package 2

My problem comes with the part of a package containig 3 or more articles and the other 2 or more as I don't know how to refer to the number of articles contained inside a package.




Also I'm having trouble with the extra part of the excersie.