The following items (see the below table) are being given away for free and you wish to maximise the value of the items that you take away. Your rucksack has a maximum weight of 15 kg and a maximum volume of 30 litres.

a) Formulate an ILP model. 

b) Solve your model.

c) Relax the integer constraints and solve. 

d) Round your solution both up and down and compare it with the integer solution. How does this compare and what can you say integer programming and linear programming problems?

Item 1 2 3 4 5 6 7 8 9 10 Weight (kg) 3.5 2.4 0.9 2.5 3.1 4.2 5.1 3.5 4.5 2.9 Volume (Litre) 4 4 5 2 11 9 11 8 11 8 Value () 52 3 1 11 7 5 2 5 7