CPLEX中整数约束与二进制约束的复杂度

Question

Recently, I have been trying to learn a bit about CPLEX and was hoping someone could help me understand the complexity when solving for integer vs. binary constraints.

For example, say we are trying to allocate a pie around 10 people for maximum utility, where each person has a utility that is linear with the amount of pie they receive. However, we want to introduce the constraint that at least 3 people have to get a bit of pie.

What's the difference between thinking of this as a single integer constraint (number_of_people_with_pie >= 3) vs. 10 binary variables (person_1_has_pie + person_2_has_pie + ... person_10_has_pie >= 3)? I would imagine the former is simplest but wonder if there is any benefits to forming the problem in terms of binary variables?

In addition to this, any recommended reading for better understanding MIP and CPLEX would be greatly appreciated, especially in better understanding where the problem becomes NP or in what situations simplex struggles to find the global minima.

Thanks!

Answer 1

I agree with Alex and Erwin's comment that this really depends on what you want to model. For this particular model I disagree with Alex: to me it makes more sense to use one decision variable per person, otherwise it may become hard to figure out which person gets how much of the pie.

A problem becomes NP hard as soon as you add integrality or SOS constraints. A good reading for MIP in general is Alex Schrijver's "Theory of Integer and Linear Programming". That should cover all the topics you need for an in-depth understanding of things.

Answer 2

It really depends on the case but in yours I would use 1 decision variable rather than 10.

Sometimes, that's not obvious and trying and measuring can prove oneself right or wrong. And that's one of the reason why using high modeling languages can help. (Abstract modeling languages such as OPL)

I recommend a MOOC on cognitive class : https://cognitiveclass.ai/courses/mathematical-optimization-for-business-problems/

and the OPL language manual : https://www.ibm.com/support/knowledgecenter/SSSA5P_12.7.0/ilog.odms.studio.help/pdf/opl_languser.pdf