Question

I’ve always been writing software to solve business problems. I came across about LIP while I was going through one of the SO posts. I googled it but I am unable to relate how I can use it to solve business problems. Appreciate if some one can help me understand in layman terms.

Answer 1

ILP can be used to solve essentially any problem involving making a bunch of decisions, each of which only has several possible outcomes, all known ahead of time, and in which the overall “quality” of any combination of choices can be described using a function that doesn’t depend on “interactions” between choices. To see how it works, it’s easiest to restrict further to variables that can only be 0 or 1 (the smallest useful range of integers). Now:

• Each decision requiring a yes/no answer becomes a variable
• The objective function should describe the thing we want to maximise (or minimise) as a weighted combination of these variables
• You need to find a way to express each constraint (combination of choices that cannot be made at the same time) using one or more linear equality or inequality constraints

Example

For example, suppose you have 3 workers, Anne, Bill and Carl, and 3 jobs, Dusting, Typing and Packing. All of the people can do all of the jobs, but they each have different efficiency/ability levels at each job, so we want to find the best task for each of them to do to maximise overall efficiency. We want each person to perform exactly 1 job.

Variables

One way to set this problem up is with 9 variables, one for each combination of worker and job. The variable x_ad will get the value 1 if Anne should Dust in the optimal solution, and 0 otherwise; x_bp will get the value 1 if Bill should Pack in the optimal solution, and 0 otherwise; and so on.

Objective Function

The next thing to do is to formulate an objective function that we want to maximise or minimise. Suppose that based on Anne, Bill and Carl’s most recent performance evaluations, we have a table of 9 numbers telling us how many minutes it takes each of them to perform each of the 3 jobs. In this case it makes sense to take the sum of all 9 variables, each multiplied by the time needed for that particular worker to perform that particular job, and to look to minimise this sum — that is, to minimise the total time taken to get all the work done.

Constraints

The final step is to give constraints that enforce that (a) everyone does exactly 1 job and (b) every job is done by exactly 1 person. (Note that actually these steps can be done in any order.)

To make sure that Anne does exactly 1 job, we can add the constraint that x_ad + x_at + x_ap = 1. Similar constraints can be added for Bill and Carl.

To make sure that exactly 1 person Dusts, we can add the constraint that x_ad + x_bd + x_cd = 1. Similar constraints can be added for Typing and Packing.

Altogether there are 6 constraints. You can now supply this 9-variable, 6-constraint problem to an ILP solver and it will spit back out the values for the variables in one of the optimal solutions — exactly 3 of them will be 1 and the rest will be 0. The 3 that are 1 tell you which people should be doing which job!

ILP is General

As it happens, this particular problem has a special structure that allows it to be solved more efficiently using a different algorithm. The advantage of using ILP is that variations on the problem can be easily incorporated: for example if there were actually 4 people and only 3 jobs, then we would need to relax the constraints so that each person does at most 1 job, instead of exactly 1 job. This can be expressed simply by changing the equals sign in each of the 1st 3 constraints into a less-than-or-equals sign.