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I heard that you can use Linear Programming for planning problems. I dont really understand how to do that because Linear Programming is optimal and planning in large scale (for example planning n jobs on m machines) has exponential difficulty.

So how can i solve for example a problem 100 jobs and 10 machines using linear programming? Can you give me some explanation or further reading?

MrWoffle
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    `Machine Scheduling` problems can be very difficult to solve to optimality. Besides Mixed Integer Programming (MIP), Constraint Programming (CP) and diverse heuristics are used to solve these problems. There is a ton of literature. – Erwin Kalvelagen Nov 09 '16 at 17:05
  • thx, is there anything literature you would recommend to start with – MrWoffle Nov 10 '16 at 12:41
  • Baker/Triesch, Principles of Sequencing and Scheduling [link](https://www.amazon.com/Principles-Sequencing-Scheduling-Kenneth-Baker/dp/0470391650) – Erwin Kalvelagen Nov 11 '16 at 00:01

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So how can i solve for example a problem 100 jobs and 10 machines using linear programming?

Generally, you can't. That isn't the sort of planning problem that Linear Programming (LP) is applicable to.

In an LP problem, you have a set of variables that you want to solve for. You have a set of linear inequalities that represent the constraints on those variables. And, you have a linear function of those variables (i.e., no exponents, no division, no "if-then-else", etc) that represents the cost (or benefit) of a given solution.

If you have a problem like that, you can use LP to efficiently generate an optimal solution. Shop floor scheduling, like what you are asking about, isn't that kind of problem.

LP tends to lend itself to "higher level" planning. Like, how much of each product should I make in each factory? In such a problem, you'll often be able to specify the constraints as linear inequalities and the cost (or benefit) as a linear function, as you must do in order to make use of LP. Notice, I said "how much of each product..." and not "how many...". Because that's another limitation of LP -- the variables must be able to take on real values. If you need your solution to give integer solutions, you're looking at an Integer Programming (or Mixed Integer Programming) problem.

Matthew McPeak
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