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I've studyed some Operations Research (maths), as well as some Project Management (for engineers) at Uni.

It is fairly well known that directed Graph with weighted edged (such as a PERT Chart, and family.) can be reduced into a series of inequalities, over the weights, which can be solved optimally via Linear Programming.

pert chart source: Wikimedia commons
Is reduced to in equalities over the times the milestones are reached.
Constraints:
t10 ≥ 0
t30 ≥ t10 + 3
t20 ≥ t10 + 4
t40 ≥ t30 + 1
t50 ≥ t30 + 3
t50 ≥ t40 + 3
t50 ≥ t20 + 3
Optimisation function: Minimise t50

Linear Programming techniques can be applied to this, the ideal ordering be determined, critical path found etc, mby using the simplex algorithm. More advanced linear programming techniques (that I can no longer remember how to do) can be applied to determine the sensitivity of the solution to changes in the constraints etc.

So this all seems very logical and great. But I've never seen the terms Operation Research or Linear Programming ever show up in my project management lectures, or in PM text books etc. Is Linear Programming actually used in real world Project Management for planning?

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Any use of linear programming within the field of project management would be wrapped within a software package. All project worthy of being called a project are simply too large to have their calculations done outside of project management software.

Finding critical paths, resource leveling, etc. are all part of software packages. While they do get taught in PM courses and in PM texts, they are normally taught on a very small scale where manual calculations do not require mathematical models.

So, in short, teaching optimizing is done on very small scales and can be done without linear programming. In real life, we use software to deal with it.

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