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Global and Local Optimizations

Is NP=P and why should we really care?

Did you notice that when something happens to you, it seems to occur to other people as well? For example when you have toddlers, suddenly you see toddlers everywhere - and naturally they are all misbehaving compared to yours J, or when you're planning your wedding, all around you, people are planning theirs, and white dresses become scarce. Lately I had this feeling myself; I was reading a David Baldacci thriller, when half way through, he introduces NP problems and the consequences of finding that P=NP. In addition, I have lately been reading many articles about Critical chain project management. I feel compelled to contribute and provide my take and insights.

There are two kinds of problems, the easy ones which we solve at school, and the hard ones. We might remember the method to solve a quadratic equation by using the x = [-b/2 ± √((b/2)^2 - ac)] /a formula. We are less sure on advising a salesperson how to plan a sales trip to 20 cities in which he can visit every city once and has a limited budget (travelling salesperson problem). When it comes to packing for the holidays we are truly baffled. How are we supposed to fit all the stuff into one checked-in luggage? After continuous deliberation, we land in the exotic resort only to find our bathing suit missing...

Unfortunately and also luckily (more on the luck aspect - later) most of the problems around us are unsolvable, or should I say - not computable in an optimal manner in a reasonable amount of time.

How is this related to project management?

The difficulties we are experiencing managing projects successfully in a resource constrained environment stem from the fact that the Resource Constrained Multi-project Scheduling Problem is very difficult to solve in an optimal manner. Moreover it can't be optimized at all. Multi Project Resource Constrained Scheduling (acronym MPRCS) is known as a NP complete (or NP hard - until this day I am not sure if I understand the exact difference) complexity type problem. In a nutshell, NP problems are very complex mathematical problems. Actually, as mentioned above, most of the problems that we are facing are of NP complexity. For example, those of you who like to travel to Alaska and need to choose equipment from a long list of possibilities, with capacity constraints, are facing an NP problem (knapsack problem).

The theory claims that if we solve one NP problem we can solve all of them. As a byproduct of the solution we would receive a noble prize, render the entire world of encryption useless, and probably get kidnapped by the NSA and never heard of again.  So far this hasn't been the case and NP problems are not computable in an optimal manner in a reasonable amount of time. Hence - luckily for us - long encryption keys are at present undecipherable in a reasonable time, which enables secure global communication networks.  Conspiracy theorists maintain that governments have already developed computational abilities that can solve NP problems - rendering encryption as we know it useless. Dan Brown in his bestseller book Digital Fortress and David Baldacci in his book Simple Genius venture down this path.

Back to project management. Since scheduling is a NP problem, we can't optimally solve a multi project resource constrained system.

Wait just a minute - am I saying that there aren't computerized tools to optimally schedule a bunch of related projects who share resources??? Truth be told, there are tools offering nearly optimal solutions and granted, while they do not provide an optimal solution they calculate, using complex algorithms, a good enough solution. The time required for providing this solution can be very long, depending on the complexity of the problem. There are environments which benefit from these sophisticated optimization tools. Consider for example a medium-size factory with 100 machines and several thousand components. Running the scheduling software for the defined production floor is lengthy and can take up to 3 days. The specific output of the software is a production plan for the explicit setting. The drawback is that once implemented, any unforeseen change in production requires: either rerunning the optimization software which doesn't make sense duration wise, or applying quick fixes which render the nearly optimal production plan obsolete. If ever you watched an operating factory, you have seen the chaotic nature and constant changes with which it is plagued.

There are industries in which nearly optimal scheduling is vital and as the environment is complex, they must invest in a sophisticated software optimization tool (do remember that the solution is nearly optimal). One example is the airline industry. They can't manually allocate numerous aircrafts and particular crews across several hundred destinations worldwide. When Lufthansa allocates specific aircrafts, along with the constraint of having each aircraft operated by a predefined crew, it makes sense to purchase an advanced and pricey optimization tool. Indeed, once a month the scheduling team runs the optimization application which serves as the basis for aircraft and crew allocation, ticket pricing, and routing worldwide. However if one of the pilots is having the shrimp in coconut sauce for dinner and suffers from food poisoning the next day, entire segments of the optimized schedule collapse and emergency escalation protocols are initiated. The central scheduling team in Frankfurt must find a quick fix to the shrimp ordeal.

Ok - for most of our projects it doesn't make sense to invest so much in building the infrastructure to solve, in a near optimal manner, the multi project resource constrained problem.  Especially as the plan is useless once changes occur. All it takes is for a critical resource to be sick one day, for the entire plan to go bust. What is then the alternative? Heuristics - or as we better know them - rules of thumb. These enable local and semi global optimizations. The two famous heuristics to resource constrained project scheduling are: Agile and critical chain. One handles the local optimization and the other semi global optimization. While Goldratt who wrote about critical chain project management knew about heuristics and global optimization, Agile proponents don't really think of it as a local optimization mechanism, but we will leave it for future articles.

More on critical chain, and Goldratt's approach to solving the complexity of resource constrained scheduling, some other time.

A FREE voice over recording of this article is found here.

You can check out my book: The Agile PMO - leading the effective, value-driven, project management office. - - where I further discuss these concepts:

The Agile PMO

More Stories By Michael Nir

Michael Nir - President of Sapir Consulting - (M.Sc. Engineering) has been providing operational, organizational and management consulting and training for over 15 years. He is passionate about Gestalt theory and practice, which complements his engineering background and contributes to his understanding of individual and team dynamics in business. Michael authored 8 Bestsellers in the fields of Influencing, Agile, Teams, Leadership and others. Michael's experience includes significant expertise in the telecoms, hi-tech, software development, R&D environments and petrochemical & infrastructure industries. He develops creative and innovative solutions in project and product management, process improvement, leadership, and team building programs. Michael's professional background is analytical and technical; however, he has a keen interest in human interactions and behaviors. He holds two engineering degrees from the prestigious Technion Institute of Technology: a Bachelor of civil engineering and Masters of Industrial engineering. He has balanced his technical side with the extensive study and practice of Gestalt Therapy and "Instrumental Enrichment," a philosophy of mediated learning. In his consulting and training engagements, Michael combines both the analytical and technical world with his focus on people, delivering unique and meaningful solutions, and addressing whole systems.