The mathematics of combining the skewed probablity distributions of each task in the critical chain are not that simple. The resulting distribution curve for the entire critical chain has an extremely long tail and a very low "body". If we look at the median of this curve then it is way to the right of the sum of the medians of the individual critical chain tasks. So just adding up the (trimmed) estimated task times does not give us anywhere near the correct answer for the duration of the project which gives us a 50-50 chance of finishing it on time, ie the project buffer should start much later than we are told. The fact that projects run using critical chain project management finish on time therefore implies that there is still too much safety in those projects.

On the other hand, if we look at the duration which gives us a 90% probability of finishing the project on time then we find it is way to the left of the the sum of the 90% times for each task. So the overall project is relatively much less risky than any individual task.

The result is that the level of safely required for the entire critical chain (ie the project buffer) is much less (proportionately) than is required for any individual project, which justifies Goldratt's claim that we can throw away much of the assumed saftey in each task. How long should the project buffer be? Unless we know the real distribution curves for the tasks involved then we cannot know and this is why we need evidence from particular industries in order to build some heuristics.

This started me thinking about what exactly is a dollar-day. I think it's simply a day in the life of a dollar bill. Imagine this drawn as a little green rectangle. Then a week in the life of that dollar bill is 7 such rectangles, which we can arrange in a line. Now, on each day the bill belongs to someone. We can mark each rectangle with that person's name. If I own the bill on Monday and Tuesday then those two rectangles show 'Tom'. If I then buy a cup of tea for a dollar from Alice then Wednesday's rectangle shows 'Alice'. In fact, Alice saves the dollar and so the remaining rectangles (as far in the future as we care to look) all contain her name.

The more rectangles which have my name on (ie the longer I possess the dollar bill), the more choice I have. My choice is the same every day: spend the dollar or save the dollar. We are calling each of these little rectangles a "dollar-day". So the more dollar-days I possess the more choice I have in my life. I could say to Alice, "You can have the dollar bill today or at the end of the week.". What will she do? She is guaranteed to take the money earlier rather than later. Why is it so important to have the money as soon as possible? It's often instructive to think of extreme cases. In one case we give a baby a dollar bill and in the other we give a dollar to a person on their death bed. In the first case we are being much more generous to the person: the baby will grow up and could choose any day during his or life to spend the dollar.

What if Alice takes a couple of days to make my cup of tea (yes, I'll get very cold tea). I wanted to pay her on Wednesday but she now has to wait until Friday to get her money. She has missed possession of two dollar-days. The dollar still exists but the dollar-days belonged to me during a time when she should have had them. From Alice's point of view these are lost throughput dollar-days. I'm annoyed because I don't have the tea when I was promised it. You might think I would be happy to have possession of two more dollar-days than I was expecting. However, my choice has been denied: I wanted to spend my dollar on Wednesday; in fact, I didn't want those two dollar-days.

The reason for lining up dollar-days as a sequence of rectangles is that we can reason about sets of dollar-days geometrically. For example, the month-long history of ten dollars would be ten sequences of small rectangles stacked on top of each other. This is a large rectangle with dimensions 10 (dollars) by 30 (days). I find it useful to teach the arithmetic of dollar-days by manipulating such large rectangles.

So, I told the client, you can't spend a dollar-day; you can either own it or not own it.

How do the goals of each of these systems, subsystems and supersystems interact? Classically, we know that measurements applied to the departments of a firm often move the firm itself away from its goal. And that's because each department's goal does not recognise the goal of the supersystem. To what extent does this happen? To what extent is it the case that a subsystem's goal is always in conflict with it's supersystem's goal? Considering a system as a network of dependencies, graph theory tells us that there are a great many possible subsystems (subgraphs) of the main system. To what extent, in reality, do each of those subgraphs represent a "real" subsystem with its own goal? At what point do we stop moving upwards, looking at higher and higher level supersystems and say: "these are separate systems for which it is not worth looking for a common goal"?

Each system is made of individuals working. Each of these people is a valid system with its own goal. If we come up with a framework for examining goal-interaction between systems and their components then it should be able to model the way people interact with each other either in the effort to reach a common goal or in the effort to attain their goal at the expense of others' goals or of the goal of the society or organization they belong to.

I think all organizations have exactly the same goal.

In order to introduce this goal, let me start by talking about money or, rather, money multiplied by time. We are familiar with the concepts of 'lost throughput dollar days' and 'inventory dollar days'. We know how important it is to know not only how long something has been happening but also how much money that thing is worth to an organization. In another thread I'll put down some thoughts about what exactly a 'dollar day' is but, for now, it represents a certain amount of choice. The more money I have and the sooner I have it, the more choice I have about what to do.

The for-profit goal, above, is all about money and time. I therefore believe it is all about choice.

On the other hand a for-purpose organization generally has more trouble articulating its goal. Sure, they have many aims. A hospital, for example, is about caring for patients (but not to the point of keeping them longer than necessary); a charity wants to do good in the world (but not to the point of being starved of cash); a school wants to prepare pupils for the world (without spending too much time on each pupil).

If we look at the most successful for-purpose organizations in each of their sectors, I'm convinced we shall discover an important property. I believe those leading organizations will each have the most choice about the way they operate and the work they do. I believe that the aims of each for-purpose organization can be translated into questions of choice. A hospital wants more choice now and in the future about which patients it treats and what it does for them; a charity wants more choice now and in the future about which people it helps and how it helps; a school wants more choice now and in the future about which pupils it takes on and what it provides for them; all the time, fulfilling legal and moral requirements.

So, maybe the goal of every organization is to have ever more choice while satisfying necessary conditions?

We must define 'choice' but I suspect it will involve the number of possible states an organization can be in while fulfilling its remit.

However, he just came back with another question: "What are the conservations laws of the Theory of Constraints?". I thought I was in for a Socratic dialogue and mentally geared up but he then professed that he had no idea what they were but that they must exist.

At this point we were being called back to the seminar he was presenting and the question just hung in the air. I mulled over it every now and again over the next few weeks and tried to understand a) what a conservation law is; b) why at least one "must" exist; c) how to find one.

The familiar high school conservation law is that of linear momentum. How many rocks have been hypothetically thrown out of the back of how many boats to show that total mass x velocity of the system remains the same. Similarly, we were told that ice skaters spin faster when they pull their arms in because they conserve angular momentum. Finally, we all know the law of conservation of energy: that energy just gets transformed but does not disappear from the system as a whole. Conservation is about something staying the same even though other things change.

Why must conservation laws exist in human organizations (the most important domain of the Theory of Constraints)? If nothing gets conserved then we surely have chaos and, in particular, unpredictability (as I'll discuss in a moment): changing one aspect of a system does not constrain another part of the system. Well, we know that's not true! We also know that humans and the organizations they exist in are far from unpredictable, which is why we are allowed to analyze them in a rigorous, scientific way. Properties of these organizations must be conserved.

How can we find TOC's versions of conservation laws? I started with a feeble approach by thinking about the logic we all rely on to exist with each other. At all times we all have a complete (ie conserved) logical foundation underpinning our beliefs about the way things are. If a fact changes to challenge those beliefs then we dismiss the fact or juggle our underlying assumptions to accommodate it. I was playing with the idea that this scales up to an organization-wide level where organizations we are trying to help predictably change the way they understand the world in the face of TOC's impeccable reasoning. However, this was all just arm-waving and not well thought-through.

So I started looking for the meaning of conservations laws in physics and came across a lovely theorem by the mathematician, Emmy Noether, who proved, in 1915, that if a system has certain simple properties that any symmetry found in that system leads to a conservation law. I'm not a pure mathematician so I haven't been through the proof or fully understand the statement of the theorem yet. Symmetry exists in a system if the system can be transformed in some way but the system appears exactly the same after the transformation as before, such as rotating a square by 90 degrees.

When we look at particular examples of her theorem, such as the law of the conservation of linear momentum, then things start to look just a bit more understandable to someone without a background in group theory. There is a something called a 'Hamiltonian' which is a description of a physical system in terms of its kinetic energy and its potential energy. If you throw a ball to me then its potential energy increases as it goes higher (it receives gravitational potential energy) and as it moves towards me its kinetic energy changes as its velocity changes. Something in physics called the 'Principle of Least Action' says that if the ball has to reach me in a certain time then the parabola it follows in the air is fixed and predictable: the difference between its potential energy and its kinetic energy is always a minimum.

Now, this language appeals to me. I'm nowhere near finding TOC analogies for these concepts but the seed is there: what is the potential or kinetic energy of a human system; is there a principle of least action which means things are highly predictable; what are the symmetries of a human system; and so on.

In snatches of spare time, I'm slowly trying to understand how these things work in physics in the hope that it gives me enough intuition to transfer the ideas to the arena we work in.