Methods: The mathematics of management

*by David Blakey*

Understanding the mathematics of management can help you be a better consultant.

[Monday 14 April 2003]

I wonder if management mathematics is still taught. It seems that many managers do not understand the mathematics of management. I think that it is essential that consultants do.

## A client compares options

Here is a real example. I discovered that one of my clients had had a decision to make about which of a number of possible options they should select. It does not matter what the options were or what the subject of the decision was. What is important is that the options all ranked roughly the same in a simple mathematical evaluation.

So, one of the managers suggested that they should work through the list of options. They should decide whether they preferred the first option or the second option, then take their choice and decide whether they preferred it or the third option, and so on. This would produce a single preferred option. I found out about this too late to prevent it, but I was able to point out the obvious flaws and get them to review all the options again. When they did so, a different one was chosen as the preferred option.

## The ice cream paradox

To explain the problem, let us consider only three options. These will be flavours of ice cream.

- Given a choice of vanilla or strawberry, I will choose vanilla.
- Given a choice of strawberry or chocolate, I will choose strawberry.
- Given a choice of chocolate or vanilla, I will choose chocolate.

Non-mathematicians are used to **transitive** sets. If **a** is greater than **b** and **b** is greater than **c**, then **a** must be greater than **c**. The options and the ice creams are **intransitive**. If I prefer **a** to **b** and I prefer **b** to **c**, it does not follow that I will prefer **a** to **c**.

Consider the ice cream example. There will be reasons for my preferring vanilla to strawberry. There will be different reasons for my preferring strawberry to chocolate. There will be a third set of reasons for my preferring chocolate to vanilla.

Now let us look at how decisions based on a transitive view will be made, in the same way as my client made their decision on the options. We shall compare the first two options, then compare the preferred option from those two with the third option, and so on. Let us look at ice cream choices in the order of vanilla, then strawberry and then chocolate.

I compare vanilla with strawberry. I prefer vanilla. Then I compare vanilla with chocolate. I prefer chocolate. My choice of ice cream will be chocolate.

If we change the order to chocolate, then strawberry and then vanilla, I start by comparing chocolate with strawberry. I choose strawberry. Then I compare strawberry with vanilla. I choose vanilla. My choice of ice cream will be vanilla.

The precedence of the individual choices remains the same. Given a choice of vanilla and strawberry, I will always choose vanilla. The problem is with the order in which I make the choices. I can change the order of the ice creams so any given flavour will be selected.

In the case of the client, their options were listed in a given order. That order may have been random. It is possible that a different final choice would have been made if the options had been in a different order. If the order of the options was indeed random, then their final choice was also random.

The real flaw is with the method. Making a choice between the first two options and then between that choice and the third option, and so on, is illogical.

## How consultants can help

Consultants can help clients who do not understand the mathematics of management. When you explain it to clients, you will notice how often they tell that it is all obvious, now that you have explained it.

I recommend that you have a good knowledge of the mathematics of management.

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