## Friday, July 1, 2011

### A brain teaser's solution

Recall that on Tuesday I posted what I described as a "brain teaser," involving the allocation of flight costs across three clients in two cities. The total cost is \$710.10. Let's begin with some suggested solutions.

Allocate evenly: one-third to each

This is clearly the easiest solution to apply, but is it the most equitable? Let's say that the normal flight to and from Boston is \$300, while it's usually \$900 to Chicago. If we divvy up the \$710.10 evenly, everyone's cost is below what it normally would be, but the Chicago client saves a much larger amount, from a proportionality perspective: is this fair? I don't think so.

Allocate based on the number of days

We're at Clients A and C one day each, and Client B for two days, thus Client B gets to pay half the fee, while the other two pay one-quarter each: rather simple to apply. However, what does it matter how long we're on the ground at each city? Let's say that I'm flying to Boston, and the normal cost is \$300, and I'm there for four days; I then fly to London, and the normal cost there is \$700, and I'm there just one day. And so, the Boston client pays four-fifths of the cost, even though a normal cost there is much lower; again, the other client benefits greatly while the other seems to be penalized.

Split the leg costs up

Here, the specific response was as follows:
• I would charge the Boston client the cost of the ticket from Newark to Boston
• I would charge the Chicago client the cost of the ticket from Chicago to Newark
• I would split the cost of the ticket from Boston to Chicago between the two clients
This requires finding out the one way costs from Newark to Boston, Boston to Chicago, and then Chicago to Newark. The charging of these fees seems fairly equitable, yes? And splitting the costs between the two legs also seems fine. I would say that this is an acceptable approach, though not what I do.

Allocate based on the normal relative round trip costs

We find out what the normal round trip cost is to and from each city. We then add these together, and determine a ratio of what they would normally pay relative to one another. We apply this ratio to the actual costs. The following table shows the details:

I think this provides an equitable allocation of the costs, but agree that the third method, too, seems worthy of consideration.

Happy 4th

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