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I have data that looks like this:

        Tell_1  Tell_2  Tell_3
        0       0      -8
       -3       0       0
        0       0       0
       -4      -2       0
        0       0      -2
      -14      -4      -1
        0       0       0
       -1       0      -1

Index:  I1      I2      I3

'----------------------------------------------

This data represents the wait time of each customer at a teller. The value 0 means the customer did not wait at all. The value -x means the customer had to wait x minutes before being served.

How can I develop an teller performance index (a mathematical function that takes the wait times an produce 1 value) that shows the teller performance ($I1$, $I2$, $I3$,...)?

One way is to just sum each column to obtain -22,-6, and -12. This is only good at showing wait times. However, this does not show how many customers were served immediately (which is a good thing). As a result this index is no good.

Another approach that I considered was to assume a max wait time value, say 100, then calculate the index per teller as $\sum(x_i+100)$ to get: 778, 794, and 788. But how good is this one?

I need the index to reflect both the wait times and the number of customers that were served immediately.

Note: This is not a homework, also, it is not a real situation.

Thanks for your help.

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    $\begingroup$ I am little confused. Did each customer go to each teller? So customer A went up to each teller and only had to wait with teller 3? $\endgroup$
    – MathIsKey
    Aug 19, 2016 at 18:31
  • $\begingroup$ You are going about this the wrong way. The correct way is to first ask a bunch of humans to rank a large number of tellers (real or fictional) based on their performance. Using this then you try to find a function that reproduces this ranking. Developing an "index" in the abstract is putting the horse before the cart. $\endgroup$
    – Willie Wong
    Aug 19, 2016 at 18:45
  • $\begingroup$ @AWashburn the customer need to go to 1 teller only. The tabular representation was not very good. I removed the left column in the hope to make it more accurate. Thanks for your comment. $\endgroup$
    – NoChance
    Aug 19, 2016 at 19:14
  • $\begingroup$ @WillieWong, asking people may result in bias. For some people 5 minutes wait may be a big deal. $\endgroup$
    – NoChance
    Aug 19, 2016 at 19:16
  • $\begingroup$ And why do you think that the formulae compiled by random strangers on the internet would perform any better than "asking people"? Your question itself already has the bias of "a good measure should show how many customers are served immediately" and it asks "how good is this one". Social interactions are complex systems; you simply cannot reduce it down to a foot race where a comparison of one single number gives an objective measure. // A version of your problem has been playing out in academia for quite a while now, where administrators prefer to convert humans to mere numbers by ... $\endgroup$
    – Willie Wong
    Aug 19, 2016 at 19:29

3 Answers 3

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I would recommend looking at the average wait time per customer. While this does not directly count the number of customers who did not have to wait at all, that number definitely affects the measure.

Teller #1 has an average wait time per customer of 2.75 minutes.

Teller #2 has an average wait time per customer of 0.75 minutes.

Teller #3 has an average wait time per customer of 1.50 minutes.

This would suggest that perhaps Teller #2 is more efficient.

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  • $\begingroup$ Thanks for your answer, sounds good. $\endgroup$
    – NoChance
    Aug 19, 2016 at 19:10
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I think you'll have a hard time finding a single number that will do what you want. Any reason why you can have a vector (more than one index)?

But this may do some of what you want.

Each customer a teller sees is worth, say, $20$ points. The wait time subtracts from this $1$ point per minute, down to zero. If the customer is seen without any wait, then there's a bonus of $10$ points.

By this formula, the three tellers in order would have a score of $178,214,188$.

So there's a premium on no wait time, and the score increases with the number of customers seen.

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  • $\begingroup$ Thanks for your answer. your answer looks good. I did not get your question though. $\endgroup$
    – NoChance
    Aug 19, 2016 at 19:12
  • $\begingroup$ Why do you have to have just a single number? Why not record number of customers seen, number seen without wait, average wait time of those who do wait, ... ? $\endgroup$
    – John
    Aug 19, 2016 at 19:18
  • $\begingroup$ If I have a single number I can sort the data, find max, min, etc. With a vector of different numbers, I need to define each of these operations 'logically'. However, it is valid to have a vector, I just thought a single value is better. $\endgroup$
    – NoChance
    Aug 19, 2016 at 19:20
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You could use the weighted average of the average wait time of those who had to wait and the proportion of customers served immediately. This would not have as nice as interpretation as others but it would be very flexible and allow you to decide the weight between the time of those who wait and number of customers served immediately.

You would need to pick $\alpha$ very carefully based on what you want.

$Index = \alpha*AVG + (1-\alpha)*p$

For example if $\alpha = .5$ then

Tel_1: $.5*22/4 + .5*4/8 = 3$

Tel_2: $.5*3/2 + .5*2/8 = .875$

Tel_3: $.5*12/4 + .5*4/8 = 1.75$

The lower the index the better where $0$ means no one had to wait

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  • $\begingroup$ Thanks for your answer. What does $\alpha$ represent? $\endgroup$
    – NoChance
    Aug 19, 2016 at 19:10
  • $\begingroup$ @NoChance $\alpha$ says how important you feel that the average wait is compared to the proportion of customers that don't wait. If $\alpha = 1$ then you don't care if any customers didn't have to wait. $\endgroup$
    – MathIsKey
    Aug 19, 2016 at 20:49
  • $\begingroup$ OK, I see, but selecting this parameter needs so thought. Thanks. $\endgroup$
    – NoChance
    Aug 19, 2016 at 21:00

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