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Sorry for the generic-sounding title.

I am currently tasked to solve a decision problem. Currently there are two factors that I must consider before making the decision.

Suppose a customer walks into a building to do whatever business he needs to do. The building is big and there are multiple entrances. There are multiple counters that are scattered around the building offering same services.

My role here is to direct this customer to the right counter based on two sets of information. Very much like those machines that issue a queuing number, but in this scenario I am giving the exact instruction on which counter to go.

The first given information is a set of fixed distance from the customer to each respective counter (note: there are multiple entrances, hence the distance to counters differs). Second is a set of randomly collected historical data that records the transaction time for each of the counter. How do I make decision based on the given data?

Ideally we prefer nearest counter, but the calculation of the distance is flawed. The calculation was a direct distance from customer to each counter. In fact, the customer can speed up the travel time through escalators, lift etc. The transaction time is always preferred, however sometimes we don't have enough collected data to make good decision.

I am currently trying putting these information to a decision-matrix, but I find it difficult to put in the values. For the distance criterion, I am putting 1 for the nearest counter, 0 otherwise. However, for the transaction time criterion, I am still trying to figure out what to fill in. I have tried putting probability of transaction time < threshold value for each counter but sometimes I am still not getting an optimal value.

This also brings in a new set of problems, how do I choose the proper weight, and what is the threshold to determine "good transaction time". I have also tried using mean and median, and scale them in range of [0, 1] too.

Just to help visualizing the problem, I am putting them into a table as follows

|           | count(good time) | count(total sample) | P(Good time) | distance   | Total                     |
|           |                  |                     | (weight=8)   | (weight=2) |                           |
|-----------|------------------|---------------------|--------------|------------|---------------------------|
| Counter 1 | 0 (no data)      | 0                   | 0            | 1          | (8 * 0.0) + (2 * 1) = 2   |
| Counter2  | 4                | 10                  | 0.4          | 0          | (8 * 0.4) + (2 * 0) = 3.2 |
| Counter3  | 12               | 20                  | 0.6          | 0          | (8 * 0.6) + (2 * 0) = 4.8 |

My question is, is decision-matrix a good way to solve this problem? if not what other approach(es) i can try?

EDIT: this is not a queue related question, assume there's no queue.

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