# How to compare utilization rates?

I have the utilization rates of several machines for each week over a year. These differ per week because of the occurrence of machine failures and changes of orders. Meaning that one week a machine could be a bottleneck and the other week not. I want to rank these machines based on the utilization and so determine their criticality. However if I calculate the mean utilization, information gets lost. The same is when I count the events the utilization is bigger than a threshold. For example;

• machine A has a utilization of 50% and 14 weeks the utlization > 90%
• machine B has a utilization of 70% and 7 weeks the utlization > 90%
• machine C has a utliziation of 68% and 7 weeks the utlization > 90%

In my opinion machine A is more critical than machine B. So ranking based on a threshold value seems reasonable. However when comparing machine B and C this is not that clear. Does anyone know a statistical/mathematical method to get a good comparison?

I found a solution:

The problem is a multi-criteria decision analysis (MCDA), as I want to rank the machines based on several (available) criteria. With intially the following criteria; downtime costs and utilization. However with utilization there was the following problem; if the events are counted that surpasses a threshold or the average utilization is used, both situations would lead to loss of information. As it is a MCDA, both criteria can be added, thus the criteria are now:

1. downtime costs per time unit
2. average utilization per timespan
3. number of events the utilization that surpasses a threshold (e.g. 95%)

As these factors are not directly commensurable and differ in importance, a weighted method is needed. To select the appropriate method 7 guidelines are used (Guitouni and Martel, 1998);

• Guideline G1: Determine the stakeholders of the decision process. If there are many decision makers (judges), one should think about group decision making methods or group decision support systems (GDSS).
• Guideline G2: Consider the Decision Maker (DM) `cognition' (DM way of thinking) when choosing a particular preference elucidation mode. If he is more comfortable with pairwise comparisons, why using tradeoffs and vice versa?
• Guideline G3: Determine the decision problematic pursued by the DM. If the DM wants to get an alternatives ranking, then a ranking method is appropriate, and so on.
• Guideline G4: Choose the Multi Criteria Analysis Problem (MCAP) that can handle properly the input information available and for which the DM can easily provide the required information; the quality and the quantities of the information are major factors in the choice of the method.
• Guideline G5: The compensation degree of the MCAP method is an important aspect to consider and to explain to the DM. If he refuses any compensation, then many MCAP will not be considered.
• Guideline G6: The fundamental hypothesis of the method are to be met (verified), otherwise one should choose another method.
• Guideline G7: The decision support system coming with the method is an important aspect to be considered when the time comes to choose a MCDA method.

The result based on all guideline except guideline G6, results in the following suitable methods: Analytic Hierarchy Process (AHP) or Promethee II. However AHP has the assumption that inner and outer criteria are independent. A correlation test showed that, there exists significant correlation between criteria 2 and 3, thus the criteria are not independent. Promethee II is therefore the appropriate method for ranking the machines given my problem situation.

Guitouni, A., Martel, J.-M., 1998. Tentative guidelines to help choosing an appropriate MCDA method. Eur. J. Oper. Res. 109, 501–521.