Possible Schedule Permutations

[wesleybankston]

I am working on a few problems for work where I need to determine the amount of schedule permutations for 3 recurring events with variable repeating periods. Let's say these three events are:

• Event A
• Event B
• Event C

Let's say events A, B and C repeat themselves every 5 years for an infinite amount of years. The events are allowed to occur in the same year (i.e. Events A, B and C can all occur in Year 1 and then Years 2, 3, 4 and 5 would all be blank).

Question 1: How many possible schedule combinations/permutations (note: I'm not sure which is the correct terminology) are there in this 5 year window? Answer attempt: My initial belief was 5! = 120 (i.e. 5 because 3 event years + 2 empty years = 5), however I don't think this accounts for the fact that events can occur in the same year. Secondary belief: I think the answer is 5*5*5 = 125?

Question 2: Same situation as above, except Event B repeats itself every 6 years (Events A and C are still on a 5 year rotation).

Question 3: Same situation as above, except Event B repeats itself every 6 years and Event C repeats itself every 7 years (Event A is still on a 5 year rotation).

Thank you for your help! Also, what branch of mathematics would this be considered?

 

[Ishuda]

I am working on a few problems for work where I need to determine the amount of schedule permutations for 3 recurring events with variable repeating periods. Let's say these three events are:

• Event A
• Event B
• Event C

Let's say events A, B and C repeat themselves every 5 years for an infinite amount of years. The events are allowed to occur in the same year (i.e. Events A, B and C can all occur in Year 1 and then Years 2, 3, 4 and 5 would all be blank).

Question 1: How many possible schedule combinations/permutations (note: I'm not sure which is the correct terminology) are there in this 5 year window? Answer attempt: My initial belief was 5! = 120 (i.e. 5 because 3 event years + 2 empty years = 5), however I don't think this accounts for the fact that events can occur in the same year. EDIT (after some more thinking): I think the answer is 5*5*5 = 125?

Question 2: Same situation as above, except Event B repeats itself every 6 years (Events A and C are still on a 5 year rotation).

Question 3: Same situation as above, except Event B repeats itself every 6 years and Event C repeats itself every 7 years (Event A is still on a 5 year rotation).

Thank you for your help! Also, what branch of mathematics would this be considered?

First, I think what you are saying is that all events in a single year would be 5 possibilities. That is, it matters which of the 5 years the 3 events would take place or, to put it another way, the year order in which the events occur matter but the order of the particulars events inside an individual year would not matter. In that case what you are asking for is the number of permutations of the combination of three events [actually 4 events but that 4th is determined by the other three].

For Question 1 your second thoughts were correct: You can put event A in any one of 5 slots (years), event B in any one of 5 slots, and event C in any one of 5 slots, so that is the 5*5*5=125. Now if it mattered which order you put them in the slot, well that's a different matter.

For Questions (2) and (3), the questions are a little unclear. That's because the repeat cycle is not the same for all three cases like it is with Question (1). For example, for Question (2) suppose you did all three events the first year. In years 6, 11, 16, 21, 26, 31, ... you would have events A and C. In years, 7, 13, 19, 25, 31, ... you would have event B. So that combined cycle is actually 30 years. However, I don't think you mean you have the complete 30 year cycle as open slots because, if you did, choosing event B to happen in the first year and events A and C to first happen in year 6 would differ only in the first year from the first example.

So, do events A and C have to occur once in the first 5 years and event B in the first six (seven) years or are they all allowed the full 30 (35) year cycle in which to start?

 

[wesleybankston]

First, I think what you are saying is that all events in a single year would be 5 possibilities. That is, it matters which of the 5 years the 3 events would take place or, to put it another way, the year order in which the events occur matter but the order of the particulars events inside an individual year would not matter. In that case what you are asking for is the number of permutations of the combination of three events [actually 4 events but that 4th is determined by the other three].

For Question 1 your second thoughts were correct: You can put event A in any one of 5 slots (years), event B in any one of 5 slots, and event C in any one of 5 slots, so that is the 5*5*5=125. Now if it mattered which order you put them in the slot, well that's a different matter.

For Questions (2) and (3), the questions are a little unclear. That's because the repeat cycle is not the same for all three cases like it is with Question (1). For example, for Question (2) suppose you did all three events the first year. In years 6, 11, 16, 21, 26, 31, ... you would have events A and C. In years, 7, 13, 19, 25, 31, ... you would have event B. So that combined cycle is actually 30 years. However, I don't think you mean you have the complete 30 year cycle as open slots because, if you did, choosing event B to happen in the first year and events A and C to first happen in year 6 would differ only in the first year from the first example.

So, do events A and C have to occur once in the first 5 years and event B in the first six (seven) years or are they all allowed the full 30 (35) year cycle in which to start?

Thank you for your response!

Maybe some context might help convey the situation for Q2 and Q3. I am dealing with 3 pieces of industrial equipment (i.e. the events) that must be taken down for maintenance every 5, 6 or 7 years.

• The current maintenance schedule corresponds to Question 1: all 3 pieces of equipment on 5-year intervals with Event A occurring in Year 1, Event B occurring in Year 3, and Event C occurring in year 4.
• In Q2, we are trying to push Event B to a 6-year maintenance interval because we think we can safely run that piece of equipment for 6 years without compromising performance.
• In Q3, we push Event B to a 6-year maintenance interval and we push Event C to a 7-year interval.

To try to answer your last question: the events must occur once in the first 5, 6 or 7 years, depending on the repeating periods in question...i.e. for Q3: we can't run Equipment A any longer than 5 years without taking it down for maintenance, Equipment B for any longer than 6 years, and Equipment C for any longer than 7 years. Also, once each event occurs for the first time, it must adhere strictly to its repeating schedule.

Hopefully that clears things up! I greatly appreciate your help.

 

[Ishuda]

Thank you for your response!

Maybe some context might help convey the situation for Q2 and Q3. I am dealing with 3 pieces of industrial equipment (i.e. the events) that must be taken down for maintenance every 5, 6 or 7 years.

• The current maintenance schedule corresponds to Question 1: all 3 pieces of equipment on 5-year intervals with Event A occurring in Year 1, Event B occurring in Year 3, and Event C occurring in year 4.
• In Q2, we are trying to push Event B to a 6-year maintenance interval because we think we can safely run that piece of equipment for 6 years without compromising performance.
• In Q3, we push Event B to a 6-year maintenance interval and we push Event C to a 7-year interval.

To try to answer your last question: the events must occur once in the first 5, 6 or 7 years, depending on the repeating periods in question...i.e. for Q3: we can't run Equipment A any longer than 5 years without taking it down for maintenance, Equipment B for any longer than 6 years, and Equipment C for any longer than 7 years. Also, once each event occurs for the first time, it must adhere strictly to its repeating schedule.

Hopefully that clears things up! I greatly appreciate your help.

The problem is then essentially the same as the first problem. A can go into 1 of 5 slots, B can go into 1 of 6 slots, and C can go into 1 of 7 slots for a total of 210 different ways.

For your particular problem though this could mean starting on a short cycle for a particular event. That is, suppose you are just at the regular maintenance time for event B (and for simplicity that the time is Q1 of year 0) and the maintenance hasn't been performed yet. You want to extend the cycle from 5 years to 6 years. You can wait up to 1 year to perform the maintenance so the only real choice you have is year 0 or year 1 (2 choices). The year zero choice is at a five year cycle and thus is a short cycle as compared to the 6 year cycle. Suppose you choose the year 0. You can now change to a six year cycle by doing another maintenance in year 1-6 (the six choice of slots above). However, if you choose other than the 6 year slot there is an elapsed time of n years (n=1, 2, 3, 4, or 5) and you will have performed an 'un-necessary' maintenance, i.e. a short cycle.