1. Handling Multiple Loaders ©Dr. B. C. Paul 2000 Revised 2008 The Binomial Formula is widely published in works on statistics. These slides contain screen shots of binomial calculations worked in Excel (a Microsoft Product) and Caterpillar’s FPC program.

2. Availability l Truck and Loader Systems require two things to work at the same time If you have one truck available 85% of the time One loader available 85% of the time Combination is available 0.85*0.85= 0.7225 or 72.25%

3. Problem Comes with Multiples l If I have 5 trucks and 1 loader (like fleet #3) I could have 5 trucks running 4 trucks running 3 trucks running 2 trucks running 1 truck running 0 trucks running l The number of trucks running raises or lowers the extend of over or undertrucking Changes the amount of wait time for a truck to load

4. Getting More Complex l With just one loader The truck fleet either can work or cannot l Gets more complicated with two or more loaders l Key to truck production is Binomial Distribution

5. Binomial Distributions l If a result is 0 or 1 and a certain proportion are 1s and a certain 0’s called a binomial distribution l Used to model truck and loader systems Let 1 mean truck or loader available Let 0 mean not available Let the proportion of 1’s be equal to the proportion of time a truck or loader is available.

6. Back to the Truck Setting l Our trucks have 85% availability We set p for our binomial distribution equal to 0.85 We set q (not available) to 0.15 l We have 6 trucks (5 for the Cable shovel case) 6 trucks running 5 trucks running 4 trucks running 3 trucks running 2 trucks running 1 truck running 0 trucks running

7. What is the Probability of having 5 trucks running? l Formula is N is total number of trucks n is the number running l Plug and Chug

8. Of Course There is More Than One Configuration for 5 Trucks Running l For simple cases we can count the number of ways it could happen l For more involved cases we can use the binomial coefficient N! means N factor For N=6, N! = 6*5*4*3*2*1

9. For Our 5 Trucks Running Case l Plug and Chug l Probability of a situation with 5 trucks running is 0.0665557 with 6 ways it could happen 0.066557 * 6 = 0.3993 or 39.93% of the time

10. Can Use the Binomial Formula to Tell How Much of Time We’ll have what ½% for 2 4.14% for 3 17.62% for 4 39.93% for 5 37.71% for 6

11. If You Suspect I Can Do This for Two Loaders Your Right 2.25% all out 25.5% just 1 72.25% all

12. So How Do I Deal With Both a Once? Cross Multiply it Out to get ratio of time you will Have any number of trucks or loaders.

13. Checking FPC You can see FPC gets The production with 6 trucks and 2 loaders

14. Good Program – Bad Screw Up Fleet Availability is 0.85 * 0.85 = 0.7225 (it is not true that all or None of the trucks are Available or all or none Of the loaders) At 100% availability Production is 7,658,623 0.7225 = 5,533,354 WOOPS

15. How Bad is the Error Proportion of time in that State X Annual Production achievable in that state = Production Achieved by that State Sum of Which is Annual Production 6,045,000 tons

16. Commentary l We Suffered a 10% Error Not negligible l Why did the answer change Because we neglected partial credit for part of the fleet being available Equipment ratio and bunching is non-linear

17. Coping l Some of Caterpillar’s Cycle times are optimistic l Program gets closer because under-estimating availability covers part of over-estimating production Over-estimated production leads to more TMPH problems being predicted

18. How to Correct FPC l Use a Binomial Probability Spreadsheet (as I did) l Run FPC at 100% availability in each state l Enter them in a table l Multiply the production table by the probability table l Sum up the answer.