1. © J. Christopher Beck 20051 Lecture 25: Workforce Scheduling 3

2. © J. Christopher Beck 2005 2 Outline Crew Scheduling Algorithm 12.6.1 Example 12.6.2

3. © J. Christopher Beck 2005 3 Crew Scheduling Given a set of flight legs Crew requirements for each flight Find an assignment of crews to flight legs so that all crews return home and they can actually crew the flights!

4. © J. Christopher Beck 2005 4 Crew Scheduling Flight legs have timings Crews need to be in the right place at the right time Crews have work regulations Break periods, maximum time without rest, etc. Crews for all flights within an airline need to be coordinated!

5. © J. Christopher Beck 2005 5 Crew Scheduling Important in transportation industry planes, trains, trucks, buses, … Often you really have a combined routing and crew scheduling problem Vehicle routing plus crew scheduling!

6. © J. Christopher Beck 2005 6 Problem Definition m jobs (e.g., a trip “leg”) Each job has a start and end place and a time interval There are n feasible combinations of legs that a crew can handle n is very large! One leg

7. © J. Christopher Beck 2005 7 Problem Definition Assume each feasible combination, j Must be a round trip start and end points are the same Has time constraints May have other constraints Rest times, etc Has a cost c j

8. © J. Christopher Beck 2005 8 Problem Definition Find the set of round trips with minimum total cost such that each leg is covered exactly once

9. © J. Christopher Beck 2005 9 More Formally (IP) a ij = 1 if leg i is part of round trip j (one crew per round trip) x j = 1 if round trip j is selected minimize subject to Known as the set partitioning problem It’s hard!

10. © J. Christopher Beck 2005 10 A Simple Example What is the minimum cost set of round trips? Construct the constraint matrix 1 2 3 4 5 6 Round trip LegsCost 13,2,110 24,2,15 33,5,610 44,5,610

11. © J. Christopher Beck 2005 11 Alg 12.6.1: A Heuristic Step 0: Pick any solution, J 1 Step 1: J 2 = {} Step 2: Find k, the round trip with max potential savings Step 3: Remove legs in k from other trips Step 4: Add k to J 2, remove trips with no legs Goto 2

12. © J. Christopher Beck 2005 12 Solving our Example Let J 1 = {1,4}, cost = 20 Find the row costs: Round trip LegsCost 13,2,110 24,2,15 33,5,610 44,5,610 such that

13. © J. Christopher Beck 2005 13 Solving our Example Round trip LegsCost 13,2,110 24,2,15 33,5,610 44,5,610 j = 1 j = 4 The current cost of leg i

14. © J. Christopher Beck 2005 14 Alg 12.6.1 Step 0: Pick any solution, J 1 Step 1: J 2 = {} Step 2: Find k, the round trip with max potential savings Step 3: Remove legs in k from other trips Step 4: Add k to J 2, remove trips with no legs Goto 2

15. © J. Christopher Beck 2005 15 Potential Savings The cost of round trip j based on current prices The real cost of round trip j Round trip LegsCostPotential Savings 13,2,1100 24,2,155 33,5,6100 44,5,6100 Calculate potential savings

16. © J. Christopher Beck 2005 16 Alg 12.6.1 Step 0: Pick any solution, J 1 Step 1: J 2 = {} Step 2: Find k, the round trip with max potential savings Step 3: Remove legs in k from other trips Step 4: Add k to J 2, remove trips with no legs Goto 2

17. © J. Christopher Beck 2005 17 Solving our Example Round trip LegsCost 13,2,110 24,2,15 33,5,610 44,5,610 Round trip LegsCost 1310 24,2,15 33,5,610 45,610 J 2 = {}J 2 = {2}

18. © J. Christopher Beck 2005 18 Back to Step 2: Potential Savings Round trip LegsCostPotential Savings 1310-6.6667 24,2,15- 33,5,6100 45,610-3.333 Calculate potential savings J 2 = {2,3} Cost = 15

19. © J. Christopher Beck 2005 19 Back to Step 1 Round trip LegsCost 13,2,110 24,2,15 33,5,610 44,5,610 Calculate row costs Calculate potential savings J 2 = {2,3} Cost = 15 J 3 = {}

20. © J. Christopher Beck 2005 20 You Should Understand Alg 12.6.1 Do Example 12.6.2 Do the problem we just did but with another round trip: 1 2 3 4 5 6 Round trip LegsCost 13,2,110 24,2,15 33,5,610 44,5,610 52,1,3,5,6,416