Lecture 23: Workforce Scheduling 1 © J. Christopher Beck 2005

Outline Bank Personnel Scheduling Demo Days-off Scheduling Example 12.2.2 Lower bounds Algorithm 12.2.1 Example 12.2.3 © J. Christopher Beck 2005

Workforce Timetabling Arrange shifts and assign people to them Constraints: Number of people per shift Minimum Days off (x/k days must be off) Weekends Nurses, call centres, hotels, restaurants, plane crew, … Bank Demo © J. Christopher Beck 2005

Days Off Scheduling (Example 12.2.2) Find minimum workforce size: W nj people required on day j Assume Sun to Sat are numbered 1 to 7 Each employee given 1 out of every 3 weekends off Each employee works exactly 5 days in 1 week (Sun – Sat) Each employee works no more than 6 consecutive days © J. Christopher Beck 2005

Days Off Scheduling (Example 12.2.2) Find lower bounds on W There are 3 that are interesting 1 of 3 weekends off day 1 2 3 4 5 6 7 Sun Mon Tue Wed Thu Fri Sat Req © J. Christopher Beck 2005

Step 1: Weekends off W = 3, number the employees Let n = max(n1,n7) Assign 1st weekend off to first W – n employees, 2nd weekend off to the next W – n and so on Employee 1 comes after employee W Cycle until you assign employee 1 to the same weekend off again © J. Christopher Beck 2005

Step 1: Weekends off W = 3 n = 2 So 1 employee is off each weekend day 4 5 6 7 Sun Mon Tue Wed Thu Fri Sat Req S M T W F 1 2 3 © J. Christopher Beck 2005

Step 1: Weekends off W = 3 n = 2 So 1 employee is off each weekend day 4 5 6 7 Sun Mon Tue Wed Thu Fri Sat Req S M T W F 1 X 2 3 © J. Christopher Beck 2005

Step 1.5: Construct Off-Day Pairs Let uj = W – nj, j = 2, … 6 = n – nj for j = 1, 7 uj is the “over-capacity” on day j Let uk = max(uj) Construct a list of day-pairs with over-capacity © J. Christopher Beck 2005

Step 1.5: Construct Off-Day Pairs Pick uk Pick another day um (m ≠ k), um > 0 If all um = 0, m ≠ k, choose m = k Add (k, m) to list and decrease uk and um by 1 Repeat n times Pairs (k,k) are called “non-distinct” pairs © J. Christopher Beck 2005

Step 1.5: Construct Off-Day Pairs W = 3 n = 2 day 1 2 3 4 5 6 7 Sun Mon Tue Wed Thu Fri Sat nj uj (2, 1) uk = u2 = 3 um = u1 = 1 © J. Christopher Beck 2005

Step 1.5: Construct Off-Day Pairs W = 3 n = 2 day 1 2 3 4 5 6 7 Sun Mon Tue Wed Thu Fri Sat nj uj (2, 1) (2, 2) uk = u2 = 2 um = u2 = 2 © J. Christopher Beck 2005

Step 1.5: Construct Off-Day Pairs W = 3 n = 2 day 1 2 3 4 5 6 7 Sun Mon Tue Wed Thu Fri Sat nj uj (2, 1) (2, 2) © J. Christopher Beck 2005

Step 2: Categorize Workers Week 1 employees fall into 4 groups: T1: WE1 off, 0 off days, WE2 off T2: WE1 off, 1 off day, WE2 on T3: WE1 on, 1 off day, WE2 off T4: WE1 on, 2 off days, WE2 on n people working each weekend, so |T3|+|T4| = n |T2|+|T4| = n So |T3| = |T2| © J. Christopher Beck 2005

Step 2: Categorize Workers n = 2 (2, 1) (2, 2) T1 = {} T2 = {1} T3 = {2} T4 = {3} day 1 2 3 4 5 6 7 Sun Mon Tue Wed Thu Fri Sat Req S M T W F 1 X 2 3 © J. Christopher Beck 2005

Step 2: Categorize Workers Assign off-day pairs to workers First to T4 – they get both days off Second to (T2,T3) pairs T3 gets earlier day T2 gets later day © J. Christopher Beck 2005

Step 3: Assign Off-Day Pairs W = 3 n = 2 (2, 1) (2, 2) T1 = {} T2 = {1} T3 = {2} T4 = {3} day 1 2 3 4 5 6 7 Sun Mon Tue Wed Thu Fri Sat Req S M T W F 1 X 2 3 © J. Christopher Beck 2005

Step 3: Assign Off-Day Pairs W = 3 n = 2 (2, 1) (2, 2) T1 = {} T2 = {1} T3 = {2} T4 = {3} day 1 2 3 4 5 6 7 Sun Mon Tue Wed Thu Fri Sat Req S M T W F 1 X 2 3 © J. Christopher Beck 2005

Step 4: Assign Off-Day Pairs for Week i Categorize workers for week i Case 1: All off-days are distinct T4(i) = T4(i-1), T3(i) = T3(i-1) T4 gets both days off, T3 gets earlier day, T2 gets later day Case 2: Not all off-days are distinct Week i schedule is identical to week 1 © J. Christopher Beck 2005

Step 4: Assign Off-Day Pairs for Week 2 (2, 1) (2, 2) T1 = {} T2 = {2} T3 = {3} T4 = {1} day 1 2 3 4 5 6 7 Sun Mon Tue Wed Thu Fri Sat Req S M T W F 1 X 2 3 © J. Christopher Beck 2005

Step 4: Assign Off-Day Pairs for Week 2 (2, 1) (2, 2) T1 = {} T2 = {2} T3 = {3} T4 = {1} day 1 2 3 4 5 6 7 Sun Mon Tue Wed Thu Fri Sat Req S M T W F 1 X 2 3 © J. Christopher Beck 2005

Step 4: Assign Off-Day Pairs for Week 3 (2, 1) (2, 2) T1 = {} T2 = {3} T3 = {1} T4 = {2} day 1 2 3 4 5 6 7 Sun Mon Tue Wed Thu Fri Sat Req S M T W F 1 X 2 3 © J. Christopher Beck 2005

Step 4: Assign Off-Day Pairs for Week 3 (2, 1) (2, 2) T1 = {} T2 = {3} T3 = {1} T4 = {2} day 1 2 3 4 5 6 7 Sun Mon Tue Wed Thu Fri Sat Req S M T W F 1 X 2 3 © J. Christopher Beck 2005

Alg 12.2.1 Overview Step 0: Calculate W Step 1: Assign Weekends Off Step 1.5: Construct Off-Day Pairs Step 2: Categorize Week 1 Workers Step 3: Assign Off Day Pairs for Week 1 Step 4: Assign Off Day Pairs for Week i © J. Christopher Beck 2005

Example 12.2.3 Find a schedule using Alg 12.2.1 day 1 2 3 4 5 6 7 Sun Mon Tue Wed Thu Fri Sat Req © J. Christopher Beck 2005