Lecture 24 Workforce Scheduling 2 J Christopher Beck

Lecture 24: Workforce Scheduling 2 © J. Christopher Beck 2005 1

Outline n Shift Scheduling n n Cyclic Staffing n n n Example 12. 3. 1 Algorithm 12. 4. 1 Example 12. 4. 2 Call Centre Scheduling © J. Christopher Beck 2005 2

In Yesterday’s Episode … n We created shift patterns and assigned one worker to each pattern n All patterns cost the same, so we just wanted to minimize the number of workers S S M T W T F S 1 X X X 2 X X X 3 X X X © J. Christopher Beck 2005 X 3

Shift Scheduling n Fixed cycle n n Predefined set of shift patterns n n n Month, week, day Each worker is assigned to exactly one pattern Each pattern has its own cost Assign workers to patterns such that Staffing requirements are met and cost is minimized © J. Christopher Beck 2005 n 4

Shift Scheduling Definition n m periods: lengths may be different bi: # people required in period i n shift patterns n nj = (a 1 j, a 2 j, …, amj) = (1, 0, 0, …, 0) n n aij = 1: workers on pattern j work period i aij = 0: workers on pattern j don’t work period i cj: cost of assigning a worker to pattern j xj: # of people assigned to pattern j © J. Christopher Beck 2005 5

IP Formulation (see Example 12. 3. 1) minimize Such an IP is hard unless there is special structure such as having no split shifts subject to Each period has enough workers © J. Christopher Beck 2005 Number workers assigned to a pattern is an positive integer or 0 6

LS Formulation n Develop a local search formulation that doesn’t depend on having no split shifts © J. Christopher Beck 2005 7

Apply Your LS to Example 12. 3. 1 Pattern Hours Total Cost Hours 1 10 AM – 6 PM 8 50 2 1 PM – 9 PM 8 60 3 12 PM – 6 PM 6 30 4 10 AM – 1 PM 3 15 5 6 PM – 9 PM 3 16 © J. Christopher Beck 2005 Hour Staff 10 AM – 11 AM 3 11 AM – 12 PM 4 12 PM – 1 PM 6 1 PM – 2 PM 4 2 PM – 3 PM 7 3 PM – 4 PM 8 4 PM – 5 PM 7 5 PM – 6 PM 6 6 PM – 7 PM 4 7 PM – 8 PM 7 8 PM – 9 PM 8 8

A Variation of Example 12. 3. 1 Pattern Hours Total Hours Cost 1 10 AM – 6 PM 8 50 2 1 PM – 9 PM 8 60 3 12 PM – 6 PM 6 30 4 10 AM – 1 PM 3 15 5 6 PM – 9 PM 3 16 6 2 PM – 5 PM, 7 PM – 9 PM 5 35 Would you solve this problem with IP or LS? © J. Christopher Beck 2005 Hour Staff 10 AM – 11 AM 3 11 AM – 12 PM 4 12 PM – 1 PM 6 1 PM – 2 PM 4 2 PM – 3 PM 7 3 PM – 4 PM 8 4 PM – 5 PM 7 5 PM – 6 PM 6 6 PM – 7 PM 4 7 PM – 8 PM 7 8 PM – 9 PM 8 9

Cyclic Staffing n n n m period cycle Each period i has requirement bi Each person works k consecutive periods and has the other m – k off cj is the cost of putting a worker on pattern j Find minimum cost schedule © J. Christopher Beck 2005 10

(5, 7)-Cyclic Staffing Problem Requirements Patterns Periods Cost 1 0 0 1 1 3 1 1 0 0 1 1 1 8 1 1 1 0 0 1 1 9 1 1 0 0 1 11 1 1 0 0 11 0 1 1 1 0 11 0 0 1 1 1 4 80 70 75 90 100 © J. Christopher Beck 2005 100 105 There are split shifts! 11

Different Special Structure Makes The Problem Easy n n n Solve LP 1 (x’ 1, x’ 2, …, x’n) If integral, you have optimal Otherwise LP 2: n n n LP 1 + If integral, you have optimal Otherwise LP 3: n n LP 1 + Always integral, so you have optimal © J. Christopher Beck 2005 12

Call Centre Scheduling: Demand demand 6 AM © J. Christopher Beck 2005 12 time

Call Centre Scheduling: Shift Patterns Breaks 6 AM © J. Christopher Beck 2005 12

Call Centre Scheduling n n Assign people to shifts to meet the demand minimize costs It gets worse: workers with different skills! n {English}, {English, French}, {French} © J. Christopher Beck 2005 15

Solution Framework (Fig 12. 4) demand Select solid tours Place breaks Modify target demand Compare fits N Close enough? Y DONE © J. Christopher Beck 2005 16

Call Centre Scheduling n n Assign people to shifts to meet the demand minimize costs It gets worse: workers with different skills! n {English}, {English, French}, {French} © J. Christopher Beck 2005 17