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# Workforce Scheduling - PowerPoint PPT Presentation

Workforce Scheduling. 1. Days-Off Scheduling 2. Shift Scheduling 3. Cyclic Staffing Problem (& extensions) 4. Crew Scheduling. Off-Days Scheduling: “Scheduling workers who fall asleep on the job is not easy.”. Topic 1. Days-Off Scheduling. Not. Days-Off Scheduling.

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### Workforce Scheduling

1. Days-Off Scheduling

2. Shift Scheduling

3. Cyclic Staffing Problem (& extensions)

4. Crew Scheduling

Off-DaysScheduling:

“Scheduling workers who fall asleep on the job is not easy.”

### Topic 1

Days-Off Scheduling

Not

• Number of workers assigned to each day

• Fixed size of workforce

• Problem: find minimum number of employees to cover a weeks operation

• Demand per day nj, j = 1,2,…,7

• k1 out of every k2 weekends (day 1 & 7) off

• Work 5 out of 7 days

• Work no more than 6 consecutive days

• Algorithm for one week

• Repeat for next week

• Cyclic schedule when repeat

• Weekend constraint

• Total demand constraint

• Maximum daily demand constraint

• Define

• First schedule weekends off (cyclic)

• Furthermore,

 Idea: Give W workers 2n days off during the week

Work both days!

Surplus when all

workers present

• Schedule weekends off

• Determine additional off days (in pairs)

• Categorize employees

• Assign off-day pairs

• Data

• Bounds:

• max(n1,...,n7) = 3, then W >= 3

• , so W >= 3

• n = max(n1, n7) = 2, k1 = 1 and k2 = 3, so

• Weekends off (one worker per weekend)

• Calculate 2n surplus days (in pairs)

• (Sun, Mon) and (Mon, Mon)

• Weekly: assign pairs to worker

(or to pair of workers)

• Week 1

1: off / on 1

2: on / off 1

3: on / on 2

### Topic 2

Shift Scheduling

• Fixed cycle of length m periods

• Have bi people assigned to ith period

• Have n shift patterns:

• Cost cj of assigning a person to shift j

• Integer decision variable: xj = # people assigned to j

• NP-hard in general

• Special structure in shift pattern matrix

• Solve LP relaxation

• Solution always integer when each column

contains a contiguous set of ones

### Topic 3

Cyclic Staffing

(& extensions)

23

1

22

2

60

42

110

34

21

116

3

24

4

3

6

3

130

20

6

3

4

18

6

2

124

20

19

6

5

2

140

The outer ring shows the average arriving intensity

at that hour.

The inner ring shows the number of centralists necessary for that particular arriving intensity.

24

7

3

6

18

6

130

4

50

6

17

4

7

110

58

6

5

16

102

5

8

80

5

5

5

100

5

5

90

15

9

96

98

72

96

14

10

13

11

12

Call center agents

# of agents needed

• An m-period cyclic schedule (e.g. 24 hours a day)

• Minimize cost

• Constraint bi for ith period

• Each worker works for k consecutive periods and is free for the next m-k

• Example: (5, 7)-cyclic staffing problem

• Shift patterns

• (5, 7) example: 7 different patterns

• Solution to LP relaxation ‘almost right’

• STEP 1: Solve LP relaxation to get

if integer STOP; otherwise continue

• STEP 2: Formulate two new LPs with

• The best integer solution is optimal

(3,5)-cyclic staffing problem

Step 1:

Solution

• No feasible solution

• Solution:

Extension 1: Days-Off Scheduling

• We can represent our days-off scheduling problem as a cyclic staffing problem as long as we can determine all the shift patterns

• Difficulty 1: unknown cycle length

• Difficulty 2: many patterns  larger problem

• Two days off in a week + no more than 6 consecutive workdays

• 24-hour operation

• 8-hour shifts with up to 8 hour overtime

• 3 shifts without overtime + 8 shifts with overtime

### Topic 44

Crew Scheduling

• Have m jobs, say flight legs

• Have n feasible combination of jobs a crew is permitted to do

2

1

5

4

3

Set partitioning

problem

6

• Cost cj of round trip j

• Define

Minimize

Subject to

• Constraints called partitioning equations

• The positive variables in a feasible solution called a partition

• NP-Hard

• Well studied like TSP, graph-coloring, bin-packing, etc.

• Say that

is a set of feasible row prices if for

• Cost of covering a job

• Let Z1 (Z2) denote the objective value of partition 1 (2)

• Then

• Potential savings of including column j is

• If all negative then optimal

• Construct a new partition as follows:

• Find the column with highest potential savings

• Include this column in new partition

• If all jobs covered stop; otherwise repeat

• 18 employees (= 15 in DH + 3 in G)

• 6 required at desk (= 5 in DH + 1 in G)

• 5 in DH (= 2 early + 3 late shift)

• Wishes (soft constraints)

• holiday

• other duties

• preference for early shif

• preference for late shift

• determine schedule for the next 8 weeks:

• that is fair

• satisfies all wishes as much as possible

Helpdesk model Groningen

• bit: person i is available at day t (no holiday)

• rit: person i has other duties at day t

• xit: person i has desk duty at day t

• All have same number of desk duties

• May conflict with other wishes, e.g. request for duty free days

• Holidays may not lead to relatively more desk duties

• Desk duties evenly spaced in time

• bit: person i is available at day t (no holiday)

• rit: person i has other duties at day t

• wij: person i prefers shift j

• xijt: person i has desk duty at day t and shift j