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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

Workforce Scheduling

1. Days-Off Scheduling

2. Shift Scheduling

3. Cyclic Staffing Problem (& extensions)

4. Crew Scheduling


Topic 1

Off-DaysScheduling:

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

Topic 1

Days-Off Scheduling

Not


Days off scheduling
Days-Off Scheduling

  • Number of workers assigned to each day

  • Fixed size of workforce

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


Constraints
Constraints

  • 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


Optimal schedule
Optimal Schedule

  • Algorithm for one week

  • Repeat for next week

  • Cyclic schedule when repeat


Lower bounds on minimum workforce w
Lower Bounds on Minimum Workforce W

  • Weekend constraint

  • Total demand constraint

  • Maximum daily demand constraint


Optimal schedule1
Optimal Schedule

  • 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


Algorithm
Algorithm

  • Schedule weekends off

  • Determine additional off days (in pairs)

  • Categorize employees

  • Assign off-day pairs


Example analysis
Example - analysis

  • Data

  • Bounds:

    • max(n1,...,n7) = 3, then W >= 3

    • , so W >= 3

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


Example solution
Example - solution

  • 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

    Topic 2

    Shift Scheduling


    Shift scheduling
    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


    Solution
    Solution

    • 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

    Topic 3

    Cyclic Staffing

    (& extensions)


    Call center agents

    24

    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


    Cyclic staffing problem
    Cyclic Staffing Problem

    • 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


    Integer program formulation
    Integer Program Formulation

    • Shift patterns

    • (5, 7) example: 7 different patterns


    Solution1
    Solution

    • 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


    Example
    Example

    (3,5)-cyclic staffing problem

    Step 1:


    Solution2

    Optimal

    Solution

    • Add together:

    • Step 2a: Add constraint:

      • No feasible solution

    • Step 2b: Add constraint:

    • Solution:


    Extension 1 days off scheduling
    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


    Example1
    Example

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


    Extension 2 cyclic staffing with overtime
    Extension 2: Cyclic Staffing with Overtime

    • 24-hour operation

    • 8-hour shifts with up to 8 hour overtime

    • 3 shifts without overtime + 8 shifts with overtime


    Topic 44

    Topic 44

    Crew Scheduling


    Crew scheduling
    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


    Notation
    Notation

    • Cost cj of round trip j

    • Define


    Integer program
    Integer Program

    Minimize

    Subject to


    Set partitioning
    Set Partitioning

    • 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.


    Row prices
    Row Prices

    • Say that

      is a set of feasible row prices if for

    • Cost of covering a job


    Change partition
    Change Partition

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

    • Then

    • Potential savings of including column j is

    • If all negative then optimal


    Heuristic
    Heuristic

    • Start with some partition

    • 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


    Helpdesk kpn
    Helpdesk (KPN)

    • 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
    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



    More wishes more constraints
    More wishes, more constraints

    • 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


    Helpdesk model leidschendam
    Helpdesk model Leidschendam

    • 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


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