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network flow formulations for a class of nurse scheduling
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Network flow formulations for a class of nurse scheduling problems - - PowerPoint PPT Presentation

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Network flow formulations for a class of nurse scheduling problems Pieter Smet Peter Brucker Patrick De Causmaecker Greet Vanden Berghe April 3, 2014 Smet et al. - Network flow formulations for a class of nurse scheduling problems 1/19

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SLIDE 1

Network flow formulations for a class

  • f nurse scheduling

problems

Pieter Smet Peter Brucker Patrick De Causmaecker Greet Vanden Berghe April 3, 2014

Smet et al. - Network flow formulations for a class of nurse scheduling problems 1/19

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SLIDE 2

Nurse rostering — The problem

E E E E Employee 1 L L L E E Employee 2 N N L L L Employee 3 N N Employee 4 N N N Employee 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Number of E shifts Number of L shifts Number of N shifts Days

  • n

4 5 5 2 3 E shift 4 2 L shift 3 3 N shift 2 2 3

Smet et al. - Network flow formulations for a class of nurse scheduling problems 2/19

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SLIDE 3

Nurse rostering — The problem

Smet et al. - Network flow formulations for a class of nurse scheduling problems 3/19

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SLIDE 4

Nurse rostering — The solution

Smet et al. - Network flow formulations for a class of nurse scheduling problems 4/19

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SLIDE 5

Nurse rostering — The solution

Smet et al. - Network flow formulations for a class of nurse scheduling problems 4/19

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SLIDE 6

Nurse rostering — The solution

Smet et al. - Network flow formulations for a class of nurse scheduling problems 4/19

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SLIDE 7

Nurse rostering — The solution

Smet et al. - Network flow formulations for a class of nurse scheduling problems 4/19

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SLIDE 8

Nurse rostering — The solution

Smet et al. - Network flow formulations for a class of nurse scheduling problems 4/19

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SLIDE 9

Nurse rostering — The solution

Smet et al. - Network flow formulations for a class of nurse scheduling problems 4/19

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SLIDE 10

Nurse rostering — The problem (revisited)

Smet et al. - Network flow formulations for a class of nurse scheduling problems 5/19

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SLIDE 11

Complexity of nurse rostering

Only three hardness proofs

  • Lau (1996)

◮ Shift succession constraints

  • Osogami and Imai (2000)

◮ Number shift types worked

  • Brunner, Bard and K¨
  • hler (2013)

◮ Number of days worked ◮ Number of consecutive days worked ◮ Number of consecutive days-off Smet et al. - Network flow formulations for a class of nurse scheduling problems 6/19

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SLIDE 12

Complexity of nurse rostering

Only three hardness proofs

  • Lau (1996)

◮ Shift succession constraints

  • Osogami and Imai (2000)

◮ Number shift types worked

  • Brunner, Bard and K¨
  • hler (2013)

◮ Number of days worked ◮ Number of consecutive days worked ◮ Number of consecutive days-off Smet et al. - Network flow formulations for a class of nurse scheduling problems 6/19

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SLIDE 13

Complexity of nurse rostering

Only three hardness proofs

  • Lau (1996)

◮ Shift succession constraints

  • Osogami and Imai (2000)

◮ Number shift types worked

  • Brunner, Bard and K¨
  • hler (2013)

◮ Number of days worked ◮ Number of consecutive days worked ◮ Number of consecutive days-off Smet et al. - Network flow formulations for a class of nurse scheduling problems 6/19

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SLIDE 14

Complexity of nurse rostering

  • Where are the easy problems?

Smet et al. - Network flow formulations for a class of nurse scheduling problems 7/19

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SLIDE 15

A nurse rostering problem P

The scheduling period T is a set of t days T = {1, ..., t}. There is a set S of s shift types S = {1, ..., s}. The workforce N is a heterogeneous set

  • f n nurses N = {1, ..., n}. On each day j and for each shift type k,

arbitrary minimum and maximum staffing demands 0 ≤ dl

jk ≤ du jk

are specified. Each nurse i has to work exactly ai days in T. Finally, each nurse i has a preference for working shift type k on day j, expressed as a cost cijk.

Smet et al. - Network flow formulations for a class of nurse scheduling problems 8/19

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SLIDE 16

A nurse rostering problem P

The scheduling period T is a set of t days T = {1, ..., t}. There is a set S of s shift types S = {1, ..., s}. The workforce N is a heterogeneous set

  • f n nurses N = {1, ..., n}. On each day j and for each shift type k,

arbitrary minimum and maximum staffing demands 0 ≤ dl

jk ≤ du jk

are specified. Each nurse i has to work exactly ai days in T. Finally, each nurse i has a preference for working shift type k on day j, expressed as a cost cijk.

Smet et al. - Network flow formulations for a class of nurse scheduling problems 8/19

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SLIDE 17

A nurse rostering problem P

The scheduling period T is a set of t days T = {1, ..., t}. There is a set S of s shift types S = {1, ..., s}. The workforce N is a heterogeneous set

  • f n nurses N = {1, ..., n}. On each day j and for each shift type k,

arbitrary minimum and maximum staffing demands 0 ≤ dl

jk ≤ du jk

are specified. Each nurse i has to work exactly ai days in T. Finally, each nurse i has a preference for working shift type k on day j, expressed as a cost cijk.

Smet et al. - Network flow formulations for a class of nurse scheduling problems 8/19

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SLIDE 18

A nurse rostering problem P

The scheduling period T is a set of t days T = {1, ..., t}. There is a set S of s shift types S = {1, ..., s}. The workforce N is a heterogeneous set

  • f n nurses N = {1, ..., n}. On each day j and for each shift type k,

arbitrary minimum and maximum staffing demands 0 ≤ dl

jk ≤ du jk

are specified. Each nurse i has to work exactly ai days in T. Finally, each nurse i has a preference for working shift type k on day j, expressed as a cost cijk.

Smet et al. - Network flow formulations for a class of nurse scheduling problems 8/19

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SLIDE 19

IP Formulation

xijk = 1 if nurse i works shift k on day j

  • therwise

min

  • i∈N
  • j∈T
  • k∈S

cijkxijk (1) s.t.

  • k∈S

xijk ≤ 1 ∀ i ∈ N, j ∈ T (2) dl

jk ≤

  • i∈N

xijk ≤ du

jk

∀ j ∈ T, k ∈ S (3)

  • j∈T
  • k∈S

xijk = ai ∀ i ∈ N (4) xijk ∈ {0, 1} ∀ i ∈ N, j ∈ T, k ∈ S (5)

Smet et al. - Network flow formulations for a class of nurse scheduling problems 9/19

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Network flow formulation

s f Source Sink

Smet et al. - Network flow formulations for a class of nurse scheduling problems 10/19

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SLIDE 21

Network flow formulation

s f Shift nodes <j,k> j = 1,...,t k = 1,...,s

.. ..

dl

jk ≤ x ≤ du jk

Source Sink

Smet et al. - Network flow formulations for a class of nurse scheduling problems 10/19

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SLIDE 22

Network flow formulation

s f Shift nodes Time nodes <j,k> <i,j> j = 1,...,t k = 1,...,s i = 1,...,n j = 1,...,t

.. .. .. ..

0 ≤ x ≤ 1 cijk dl

jk ≤ x ≤ du jk

Source Sink

Smet et al. - Network flow formulations for a class of nurse scheduling problems 10/19

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SLIDE 23

Network flow formulation

s f Shift nodes Time nodes <j,k> <i,j> i j = 1,...,t k = 1,...,s i = 1,...,n j = 1,...,t i = 1,...,n Nurse nodes

.. .. .. .. .. ..

0 ≤ x ≤ 1 x = ai cijk dl

jk ≤ x ≤ du jk

0 ≤ x ≤ 1 Source Sink

Smet et al. - Network flow formulations for a class of nurse scheduling problems 10/19

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SLIDE 24

Network flow formulation

Theorem

An optimal integer minimum cost flow in the network G corresponds to an optimal solution for problem P.

Smet et al. - Network flow formulations for a class of nurse scheduling problems 11/19

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Network flow formulation — An example

  • There are three days: Monday (M), Tuesday (T) and Wednesday

(W).

  • There are two shifts: early (e) and late (l).
  • There are five nurses.
  • Each nurse has to work two days.

The following staffing demands are required Monday Tuesday Wednesday early late early late early late [2,3] [1,2] [2,3] [1,2] [1,2] [1,2]

Smet et al. - Network flow formulations for a class of nurse scheduling problems 12/19

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Network flow formulation — An example

Monday Tuesday Wednesday early late early late early late [2,3] [1,2] [2,3] [1,2] [1,2] [1,2] s

M e M l T e T l W e W l M 1 M 2 M 3 M 4 M 5 T 1 T 2 T 3 T 4 T 5 W 1 W 2 W 3 W 4 W 5 1 2 3 4 5

f

{2,3} {1,3} {2,3} {1,2} {1,2} {1,2} {2} {2} {2} {2} {2}

Smet et al. - Network flow formulations for a class of nurse scheduling problems 13/19

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SLIDE 27

Network flow formulation — An example

Monday Tuesday Wednesday early late early late early late [2,3] [1,2] [2,3] [1,2] [1,2] [1,2]

2 2 2 1 1 2

s

2 2 2 2 2

M e M l T e T l W e W l M 1 M 2 1 2 3 4 5

f

M 5 T 1 T 2 T 3 T 4 T 5 W 1 W 2 W 3 W 4 W 5 M 3 M 4

Smet et al. - Network flow formulations for a class of nurse scheduling problems 13/19

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SLIDE 28

Network flow formulation — An example

Monday Tuesday Wednesday early late early late early late [2,3] [1,2] [2,3] [1,2] [1,2] [1,2]

2 2 2

s

M e M l T e T l W e W l M 1 M 2 M 3 M 4 M 5 T 1 T 2 T 3 T 4 T 5 W 1 W 2 W 3 W 4 W 5 1 2 3 4 5

f Smet et al. - Network flow formulations for a class of nurse scheduling problems 13/19

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SLIDE 29

Network flow formulation — An example

Monday Tuesday Wednesday early late early late early late [2,3] [1,2] [2,3] [1,2] [1,2] [1,2]

W 5

2 2 2

s

M e M l T e T l W e W l M 1 M 2 M 3 M 4 M 5 T 1 T 2 T 3 T 4 T 5 W 1 W 2 W 3 W 4

e l Employee 1

1 2 3 4 5

f Smet et al. - Network flow formulations for a class of nurse scheduling problems 13/19

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SLIDE 30

Network flow formulation — An example

Monday Tuesday Wednesday early late early late early late [2,3] [1,2] [2,3] [1,2] [1,2] [1,2]

W 5

2 2 2

s

M e M l T e T l W e W l M 1 M 2 M 3 M 4 M 5 T 1 T 2 T 3 T 4 T 5 W 1 W 2 W 3 W 4

e e Employee 3

1 2 3 4 5

f Smet et al. - Network flow formulations for a class of nurse scheduling problems 13/19

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SLIDE 31

Computational experiments — Setup

Problem P corresponds to a subset of instances from NSPLib.

  • Number of days is 7
  • Number of shifts is 4
  • Number of nurses is 25, 50, 75, 100
  • Number of days worked is 5

Three solution techniques compared

  • Best known results from the literature
  • IP formulation solved with CPLEX 12.5
  • Network flow formulation solved with LEMON 1.3

Smet et al. - Network flow formulations for a class of nurse scheduling problems 14/19

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SLIDE 32

Computational experiments — Setup

Problem P corresponds to a subset of instances from NSPLib.

  • Number of days is 7
  • Number of shifts is 4
  • Number of nurses is 25, 50, 75, 100
  • Number of days worked is 5

Three solution techniques compared

  • Best known results from the literature
  • IP formulation solved with CPLEX 12.5
  • Network flow formulation solved with LEMON 1.3

Smet et al. - Network flow formulations for a class of nurse scheduling problems 14/19

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SLIDE 33

Computational experiments — Setup

Problem P corresponds to a subset of instances from NSPLib.

  • Number of days is 7
  • Number of shifts is 4
  • Number of nurses is 25, 50, 75, 100
  • Number of days worked is 5

Three solution techniques compared

  • Best known results from the literature
  • IP formulation solved with CPLEX 12.5
  • Network flow formulation solved with LEMON 1.3

Smet et al. - Network flow formulations for a class of nurse scheduling problems 14/19

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SLIDE 34

Computational experiments — Results

Best known CPLEX LEMON Days Nurses

  • Avg. cost

Feasible

  • Avg. cost

Feasible

  • Avg. cost

Feasible 7 25 305.11 88.27% 245.41 100.00% 245.41 100.00% 50 587.07 90.03% 489.77 100.00% 489.77 100.00% 75 912.86 88.70% 740.11 100.00% 740.11 100.00% 100 1389.23 90.49% 1191.19 100.00% 1191.19 100.00%

Table: Comparison of solution quality.

Smet et al. - Network flow formulations for a class of nurse scheduling problems 15/19

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SLIDE 35

Computational experiments — Results

Days Nurses Best known CPLEX LEMON 7 25 1.6636 0.0206 0.0014 50 4.4157 0.0324 0.0031 75 12.4680 0.0447 0.0062 100 20.0003 0.0579 0.0102 Table: Comparison of calculation time (in seconds).

Smet et al. - Network flow formulations for a class of nurse scheduling problems 16/19

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SLIDE 36

Extensions to the problem description

  • Unavailabilities

◮ For days ◮ For shifts

  • Fixed assignments

◮ For days ◮ For shifts

  • Daily employment cost
  • Small variations on the horizontal constraint definition

Smet et al. - Network flow formulations for a class of nurse scheduling problems 17/19

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SLIDE 37

Extensions to the problem description

  • Unavailabilities

◮ For days ◮ For shifts

  • Fixed assignments

◮ For days ◮ For shifts

  • Daily employment cost
  • Small variations on the horizontal constraint definition

Smet et al. - Network flow formulations for a class of nurse scheduling problems 17/19

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SLIDE 38

Extensions to the problem description

  • Unavailabilities

◮ For days ◮ For shifts

  • Fixed assignments

◮ For days ◮ For shifts

  • Daily employment cost
  • Small variations on the horizontal constraint definition

Smet et al. - Network flow formulations for a class of nurse scheduling problems 17/19

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SLIDE 39

Extensions to the problem description

  • Unavailabilities

◮ For days ◮ For shifts

  • Fixed assignments

◮ For days ◮ For shifts

  • Daily employment cost
  • Small variations on the horizontal constraint definition

Smet et al. - Network flow formulations for a class of nurse scheduling problems 17/19

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SLIDE 40

Future research

  • More easy problems?

◮ What about number of shift types worked?

→ No, see Osogami and Imai (2000)

◮ What about consecutiveness constraints?

→ . . .

Smet et al. - Network flow formulations for a class of nurse scheduling problems 18/19

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SLIDE 41

Future research

  • More easy problems?

◮ What about number of shift types worked?

→ No, see Osogami and Imai (2000)

◮ What about consecutiveness constraints?

→ . . .

Smet et al. - Network flow formulations for a class of nurse scheduling problems 18/19

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SLIDE 42

Future research

  • More easy problems?

◮ What about number of shift types worked?

→ No, see Osogami and Imai (2000)

◮ What about consecutiveness constraints?

→ . . .

Smet et al. - Network flow formulations for a class of nurse scheduling problems 18/19

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SLIDE 43

Future research

  • More easy problems?

◮ What about number of shift types worked?

→ No, see Osogami and Imai (2000)

◮ What about consecutiveness constraints?

→ . . .

Smet et al. - Network flow formulations for a class of nurse scheduling problems 18/19

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SLIDE 44

Future research

  • More easy problems?

◮ What about number of shift types worked?

→ No, see Osogami and Imai (2000)

◮ What about consecutiveness constraints?

→ . . .

Smet et al. - Network flow formulations for a class of nurse scheduling problems 18/19

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Thank you!

Smet et al. - Network flow formulations for a class of nurse scheduling problems 19/19