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
Nurse rostering — The problem Days E L N on shift shift shift Employee 1 4 4 0 0 E E E E Employee 2 L L L E E 5 2 3 0 Employee 3 N N L L L 5 0 3 2 Employee 4 N N 2 0 0 2 Employee 5 N N N 3 0 0 3 Number of E shifts 0 1 1 1 1 1 1 Number of L shifts 1 1 1 0 1 1 1 Number of N shifts 1 1 1 1 1 1 1 Smet et al. - Network flow formulations for a class of nurse scheduling problems 2/19
Nurse rostering — The problem Smet et al. - Network flow formulations for a class of nurse scheduling problems 3/19
Nurse rostering — The solution Smet et al. - Network flow formulations for a class of nurse scheduling problems 4/19
Nurse rostering — The solution Smet et al. - Network flow formulations for a class of nurse scheduling problems 4/19
Nurse rostering — The solution Smet et al. - Network flow formulations for a class of nurse scheduling problems 4/19
Nurse rostering — The solution Smet et al. - Network flow formulations for a class of nurse scheduling problems 4/19
Nurse rostering — The solution Smet et al. - Network flow formulations for a class of nurse scheduling problems 4/19
Nurse rostering — The solution Smet et al. - Network flow formulations for a class of nurse scheduling problems 4/19
Nurse rostering — The problem (revisited) Smet et al. - Network flow formulations for a class of nurse scheduling problems 5/19
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¨ ohler (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
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¨ ohler (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
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¨ ohler (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
Complexity of nurse rostering • Where are the easy problems? Smet et al. - Network flow formulations for a class of nurse scheduling problems 7/19
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 of n nurses N = { 1 , ..., n } . On each day j and for each shift type k , arbitrary minimum and maximum staffing demands 0 ≤ d l jk ≤ d u jk are specified. Each nurse i has to work exactly a i days in T . Finally, each nurse i has a preference for working shift type k on day j , expressed as a cost c ijk . Smet et al. - Network flow formulations for a class of nurse scheduling problems 8/19
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 of n nurses N = { 1 , ..., n } . On each day j and for each shift type k , arbitrary minimum and maximum staffing demands 0 ≤ d l jk ≤ d u jk are specified. Each nurse i has to work exactly a i days in T . Finally, each nurse i has a preference for working shift type k on day j , expressed as a cost c ijk . Smet et al. - Network flow formulations for a class of nurse scheduling problems 8/19
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 of n nurses N = { 1 , ..., n } . On each day j and for each shift type k , arbitrary minimum and maximum staffing demands 0 ≤ d l jk ≤ d u jk are specified. Each nurse i has to work exactly a i days in T . Finally, each nurse i has a preference for working shift type k on day j , expressed as a cost c ijk . Smet et al. - Network flow formulations for a class of nurse scheduling problems 8/19
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 of n nurses N = { 1 , ..., n } . On each day j and for each shift type k , arbitrary minimum and maximum staffing demands 0 ≤ d l jk ≤ d u jk are specified. Each nurse i has to work exactly a i days in T . Finally, each nurse i has a preference for working shift type k on day j , expressed as a cost c ijk . Smet et al. - Network flow formulations for a class of nurse scheduling problems 8/19
IP Formulation � 1 if nurse i works shift k on day j x ijk = otherwise 0 � � � (1) min c ijk x ijk i ∈ N j ∈ T k ∈ S � x ijk ≤ 1 ∀ i ∈ N, j ∈ T (2) s.t. k ∈ S d l � x ijk ≤ d u jk ≤ ∀ j ∈ T, k ∈ S (3) jk i ∈ N � � ∀ i ∈ N (4) x ijk = a i j ∈ T k ∈ S x ijk ∈ { 0 , 1 } ∀ i ∈ N, j ∈ T, k ∈ S (5) Smet et al. - Network flow formulations for a class of nurse scheduling problems 9/19
Network flow formulation Source Sink s f Smet et al. - Network flow formulations for a class of nurse scheduling problems 10/19
Network flow formulation Source Shift nodes Sink j = 1,...,t k = 1,...,s .. d l jk ≤ x ≤ d u jk s <j,k> f .. Smet et al. - Network flow formulations for a class of nurse scheduling problems 10/19
Network flow formulation Source Shift nodes Time nodes Sink j = 1,...,t k = 1,...,s i = 1,...,n j = 1,...,t .. .. d l jk ≤ x ≤ d u 0 ≤ x ≤ 1 jk s <j,k> <i,j> f c ijk .. .. Smet et al. - Network flow formulations for a class of nurse scheduling problems 10/19
Network flow formulation Source Shift nodes Time nodes Nurse nodes Sink j = 1,...,t k = 1,...,s i = 1,...,n j = 1,...,t i = 1,...,n .. .. .. d l jk ≤ x ≤ d u 0 ≤ x ≤ 1 0 ≤ x ≤ 1 x = a i jk s <j,k> <i,j> i f c ijk .. .. .. Smet et al. - Network flow formulations for a class of nurse scheduling problems 10/19
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
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
Network flow formulation — An example M 1 Monday Tuesday Wednesday M early late early late early late 2 [2,3] [1,2] [2,3] [1,2] [1,2] [1,2] M 3 M 4 M e M 5 M T l 1 1 {2,3} T {1,3} 2 {2} T 2 e {2} {2,3} T s f 3 {2} 3 {1,2} {2} T l T 4 {2} {1,2} 4 {1,2} T W 5 5 e W 1 W l W 2 W 3 W 4 W 5 Smet et al. - Network flow formulations for a class of nurse scheduling problems 13/19
Network flow formulation — An example M 1 Monday Tuesday Wednesday M early late early late early late 2 [2,3] [1,2] [2,3] [1,2] [1,2] [1,2] M 3 M 4 M e M 5 M T l 1 1 2 T 2 2 2 T 2 e 2 2 T s f 3 2 3 1 2 T l T 4 2 1 4 2 T W 5 5 e W 1 W l W 2 W 3 W 4 W 5 Smet et al. - Network flow formulations for a class of nurse scheduling problems 13/19
Network flow formulation — An example M 1 Monday Tuesday Wednesday M early late early late early late 2 [2,3] [1,2] [2,3] [1,2] [1,2] [1,2] M 3 M 4 M e M 5 M T l 1 1 2 T 2 2 T 2 e T s f 3 3 T l T 4 4 2 T W 5 5 e W 1 W l W 2 W 3 W 4 W 5 Smet et al. - Network flow formulations for a class of nurse scheduling problems 13/19
Network flow formulation — An example M 1 Monday Tuesday Wednesday M early late early late early late 2 [2,3] [1,2] [2,3] [1,2] [1,2] [1,2] M 3 M 4 M e M 5 M T l 1 1 2 T 2 2 T 2 e T s f 3 3 T l T 4 4 2 T W 5 5 e W 1 W l W 2 W 3 W Employee 1 4 e l W 5 Smet et al. - Network flow formulations for a class of nurse scheduling problems 13/19
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