The overlapping case in an integrated staffing and rostering formulationKomarudin, M.-A. Guerry, G. Vanden Berghe, and T. De Feyter

Contents Introduction

Personnel structure Nurse rostering problem

Literature review Roster Quality Staffing (RQS) Problem Subgroup formulation Overlapping rostering constraint Illustration Conclusion and future research

Introduction - Personnel structure represents the number and the

composition of distinct personnel subgroups

used for (long term) manpower planningHead Nurse

Nurse with Skill

1

Nurse with Skill

2

Nurse with Skill

3

Trainee Nurse

1 12 15 5 7

Introduction - Nurse Rostering concerns arranging the available nursing

personnel to work requirements. Short to medium planning period

M: Morning (6-14)D: Day (14-22)N: Night (22-6)

S1: Skill-1S2: Skill-2

Nurse

December 201118 19 20 …

A M M MB D D DC N N DD NE M M…

Reqs. 5M6D3N

4-S13-S2...

5M7D2N

6-S15-S2

…

7M5D4N

5-S14-S2

…

Nurse

December 201118 19 20 …

A M M MB D D DC N N DD NE M M…

Reqs.

Nurse Rostering Problem

Many constraints Quality = the violation of soft constraints

The lower the betterM = Morning (6-14), D = Day (14-22), N = Night (22-6)

Day-shift after

night shift

Nurse Rostering Problem -More on Soft Constraints

• NP-hard Problem Used metaheuristics (VNS & ALNS )

Work requirements

Coverage constraint

s

Secondary skill

constraintsPersonal requests

C asks days off at

20/12

Series constraint

sMax

consecutive nights

Counter constraint

s40 hours

/week

Successive series

Refer to Bilgin et al. (2010) for more detail

Motivation

Roster quality

Rostering problemMetaheuristics …

Personnel structure (PS)Nurse to patient ratio

Work requirementsStatistical forecasting

Usually fixed

Motivation

Roster quality

Rostering problemMetaheuristics …

Personnel structure (PS)Nurse to patient ratio

Work requirementsStatistical forecasting

Usually fixed

• Understaffing• Overstaffing• Conflicting

constraints

Undesired condition

How we can determine PS good quality roster?

Feed

back

Literature reviewNo Approach Paper Review

1 Staffing and rostering sequentially

• Abernathy, et al. (1973)

• Ozcan (2009)

• restrict the possibilities at the rostering level suboptimal solutions

2 Integrated staffing and rostering

• Venkataraman & Brusco (1996)

• Mundschenk & Drexl (2007)

• Li et al. (2007)• Beliën et al. (2011)

• Constraints is rather limited, compared to nurse rostering

• (some) only tries to meet the aggregate requirements, without actually produce schedule

3 Part-time/ Annualized workers

• Bard & Purnomo (2005)

• Corominas, et al. (2007)

• only determine the number of additional workers

• May not available

4. Staff allocation

• Haspeslagh (2012)• Maenhout &

Vanhoucke (2013)

• Focus on staff allocation, not determining the personnel structure

How to investigate the suitability of a personnel structure while considering the

roster quality?𝑞𝑣 (𝑅 (𝑛∗))≤𝑞𝑣 (𝑅(𝑛))

Personnel structureRoster

Roster quality

Roster Quality Staffing (RQS) Problem statement

Three-step approach1. Static check2. Roster quality component examination3. Simple neighbourhood examination

Roster Quality Staffing (RQS) Problem statement

How to investigate the suitability of a personnel structure while considering the

roster quality?𝑞𝑣 (𝑅 (𝑛∗))≤𝑞𝑣 (𝑅(𝑛))

Personnel structureRoster

Roster quality

New formulation1. Subgroup formulation2. Overlapping rostering

constraint

Defining subgroups on RQS Characteristics to differentiate

subgroups

: the set of characteristics for subgroup is the -th characteristic of subgroup .

Example:

𝜁 𝑖={𝜁1𝑖 ,𝜁 2

𝑖 ,𝜁 3𝑖 ,…,𝜁 𝑙

𝑖 }

1

Subgroup Characteristics1 {ward X, skill B, 100% FTE}2 {ward Y, skill A & B, 85%

FTE}3 {ward X & Y, skill B}… …

Subgroup formulations on RQS1. Exactly-same

characteristics based

,

2. Main-characteristic based

,beneficial if the main-characteristic has a high penalty weight.

Subgroup

Characteristics

1 {A, B, C}2 {B}3 {B, C}4 {C}Subgroup

Characteristics

1 {A}2 {B}3 {C}

Subgroup formulation on RQS (application)1. Multiple wards2. Float unit personnel3. Cross-trained personnel

Example:Subgroup Characteristics1 {ward X, skill B, 100% FTE}2 {ward Y, skill A & B, 85%

FTE}3 {ward X & Y, skill B}… …

Overlapping personnel constraints Subgroup-specific vs.

overlapping Overlapping: associated with

two or more subgroups. Example:

coverage requirements for skill C

tutorship, etc.

2

Sub-group

Skill characteristics

1 {A, B, C}2 {B}3 {B, C}4 {C}

Sub-group

Characteristics

1 {Trainer}2 {Trainee}3 {B, C}

G1 G2 Tutorship constraint

Yes Yes YesOther situation No

Models for overlapping rostering constraints1. possessed by at least two subgroups,

e.g. coverage reqs. for float unit, , and .

2. partially possessed by at least two subgroups, e.g. tutorship

, , , and.

Characteristics of overlapping rostering constraint l

Characteristics of subgroup i

2 – Complementary(AND)

1 – Substitution(OR)

Application on three-step approachThree proposed examinations:1. Static check2. Roster quality component examination

Overlapping rostering constraints3. Simple neighbourhood examination

Subgroup formulation three examinations

Illustration Emergency ward taken from Bilgin et al.

(2012) 4 weeks

Ward Skill 1 Skill 2 Skill 3 Skill 4 Total Nurse Emergency Primary skill 1 16 4 6 27 Secondary skill 20

Subgroup Pattern (Primary skill – secondary skill)

No Of Nurses

Subgroup 1 (G1) Skill 1 1 Subgroup 2 (G2) Skill 2 – Skill 4 16 Subgroup 3 (G3) Skill 3 – Skill 4 4 Subgroup 4 (G4) Skill 4 6 Total 27

Exactly-same subgroup based

Emergency ward(1) static check & (2) roster quality

Ratio=𝑊𝑜𝑟𝑘𝑟𝑒𝑞𝑢𝑖𝑟𝑒𝑚𝑒𝑛𝑡𝑠𝑅𝑒𝑠𝑜𝑢𝑟𝑐𝑒𝑠

1G1-Skill

1G2-Skill

2G3-Skill

3G4-Skill

40.95 0.85 0.85 0.85

Reasonable, provide some extra time for vacation/illness

Emergency ward(1) static check & (2) roster quality

Ratio=𝑊𝑜𝑟𝑘𝑟𝑒𝑞𝑢𝑖𝑟𝑒𝑚𝑒𝑛𝑡𝑠𝑅𝑒𝑠𝑜𝑢𝑟𝑐𝑒𝑠

1G1-Skill

1G2-Skill

2G3-Skill

3G4-Skill

40.95 0.85 0.85 0.85

Reasonable, provide some extra time for vacation/illness

Coverage constraint Secondary skill Counter SeriesSkill

1Skill 2

Skill 3

Skill 4

Skill 1

Skill 2

Skill 3

Skill 4

1.6% 2.2% 64.0%

16.1%

10.2% 2.9%

Skill 4 understaffed

Soft constraint violations2 The lower

the better

G2, G3 & G4

Substitution overlapping constraint

The lower the better

Emergency ward(3) Simple neighbourhood examination3 Ward   G1 G2 G3 G4

Emergency (1) First Skill 1 16 4 6

Emergency ward(3) Simple neighbourhood examination3 Ward   G1 G2 G3 G4

Emergency (1) First Skill 1 16 4 6

Add G2, G3 & G4

Wilcoxon test produces very low p-values, close to zero.

Difference significant

Skill 4 understaffed

ConclusionConclusions:1. Subgroup formulation provides more

general formulation: multi-ward, float unit, cross-trained, etc.

2. Overlapping rostering constraint: clarifies the source of a low roster quality

3. RQS problem can be decomposed into smaller problems

Future directions:4. develop an optimization approach to find

the most preferable personnel structure5. Integrate with manpower planning

References Abernathy, W. J., Baloff, N., Hershey, C. J. & Wandel, S., 1973. A

Three-Stage Manpower Planning and Scheduling Model-A Service Sector Example. Operations Research, 21(3), pp. 693-711.

Bard, J. F. & Purnomo, H. W., 2005. A column generation-based approach to solve the preference scheduling problem for nurses with downgrading. Socio-Economic Planning Sciences, 39(3), pp. 193-213.

Beliën, J., Cardoen, B. & Demeulemeester, E., 2011.. Improving workforce scheduling of aircraft line maintenance at Sabena Technics. Interfaces. In Press.

Bilgin, B., De Causmaecker, P., Rossie, B. & Vanden Berghe, G., 2010. Local search neighbourhoods for dealing with a novel nurse rostering model. Annals of Operations Research, pp. 1-25.

Broos Maenhout and Mario Vanhoucke. An integrated nurse stang and scheduling analysis for longer-term nursing sta allocation problems. Omega, 41(2):485{499, 2013. 429

Corominas, A., Lusa, A. & Pastor, R., 2007. Planning production and working time within an annualised hours scheme framework. Annals of Operations Research, 155(1), pp. 5-23.

Li, Y., Chen, J. & Cai, X., 2007. An integrated staff-sizing approach considering feasibility of scheduling decision. Annals of Operations Research, 155(1), pp. 361-390.

Mundschenk, M. & Drexl, A., 2007. Workforce planning in the printing industry. International Journal of Production Research, 45(20), pp. 4849-4872.

Ozcan, Y. A., 2009. Quantitative Methods in Health Care Management: Techniques and Applications. 2nd ed. San Fransisco: Jossey-Bass.

Stefaan Haspeslagh. Negotiation at the short and mid term level supported by analysis of nurse rostering problems. PhD thesis, Computer Science, Kulak, 2012.

Venkataraman, R. & Brusco, M. J., 1996. An integrated analysis of nurse staffing and scheduling policies. 24(1), pp. 57-71.

References