UNIVERSITEIT GENT FACULTEIT ECONOMIE EN...

UNIVERSITEIT GENT

FACULTEIT ECONOMIE EN BEDRIJFSKUNDE

ACADEMIEJAAR 2012 - 2013

INTEGER PROGRAMMING FOR THE TOUR

SCHEDULING PROBLEM: MODELLING AND

MATHEMATICAL PROGRAMMING

Masterproef voorgedragen tot het bekomen van de graad van

Master of Science in de Toegepaste Economische Wetenschappen: Handelsingenieur

Stijn IMMESOETE

Onder leiding van

Prof. Dr. Broos Maenhout

ii

Permission

Ondergetekende verklaart dat de inhoud van deze masterproef mag geraadpleegd en/of

gereproduceerd worden, mits bronvermelding.

Stijn Immesoete

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Preface

Het opmaken van een thesis is een werk van lange adem. Gelukkig waren er verscheidene personen

die mij hierin sterk hebben ondersteund.

Allereerst wil ik dan ook een dankwoord uitbrengen aan Prof. Dr. Broos Maenhout, begeleider en

tevens promotor van deze thesis. Wat ik van harte heb kunnen waarderen tijdens onze anderhalf jaar

durende samenwerking is uw enorme paraatheid en bereidwilligheid. In eerste instantie denk ik

hierbij aan de telkenmale erg snelle respons bij de vele vragen die ik had voor of na één van onze

vergaderingen. Bovendien had u ook een sterk inhoudelijke en efficiënte aanpak indien er zich een

probleem stelde. Dit leidde niet alleen tot significante tijdswinst voor beide partijen maar ook tot een

meer gefundeerde thesis.

Ten tweede zou ik een algemeen woord van dank willen uitbrengen aan de faculteit economie en

bedrijfskunde (FEB). Gedurende mijn vijf jaar lange opleiding hebben zij mij de gelegenheid gegeven

om mijn interesseveld sterk te verdiepen en te verruimen. Deze extra bagage bleek recentelijk nog

een ideale doorstart te vormen naar het verdere bedrijfsleven. Daarenboven heb ik jarenlang mogen

gebruik maken van de prachtige bibliotheek die de FEB tot zijn bezit heeft. Meerdere malen heeft

deze ruimte mij kunnen inspireren en aanzetten tot het voltooien van de talloze groepswerken.

Ten laatste zou ik mijn familie en vrienden willen bedanken waar ik ettelijke malen beroep op heb

gedaan. Dank u allen voor uw blijvende steun.

Stijn Immesoete

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Content

Abbreviations ........................................................................................................................................ vi

List with figures .................................................................................................................................... vii

List with tables .................................................................................................................................... viii

1. Introduction ................................................................................................................................... 1

1.1. Subject motivation ................................................................................................................. 1

1.2. Context of the problem .......................................................................................................... 1

1.3. Focus of the research ............................................................................................................. 2

1.3.1. The personnel planning pyramid .................................................................................... 2

1.3.2. The relation between the staffing- and scheduling decision .......................................... 4

1.3.3. The tour scheduling problem .......................................................................................... 5

1.4. Outline .................................................................................................................................... 5

2. Literature Study: the Tour Scheduling Problem ............................................................................. 7

2.1. Visualising a tour .................................................................................................................... 8

2.2. Mathematical formulation: a standardized approach ............................................................ 9

2.2.1. A set-covering model (SCM) ........................................................................................... 9

2.2.2. An implicit model (IM) .................................................................................................. 11

2.2.3. A derivative model ........................................................................................................ 13

2.3. Different aspects of a tour scheduling model: Literature review ......................................... 15

2.3.1. Objective function(s) .................................................................................................... 16

2.3.2. Decision variables ......................................................................................................... 20

2.3.3. Personnel characteristics .............................................................................................. 22

2.3.4. Shift definitions............................................................................................................. 23

2.3.5. Constraints.................................................................................................................... 27

2.4. The link between theory and practice .................................................................................. 44

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3. Programming the Tour Scheduling Problem: Modelling and Analysis .......................................... 45

3.1. Model formulations .............................................................................................................. 46

3.1.1. Model 1: Set-covering model (SCM) ............................................................................. 46

3.1.2. Model 2: Implicit model (IM) ........................................................................................ 51

3.1.3. Model 3: Hybrid model (HM) ........................................................................................ 55

3.2. Solution methodology .......................................................................................................... 61

3.2.1. Settings ......................................................................................................................... 62

3.2.2. Demand pattern generator ........................................................................................... 64

3.2.3. Tour generator.............................................................................................................. 66

3.2.4. Cost Generator ............................................................................................................. 76

3.2.5. Optimization ................................................................................................................. 77

3.3. Results .................................................................................................................................. 78

3.3.1. Baseline model (BM)..................................................................................................... 79

3.3.2. Scenario 1: Objective function modification: minimize the workforce size .................. 83

3.3.3. Scenario 2: Break placement ........................................................................................ 85

3.3.4. Scenario 3: Intra-tour start-time flexibility (IM only) .................................................... 87

3.3.5. Scenario 4: High- and low skilled employees ................................................................ 89

3.3.6. Scenario 5: demand pattern modification & the relationship with PT employees ....... 91

4. Conclusion .................................................................................................................................... 99

5. Limitations and Future Recommendations................................................................................. 102

References ............................................................................................................................................. x

Appendix A: Overview table of tour scheduling Characteristics ........................................................... xv

Appendix B: Detailed setup for the Baseline model .......................................................................... xviii

Appendix C: Results of changing the FT/PT ratio of the BM ................................................................. xx

Appendix C: Results of changing the cost PT ratio of the BM .............................................................. xxi

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Abbreviations

BM Baseline Model

BR Break Window

BW Band Width

FT Full-time

HM Hybrid Model

IM Implicit Model

PT Part-time

SCM Set-covering Model

TSM Tour Scheduling Model

TSP Tour Scheduling Problem

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List with figures

Figure 1: The Personnel Planning Pyramid ............................................................................................. 3

Figure 2: Tour Scheduling Characteristics: ‘Roadmap’ ......................................................................... 16

Figure 3: Main components of the tour scheduling optimization system ............................................ 62

Figure 4: Subdivisions of the component ‘Settings’.............................................................................. 62

Figure 5: Lower- and upper bound for the random bounded demand generator ................................ 66

Figure 6: Set-covering method: subdivisions of the ‘Tour Generator’ component .............................. 67

Figure 7: Implicit method: Subdivisions of the component ‘Tour Generator’ ...................................... 71

Figure 8: Hybrid method: Subdivisions of the component ‘Tour Generator’ ....................................... 74

Figure 9: Cost generator: transformation process ................................................................................ 76

Figure 10: Default cost per period per day ........................................................................................... 76

Figure 11: Subdivisions of the component ‘Optimization’ .................................................................... 77

Figure 12: BMs – coverage patterns ..................................................................................................... 82

Figure 13: Scenario 1 – Total work schedule cost and -workforce size ................................................ 84

Figure 14: Scenario 1 – Average daily surplus coverage ....................................................................... 84

Figure 15: Scenario 2 – Rows and columns .......................................................................................... 85

Figure 16: Scenario 2 – Total work schedule cost and -workforce size ................................................ 87

Figure 17: Scenario 3 – Total work schedule costs ............................................................................... 88

Figure 18: Scenario 4 – Workforce distribution and total work schedule cost for the IM .................... 91

Figure 19: Scenario 5 – Demand patterns considered .......................................................................... 93

Figure 20: Scenario 5 – Total work schedule cost with and without PT employees ............................. 94

Figure 21: Scenario 5 – Difference in total work schedule cost with and without PT employees ........ 97

Figure 22: Scenario 5 – Absolute and relative cost difference per demand pattern ............................ 98

viii

List with tables

Table 1: Visualisation of a tour – representation 1: one shift per day .................................................... 8

Table 2: Visualisation of a tour – representation 2: all shifts per day .................................................... 8

Table 3: Visualisation of a tour – representation 3: binary representation ............................................ 8

Table 4: A standard Tour scheduling problem: Set-covering Model..................................................... 10

Table 5: Objective Function(s) .............................................................................................................. 20

Table 6: Type of Decision Variable(s) ................................................................................................... 22

Table 7: Personnel characteristics ........................................................................................................ 23

Table 8: Shift definitions + Break placement ........................................................................................ 26

Table 9: Coverage constraints .............................................................................................................. 29

Table 10: Assignment constraints ........................................................................................................ 30

Table 11: Consecutiveness constraints ................................................................................................. 32

Table 12: Working hour constraints ..................................................................................................... 34

Table 13: Holiday/days off constraint .................................................................................................. 34

Table 14: Work pattern constraints...................................................................................................... 35

Table 15: Weekend constraints ............................................................................................................ 37

Table 16: Ratio constraint .................................................................................................................... 38

Table 17: Balancing constraint and Tutorship ...................................................................................... 39

Table 18: Constraints – Overview ......................................................................................................... 42

Table 19: Frequency of constraint-usage ............................................................................................. 43

Table 20: Trend generation .................................................................................................................. 65

Table 21: Set-covering method: output of step 1 of the ‘Tour Generator’ component ....................... 67

Table 22: Set-covering method: impact of 14 time-related constraints ............................................... 68

Table 23: Set-covering method: feasible tours after step 1 of the ‘Tour Generator’ component ........ 69

Table 24: Set-covering method: transforming a single day from step 1 to step 2 ................................ 70

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Table 25: Set-covering method: break placement for a shift of 10 hours with a BR of 5 ..................... 70

Table 26: Implicit method: output of the days-off scheduling subcomponent .................................... 72

Table 27: Implicit method: output of the shift scheduling subcomponent .......................................... 73

Table 28: Implicit method: break placement........................................................................................ 74

Table 29: Comparison of the variables under implicit- and hybrid modelling ...................................... 75

Table 30: Hybrid method: output of the shift scheduling subcomponent ............................................ 75

Table 31: Cost Generator: Transformation formulas ............................................................................ 77

Table 32: BMs – summary of the results .............................................................................................. 80

Table 33: Scenario 1 – summary of the results..................................................................................... 83




Table 37: Mean and standard deviation for low and high variable demand patterns .......................... 94

Table 38: Scenario 5 – summary of the results (a) ............................................................................... 95

Table 39: Scenario 5 – summary of the results (b) ............................................................................... 96

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Chapter 1

1. Introduction

1.1. Subject motivation

This thesis is built around a wide-ranging literature study which is the starting point of the

development and analysis of a mathematical programming formulation. In detail, this thesis is about

the tour scheduling problem (TSP) which is on turn a very important field of study within the domain

of personnel planning. The goal is to gain increasing knowledge in this specific part of personnel

planning to develop a mathematical integer programming formulation that is able to analyse all the

different aspects included in tour scheduling.

This combination of studying literature which is rich of mathematics and particularly the possibility to

program an optimization model has encouraged me to put an extensive amount of time and effort in

this thesis. Above that, the subject perfectly fits with my studies as a business engineer specialised in

operational management. In this respect, I am convinced that this work will help me to develop new

skills and acquire the necessary experience to start with my future career as a consultant.

1.2. Context of the problem

In many industries or branches, like services (e.g. health care, call centers, etc.), transportation (e.g.

airline, railway, bus, etc.) or retail, personnel is one of the main cost drivers. Hence, ensuring that the

workforce is efficiently and effectively allocated is an absolute must, especially in profit

organisations. However, this allocation is not as easy as it seems and can be very complex.

On the one hand, the complexity in allocating personnel is due to the differences between persons

and goods. As a case in point, the volatility of demand is a much bigger problem for service

organisations than for product manufacturers. While goods can be kept in stock, peak -or near-peak

demand conditions in service organisations must be compensated with expensive overtime hours,

extra temporarily employees or reallocation of the personnel. On the other hand, both employers

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and employees impose several restrictions or rules which have to be fulfilled simultaneously. The

following topics are often included in the discussion about staffing the demand pattern: length and

number of shift types, number of consecutive days on/off, rest time between assignments, free days,

weekend regulations, fairness among the employees (balancing the workload), and so on.

Furthermore, in every country numerous working rules and laws are imposed by local governments,

labor unions and Collective Bargaining Agreements (CBA’s).

In order to deal with all these rules and restrictions professors, scholars, researchers, engineers,

mathematicians and other people specialized in the field of personnel planning have tried to develop

adequate mathematical formulations which could be solved by optimisation software. Main purpose

is to both fulfil the objectives of the organisation and satisfy those imposed by the employees.

Throughout the past decades, formulations have become very complex because more and more

restrictions and rules were taken into account. This, of course, results in a more adequate personnel

planning model but also has it disadvantages. One of these setbacks is the larger CPU time to solve

the master scheduling problem.

1.3. Focus of the research

The objective of this thesis is to gain deeper understanding in the TSP. But before I explain this

specific type of personnel planning, it is important to understand the overall decision framework and

its characteristics.

1.3.1. The personnel planning pyramid

As mentioned before, a personnel planning is for many organisations a crucial aspect, especially in

combination with the continuously increasing cost consciousness in our current global business

environment. Therefore, it is necessary to understand personnel planning in all of its aspects.

Personnel planning has many definitions, one of them is cited below:

“The process of constructing work timetables for its staff so that an organisation

can satisfy the demand for its goods or services.” (Ernst et al., 2004a)

This process is divided in three interrelated decision domains commonly spanning a horizon of one

year (Warner, 1976a and Maier-Rothe et al., 1973) (See Figure 1):

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Figure 1: The Personnel Planning Pyramid

The first decision or staffing decision has a long term time scope and is also known as demand

modelling or the strategic phase of personnel planning. People of the organisation determine

annually or semi-annually the number of staff members of each skill class. Based on the way this

number is calculated, we distinguish three categories: task based demand, flexible demand and shift

based demand. For a more detailed explanation of these categories I refer the interested reader to

Ernst et al. (2004a).

The scheduling decision is the next step in the pyramid of personnel planning. This step is consistent

with work schedule modelling and corresponds to the tactical phase of personnel planning, which

has a medium term time scope. Every month (or other similar time period) each staff member needs

a schedule that sets his/her working hours, breaks and off days. This decision is often strongly

influenced by the preferences of the employee. Three broad categories are commonly stated here:

days-off scheduling, shift scheduling and line of work construction or tour scheduling:

- Days-off scheduling handles the determination of nonwork days in the schedule. This decision

should not be taken overnight because “the pattern of the nonwork days in the working period

is highly related to staff satisfaction” (Felici et al., 2004). The problem itself is about matching

the working week of an employee, often four or five days, with the operating week in the

facility (mostly seven days).

Staffing decision - Strategic phase

Scheduling decision - Tactical phase

Allocation decision - Operational phase

Feedback

Feedback

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- “Shift scheduling, also called time-of-day scheduling, deals with the problem of selecting, from

a potentially large pool of candidates, what shifts are to be worked, together with an

assignment of the number of employees to each shift, in order to meet demand” (Ernst et al.,

2004a). In general, two types are often distinguished: a schedule with non-overlapping shifts,

which is the simplest approach, and another one which studies the shift scheduling problem

when shifts do overlap.

- Line of work construction (often referred to as tour scheduling when dealing with flexible

demand) is nothing more than the design of a proper work schedule. It is a generalized form of

the previous two categories, so both dealing with days-off scheduling and shift scheduling.

The final step in the personnel planning pyramid is the allocation decision. This step corresponds to

the operational phase and is nothing more than the daily ‘fine tuning’ of the work schedule or

matching the daily variability in supply and demand.

Some papers mention the existence of a fourth decision that needs be the taken into account,

namely the assignment decision (Warner, 1976a). However, this and the previous step do not lie in

the scope of this thesis and will not be cited any further.

1.3.2. The relation between the staffing- and scheduling decision

The vast majority of papers dealing with the staffing- or scheduling decision can be placed within the

personnel planning pyramid mentioned above. Some articles only deal with the staffing- or

scheduling decision, while others handle both.

One reason for the decomposition of the rostering problem can be that the staffing decision is often

made in conjunction with the budgeting- or finance office, while scheduling decisions are typically

made by employees who stand closer to the labourers. Another, more practical, reason is the

computational effort. Logically, models that try to solve both the staffing –and scheduling decision

will take more time than if you would solve them separately. On top of that, “a common problem

associated with the integration of staffing and scheduling decisions is that strict enforcement of

aggregate staffing decisions can lead to infeasibility at the scheduling phase” (Venkataraman et al.,

1996). In literature, this problem is frequently solved by introducing hard –and soft constraints. In

the next chapter I will go into more detail about the difference between these two types of

constraints.

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Nevertheless, combining both the staffing –and scheduling decision also has its advantages. In terms

of creating the best work schedules, it is better to create models that deal both with staffing and

scheduling. Furthermore, inefficient staffing decisions can lead to expensive scheduling decisions

which are based on more accurate forecasts (Abernathy et al., 1973). Also, every scheduling decision

“requires the support of the longer-term aggregate staffing decision since labor to be scheduled in

each period reflects the current staffing capabilities” (Venkataraman et al., 1996) and future

demands.

1.3.3. The tour scheduling problem

The purpose of this thesis is to focus only on one particular domain of the personnel planning

pyramid, namely the TSP. This particular field of personnel planning is gaining a lot of interest in the

past couple of decades, especially because of its wide usage domain in many kinds of application

areas, “potential benefits of lower costs and improved productivity which can be realized from the

efficient utilization of labor” (Brusco et al., 1995).

“The standard tour scheduling problem is to simultaneously determine daily shift start times and days

worked patterns that minimize cost over a one week planning cycle while meeting

labor coverage requirements for each planning period” (Isken, 2004).

As stated in this definition, the TSP covers both the shift- and days-off scheduling problem, both part

of the tactical phase of personnel planning. Logically, when the rostering horizon covers only one

day, the TSP converts into shift scheduling. If there is only a single shift scheduled each day, the TSP

is equivalent to the days-off scheduling problem. In turn, “a tour is defined by a set of work days, and

the daily shifts for each work day, for a planning horizon of one or more weeks” (Brusco, 1998).

1.4. Outline

The remainder of this thesis is organized as follows. Chapter 2 captures a wide ranging literature

study about tour scheduling. First, the core of the TSP is revealed by providing a definition and a clear

visual representation of a tour. Secondly, I will discuss the different mathematical approaches to

solve the TSP. These standardized approaches are very limited which makes it possible to make an in-

depth study. Thirdly, a ‘roadmap’ is demonstrated summarizing all the different tour scheduling

categories. As the reader will find out later on, there are five main categories: the objective

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function(s), the decision variable(s), the personnel characteristics, the shift definitions and the

constraints. Each tour scheduling category is then further subdivided and studied extensively. This

study is performed by comparing the subcategories for more than 40 papers coping with the TSP.

The last section of chapter 2 points out how the tour scheduling categories will be incorporated in

the next chapter. The latter is built around programming the two basic tour scheduling approaches,

an (explicit) set-covering model (SCM) and an implicit model (IM). Above that, a third model is

suggested which combines the tour generation steps of previous two models. The first section of

chapter 3 provides for each model a complete mathematical programming formulation. The latter is

based on their respective basic models explained in chapter 2 and all of the different tour scheduling

characteristics found in the existing literature. In section 3.2 I will discuss the solution methodology

of the programs. We can distinguish five main components: the settings, a demand pattern

generator, a tour generator, a cost generator and a final optimization step. For each component, the

differences and similarities between the three models are argued into detail. The last section of

chapter 3 presents for each model the (computational) results for a specific scenario. These scenarios

give more insight in the impact of modifying the subcategories of the 5 main tour scheduling

components identified in chapter 2. In chapter 4, general conclusions are drawn. Chapter 5 captures

the limitations of this thesis and directions for future research are given.

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Chapter 2

2. Literature Study: the Tour Scheduling Problem

The amount of literature studying the art of personal planning has exploded the past couple of

decades. Two reasons are unquestionably part of the root cause of this phenomenon. First,

organisations are faced with a continuously rising cost per person. For this reason, it is evident that,

certainly profit organisations, are exploring any kind of method which could help in reducing their

total personnel cost. A more efficient and effective personnel planning method is definitely part of

this. As a case in point, it is often seen that companies hire specialized institutions or ask universities

to develop state-of-the-art personnel scheduling software. Another reason is found in the world of

information technology. Significant improvements are made in computer software which makes it

possible to include more specifications and, hence, better align the requirements of both the

employer and the employees. This, in turn, broadens the playing field for researchers to study the art

of personnel planning.

Being part of personnel planning, tour scheduling has seen the same expansion in literary work.

Comprehensive surveys of the TSP in the early years of research are provided in Baker (1976), and

Tien and Kamiyama (1982). For the period 1990-2001, a complete survey is found in Alfares (2004).

Ernst et al. (2004b) covers both time periods starting from 1970 until 2004. For the period 2004 until

July, 1st 2012, J. Van den Bergh et al. (2012) identify 291 articles about personnel scheduling, mostly

dealing with shift scheduling and tour scheduling. This is not a complete set of articles on personnel

scheduling, but the authors state that “they have identified a set that is representative of the work

carried out by the research community”. Almost every paper referred to in the next paragraphs

and/or chapters, is part of at least one of these review articles stated above. Other, more general,

interesting review articles are Burke et al. (2004) and Ernst et al. (2004a).

In what follows I will try to make the reader comfortable in the wide spectrum of tour scheduling

literature. First, a tour is defined and three different visualisation techniques are illustrated. Next,

mathematical formulations are introduced, together with a short explanation of key papers

addressing the respective TSP. Finally, a distinction is made between five tour scheduling categories.

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2.1. Visualising a tour

In literature, there are several commonly used visualisation techniques to represent a tour.

Remember, “a tour is defined by a set of work days, and the daily shifts for each work day, for a

planning horizon of one or more weeks” (Brusco, 1998). Let us now assume that a work week

consists of seven days and each day there are three working shifts: the early shift (E) from 6am until

2pm, the late shift (L) from 2pm until 10pm and the night shift (N) from 10pm until 6am. One

possible method to represent a tour is by mentioning for each worker and for each day, the shifts

they have worked. For example, in table 1 each row represents one single tour. The ‘F’ stands for a

free shift or in other words for a nonworking day:

Monday Tuesday Wednesday Thursday Friday Saturday Sunday

Worker 1 E E F F L L L

Worker 2 L L E E E F F

Worker 3 N N L L F F N

Table 1: Visualisation of a tour – representation 1: one shift per day

Another, very similar, representation method makes use of crosses to link each worker to a certain

shift. This method has the advantage that it is very easy to interpret and analyze (see table 2):

Monday Tuesday Wednesday Thursday Friday Saturday Sunday

E L N F E L N F E L N F E L N F E L N F E L N F E L N F

Worker 1 x x x x x x x



Table 2: Visualisation of a tour – representation 2: all shifts per day

The final method or binary representation method in table 3 is less user friendly but easier to cope

with if the data has to be used by programmers:

Worker 1 1000 – 1000 – 0001 – 0001 – 0100 – 0100 – 0100

Worker 2 0100 – 0100 – 1000 – 1000 – 1000 – 0001 – 0001

Worker 3 0010 – 0010 – 0100 – 0100 – 0001 – 0001 – 0010

Table 3: Visualisation of a tour – representation 3: binary representation

9

Minimize

S.t.

2.2. Mathematical formulation: a standardized approach

The TSP is in its most basic approach always represented as a set-covering formulation. The latter is a

type of model in which the objective is to cover the demand for staff (Ernst et al., 2004b). In what

follows, a distinction is made between an explicit- and an implicit set-covering formulation. The

former represents each tour by a separate integer variable (Easton and Rossin, 1991a). This integer

variable consists of a set of specific days worked during the week (days-off scheduling), together with

a pattern of shifts or shift starting times (shift scheduling). The implicit approach uses a combination

of variables whereby shifts are represented by “starting-time bands” and days-on patterns. In

addition, tours are formed after optimizing the problem. Because the explicit approach is the oldest

and most basic one, researchers often assume that the default setting for TSPs is an explicit set-

covering one. In other words, no distinction is made any more between explicit and implicit set-

covering formulations, but between set-covering- and implicit modelling. In what follows we will also

see that there is a commonly used derivative tour scheduling model (TSM), especially applied in

nurse scheduling.

2.2.1. A set-covering model (SCM)

Based on the general SCM of Dantzig (1954) and the firstly developed integer programming

formulation for TSPs of Morris and Showalter (1983), we can make a clear mathematical distinction

between shift scheduling, days-off scheduling and tour scheduling:

∑( )

∑( )

p = 1, 2, …, P;

j = 1, 2, …, J;

with 1 ≤ J ≤ 7

t = 1, 2, …, T.

10

Shift scheduling Days-off scheduling Tour scheduling

xt Number of employees

assigned to daily shift t

Number of employees

assigned to days-off pattern t

Number of employees

assigned to tour t

ct Cost of shift t Cost of days-off pattern t Cost of tour t

dpj

Required staffing level for

time period p (day j is not

considered)

Required staffing level for day

j (period p is not considered)

Required staffing level for

time period p, day j

apjt

1 if time period p is a work

period in daily shift type t, 0

otherwise (day j is not

considered)

1 if day j is a work day in the

days-off pattern t, 0

otherwise (period p is not

considered)

1 if time period p is a work

period and day j is a work day

in tour t, 0 otherwise

T

(index t) Set of shift types Set of days-off patterns Set of tour types

P

(index p)

Set of time periods to be

scheduled over a single day

Set of planning periods to be

scheduled over the week,

where the planning period is

defined by the period-day

pair (p, j)

J

(index j)

Set of days per week that the

system operates

Table 4: A standard Tour scheduling problem: Set-covering Model

The most important difference between these three scheduling problems is hidden under the

interpretation of the indices, as can be seen in table 4. For example, index ‘t’ represents a shift, a

days-off pattern and a tour in the shift-, days-off- and tour scheduling problem, respectively. The

objective function of this standard TSP is to minimise the total cost of employees working a certain

tour type t. In reality, however, the objective of the TSP often consists of multiple goals, which is also

called goal modelling. A more detailed explanation of the objective functions addressed in tour

scheduling is provided in paragraph 2.3.1. Dantzig’s SCM only adds one constraint, a so called

coverage constraint, which states that there should be a minimum staffing level dpj for each time

period p on day j. The fact that there is only one simple constraint makes this model very basic and

not usable in practice. Many others must be added to make sure that it becomes a model which is

capable to cope with the needs of both the employer and the employee.

Many researchers have used Dantzig’s SCM as the starting point for their analysis about tour

scheduling. In the latest years of the 20th century, it was especially Brusco and Jacobs that published

a lot of research. For example, Brusco and Jacobs (1998a) is built upon one of their previously

released papers, namely Brusco and Jacobs (1995). The paper studies the restricted starting-time TSP

and solves this by a two-stage heuristic solution. To evaluate their solution, numerous experiments

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are conducted with data from United Airlines airport ground stations. Brusco and Johns (1996)

present a sequential integer programming heuristic to develop a personnel schedule. One of the

steps in the heuristic incorporates the general SCM of Dantzig (1954). At the end, an experimental

study is performed with a full-time (FT) workforce and a mixed workforce. Brusco and Jacobs (1998b)

discuss the main problem of the SCM for tour scheduling. Namely, often too many variables/tours

are included which makes it not possible for modern integer program solvers to find an optimal

solution. Strategies are with success developed to eliminate redundant columns. The research

conducted in Brusco and Jacobs (2001) is in line with the topics of Brusco and Jacobs (1998a - 1998b).

First, they suggest an integer lineair programming formulation for the limited starting-time TSP. Next,

an experimental analysis and case study are performed in order to investigate the impact of the

number of daily starting periods, S, on the probability of finding an optimal solution. It turned out

that only a very small portion of S was capable of providing the minimum workforce size. More

recently, in Alfares (2007) an explicit IP model is solved to determine tour schedules for IT help desk

operators. Costs need to be minimized while incoming calls need to be answered 24 hours a day. The

workforce size, employee utilisation and service level are other determinants to evaluate alternative

schedules. Ertogral and Bamuqabel (2008) study the art of developing personnel schedules for a

bilingual telecommunication call center. Two models are created, one for no flexible workers and

one for the case with flexible workers. The former is a simple adaption of Dantzig’s SCM.

2.2.2. An implicit model (IM)

Next to the (explicit) SCM of Dantzig, an implicit modelling approach has also been developed. Bailey

was the first to implicitly integrate the shift scheduling problem with an explicit representation of

days-off patterns (Bailey 1985). In addition, substantially fewer variables are needed, which is of

course a big advantage compared to Dantzig’s SCM. Often we see that so-called start-time bands are

used when the implicit modelling approach is applied. These bands are nothing more than a range,

known as the band width (BW), in which starting times can vary within an individual tour:

12

Minimize

s.t.

∑ ∑

∑ ∑

∑ ∑

where

= number of employees assigned to start-time band s and days-off pattern f;

= 1 if period p is a work period in a shift in start-time band s beginning in hour h, 0

otherwise;

= 1 if day j is a work day for days-off pattern f, 0 otherwise;

= number of employees assigned to a shift on day j in start-time band s beginning in

period h;

= minimum number of employees required in time period p on day j;

= number of start-time bands (index s);

= number of days-off patterns (index f);

= number of shift start times per start-time band (index h), also known as the BW;

= number of planning time periods per operating day (index p);

= number of work days per operating week (index j).

As a result of the more efficient approach that implicit modelling seemed to be, more and more

researches started to study and extend this method. For example, Jacobs and Bechtold published

two papers in 1993 addressing an implicit TSP (Jacobs & Bechtold, 1993a and Jacobs& Bechtold,

1993b). Their purpose was to investigate the effect of multiple labor scheduling flexibility

alternatives, as there are, for example, shift length, tour length, meal break, shift start-times, and so

forth. Jarrah et al. (1994) optimizes an implicit TSM which is based on a combination of the implicit

break-flexibility model of Bechtold and Jacobs (1990) and the days-on algorithms developed by Burns

and Carter (1985). The BW of the model was set equal to the number of starting times in the

operating day. Jacobs and Brusco (1996) present an implicit TSM with overlapping start-time bands.

p = 1, 2, …, P;

j = 1, 2, …, J;

j = 1, 2, …, J;

s = 1, 2, …, S;

j, s, f, h.

13

Minimize

s.t.

Problems with more than two million variables using a SCM were tested for this IM. The results

indicated that implicit modelling can provide an important improvement in scheduling efficiency

compared to SCM. Brusco and Jacobs (2000) integrates and extends the models of Bechtold and

Jacobs (1990) and Jacobs and Brusco (1996). Namely, both start-time and meal-break flexibility are

incorporated in an implicit TSM for a continuous environment. Isken (2004) proposes on his turn a

general and a more extended implicit TSM that complements the one developed in Brusco and

Jacobs (2000). In the past decade, Bard J. has published several papers dealing with the implicit TSM,

for example, Bard (2004a), Bard (2004b) and Bard et al. (2007). Eitzen et al. (2004) performs an

analysis on the size of an implicit general SCM for a non-hierarchical multiple-skilled workforce.

Results are presented for up to 110 employees, 9 different skill levels and three demand patterns.

Moreover, the authors proposed these results for three column-based solution techniques: column

expansion, column subset, and branch-and-price. Seckiner et al. (2007) compose an implicit TSM

based on an IP formulation developed by Billionnet (1999). The main difference with the latter is that

workers can be assigned to alternative shifts in a day during the entire scheduling horizon, whereas

Billionnet’s model restricted workers to be assigned to one shift type. Recently, Rong (2010)

developed two-monthly TSMs with mixed skills considering the weekend off requirements.

2.2.3. A derivative model

Set-covering and implicit modelling are the two standard approaches to solve the TSP. Both models

use a decision variable (represented by ) that captures the number of employees assigned to a

tour or to a start-time band and days-off pattern, respectively. However, when the number of

employees is known, for example in a nursing unit, the decision variable transforms to a binary

variable which is 1 if, for example, nurse i has to work shift k on day j, 0 otherwise. The TSP with a

binary decision variable has in many cases the following structure:

∑ ∑

∑

( )

{ }

i, j, k.

14

where

= 1, if nurse i is assigned to shift k on day j, 0 otherwise;

= number of personnel above the required coverage djk on day j, shift k;

= minimum number of employees required on day j, shift k;

= set of employees (index i);

= set of days in the scheduling horizon (index j);

= set of shift types: K = {1, 2, 3, 4} indicating an early-, day-, late- or

night shift, respectively (index k).

The objective here is not longer to minimize the number of employees but the level of overstaffing,

represented by the variable . The only constraint added to the model is very similar to the

previous two models, except for the subtraction of the overstaffing variable. It is important to

understand that this method is a very different approach of the TSP. Namely, in the SCM and the

implicit modelling approach, new employees are added to the model and assigned to a tour or a

(non-)working pattern until labor demand is met. In this third tour scheduling model (TSM), however,

each individual employee/nurse is assigned to a series of shifts spread over the scheduling horizon.

As a result, complete tours or (non-) working patterns are created only when the full schedule has

been set up. In other words, employees or not assigned to an existing tour, but rather to

independent shifts and (non-) working days that are combined to a complete tour for each employee

separately.

Warner (1976b) was one of the first to introduce an IP formulation for the nurse scheduling problem

based on this derivative model. The formulation considers different nurse grades and each nurse has

to fill in a sheet in which they can rank their aversions to certain shift patterns. Kostreva and Jennings

(1991) presented a very similar IP and try to solve this on a microcomputer with the help of Benders

decomposition technique. Beaulieu et al. (2000) use a mathematical programming approach for

scheduling physicians in the emergency room. Moz and Pato (2004) tackle the nurse rerostering

problem for a public hospital in Portugal. Existing labor rules and organisational policies need to be

taken into account and are regarded as hard constraints (cf. infra). Integer linear- and constraint

programming are the two methods proposed in Trilling et al. (2006) to solve an anaesthesiology

nurse scheduling problem. De Grano et al (2009) describe the typical nurse scheduling problem with

both nurse preferences and hospital constraints. A two-stage approach is applied, whereby the first

stage allocates nurses to their preferred shifts and rest days based on a bidding procedure and the

second stage fills the remaining shifts by allocating them to nurses who have not yet met their

15

minimum required hours. Burke et al. (2010) study a highly-constrained nurse scheduling problem.

The latter is solved by combining the advantages of IP and variable neighbourhood search. Ronnberg

and Larsson (2010) try to find out “if it is possible to create an optimisation tool that automatically

delivers a usable schedule based on the schedules proposed by the nurses.” (Ronnberg and Larsson,

2010) Valouxis et al. (2012) present a two phase approach for the nurse rostering problem,

sequentially solved using integer mathematical programming. In the first phase, the demand that

needed to be covered by a particular nurse was decided for each day of the week. In the second

phase, the nurses were assigned to specific daily shifts. Apart from nurse scheduling, the derivative

model is also often applied in transit operator scheduling, telecommunicating and airline crew

scheduling (Nanda and Browne, 1992). Two examples of the latter are Felici and Gentile (2004) and

Stolletz (2010). Felici and Gentile, 2004 starts from a general IP model to schedule workers who

provide continuous services. The goal of the model is to maximize staff satisfaction measured by

positive weights for pairs of shifts in consecutive days. Stolletz (2010) creates a binary linear

programming formulation for check-in counters at airports. The impact of the degree of flexibility

and economies of scale are analysed by a numerical study.

The advantage of tour scheduling, so combining shift- and days-off scheduling, over single shift- or

days-off scheduling is that the model stands much closer to reality and is therefore faster

implementable in practice. The logical disadvantage, however, is a significant increase in complexity

regarding the number of variables, parameters and constraints. This increase is not clear after

studying the basic TSMs introduced before, but will be after reading the next chapter.

2.3. Different aspects of a tour scheduling model: Literature review

Papers dealing with the TSP distinct themselves based on five broad categories: the objective

function(s), the decision variable(s), the personnel characteristics, the shift definitions and the

constraints. Figure 2 acts as a ‘roadmap’ by summarising these categories and indicating the

respective paragraphs throughout this chapter. For each category the same 43 tour scheduling

papers are compared. Most of them have already been discussed in one of the previous paragraphs.

It should also be clear by now that this thesis only focuses on integer programming and no other

solution method. Remember the definition of Easton and Rossin (1991a) where they state that “Tour

scheduling problems are linear integer programs”, so this limitation is nothing more than a logical

step.

16

Tour Scheduling Categories

Figure 2: Tour Scheduling Characteristics: ‘Roadmap’

2.3.1. Objective function(s)

The most frequently used objective function in tour scheduling is minimizing an organisations total

personnel cost. Its most basic mathematical formulation, as seen in the standardized approaches of

tour scheduling, is composed of a cost variable (i.e. ) and a decision variable (i.e. ). The cost

variable often represents the wage to employ a labourer on a certain shift and day. However, there

are many other cost variables, such as different costs per skill category (e.g. greater pay grade for

highly-skilled workers), a cost depending on the day of the week (e.g. weekend days are more

• Cost Objective

• Workforce Size Objective

• Other

Objective Function(s): 2.3.1

• Number of employees assigned to tour t

• Number of employees assigned to start-time band s and days-on pattern f

• 1, if employee (~nurse) i is assigned to work shift j on day k

Decision Variable(s): 2.3.2

• Labor contract: Full time (FT) an d/or Part time (PT)

• Skills

Personnel Characteristics: 2.3.3

• Overlap

• Start times: fixed and/or definable

• Length: fixed and/or definable

• Breaks

Shift Definitions: 2.3.4

• Coverage Constraints

• Time-related Constraints

• Balancing Constraints

Constraints: 2.3.5

17

expensive for the organisation), overtime costs, outsource costs, costs of executing tasks, different

costs for PT and FT employees, travel costs, production costs, and so forth. Minimizing the total

workforce size is also a popular objective function. Mathematical, this objective function differs from

the previous one because there is no cost variable present, hence, only a decision variable. The

reason why this type of objective function is slightly less popular is because minimizing the personnel

cost brings along more possibilities. Namely, “when using personnel cost rather than an explicit

number of employees, one can make a trade-off between hiring employees, overtime, casual

workers, etc., by assigning a (relative) cost to all of these factors” (J. Van den Bergh et al., 2012).

Besides minimizing the personnel cost or the total workforce size, many other types and

combinations of objective functions are used within tour scheduling (more recent objectives are

added to the list of Bechtold, Brusco and Showalter (1991)):

- Fairness among employees/balancing the workload (e.g. balance between workers in the

number of weekends off and the number of night shifts);

Minimization:

- Minimize the total number of overstaffed shifts or, in general, the level of overstaffing;

- Minimize the total number of understaffed shifts or, in general, the level of understaffing;

- Minimize personnel dissatisfaction;

- Minimize (mean) lateness of the completion of tasks;

- Minimize the number of different work schedules used;

- Minimize the variation of staff surplus over a certain time period.

Maximization:

- Maximize personnel satisfaction;

- Maximize the number of workers who receive a long weekend;

- Maximize the number of schedules with consecutive days-off;

- Maximize total profit;

- Maximize the long-run benefit (net present value (NPV)) of providing good service.

Most of these objectives are formulated when the TSP is solved by a goal programming approach.

This multi-objective method is used when researchers try to solve a problem which addresses the

needs of as much stakeholders as possible. For example, on the one hand, operation managers may

want to limit the number of casual labourers because of continuity reasons or availability of

knowhow. But on the other hand, the organisation itself could insist on minimizing the personnel

cost. This multi-objective problem is solved by adding penalties or weight factors to each individual

18

Minimize

goal so as to combine these goals into one single objective function. Of course, the question then

arises: which goal is more important and, thus, gets the highest weight factor? A small example is

shown below:

∑( )

∑(

)

where

= labor understaffing and overstaffing in period p, respectively;

= penalty for understaffing and overstaffing in period p, respectively.

The first part of this objective is equal to the SCM shown in paragraph 2.2. The second part, however,

adds under- and overstaffing variables multiplied with their respective penalties.

To end this paragraph, table 5 lists more than forty papers dealing with an integer programming

approach for the TSP. For each paper, the respective objective function is indicated by an ‘x’. A

distinction is made between the following (sub-) categories:

Cost objective:

(1) Personnel cost (regular wage cost);

(2) Different cost per day;

(3) Different cost per skill;

(4) Overtime cost;

(5) Outsource cost;

(6) Cost of executing tasks;

(7) Different cost for PT and FT employees;

Workforce size objective:

(8) Minimizing the workforce size;

Other objectives:

(9) Minimize Understaffing;

(10) (Dis-) Satisfaction of employees;

(11) Balancing the workload;

(12) Multi-objective model;

(13) Other* (further explanation below the table)

19

Cost objective Workforce

size Other

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)

Alfares, 2007; x x Al-Yakoob and Sherali,

2007; x x x x

Bard, 2004a; x

Bard, 2004b; x x x

Bard and Wan, 2008; x

Bard et al., 2003; x x

Bard et al., 2007; x

Beaulieu et al., 2000; x x

Berrada et al., 1996; x

Brusco and Jacobs, 1998a; x

Brusco and Jacobs, 1998b; x

Brusco and Jacobs, 2000; x

Brusco and Jacobs, 2001; x

Brusco and Johns, 1996; x

Burke et al., 2010; x

Chen et al., 2010; x x x

Choi et al., 2009; x x x

De Grano et al., 2009; x

De Matta and Peters, 2009; x x

Eitzen et al., 2004; x Ertogral and Bamuqabel,

2008; x x

Felici and Gentile, 2004; x

Isken, 2004; x

Jacobs and Brusco, 1996; x

Jarrah et al., 1994; x Kostreva and Jennings, 1991; x

Lezaun et al., 2006; x x1

Lezaun et al., 2007; x2

Lezaun et al., 2010; x3

Love and Hoey, 1990; x x x x

Mohan, 2008; x

Moz and Pato, 2004; x4

Ozkarahan, 1989; x x

Rong, 2010; x x Ronnberg and Larsson,

2010; x x x

20

Seckiner et al., 2007; x x

Stolletz, 2010; x

Thompson, 1995; x x5

Topaloglu and Ozkarahan

(2004); x x

Trilling et al., 2006; x

Valouxis and Housos, 2000; x x

Valouxis et al., 2012; x6

Warner, 1976b; x

Table 5: Objective Function(s)

1

Lezaun et al. (2006): One model is split up in several sub models all with their own objective. In one model is the

goal to plan reserve weeks immediately before or after vacation. Another model seeks to

select the most attractive patterns according to the preferences of the drivers. 2 Lezaun et al. (2007): No real objective function is suggested. The goal was to set up a model that fulfils the

constraints and assigns patterns according to the employees’ preferences. 3 Lezaun et al. (2010): The main objective of this project is to draw up the annual roster for all working shifts at

stations, taking into account staff seconded from one station to another in such a way that the

final result is fair and the distances travelled are minimized. 4 Moz and Pato (2004): The objective function is to minimize the total cost of a multicommodity flow model.

5 Thompson (1995): The objective function in a New Formulation for the weekly Labor Scheduling Problem (NFLSP)

is to seek a trade-off between the cost of tours and the effect of different staffing levels on

customer service-related profits (NPV profits). 6 Valouxis et al. (2012): The goal here is to satisfy as many soft constraints as possible, but all variables and their

penalties are summed into one non-financial cost variable.

Table 5 points out that cost, (dis-) satisfaction with the work schedule and multiple goal models are

commonly used to formulate the objective function of the TSP. In detail, cost objectives are often

formulated as personnel costs. Hereby, a cost variable is multiplied with the decision variable of the

respective model. Furthermore, the degree of satisfaction is included based on a survey held by the

employer. A score is then given by the employee to work a particular shift on a certain day and this

score or penalty is, just like the cost variable, multiplied with the decision variable in the objective

function. Lastly, goal modelling is, as said before, a combination of objectives.

2.3.2. Decision variables

The reasoning followed here is built upon the discussion of decision variables started in paragraph

2.2. In this paragraph two standard models and one derivative model were proposed. The first model

was called the SCM and the second one suggested an implicit modelling approach for the TSP. Both

21

models are built around a decision variable that captures the number of employees. In detail, the

following two decision variables are used:

- Number of employees assigned to tour t;

- Number of employees assigned to start-time band s and days-on pattern f.

The derivative model that was explained, dealt with the TSP when the workforce size was already

known. This transformed the decision variable into a binary variable:

- 1 if employee (~nurse) i is assigned to work shift j on day k.

These three types of decision variables are commonly used in tour scheduling literature. Table 6

indicates for each paper to what type their decision variable(s) belong(s) to. We make a distinction

between the following types:

- Type1: Set-covering:

The model formulates (complete) tours before optimization is performed. Every tour is

explicitly defined by a separate integer variable;

- Type 2: Implicit:

The model formulates (complete) tours after optimization is performed. Often, the model

requires several steps before and after optimization.

- Type 3: Derivative:

Each employee is unique and restricts the way by which they can be scheduled. The

number of the employee is used as an index in at least one (of the) decision variable(s).

Type of the Decision

Variable(s)

Type of the Decision

Variable(s)

Alfares, 2007; 1 Isken, 2004; 2

Al-Yakoob and Sherali, 2007; 3 Jacobs and Brusco, 1996; 2

Bard, 2004a; 2 Jarrah et al., 1994; 2

Bard, 2004b; 2 Kostreva and Jennings, 1991; 3

Bard and Wan, 2008; 3 Lezaun et al., 2006; 3

Bard et al., 2003; 2 Lezaun et al., 2007; 3

Bard et al., 2007; 2 Lezaun et al., 2010; 3

Beaulieu et al., 2000; 3 Love and Hoey, 1990; 2

Berrada et al., 1996; 3 Mohan, 2008; 3

Brusco and Jacobs, 1998a; 1 Moz and Pato, 2004; 3

Brusco and Jacobs, 1998b; 1 Ozkarahan, 1989; 2

Brusco and Jacobs, 2000; 2 Rong, 2010; 2

22

Brusco and Jacobs, 2001; 1 Ronnberg and Larsson, 2010; 3

Brusco and Johns, 1996; 1 Seckiner et al., 2007; 2

Burke et al., 2010; 3 Stolletz, 2010; 3

Chen et al., 2010; 2 Thompson, 1995; 2

Choi et al., 2009; 1 Topaloglu and Ozkarahan

(2004); 2

De Grano et al., 2009; 3 Trilling et al., 2006; 3

De Matta and Peters, 2009; 1 Valouxis and Housos, 2000; 3

Eitzen et al., 2004; 2 Valouxis et al., 2012; 3

Ertogral and Bamuqabel, 2008; 1 Warner, 1976b; 3

Felici and Gentile, 2004; 3

Table 6: Type of Decision Variable(s)

2.3.3. Personnel characteristics

When a personnel scheduling model has the goal to allocate workers to a certain shift or

workstation, one should take into account the different characteristics of its workforce. In tour

scheduling literature, multiple personnel characteristics are often applied (J. Van den Bergh et al.,

2012), such as an individual productivity level, seniority level, skills and whether he or she is a full- or

part time employee. This classification is necessary to align the theoretical TSM as much as possible

with reality. As a case in point, we can think of young graduates who do not yet have the required

skills to exercise a certain job. For example, in hospitals you often see that young nurses are

scheduled at the same time with more senior nurses, so that they can learn and gain the necessary

experience. If employees are scheduled to shifts or workstations for which they do not have the right

skills, this could entail a lower profit margin because of a less effective and efficient production.

Often we see that this ‘misplacement’ of workers is represented by a penalty weight in the objective

function (cf. supra). Classifications based on the productivity- and/or seniority level are closely linked

with those of skill categories. However, in tour scheduling literature productivity and seniority of

employees is of less importance. The classification of full- and part time workers is often not enough

to fulfil the required coverage. Therefore, so-called casual workers are also included in the model.

This specific group type of workers consists of, for instance, another company department or interim

workers. A last type of classification is based on the grouping of employees. It may be necessary to

assign a crew (or team) instead of scheduling each worker on its own. A good example is found in the

transportation area, but this is not in line with this thesis.

Table 7 indicates for each paper whether a distinction is made between full- and part time

employees and if skill levels are adopted into the model:

23

Labor Contract

Skills

Labor Contract

Skills Full time

Part time

Full time

Part time

Alfares, 2007; x Isken, 2004; x x

Al-Yakoob and Sherali, 2007; x Jacobs and Brusco, 1996; x

Bard, 2004a; x x x Jarrah et al., 1994; x x

Bard, 2004b; x x x Kostreva and Jennings, 1991; x

Bard and Wan, 2008; x x x Lezaun et al., 2006; x

Bard et al., 2003; x x Lezaun et al., 2007; x

Bard et al., 2007; x x Lezaun et al., 2010; x

Beaulieu et al., 2000; x Love and Hoey, 1990; x x x

Berrada et al., 1996; x x x Mohan, 2008; x

Brusco and Jacobs, 1998a; x x Moz and Pato, 2004; x

Brusco and Jacobs, 1998b; x x Ozkarahan, 1989; x x x

Brusco and Jacobs, 2000; x Rong, 2010; x x

Brusco and Jacobs, 2001; x Ronnberg and Larsson, 2010; x x x

Brusco and Johns, 1996; x x x Seckiner et al., 2007; x x

Burke et al., 2010; x x Stolletz, 2010; x

Chen et al., 2010; x x Thompson, 1995; x

Choi et al., 2009; x x Topaloglu and Ozkarahan (2004); x x

De Grano et al., 2009; x x Trilling et al., 2006; x x

De Matta and Peters, 2009; x Valouxis and Housos, 2000; x

Eitzen et al., 2004; x x x Valouxis et al., 2012; x x x

Ertogral and Bamuqabel, 2008; x x x Warner, 1976b; x x

Felici and Gentile, 2004; x

Table 7: Personnel characteristics

Analyzing table 7 teaches us that half of the papers incorporate a labor contract with both full- and

part time employees. Moreover, if skills are integrated, one can be sure that the model also offers a

distinction between full- and part time employees.

2.3.4. Shift definitions

Next to deciding which of the personnel characteristics should be taken into account, the modeller

should also take some key decisions regarding shift characteristics. There are three types of decisions

that should be discussed when creating a TSM: the length of a shift, shift starting times and whether

shifts may overlap or not. The attentive reader will already have noticed that these decisions are

mainly part of shift scheduling rather than both shift- and days-off scheduling, as is the case for tour

24

scheduling. The ability to determine the length of a shift gives the scheduler a lot of flexibility.

However, there are some unwritten rules that the length of each shift should be at least 4 hours and

maximum 10 hours. Shifts of 12 hours do still exist, but they are more and more banned mostly

because of humanity reasons. A shift length of 8 hours is most common.

Besides the flexibility in scheduling shifts coming from a wide range of shift lengths, the modeller is

also free in his/her selection of shift starting times. “The selection of daily shift starting times consists

of two interrelated decisions: the number, S, of shift starting times permitted during an operating

day, and the determination of precisely which of the S planning periods of the day will be allocated

as starting times” (Brusco and Jacobs, 2001). It is clear that there should be a trade-off regarding the

value of S. Namely, increasing the value of S has both its advantages and disadvantages:

Advantages:

- It is possible to better align the supply and demand of required labour units, even with fewer

workers;

- Because of this higher flexibility at the coverage aspect, there is a lower probability of

overstaffing the coverage requirements. For this reason, an organisation might expect greater

labor productivity.

- More starting times should increase the fairness of the schedule and reduce shortages, which

may result in less turnover and absenteeism.

Disadvantages:

- A significantly higher administrative burden;

- There are more time periods at which employees can start working, take a break, go back to

work and will stop working. In addition, it will be more difficult to monitor employee

punctuality. On top of that, there might be a serious loss of labor productivity.

- The effectiveness of (de-)briefings could decrease because it will be harder to schedule these

moments and address as much employees as possible at the same time;

- Finally, an increase in starting times makes it more difficult to take advantage of teamwork.

In table 8, a distinction is made between fixed and definable start times and shift lengths. These

subcategories are an indication for the scheduler’s freedom. An example that you will often find in

tour scheduling literature is fixed start times and –shift lengths for full time employees (e.g. three 8-

hour shifts), while part time employees have more variable starting times and shift lengths (e.g.

possibility to start on every hour of the day).

25

A final decision that the modeller should handle, is the question whether shifts may overlap or not.

When the demand for labor units is very irregular, for instance when working in a call center,

multiple shifts covering the same time period is definitely something worth questioning. The main

reason for this is that overlapping shifts give the scheduler the possibility to increase the number of

employees at certain (peak) times. What’s more, “one can deal with demand peaks without being

forced to schedule overtime or to hire extra employees and one can avoid excess staff during low

demand periods” (J. Van den Bergh et al., 2012). Nevertheless, it is not always specifically mentioned

in the existing literature if shifts do overlap, and if they do, to what extent.

Not really part of the discussion about shift definitions is the adoption of break time flexibility.

However, break placement is definitely more part of shift scheduling than of days-off scheduling.

Therefore, table 8 not only provides information about shifts overlap, -start times and -lengths but

also break flexibility is adopted into the table. Important to note is that in some papers, the

scheduler/operator has to select tours or working patterns out of a predefined set of tours/patterns.

The latter could, however, include breaks, but since no information is provided I was not able to

include this into the table. Moreover, the reader should be aware of the fact that both meal-and rest

breaks are considered. A distinctive overview of break scheduling for more than sixty papers is

provided in Thompson and Pullman (2007).

Overlap Start times Length

Breaks Fixed Definable Fixed Definable

Alfares, 2007; x x x x

Al-Yakoob and Sherali, 2007; x x

Bard, 2004a; x x x x x x

Bard, 2004b; x x x x x

Bard and Wan, 2008; x x x x x x

Bard et al., 2003; x x x x x x

Bard et al., 2007; x x x x x x

Beaulieu et al., 2000; x x

Berrada et al., 1996; x x

Brusco and Jacobs, 1998a; x x x

Brusco and Jacobs, 1998b; x x x

Brusco and Jacobs, 2000; x x x x

Brusco and Jacobs, 2001; x x x x x

Brusco and Johns, 1996; x x x x

Burke et al., 2010; x x x

Chen et al., 2010; x x x

26

Choi et al., 2009; x x x

De Grano et al., 2009; x x x

De Matta and Peters, 2009; x x x

Eitzen et al., 2004; x x

Ertogral and Bamuqabel, 2008; x x x x x

Felici and Gentile, 2004; x x x

Isken, 2004; x x x

Jacobs and Brusco, 1996; x x x

Jarrah et al., 1994; x x x x x

Kostreva and Jennings, 1991; x x

Lezaun et al., 2006; x x

Lezaun et al., 2007; x x

Lezaun et al., 2010; x x x

Love and Hoey, 1990; x x x

Mohan, 2008; x x x

Moz and Pato, 2004; x x

Ozkarahan, 1989; x x x

Rong, 2010; x

Ronnberg and Larsson, 2010; x x

Seckiner et al., 2007; x x x

Stolletz, 2010; x x x

Thompson, 1995; x x x x

Topaloglu and Ozkarahan (2004); x x x x

Trilling et al., 2006; x x x

Valouxis and Housos, 2000; x x

Valouxis et al., 2012; x x x

Warner, 1976b; x x

Table 8: Shift definitions + Break placement

After taking a closer look at table 8, we can summarize a couple of findings. First, overlaps are

possible in two thirds of the papers and seem to be a result of definable shift start times, which is no

more than logical. Furthermore, fixed start times and shift lengths are still a very popular

combination. This, however, does not give the scheduler a lot of flexibility. Around one third of the

papers adjust its model to include breaks. What’s more, if breaks are added to the model, this is

often done by creating a new decision variable. For example, in most of the papers of Bard, one of

the decision variables captures the number of breaks initiated in period t on day d. Also binary

decision variables, to indicate whether an employee takes a break, are a couple of times added to

the set of decision variables.

27

2.3.5. Constraints

Constraints set the minimum and maximum boundaries for the combination of a set of variables. If

there would only be an objective function without any type of constraint, the model would minimize

or maximize each variable to an unfeasibly low, respectively high, value.

Before discussing the different types of constraint categories, it is important to explain the difference

between hard- and soft constraints. Hard constraints are those that must be satisfied at all costs,

while soft constraints only express a preference of a certain outcome over other ones. “The goal is

always to schedule resources to meet the hard constraints while aiming at a high quality result with

respect to soft constraints” (Cheang et al., 2003). Fact that most soft constraints are not equally

important, penalties or weights or added to the constraints which must be minimised or maximised

by the objective function of the model (cf. supra).

The set of restrictions or constraints used in tour scheduling can be subdivided in three broad

categories: coverage- , time-related- and balancing- and preference constraints. In what follows I will

provide a detailed overview of the most common constraints per category applied in the area of tour

scheduling. Furthermore, each constraint will be represented on a mathematical manner (both hard

and soft) so that the TSP becomes more complete.

Most of the constraints presented can be used in two ways. First, the scheduler can make a set of

tours, without looking at any type of feasibility or constraint. After that, the constraints can be added

to the model in such a way that only a small set of workable tours remains. In the other, second,

method, the constraints can be used to create from scratch a workable set of tours. In that way, the

constraints or not used to select tours, but to make tours.

To understand all of the mathematical equations, it is important to clearly enumerate the variables,

parameters and sets utilized in the model. Below are a decision variable and some sets defined that

form the backbone of all the equations presented in this chapter:

Decision variable:

- xt number of employees assigned to tour t;

Other variable:

- ajkt 1, if shift k on day j in tour t is a working shift, 0 otherwise;

Sets:

- T set of tours (index t);

28

- J set of days in the scheduling horizon (index j);

- K set of shift types: K = {Early ‘E’, Day ‘D’, Late ‘L’, Night ‘N’} (index k).

2.3.5.1. Coverage constraints

When talking about coverage constraints in the area of personnel scheduling, one suggests that

there should be a minimum or maximum number of personnel present to deliver the necessary

service or staff a certain work unit. When hard coverage constraints are added to the model,

understaffing is not allowed (see left part of table 9). However, it is possible to schedule some excess

staff to anticipate any irregularities in labor demand. Van den Bergh et al. (2012) found that the hard

coverage constraint has a presence at a level of almost 75% in the 291 papers that they have studied

about personnel scheduling. On top of that, Van den Bergh et al. (2012) also suggests that the soft

coverage constraint, which allows both under- and overstaffing, is a central constraint in many

personnel scheduling models. An example of such a soft constraint can be found in the rightmost

column of table 9.

A point of discussion when setting up the coverage constraints is the handling of over- and

understaffing. Namely, schedulers often have the possibility to hire, fire and assign more or less

hours to certain individuals. These options increase the flexibility of the modeller, but one should

also take into account the costs associated with these practices.

Coverage constraints often set barriers to the extent at which it is possible to mix a group of workers

with different personnel characteristics. Remember that the usage of personnel characteristics, such

as an individual productivity level, seniority level, skills, etc., is an important topic in tour scheduling.

For instance, if skill categories are a way to classify the organisation’s staff, hard coverage constraints

can impose a minimum of highly-skilled workers needed at a specific time period. Another example is

the ability of soft coverage constraints to schedule highly-skilled workers, at a certain penalty rate in

the objective function, to a job that only requests low-skilled workers.

Notation:

Slack/surplus variables:

- # personnel staffed below the required coverage djk on day j, shift k (understaffing);

- # personnel above the required coverage djk on day j, shift k (overstaffing).

29

Other variable:

- djk coverage requirement of shift type k on day j.

Hard Soft

Coverage

requirements

Daily coverage requirement of each shift type

(no over- and understaffing allowed):

∑

{ }

Daily coverage requirement of each shift type

(both over- and understaffing allowed):

∑

{ }

Table 9: Coverage constraints

2.3.5.2. Time-related constraints

This type of constraints imposes restrictions that are related to a specific time period. In other words,

these are rules that enforce minima, maxima, exact numbers, successions, patterns, and so forth

within the scheduling period. In literature, a distinction is made between the following sub-

categories:

Assignment constraints:

Limiting the number of working days or assignments per day, week, etc., is a good measure to attract

workers. Furthermore, it avoids that an employee has to work many short shifts. Interesting to

notice, the minimization or maximization of the number of assignments is closely related to the

number of (consecutive) free days imposed each week. The maximization of the number of

assignments to a certain shift type is beneficial for the employee because it ensures that the

variation in shifts types is reduced to a minimum. Examples of assignment constraints are provided in

table 10.

Notation:

Parameters:

- A maximum number of assignments within the scheduling period;

- B maximum number of assignments during a time period of 7 days;

- C maximum number of shifts worked during the day;

- D minimum number of assignments within the scheduling period;

- E minimum number of assignments during a time period of 7 days;

- F maximum number of night shifts within the scheduling period.

30


- number of assignments in tour t above A;

- btw number of assignments in tour t above B, starting at day w;

- cjt number of shifts on day j in tour t above C;

- dt number of assignments in tour t below D;

- etw number of assignments in tour t below E, starting at day w;

- ft number of night shifts in tour t above F.

Hard Soft

Max #

assignments

Max # assignments during the scheduling

period:

∑ ∑

Max # assignments during a 7-day period:

∑ ∑

{ }

For each day, an employee cannot start more

than C shift(s) (in practice: C = 1):

∑

; j {1,…,J}.

Max # assignments during the scheduling

period:

∑ ∑

Max # assignments during a 7-day period:

∑ ∑

{ }

For each day, an employee should not start

more than C shift(s) (in practice: C = 1):

∑

; j {1,…,J}.

Min #

assignments

Min # assignments during the scheduling

period:

∑ ∑

Min # assignments during a 7-day period:

∑ ∑

{ }

Min # assignments during the scheduling

period:

∑ ∑

Min # assignments during a 7-day period:

∑ ∑

{ }

Max #

assignments

to a shift type

Max # night shifts during the scheduling

period:

∑ ( ) .

Max # night shifts during the scheduling

period:

∑ ( ) .

Table 10: Assignment constraints

31

Consecutiveness constraints:

Maximizing the number of consecutive days prevents the on- and off switching between working-

and non-working patterns. When the minimum number of consecutive days is equal to two, the

constraint prevents the modeller to schedule a working day between two non-working days (or a day

off between two working days). Another, very similar, constraint is a maximum number to the

consecutiveness of shifts of the same type. Table 11 illustrates the mathematical formulations of six

hard- and soft constraints.

Notation:

Parameters:

- G1 maximum number of consecutive working days within the scheduling period;

- G2 minimum number of consecutive working days within the scheduling period;

- maximum number of consecutive assignments of shift type k;

- minimum number of consecutive assignments of shift type k;

- O1 maximum number of consecutive days off;

- O2 minimum number of consecutive days off;


- gtr 1, if the constraint maximum G1 consecutive number of working days is violated for

tour t (with r as the first day of the consecutive series), 0 otherwise;

- htrk 1, if the constraint maximum consecutive number of assignments of shift type k is

violated for tour t (with r as the first day of the consecutive series), 0 otherwise;

- mtj 1, if the constraint minimum G2 consecutive number of working days is violated for

tour t (with r as the day after the consecutive series), 0 otherwise;

- ntjk 1, if the constraint minimum consecutive number of assignments of shift type k is

violated for tour t (with r as the first day of the consecutive series), 0 otherwise;

- otr 1, if the constraint maximum O1 consecutive number of days off is violated for tour t

(with r as the first day of the consecutive series), 0 otherwise;

- ptj 1, if the constraint minimum O2 consecutive number of days off is violated for tour t

(with r as the day after the consecutive series), 0 otherwise;

Other variables:

- 1, if day r is a day off and day (r-1) a working day, 0 otherwise;

- 1, if (r- G2) ≥ 1 (the value of the first day in the scheduling period), 0 otherwise;

- 1, if shift k on day r is a shift off and shift k on day (r-1) a working shift, 0 otherwise;

- 1, if (r-

) ≥ 1 (the value of the first day in the scheduling period), 0 otherwise;

32

- 1, if day r is a day on and day (r-1) a day off, 0 otherwise;

- 1, if (r- O2) ≥ 1 (the value of the first day in the scheduling period), 0 otherwise.

Hard Soft

Max #

consecutive

days/shifts

Maximum # consecutive working days during

the scheduling period:

∑ ∑

;

r {1,…,J- }.

Maximum # consecutive assignments of shift

type k:

∑

;

r {1,…,J- }.

Maximum # consecutive working days during


∑ ∑

;

r {1,…,J- }.

Maximum # consecutive assignments of shift

type k:

∑

;

r {1,…,J- }.

Min #

consecutive

days/shifts

Minimum # consecutive working days during


(∑ ∑ )

( )

( )

; r {2,…,J}.

Minimum # consecutive assignments of shift

type k during the scheduling period:

(∑ )

(

) (

)

; r {2,…,J}.

Minimum # consecutive working days during


(∑ ∑ )

( )

( )

; r {2,…,J}.

Minimum # consecutive assignments of shift

type k during the scheduling period:

(∑ )

(

) (

)

; r {2,…,J}.

Max #

consecutive

days off

Maximum # of consecutive days-off:

∑ ∑ ;

r {1,…,J- }.

Maximum # of consecutive days-off:

∑ ∑ ;

r {1,…,J- }.

Min #

consecutive

days off

Minimum # consecutive days off during the

scheduling period:

(∑ ∑ )

( ) (

)

; r {2,…,J}.

Minimum # consecutive days off during the

scheduling period:

(∑ ∑ )

( ) (

)

; r {2,…,J}.

Table 11: Consecutiveness constraints

33

Working hour constraints:

Restrictions concerning the number of working hours are often dictated by government institutions

or other official organisations. However, one is able to schedule overtime hours to a certain extent.

Table 12 not only formulates hard- and soft constraints for the maximum- and minimum number of

working hours during the scheduling period, but also these for the minimum number of hours

between two assignments. The latter is only formulated as a hard constraint because penalizing this

constraint is often not legal.

Notation:

Parameters:

- Q maximum number of working hours within the scheduling period;

- R minimum number of working hours within the scheduling period;

- S minimum time between two assignments;


- qt number of hours in tour t that are worked above Q;

- rt number of hours in tour t that are worked below R;

Other variables:

- denotes the length in hours of shift k on day j;

- denotes the starting hour of shift k on day j;

- denotes the total number of working hours per day;

- 1, if day j is a day off, 0 otherwise.

Hard Soft

Max # hours &

overtime

Maximum # working hours during the

scheduling period:

∑ ∑

Maximum # working hours during the

scheduling period:

∑ ∑

Min # hours

Minimum # working hours during the

scheduling period:

∑ ∑

Minimum # working hours during the

scheduling period:

∑ ∑

34

Time between

assignments

Minimum # hours between two assignments:

( ) ( ∑ ( ) ( ( )

) )

( ) ( ∑ (

) )

{ }

/

Table 12: Working hour constraints

Holiday/days off constraint:

Each personnel member is able to request specific off days, holidays or shifts at which they are

preferably not scheduled. Again, rules imposed by official institutions have to be followed.

Let it be clear that this type of constraint is more useful when we work with the third model of

paragraph 2.2. Namely, holiday- and/or days off constraints are more useful when working with a

known workforce for which the individuals all have their own preferences than when we have to

assign tours to anonymous workers. For this reason, the decision variable is changed to a binary

variable for a known workforce size. An example is provided in table 13.

Notation:

Decision variable:

- xijk 1, if employee i has to work shift k on day j;

Surplus variable:

- tijk 1, if employee i has to work shift k on day j;

Hard Soft

Days/shifts

on/off

Employee i cannot work shift k:

{ }

Employee i should not work shift k:

{ }

Table 13: Holiday/days off constraint

Work pattern constraints:

When an employer wants to keep his staff satisfied, he/she would better not use certain work

patterns. For example, patterns that schedule a work shift after another work shift, with only a

minimum amount of rest time, are not beneficial both for the employer and the employee.

Therefore, one should prohibit certain shift sequences and make sure that there is always enough

time between two assignments. Table 14 provides two examples regarding the scheduling of a night

shift.

35

Patterns, consecutive

shifts and sequence of

shift type

A night shift cannot be followed by an early shift and/or a day shift:

( ) ( )( ) ( )( ) ; j {1,…,J-1}.

Free days after a series

of night shifts

A constraint of minimum 2 days off after a series of night shifts is

equivalent to the following three sub-constraints (‘N’ denotes a night

shift, ‘0’ a day-off and ‘1’ a working day):

- Prohibit a sequence of ‘N01’:

( )( ) ∑ ∑ ( ) ( ) ( )

( ) ( )

; j {2,…,J-1};


( )( ) ∑ ∑ ( ) ( ) ( )

( ) ( )

; j {2,…,J-1};


( )( ) ∑ ∑ ( ) ( ) ( )

( ) ( )

; j {2,…,J-1}.

Table 14: Work pattern constraints

Weekend constraints:

An important aspect for workers to consider applying, are the weekend constraints imposed by the

organisation. Employees often expect to get something in return when he/she has worked during the

weekend. For instance, when somebody has worked two full weekends in a row, he/she might ask

not to work the upcoming weekend. The most vital constraint of the ones enumerated in table 15 is

unquestionable the complete (and extended) weekend constraint. An extended weekend constraint

schedules automatically Monday as a day off when both the preceding weekend and the following

day are scheduled as off days. A final popular weekend constraint is the prohibition of a night shift

before an on-duty weekend.

Notation:

Sets:

- J1 set of indices of the last day of each week within the scheduling horizon = {7, 14, …}

(index j1);

Parameters:

- U maximum number of on-duty weekends during the scheduling period;

36

- Z minimum number of consecutive on-duty weekends during the scheduling period;


- ut number of on-duty weekends in tour t above U;

- 1, if in tour t day j1 is a work day but not day (j1-1), 0 otherwise;

- 1, if in tour t day (j1-1) is a work day but not day j1, 0 otherwise;

- 1, if in tour t day (j1+1) is a work but not day j1 and day (j1+2), 0 otherwise;

- ztr 1, if the constraint maximum number of consecutive weekends is violated in tour t

(with r as the first day of the consecutive series), 0 otherwise;

-

1, if in tour t shift k is a work shift on day (j1-1) but not on day j1, 0 otherwise;

-

1, if in tour t shift k is a work shift on day j1 but not on day (j1-1), 0 otherwise;

-

1, if the constraint no night shift before a free weekend is violated in tour t on week-

end day j1, 0 otherwise;

Hard Soft

Max # weekends in

# weeks

Maximum # of on-duty weekends during


∑ ∑

(Must be combined with the complete

weekend constraint)

Maximum # of on-duty weekends during


∑ ∑


weekend constraint)

Complete (and

extended) weekends

There must be either no shifts or two

shifts in weekends:

∑ ( )

; J1.

Monday must be a day off if the

preceding weekend and the following day

are off days:

∑ ( ) ( )

; {7,…, J1-7}.


weekend constraint)

There should be either no shifts or two

shifts in weekends:

∑ ( )

; J1.

Monday should be a day off if the

preceding weekend and the following day

are off days:

∑ [ ( ) ( ) ]

; {7,…, J1-7}.


weekend constraint)

37

Max # consecutive

weekends

Maximum # consecutive on-duty week-

ends during the scheduling period:

∑ ∑

; r {7,…, J1-7Z}.


weekend constraint)

Maximum # consecutive on-duty week-

ends during the scheduling period:

∑ ∑

; r {7,…, J1-7Z}.


weekend constraint)

Identical weekends

If an employee works during the week-

end, than he/she must do the same shift

on both days:

( )

; ;

If an employee works during the weekend,

than he/she should do the same shift on

both days:

( ) +

; ;

No night shift before

a free weekend

If an employee has a free weekend,

than he/she cannot do a night shift on

Friday:

( )( ) ∑

; .

If an employee has a free weekend, than

he/she should not do a night shift on

Friday:

( )( ) ∑

; .

Table 15: Weekend constraints

Other constraints:

Sometimes, jobs need to be executed in a certain time window or before a specific date. If this is the

case, the scheduler should assign the employees in such a manner that those requirements are

always fulfilled. Another aspect in many organisations is the availability of resources, such as work

tools, machines, etc. Overstaffing a work unit because there is a lack of machinery is detrimental for

any type of organisation. A final constraint used in tour scheduling is concerned with the ratios

between groups of workers. This constraint limits the relative share of a work unit, for example

because of cost-, stakeholder- or learning aspects. In table 16 two labor ratio constraints are

formulated: the first sets a bound to the amount of full-time (FT) to part-time (PT) employees for

every day and shift, while the second one is very similar but limits the number of low- to high skilled

employees.

Notation:

Sets:

- I1 set of FT employees (index i1);

- I2 set of PT employees (index i2);

38

- I3 set of high skilled employees (index i3);

- I4 set of low skilled employees (index i4);

Parameters:

- α FT to PT labor ratio

- β high skilled to low skilled labor ratio

Variables:

- 1 if FT employee i1 is assigned to shift type k for day j, 0 otherwise;

- 1 if PT employee i2 is assigned to shift type k for day j, 0 otherwise;

- 1 if high skilled employee i2 is assigned to shift type k for day j, 0 otherwise;

- 1 if low skilled employee i2 is assigned to shift type k for day j, 0 otherwise;

Ratio groups of personnel

For every shift in the scheduling horizon, the number of FT to PT

employees must be bigger than α:

∑

∑

⁄ α { }

For every shift in the scheduling horizon, the number of high skilled

to low skilled employees must be bigger than β:

∑

∑

⁄ β { }

Table 16: Ratio constraint

2.3.5.3. Balancing- and preference constraints

Fairness among an organisation’s staff is a crucial aspect in the eyes of the employees. Therefore, it is

very likely that the modeller has to include constraints that balance the workload, the number of

assignments in a specified time-period, the number of assignments to a particular shift type, the time

between two assignments, and so forth. Moreover, one should also address the preferences of each

individual worker. These preferences could imply a willingness to work on a specific location, day,

shift, etc., a preference to work together with somebody else (tutorship), and so on. Worth noticing

is that both balancing- and preference constraints are often included as soft constraints, that

balancing constraints are very closely related to the assignment constraints and the preference

constraints to the day/shift on/off constraint(s).

39

Notation:

Parameters:

- M1 minimum number of early shift days during the scheduling period;

- M2 maximum number of early shift days during the scheduling period.

Variables:

- 1 if employee i1 has to work day j shift k without employee i2, 0 otherwise;

- 1 if employee i2 has to work day j shift k without employee i1, 0 otherwise;

Balancing constraints

In all tours there must be between M1 and M2 early shifts during the

entire scheduling horizon:

∑ ( )

Tutorship

Employees i1 and i2 would like to work together:

( ) (

)

{ }

Table 17: Balancing constraint and Tutorship

2.3.5.4. Overview

Table 18 indicates for each paper which constraint is adopted into the respective model. The

following list of constraints is been used:

Coverage Constraints:

(1) Hard coverage constraint;

(2) Soft coverage constraint;

(3) Coverage constraint which allows understaffing;

(4) Coverage constraint which allows overstaffing;

Assignment Constraints:

(5) Maximum number of assignments;

(6) Maximum number of assignments to a shift type;

Consecutiveness:

(7) Maximum number of consecutive days/shifts;

(8) Minimum number of consecutive days/shifts;

(9) Maximum number of consecutive days off;

40

(10) Minimum number of consecutive days off;

(11) Maximum number of hours & overtime;

(12) Minimum number of hours;

(13) Days/shifts on/off;

(14) Time between assignments;

(15) Patterns, consecutive shifts and sequence of shift type;

(16) Free days after a night shift;

Weekend:

(17) Maximum number of weekends in x-number of weeks;

(18) Complete (and extended) weekends;

(19) Maximum number of consecutive weekends;

(20) Identical weekends;

(21) Ratio groups of personnel;

(22) Balancing constraint(s).

41

Coverage constraint

Time-related constraints Balancing

Assignment Consecutiveness Weekend

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22)

Alfares, 2007; x x x x Al-Yakoob and Sherali,

2007; x x x x x x x x

Bard, 2004a; x x x x x x x x

Bard, 2004b; x x x x

Bard and Wan, 2008; x x x x x x

Bard et al., 2003; x x x x

Bard et al., 2007; x x x x x

Beaulieu et al., 2000; x x x x x x x x x x

Berrada et al., 1996; x x x x x x x x

Brusco and Jacobs, 1998a; x x x

Brusco and Jacobs, 1998b; x x

Brusco and Jacobs, 2000; x x

Brusco and Jacobs, 2001; x x

Brusco and Johns, 1996; x x x

Burke et al., 2010; x x x x x x x x x x x

Chen et al., 2010; x x

Choi et al., 2009; x x x x x

De Grano et al., 2009; x x x x x x x x x x

De Matta and Peters, 2009; x x x x x

Eitzen et al., 2004; x x x x x x x x x x Ertogral and Bamuqabel,

2008; x x

Felici and Gentile, 2004; x x x x x x

42

Coverage constraint

Time-related constraints Balancing

Assignment Consecutiveness Weekend

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22)

Isken, 2004; x x x

Jacobs and Brusco, 1996; x x

Jarrah et al., 1994; x x x Kostreva and Jennings,

1991; x x x x x x x x

Lezaun et al., 2006; x x x x x x x x x

Lezaun et al., 2007; x x x x x x x

Lezaun et al., 2010; x x x x x x x x x

Love and Hoey, 1990; x x x x

Mohan, 2008; x x x

Moz and Pato, 2004; x x x x x

Ozkarahan, 1989; x x x

Rong, 2010; x x x x Ronnberg and Larsson,

2010; x x x x x x x x x x

Seckiner et al., 2007; x x x

Stolletz, 2010; x x x x x

Thompson, 1995; x Topaloglu and Ozkarahan

(2004); x x x x x x x x

Trilling et al., 2006; x x x x x x x x

Valouxis and Housos, 2000; x x x x x x x x

Valouxis et al., 2012; x x x x x x x x x x x x

Warner, 1976b; x x x x x x

Table 18: Constraints – Overview

43

A couple of conclusions can be drawn after analyzing table 18. Below, a frequency table is set up to give the

reader a clear overview of the most important constraints.

Constraint Percentage

Hard coverage constraint 78%

Coverage constraint which allows overstaffing 61%

Maximum number of assignments 41%

Maximum number of consecutive days/shifts 39%

Patterns, consecutive shifts and sequence of shift type 33%

Days/shifts on/off 30%

Time between assignments 30%

Maximum number of hours & overtime 28%

Minimum number of consecutive days off 24%

Ratio groups of personnel 24%

Soft coverage constraint 20%

Complete (and extended) weekends 20%

Balancing constraint(s) 20%

Maximum number of assignments to a shift type 17%

Minimum number of consecutive days/shifts 17%

Maximum number of weekends in x-number of weeks 15%

Free days after a night shift 11%

Coverage constraint which allows understaffing 9%

Minimum number of hours 7%

Maximum number of consecutive weekends 7%

Maximum number of consecutive days off 4%

Identical weekends 2%

Table 19: Frequency of constraint-usage

Immediately we can see that one constraint is by far the most popular one, namely the hard coverage

constraint. The frequency percentage of the hard coverage constraint is in line with the percentage found

by J. Van den Bergh et al. (2012). A hard coverage constraint in combination with overstaffing occurs 52%

of the times. Understaffing, on the other hand, is not so much added to the IP TSM. Also popular are the

maximum number of assignments and consecutive days/shifts. Most of the time, these constraints are

imposed by the government, together with the ‘time between assignments’ constraint. Remarkable is the

percentage of the ‘days/shifts on/off constraint’. This constraint allows workers to ask for a certain

day/shift off. In addition, adding this constraint makes the schedule more popular in the eyes of the

employees. In about one third of the papers, patterns and sequence of shift types are included. Not so

44

popular are the consecutiveness constraints that impose a maximum, namely the maximum number of

consecutive weekends and days off. Only the paper of Valouxis et al., 2012 adds the identical weekend

constraint. Also the other weekend constraints are ranked at the bottom.

2.4. The link between theory and practice

To finalize this chapter I would like to come back on some key findings that will be of interest for the

remainder of this paper. Namely, in the following chapter three TSMs will be mathematically formulated,

programmed and discussed. The order of discussion follows the ‘roadmap’ of figure 2.

First of all, the objective function used for the baseline models (BMs) of chapter 3 is in line with the most

popular one in existing tour scheduling, i.e. cost minimization. The cost variable created captures numbers

(1), (2), (3) and (7) defined in paragraph 2.3.1. At the end of chapter 3, objective (8), minimizing the

workforce size, will also be studied briefly.

Secondly, decision variable types 1 and 2 will be applied extensively. Type 3, mostly used in nurse

scheduling, will be dropped and left open for further research.

Thirdly, the BMs introduced in paragraph 3.3.1 will make a distinction between full- and part-time

employees. A small study will be performed on the impact of the ratio setting the boundaries for part-

timers compared to the FT workforce. Skill levels are introduced in one of the scenarios adapting the BMs.

The fourth tour scheduling category discussed shift overlap, -start-times, -lengths, and break placement. All

of them introduce some sort of flexibility into the TSM. Therefore, the field of study should be carefully

defined which, in addition, delivers more in-depth and valuable results. In extension, the influence of

varying shift start times and break placement will be investigated.

The last category of the tour scheduling roadmap captures a wide range of constraints applied to TSMs.

Almost every constraint discussed here is added to each of the upcoming three models. The parameters of

the respective constraints are fixed during the entire analysis.

45

Chapter 3

3. Programming the Tour Scheduling Problem: Modelling and Analysis

Due to the size, complexity and pure integer nature of the TSP, it has been proven difficult to solve

optimally. In other words, the higher the variety in the workforce structure/personnel characteristics (e.g.

an individual productivity- and seniority level, skills, etc.), the work rules/shift definitions (e.g. length of

each shift, frequency and duration of meal and rest breaks, allowable shift starting times, limitations to the

number of consecutive workdays, etc.) and the longer the length of the minimum planning interval, the

harder it is and the longer it will take to find a feasible solution (cfr. supra).

To confirm these general findings, a second part of this thesis consists of modelling and analyzing the

standard TSPs. Three models are developed and described in more detail: a set-covering model (SCM), an

implicit model (IM) and a hybrid model (HM). The third model that was explained in paragraph 2.2 and

mostly used in nurse scheduling is left open for further research. The main reason to focus only on the SCM

and IM is the similarity in their main decision variable. Namely, the variable represents a number of

employees assigned to a certain period or shift per day. On the contrary, the decision variable of the third

model is a binary variable which is equal to 1 if a worker (~nurse) is scheduled to a particular shift. This

difference requires a completely other programming approach and is therefore not taken under

investigation any further. Above that, the in- and output requirements for this third, derivative, model are

not comparable to those of the set-covering- and implicit tour scheduling programs. For example, on the

one hand, demand patterns are loaded into a set-covering- and implicit program and act as the basis to

determine the workforce level. On the other hand, the workforce level is the main input in nurse scheduling

programs and is a key parameter in generating coverage patterns (Vanhoucke and Maenhout, 2009).

Chapter 3 is organized as follows. First, the SCM, IM and HM are mathematically formulated as mixed

integer programs. The latter is then used to set up these models in C++ for which the solution methodology

is thoroughly discussed. Lastly, I will analyze and compare the (computational) results of varying in- and

outputs of the programs. More precisely, a basic model is defined and the impact on the results for five

different scenarios is studied.

46

3.1. Model formulations

The basic TSM takes a set of daily demands as input and tries to find a personnel schedule that minimizes

or maximizes an objective while satisfying all labor rules and company policies. There are several methods

to create this personnel schedule. Two standard types are set-covering and implicit modelling. A brief

mathematical formulation has already been shown in paragraph 2.2. In this paragraph, however, the

models will be completed and made more flexible. This is done by adopting all the different tour scheduling

characteristics listed in figure 2 into the models. More details about this will be provided in paragraph 3.2

discussing the solution methodologies. Besides the SCM and IM, a HM is set up. The reason why it is called

a ‘hybrid’ model will become clear after comparing its parameters and (decision) variables with the two

basic TSMs.

In each of the following three models, two similar conditions will be imposed. Both are related to

paragraph 2.3.3 discussing the personnel characteristics. First, there are two types of workers: FT = full-

time and PT = part-time workers. The user is able to define, with the parameter α, to what extent PT

workers can contribute to cover the demand. If α is set equal to 0, PT workers are excluded from the

model. The same reasoning can be followed for the second condition which makes a distinction between

two types of skill levels: H = higher level and L = lower level. The parameter β is used to limit the number of

lower level employees. Besides the similarities between these two conditions, there is also an important

difference. Namely, α is used as a ratio for bounding the total PT workforce over the entire scheduling

horizon to a percentage of the FT workforce. β, however, restricts on every period of the day the number of

lower level employees to a certain part of the number of higher level employees working that same period.

This difference in time scope is needed because it doesn’t matter whether demand is fulfilled by FT or PT

employees, but not every ‘unit’ of demand can be handled by low skilled employees. A ratio based on the

total periods worked instead of the number of employees might be a good alternative (Bard et al., 2003).

3.1.1. Model 1: Set-covering model (SCM)

In this paragraph a mathematical problem formulation is provided for creating a work staff that fully covers

a certain demand pattern by assigning employees to feasible working tours. The tours are defined down to

the level of single working periods.

Notation:

Sets:

- J set of days in the scheduling horizon (index j)

47

- W set of weekends within the scheduling horizon (index w)

- set of days that correspond to the last day of each week within the scheduling

horizon (index ), i.e.

, , etc. (w W)

- K set of shift types: K = {Early ‘E’, Day ‘D’, Late ‘L’, Night ‘N’} (index k)

- LFT set of FT shift lengths (index lFT)

- LPT set of PT shift lengths (index lPT)

- P set of periods per day (index p)

- set of FT start times (index )

- set of PT start times (index )

- TFT set of feasible FT tours, expressed in shifts and days (index tFT)

- TPT set of feasible PT tours, expressed in shifts and days (index tPT)

- set of feasible FT tours, expressed in days and periods (index )

- set of feasible PT tours, expressed in days and periods (index )

Parameters:

- maximum number of assignments for FT employees, PT employees

respectively, within the scheduling period

- minimum number of assignments for FT employees, PT employees


- maximum number of night shifts for FT employees, PT employees


-

maximum number of consecutive working days for FT employees, PT

employees respectively, within the scheduling period

-

minimum number of consecutive working days for FT employees, PT


-

maximum number of consecutive assignments of shift type k for FT

employees, PT employees respectively

-

minimum number of consecutive assignments of shift type k for FT


-

maximum number of consecutive days off for FT employees, PT employees


-

minimum number of consecutive days off for FT employees, PT employees


- maximum number of working periods within the scheduling period

48

- minimum number of working periods within the scheduling period

- S minimum time between two assignments

- shift length, in periods, for FT shift length lFT (index

) ( lFT LFT )

- shift length, in periods, for PT shift length lPT (index

) ( lPT LPT )

- start period for shift type k corresponding to FT start time (index

) ( stFT STFT ) ( k K )

- start period for shift type k corresponding to PT start time (index

) ( stFT STFT ) ( k K );



- γ percentage pay for low skilled employees compared to high skilled employees

- δ percentage pay for PT employees compared to FT employees

Decision variables:

-

number of high skilled employees assigned to FT tour

-

number of low skilled employees assigned to FT tour

-

number of high skilled employees assigned to PT tour

-

number of low skilled employees assigned to PT tour

Other variables:

- 1, if shift k on day j in FT tour tFT , PT tour tPT respectively, is a working

shift, 0 otherwise

-

1 if time period p is a work period in FT tour , PT tour

respectively, 0 otherwise

- , the total cost of working FT tour , PT tour , respectively

- minimum number of employees required in time period p on day j

- 1, if day r is a day off and day (r-1) a working day, 0 otherwise (r {2,…,J})

- 1, if (r- G2) ≥ 1, with 1 the value of the first day of the scheduling horizon,

0 otherwise (r {2,…,J})

- 1, if shift k on day r is a shift off and shift k on day (r-1) a working shift, 0 otherwise

(r {2,…,J})

- 1, if (r-

) ≥ 1, with 1 the value of the first day of the scheduling horizon,


- 1, if day r is a day on and day (r-1) a day off, 0 otherwise (r {2,…,J})

49

- 1, if (r- O2) ≥ 1, with 1 the value of the first day of the scheduling horizon,


Model formulation

MIN: (

) (

) (

) (

) (1)

∑ (

)

( ( ) ) + ∑ (

)

( ( ) )

{ }

(2)

∑ ∑

(3)

∑ ∑

(4)

∑ ( ) (5)

∑ ∑

; r { - }. (6)

(∑ ∑ )

(

) (

) ; r { } (7)

∑

; r { -

}. (8)

(∑ )

(

) (

) ; r { } (9)

∑ ∑

. (10)

(∑ ∑ )

( )

( )

; r {2,…,J}. (11)

∑ ∑

(12)

∑ ∑

(13)

( ∑ ( ) ( )) ( (∑ ))

( ∑ (

))

{ } (14)

( ) ( )( ) ( )( ) ; j {1,…,J-1}. (15)

50

( )( ) ∑ ∑ ( ) ( ) ( )

( ) ( ) ; j {2,…,J-1};

( )( ) ∑ ∑ ( ) ( ) ( )

( ) ( ) ; j {2,…,J-1};

( )( ) ∑ ∑ ( ) ( ) ( )

( ) ( ) ; j {2,…,J-1}. (16)

∑ ( )

; . (17)

( )

; ; (18)

( )( ) ∑

; (19)

Equation (3) until (19) is repeated here with superscript PT instead of FT

∑ (

) ∑ (

)

(20)

∑ (

( ( ) ) ) ∑ (

( ( ) ) )

∑ (

( ( ) ) ) ∑ (

( ( ) ) ) { }

(21)

,

,

,

i eger

Eq. (1) is the objective function, minimizing the total cost. Cost ratio indicators are added for PT- and low

skilled employees. Eq. (2) is a coverage constraint and indicates the minimum number of workers that

needs to be scheduled on a certain day and period. Eqs. (3), (4) and (5) are assignment constraints. Eqs. (3)

and (4) regulate the maximum and minimum number of assignments during the scheduling horizon. Eq. (5)

limits the number of assignments to a night shift. Eqs. (6), (7), (8), (9), (10) and (11) are consecutiveness

constraints. Eqs. (6) and (7) express the restriction on the maximum and minimum number of consecutive

working days. Eqs. (8) and (9) follow the same principle but than for the number of consecutive

assignments of shift type k. Eqs. (10) and (11) restrict the number of consecutive days off to a maximum

and a minimum, respectively. Eqs. (12) and (13) limit the maximum and minimum number of working

periods during the entire scheduling horizon. Eq. (14) ensures that there is always enough time between

two assignments. Eq. (15) prohibits the sequence of a night shift and an early or day shift. Eq. (16)

guarantees a minimum number of two days off after a series of night shifts. Eqs. (17), (18) and (19) are

weekend constraints. Eq. (17) ensures that employees have to work the complete weekend or not. Eq. (18)

imposes that the assignments of employees in the weekends need to be identical. Eq. (19) restricts

scheduling a night shift before a weekend off. Eq. (20) and (21) are labor ratio constraints. The first limits

51

the number of FT to PT employees, while the second imposes a limit to the number of low to high skilled

employees.

3.1.2. Model 2: Implicit model (IM)


a certain demand pattern by assigning employees to a combination of feasible days-off patterns and -shift

patterns. The latter is defined by a set of start-time bands and band widths (BWs).

Notation:

Sets:

- set of bands for FT shift length lFT (index )

- set of bands for PT shift length lPT (index )

- BR a set of consecutive periods during which a break may be given (index br)

- BW band width, a range in which starting times can vary within a band (index bw)

- DOPFT set of feasible FT days-off patterns (index )

- DOPPT set of feasible PT days-off patterns (index )





, , etc. (w W)

- LFT number of FT shift lengths (index lFT)

- LPT number of PT shift lengths (index lPT)


) ( lFT LFT )


) ( lPT LPT )


Parameters:





-



52

-



-

maximum number of consecutive days off for FT employees, PT employees


-

minimum number of consecutive days off for FT employees, PT employees


- M a big number







Decision variables:

-

r

number of high skilled employees assigned to FT days-off pattern dopFT, shift

length number lFT, start-time band and break-period br

-

r

number of low skilled employees assigned to FT days-off pattern dopFT, shift


-

r

number of high skilled employees assigned to PT days-off pattern dopFT,

shift length number lFT, start-time band and break-period br

-

r

number of low skilled employees assigned to PT days-off pattern dopFT, shift


-

r

number of high skilled, FT employees assigned to a shift on day j, shift length

number lFT, start-time band with band width bw and break-period br

-

r

number of low skilled, FT employees assigned to a shift on day j, shift length

number lFT, start-time band with band width bw and break-period br

-

r

number of high skilled, PT employees assigned to a shift on day j, shift

length number lPT, start-time band with band width bw and break-period br

-

r

number of low skilled, PT employees assigned to a shift on day j, shift

length number lPT, start-time band with band width bw and break-period br

53

Surplus variables:

- number of working periods above for FT shift length number and

days-off pattern

- number of working periods above for PT shift length number lPT and

days-off pattern

- number of working periods below for FT shift length number lFT and

days-off pattern

- number of working periods below for PT shift length number lPT and

days-off pattern

Other variables:

-

1, if period p is a working period for shift length number lFT, start-time band

with band width bw and break-period br, 0 otherwise

-

1, if period p is a working period for shift length number lPT, start-time band


- 1, if day j in FT days-off pattern is a working day, 0 otherwise

- 1, if day j in PT days-off pattern is a working day, 0 otherwise

-

r the average cost of working FT days-off pattern dopFT, shift length number lFT,

start-time band and break-period br

- r the average cost of working PT days-off pattern dopPT, shift length number

lPT, start-time band and break-period br








- 1, if

or

> 0 for FT shift length number and

days-off pattern , 0 otherwise

- 1, if

or

> 0 PT shift length number and

days-off pattern , 0 otherwise

54

Model formulation:

MIN: (

*

) (

*

) +

(

* ) (

*

) (1)

∑ ∑ ∑ ∑

( (

))

∑ ∑ ∑ ∑ (

(

))

{ } (2a)

∑ ∑

– ∑ ∑

(

)

{ }

∑ ∑

– ∑ ∑

(

)

{ } (2b)

∑ (3)

∑ (4)

∑

; r { - }. (5)

(∑

(

) (

) ; r { } (6)

∑

. (7)

(∑

( )

( )

; r {2,…,J}. (8)

∑ ( )

(9)

∑ ( )

(10)

( )

r (11)

( )

;

. (12)

Equation (2b) until (12) is repeated here with superscript PT instead of FT

55

∑ ∑ ∑ ∑ (

) *

∑ ∑ ∑ ∑ (

)

(13)

∑ ∑ ∑ ∑ (

) +

∑ ∑ ∑ ∑ (

)

∑ ∑ ∑ ∑ (

)

∑ ∑ ∑ ∑ (

) { } (14)

,

,

,

i eger

,

,

,

i eger

Eq. (1) of this problem formulation minimizes the total cost. PT and low skilled employees are paid less,

indicated by the parameters δ and γ, respectively. Eq. (2a) is a coverage constraint and indicates the

minimum number of workers that needs to be scheduled on a certain day and period. Eq. (2b) ensures that

the number of workers assigned to a particular working day is equal to the total amount of workers

assigned to a band on that specific day. Eqs. (3) and (4) regulate the maximum and minimum number of

assignments during the scheduling horizon. Eqs. (5), (6), (7) and (8) are consecutiveness constraints. Eqs. (5)

and (6) express the restriction on the maximum and minimum number of consecutive working days. Eqs.

(7) and (8) restrict the number of consecutive days off to a maximum and a minimum, respectively. Eqs. (9)

and (10) limit the maximum and minimum number of working periods during the entire scheduling horizon.

Eq. (11) ensures that the previous two equations are included in the optimization model. More details

about this working method follow in paragraph 3.2 discussing the solution methodology. Eq. (12) ensures

that employees have to work the complete weekend or not. Eq. (13) and (14) are labor ratio constraints.

The first limits the number of FT to PT employees, while the second imposes a limit to the number of low to

high skilled employees.

3.1.3. Model 3: Hybrid model (HM)


a certain demand pattern by assigning employees to a combination of feasible tours and - shift patterns.

The former is expressed in shifts and days while the latter is defined by a string of periods covering a full

56

day for each shift. The method holds the middle between the previous two standard tour scheduling

techniques.

Notation:

Sets:

- BR a set of consecutive periods during which a break may be given (index br)





, , etc. (w W)

- K set of shift types: K = {Early ‘E’, Day ‘D’, Late ‘L’, Night ‘N’} (index k)

- LFT number of FT shift lengths (index lFT)

- LPT number of PT shift lengths (index lPT)


) ( lFT LFT )


) ( lPT LPT )


- number of FT start times (index )

- number of PT start times (index )

- start period for shift type k corresponding to FT start time (index

) ( stFT STFT ) ( k K )

- start period for shift type k corresponding to PT start time (index

) ( stFT STFT ) ( k K );

- TFT set of feasible FT tours (index tFT)

- TPT set of feasible PT tours (index tPT)

Parameters:





- maximum number of night shifts for FT employees, PT employees


-



57

-



-

maximum number of consecutive assignments of shift type k for FT


-

minimum number of consecutive assignments of shift type k for FT


-

maximum number of consecutive days off for FT employees, PT


-

minimum number of consecutive days off for FT employees, PT




- S minimum time between two assignments





Decision variables:

-

number of high skilled employees assigned to FT tour tFT with start-time

number , shift length number and break-period br

-

number of low skilled employees assigned to FT tour tFT with start-time


-

number of high skilled employees assigned to PT tour tPT with start-time


-

number of low skilled employees assigned to PT tour tPT with start-time


Surplus variables:

- ,

number of working periods above for FT shift length number lFT and

tour , PT shift length number lPT and tour respectively

- ,

number of working periods below for FT shift length number lFT and

tour , PT shift length number lPT and tour respectively

58

-

number of working periods below for FT shift length number lFT , start time

number and tour

-

number of working periods below for PT shift length number lPT , start

time number and tour

Other variables:

- 1, if shift k on day j in FT tour r i e re e i e is a working

shift, 0 otherwise

- 1, if period p for shift type k with start time number , shift length number

and break-period br is a working shift, 0 otherwise

- 1, if period p for shift type k with start time number , shift length number

and break-period br is a working shift, 0 otherwise

- the total cost of working FT tour tFT with start-time number , shift length

number and break-period br

- the total cost of working PT tour tPT with start-time number , shift

length number and break-period br





- 1, if shift k on day r is a shift off and shift k on day (r-1) a working shift, 0 otherwise

(r {2,…,J})

- 1, if (r-

) ≥ 1, with 1 the value of the first day of the scheduling horizon,





- 1, if

or > 0 for FT shift length number lFT and tour ,

0 otherwise

- 1, if

or > 0 for PT shift length number lPT and tour ,

0 otherwise

- 1, if

> 0 for FT shift length number lFT, start time number

and tour , 0 otherwise

59

- 1, if

> 0 for PT shift length number lPT, start time number

and tour , 0 otherwise

Model formulation

MIN: ( ) + (

)

( * ) (

) (1)

∑ (

) +

∑ (

)

{ } (2)

∑ ∑

(3)

∑ ∑

(4)

∑ ( ) (5)

∑ ∑

; r { - }. (6)

(∑ ∑ )

(

) (

) ; r { } (7)

∑

; r { -

}. (8)

(∑ )

(

) (

) ; r { } (9)

∑ ∑

. (10)

(∑ ∑ )

( )

( )

; r {2,…,J}. (11)

∑ ∑ ( )

(12)

∑ ∑ ( )

(13)

( )

r (14)

60

( ∑ ( ) ( )) ( (∑ ))

( ∑ (

))

{ } (15)

( )

r (16)

( ) ( )( ) ( )( ) ; j {1,…,J-1}. (17)

( )( ) ∑

∑ ( )

( ) ( )

( ) ( ) ; j {2,…,J-1};

( )( ) ∑ ∑ ( ) ( ) ( )

( ) ( ) ; j {2,…,J-1};

( )( ) ∑ ∑ ( ) ( ) ( )

( ) ( ) ; j {2,…,J-1}. (18)

∑ ( )

; . (19)

( )

; ; (20)

( )( ) ∑

; (21)

Equation (3) until (21) is repeated here with superscript PT instead of FT

∑ ∑ ∑ ∑ (

) *

∑ ∑ ∑ ∑ (

)

(22)

∑ ∑ ∑ ∑ ∑ (

) +

∑ ∑ ∑ ∑ ∑ (

)

∑ ∑ ∑ ∑ ∑ (

∑ ∑ ∑ ∑ ∑ (

)

{ } (23)

,

, ,

i eger

61

Eq. (1) set the objective function of this problem formulation. The goal is to minimize the total cost while

incorporating the necessary skill- and PT cost ratios. Eq. (2) is a coverage constraint and indicates the

minimum number of workers that needs to be scheduled on a certain day and period. Eqs. (3), (4) and (5)

are assignment constraints. Eqs. (3) and (4) regulate the maximum and minimum number of assignments

during the scheduling horizon. Eq. (5) limits the number of assignments to a night shift. Eqs. (6), (7), (8), (9),

(10) and (11) are consecutiveness constraints. Eqs. (6) and (7) express the restriction on the maximum and

minimum number of consecutive working days. Eqs. (8) and (9) follow the same principle but than for the

number of consecutive assignments of shift type k. Eqs. (10) and (11) restrict the number of consecutive

days off to a maximum and a minimum, respectively. Eqs. (12) and (13) limit the maximum and minimum

number of working periods during the entire scheduling horizon. Eq. (14) ensures that the previous two

equations are included in the optimization model. More details about this working method follow in

paragraph 3.2 discussing the solution methodology. Eq. (15) ensures that there is always enough time

between two assignments. Eq. (16) has the same purpose as eq. (14) but is further detailed down to the

level of starting times. Eq. (17) prohibits the sequence of a night shift and an early of day shift. Eq. (18)

guarantees a minimum number of two days off after a series of night shifts. Eqs. (19), (20) and (21) are

weekend constraints. Eq. (19) ensures that employees have to work the complete weekend or not. Eq. (20)

imposes that the assignments of employees in the weekends need to be identical. Eq. (21) restricts

scheduling a night shift before a weekend off. Eq. (22) and (23) are labor ratio constraints. The first limits

the number of FT to PT employees, while the second imposes a limit to the number of low to high skilled

employees.

3.2. Solution methodology

Starting from their mathematical problem formulations, a set-covering-, implicit- and hybrid model are set

up in the integrated development environment (IDE) Microsoft Visual C++ 2010 Express.

Each program can be divided in five main components: ‘The Settings’, a ‘Demand Pattern Generator’, a

‘Tour Generator’, a ‘Cost Generator’ and ‘The Optimization’. These steps can also be found in the programs

themselves under the source file ‘main.cpp’. Figure 3 shows a visual representation of these five

components:

62

Figure 3: Main components of the tour scheduling optimization system

Next, each component is described into more detail. The differences between and for each of the three

models is thoroughly explained, supported with a couple of examples. Also, if relevant, an enumeration is

provided of sets/parameters/variables introduced in the mathematical problem formulations and used in

the respective component or subcomponent.

3.2.1. Settings

To increase the user friendliness of the programs, all user definable variables are listed in the source file

‘Settings.cpp’, the first component of figure 4. This file is subdivided into seven subcomponents:

Figure 4: Subdivisions of the component ‘Settings’

1) General settings: Besides the path to write the output to, this section enables the user to choose

the number of shifts, days and periods. In the IM, the parameter ‘shifts’ is replaced by the

parameter BW. This denotes the number of starting times within a band. The calculation of the

number bands will be explained in paragraph 3.2.3.2.

Sets/Parameters/Variables: J, K, P, BW

2) Demand generator: The user is able to choose between two demand generators. The first one is

called the nurse scheduling problem generator (NSPGen) and is developed by Prof. dr. Mario

Vanhoucke and Prof. dr. Broos Maenhout. Although it has been developed in light of the nurse

scheduling problem, it can also be used to generate demand patterns for the TSP. The second

demand generator uses an upper and lower bound to randomly generate feasible demand

Settings Demand Pattern

Generator

Tour Generator

Cost Generator

Optimization

Settings

General Settings

Demand Generator

Break Placement

FT Personnel

PT Personnel

Skill Levels Objective function

63

patterns. More information about these two demand generators is provided in the next paragraph,

3.2.2.

Sets/Parameters/Variables:

3) Break placement: Two variables need to be defined. The first is the break window (BR), a set of

consecutive periods during which a break may be given. If this variable is set to 0, no breaks are

included into the model. If it’s greater than or equal to 1, as many work patterns are created as the

width of the BR. Each work pattern than contains one different single break-period, all

concentrated around the middle of the respective shift. A visual representation of this break

placement method will follow in paragraph 3.2.3. Finally, the second variable is the minimum shift

length, expressed in periods, required for assigning a break.

Sets/Parameters/Variables: BR

4) FT personnel: This section defines all the variables influencing the creation of FT work patterns. In

detail, parameters needed for the minimum and maximum of time-related constraints, the number

of shift lengths and start times per shift, the lengths themselves expressed in periods and the

starting period(s) per shift. The number of start times and the staring period(s) are, however, only

used in the set-covering- and hybrid tour scheduling approach.

Sets/Parameters/Variables: , ,

, ,

, ,

, , S, LFT,

, ,

5) PT personnel: Sets the exact same variables as for FT personnel but now for PT workers. One

variable, however, is added and defines the percentage of PT workers to FT workers. The attentive

reader will recognise this as the parameter ‘α’ used in the mathematical formulations of the

models.

Sets/Parameters/Variables: , ,

, ,

, ,

, , S, LPT,

, ,

, α

6) Skill levels: Contains only one variable, namely the variable ‘β’. This ratio defines the maximum

percentage of low skilled workers per period per day based on the number of high skilled

employees working that same period.

Sets/Parameters/Variables: β

7) Objective function: The standard objective function of each mathematical problem formulation and

of the corresponding optimization program minimizes the total workforce cost. However, the user

also has the ability to choose between three other objectives: minimizing the total workforce size,

the level of overstaffing and –understaffing. To increase the flexibility of the program, two

additional parameters can be defined: a ratio which sets the cost percentage for PT employees and

64

another ratio, with the same purpose, but for low skilled employees. These ratios are only relevant

when the cost objective function is applied.

Sets/Parameters/Variables: γ, δ

Notice that nothing is or has been calculated after this first component. The source file ‘Settings.cpp’ is

purely meant as a medium between the user and the tour scheduling optimization system.

3.2.2. Demand pattern generator

Two demand generators are integrated into the program: ‘NSPGen’ and a demand generator based on

bounded random numbers.

‘NSPGen’ is a useful tool to generate several nurse scheduling problem instances under a controlled design.

The generation is based on a set of complexity indicators, discussed into great detail in Vanhoucke and

Maenhout (2009). Section 3 of this working paper presents three classes of indicators measuring the size,

preferences and coverage requirements of a nurse scheduling problem instance, respectively. However, to

generate a demand pattern for one of our three tour scheduling approaches, we do not need the second

class in our program. Namely, preferences can only be included if we start from an existing workforce, as is

the case in the nurse scheduling problem. In the introduction of chapter 3, I have already explained that

this approach is not in line anymore with this thesis. The three complexity indicators of the first- and the

third class, respectively, are used as input parameters to generate a particular demand pattern. The

number of nurses, -shifts and -days are three indicators who belong to the first class. Their functionality is

obvious and doesn’t need any further explanation. The three complexity indicators of the third class are the

total-coverage constrainedness (TCC), the day-coverage distribution (DCD) and the shift-coverage

distribution (SCD). “The first indicator, TCC, serves as an indicator to generate the total number of nurses

required for the complete scheduling horizon. The second indicator, DCD, assigns requirements to

individual days and measures whether the daily coverage is distributed equally over all days. The last

indicator, SCD, has the same functionality as the previous indicator, but on a shift level. More precisely, the

SCD measures the distribution of the coverage requirements over all shifts for a particular day j.”

(Vanhoucke and Maenhout, 2009) Further explanation about these classes and their complexity indicators

is provided in Vanhoucke and Maenhout (2009). Also, the complete ‘NSPGen’ program and its tutorial can

be downloaded from www.projectmanagement.ugent.be/nsp.php.

http://www.projectmanagement.ugent.be/nsp.php

65

The second generator creates demand patterns based on random numbers which are bounded between

two figures for each period of the day. The user of the program has to define four parameters: the lowest-

and the highest starting demand, a multiple and a trend. The first two give an indication of the workforce

size needed at the first period of each day. Next, the multiple determines the absolute value of the increase

or decrease in workforce size compared to the previous period. Lastly, the trend is a measure of seasonality

per day. If the trend is set equal to 0, no significant peaks are scheduled. However, if the trend equals 1 or

more, there will be an equal number of peaks scheduled within a single day. To incorporate this trend

within the program, a string of binary values is generated. The length of this string is equal to the number

of periods per day. Table 20 shows per trend its respective string of binary values for a 16-period-day:

Trend Period

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0 0 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1

1 0 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1

2 0 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1

Table 20: Trend generation

Hence, the demand for a particular period p is given by the following formula:

This process is repeated for every day of the week. Figure 5 shows the lower and upper bound for each

period of the day. The following setup was used: P = 16, lowest possible demand (1) = 20, highest possible

demand (1) = 25, Multiple = 4, Trend = 1.

Demand in period p = random number between [ lowest possible demand(p) , highest possible demand(p) ]

With:

lowest possible demand(p) = lowest possible demand(p-1) + Trend(p) * Multiple

highest possible demand(p) = highest possible demand(p-1) + Trend(p) * Multiple

lowest possible demand(1) = defined by the user;

highest possible demand(1) = defined by the user.

66

The demand for each period is now generated by selecting a random number that is located between the

green and the red line.

3.2.3. Tour generator

This component is the core of each program and creates work patterns (Tours) based on a set-covering-, an

implicit- or a hybrid method.

3.2.3.1. Set-covering method

Tour generation is in its purest form under the set-covering formulation originally suggested by Dantzig

(1954), see paragraph 2.2. This general formulation is, however, far from realistic and needs to be

completed with at least a number of time-related constraints. This task has been done in paragraph 3.1.1,

suggesting a much more realistic TSM. The attentive reader should have noticed that the model is built

around two variables:

- 1, if shift k on day j in FT tour tFT , PT tour tPT respectively, is a working

shift, 0 otherwise

-

1 if time period p is a work period in FT tour , PT tour

respectively, 0 otherwise

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Upper bound 25 29 33 37 41 45 49 53 49 45 41 37 33 29 25 21

Lower bound 20 24 28 32 36 40 44 48 44 40 36 32 28 24 20 16

0

10

20

30

40

50

60Demand

Figure 5: Lower- and upper bound for the random bounded demand generator

67

Both variables are the basis in creating feasible tours. Namely, the first variable is used in the source file

‘TourGeneration_Step1.cpp’ and the second one in ‘TourGeneration_Step2.cpp’. The tour generator

component can be subdivided as follows:

Figure 6: Set-covering method: subdivisions of the ‘Tour Generator’ component

In step 1, tours are generated as seen in table 3. In other words, per day and per shift a binary

representation is used to point out if it is a working shift or not. Important to mention is that a free shift is

included, which means that every day exactly one of the shifts is equal to 1. If the scheduling horizon is set

at 7 days and there are 3 working shifts and 1 free shift (the last value), the first 10 tours look as follows:

Tour Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7

1 1000 1000 1000 1000 1000 1000 1000

2 0100 1000 1000 1000 1000 1000 1000

3 0010 1000 1000 1000 1000 1000 1000

4 0001 1000 1000 1000 1000 1000 1000

5 1000 0100 1000 1000 1000 1000 1000

6 0100 0100 1000 1000 1000 1000 1000

7 0010 0100 1000 1000 1000 1000 1000

8 0001 0100 1000 1000 1000 1000 1000

9 1000 0010 1000 1000 1000 1000 1000

10 0100 0010 1000 1000 1000 1000 1000

Table 21: Set-covering method: output of step 1 of the ‘Tour Generator’ component

Tour Generator

Step 1

Feasibility Check

Step 2

Feasibility Check

68

The total number of tours will be equal to 16.384 (= 4^7). This is, however, without any restriction. In

reality, the first tour, for example, will never be worked because it is not beneficial for the company and the

person itself to work 7 days in a row. Therefore, (time-related) constraints are added to the model. In

detail, every tour created in step 1 runs through 14 feasibility checks: constraints (3) – (11) and (15) – (19)

as seen in model 3.1.1. The impact of each individual constraint on the total number of feasible tours varies

drastically. Suppose every day there are 4 working shifts (an early-, day-, late-, and night shift) and one

additional free shift during a scheduling horizon of 7 days. Table 22 lists for each of the 14 constraints the

remaining number of tours, the corresponding parameter (only for FT employees) used in the mathematical

formulation and the value of this variable:

Table 22: Set-covering method: impact of 14 time-related constraints

The percentages give the user a good idea of the impact of each individual constraint. For the assignment-

and consecutiveness constraints, however, these percentages change when the value of the corresponding

Tours % Constraint Var. Value

78125 100,00% Maximum (5 shifts, 7 days => 78125 tours)

Assignment constraints:

33069 42,33% Maximum number of assignments AFT 5

77760 99,53% Minimum number of assignments DFT 3

75520 96,67% Maximum number of assignments to a night shift FFT 3

Consecutiveness constraints:

53549 68,54% Maximum number of consecutive days G1FT 5

33089 42,35% Minimum number of consecutive days G2FT 3

78089 99,95% Maximum number of consecutive shifts Hk,1FT 5

841 1,08% Minimum number of consecutive shifts Hk,2FT 2

77700 99,46% Maximum number of consecutive days off O1FT 3

24613 31,50% Minimum number of consecutive days off O2FT 2

Work pattern constraints:

45465 58,20% No early- and day shift after a night shift

76700 98,18% Minimum 2 days off after a series (>3) of night shifts

Weekend constraints:

53125 68,00% Complete weekends

72500 92,80% No night shift before a free weekend

15625 20,00% Identical weekends

69

parameter is set higher or lower. When all the constraints with the values used in table 22 are applied as a

feasibility check, only 67 tours remain. Table 23 lists the first 5 feasible tours after step 1:

Tour Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7

1 00001 00001 10000 10000 10000 10000 10000

2 00001 00001 00001 10000 10000 10000 10000

3 00001 00001 01000 01000 10000 10000 10000

4 00001 00001 00100 00100 10000 10000 10000

5 00001 00001 01000 01000 01000 10000 10000

Table 23: Set-covering method: feasible tours after step 1 of the ‘Tour Generator’ component

To end the discussion of tour generation step 1, a small word should be said about the variable

. This variable is now completely filled with feasible binary variables per shift, per day and per

tour. Next to Table 23, the values for this variable are shown for day 1, tour 1 and day 7, tour 2.

In step 2, each day per tour is levelled down to the number of periods per day, i.e. if P = 24, a string of 24

binary variables forms one day of a tour. In other words, the second variable is an extrapolation of the first

variable. The place where a work period is planned, i.e. a value of 1, is determined by the type of shift

scheduled in step 1, the start time(s) of that shift, the possible shift lengths and whether break placement is

included into the model or not. An example will provide the necessary clarification:

Example (for FT personnel): Consider day 3 of tour 1 in table 23, where an early shift is scheduled. Also,

define the following sets and parameters:

- LFT (set of shift lengths) = 2;

- (length, in periods, for shift length 1) = 6;

- (length, in periods, for shift length 2) = 8;

- (set of start-times) = 2;

- (start-time, in periods, for start-time 1 of shift type ‘early’) = 0;

- (start-time, in periods, for start-time 2 of shift type ‘early’) = 3;

- BR (a set of consecutive periods during which a break may be given) = 2;

- Minimum shift length to schedule a break = 6;

After considering all of the sets and parameters defined above, we find the following 8 (= LFT * * BR)

strings of each 24 binary values for day 3 of tour 1 after tour generation step 2:

a111 = 0 a121 = 0 a131 = 0 a141 = 0 a151 = 1

a712 = 1 a722 = 0 a732 = 0 a742 = 0 a752 = 0

70

Day 3 Day 3

1 110111000000000000000000 5 000110111000000000000000

2 111011000000000000000000 6 000111011000000000000000

3 111011110000000000000000 7 000111011110000000000000

4 111101110000000000000000 8 000111101110000000000000

Table 24: Set-covering method: transforming a single day from step 1 to step 2

The first two tours and tours 5 and 6 have a shift length of 6 periods (= ). The other tours have a shift

length of 8 periods (= ). Starting times are the same for the first 4 tours (

= 0) and tours 5 until

8 ( = 3). Because the BR was set equal to 2 (= BR), each possible start and shift length

combination is further divided into 2 strings. Each string has a single break period scheduled around the

middle of the shift. Table 25 shows how the program would schedule breaks into a 10-hour shift with a

beak-window equal to 5:

1 1110111111

2 1111011111

3 1111101111

4 1111110111

5 1111111011

Table 25: Set-covering method: break placement for a shift of 10 hours with a BR of 5

The feasibility check of step 2 concerns three working hour constraints: (12) until (14) of model 3.1.1. These

constraints couldn’t be tested in step 1 because these tours were not yet generated down to the level of

working periods/hours. The first two constraints check, for each generated tour, whether the total amount

of working hours falls between a predefined minimum ( ) and maximum ( ). The third constraint

checks whether the time between two assignments is at least equal to ‘S’, with ‘S’ expressed in periods.

After this feasibility check, variable

is now completely filled with feasibly tours. If J = 7 and P =

24, each tour consists of 7 x 24 binary variables. In addition, the total number of binary variables used for

optimization could easily add up to a few billion. Next to table 24, a couple of values for are shown.

Remark that the example is given for day 3, so we start at period 49 (= 2*24+1).

a49,7 = 0 a50,7 = 0 a51,7 = 0 a52,7 = 1 a53,7 = 1 a54,7 = 1

71

3.2.3.2. Implicit method

The basic implicit TSM is shown in paragraph 2.2. This is, just as for the SCM, a far too basic approach to

generate personnel schedules. Therefore, the model has been completed with time-related constraints,

which is shown in paragraph 3.1.2. The implicit TSM is built around the following two variables (only FT

variables are considered):

- 1, if day j in FT days-off pattern is a working day, 0 otherwise

-

1, if period p is a working period for shift length number lFT, start-time band


Variable is filled with days-off patterns and

with shift schedules. Unlike the SCM,

where the second variable is an extrapolation of the first one, both variables are independent from each

other and both are needed in the optimization step. The tour generator component of figure 3 can be

subdivided as follows:

Figure 7: Implicit method: Subdivisions of the component ‘Tour Generator’

The methodology followed for the subcomponent days-off scheduling is exactly the same as step 1 of the

set-covering method. This means that a pattern is created immediately followed by a feasibility check. The

content is, however, slightly different. Namely, for the days-off scheduling part, each day consists of only

one value, i.e. 1 for a day on and 0 for a day off. Remember that in the set-covering method each day

consisted of a string of binary values where each value corresponded to a particular shift. Because of this

limited number of combinations (only two per day, i.e. 0 or 1), the possible number of work patters that

could be created is much lower. For example, a scheduling horizon of 7 days can only generate 128 (= 2^7)

Tour Generator

Days-off scheduling

Feasibility Check

Shift scheduling

Feasibility Check

72

days-off schedules. Table 26 shows the first 8 days-off patterns created by the source file

‘DaysOffPattrenGeneration.cpp’:

Days-off pattern

Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7

1 0 0 0 0 0 0 0

2 1 0 0 0 0 0 0

3 0 1 0 0 0 0 0

4 1 1 0 0 0 0 0

5 0 0 1 0 0 0 0

6 1 0 1 0 0 0 0

7 0 1 1 0 0 0 0

8 1 1 1 0 0 0 0

Table 26: Implicit method: output of the days-off scheduling subcomponent

When a days-off pattern is created, it runs through 7 constraints to check whether it is a feasible pattern or

not. These constraints are mathematically formulated in paragraph 3.1.2 and correspond to the numbers

(3) – (8) and (12). These 7 constraints are a clear difference with the 14 constraints of step 1 of the tour

generation component of the set-covering method. The cause of this difference lies in the fact that 7

constraints are so-called shift constraints and therefore not applicable in this implicit modelling approach.

The impact of the remaining 7 constraints is about the same as found in table 22. However, this is only

relatively speaking, not in absolute terms. To close the discussion of the days-off scheduling

subcomponent, it is worth mentioning that next to table 26 a text area shows a couple of values for the

variable .

The next step is to generate the values of the variable

. This is done in the source file

‘ShiftPatternGeneration.cpp’. The output of this file is a bunch of binary strings with a length equal to the

number of periods per day. The number of strings and their content is determined by the sets BW and L(FT).

These sets represent the BW, a range in which starting times can vary within a band, and the number of

(FT) shift lengths, respectively. “A start-time band is defined as a range in which starting times can vary

within an individual tour” (Jacobs and Brusco, 1996). In detail, the number of bands for a particular shift

length is defined by the following equation:

= P - - BW + 2

b3,1 = 0 b3,2 = 1 b4,1 = 1 b4,2 = 1

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The formula is exactly equal to one of the formulas in Jacobs and Brusco (1996), studying the influence of

overlapping start-time bands in implicit tour scheduling. Table 27 shows the strings for the first three bands

of a 24 period day with shift length 8 and BW 4:

Band 1

111111110000000000000000

011111111000000000000000

001111111100000000000000

000111111110000000000000

Band 2

011111111000000000000000

001111111100000000000000

000111111110000000000000

000011111111000000000000

Band 3

001111111100000000000000

000111111110000000000000

000011111111000000000000

000001111111100000000000

Table 27: Implicit method: output of the shift scheduling subcomponent

The total number of bands under these conditions (P = 24, = 8 and BW = 4) is equal to 14. This is,

however, without break placement. If the user includes breaks by, for instance, setting BR equal to 3, each

string per band transforms into three strings, as seen in table 25. An important remark here is that the

maximum BR is equal to 5. The reason for this is to respect the minimum time between assignments and to

prevent the user of introducing unrealistic high break scheduling flexibility. An example for band 1 of table

27 is provided in table 28. Next to the table, a couple of values for the variable

are shown

(assume that there is only 1 shift length, hence = 1).

Band 1

111011110000000000000000

111101110000000000000000

111110110000000000000000

011101111000000000000000

011110111000000000000000

011111011000000000000000

Band width (BW) = 4

a1,1,2,3,1 = 0 a1,1,2,3,2 = 1 a1,1,2,3,3 = 1 a1,1,2,3,4 = 1 a1,1,2,3,5 = 1

74

Table 28: Implicit method: break placement

The final step of the tour generator component under the implicit modelling approach is a feasibility check.

This check corresponds to constraints (9) - (11) of model 3.1.2. Equation (9) sets a maximum to the number

of working periods and equation (10) sets the minimum. This way, every days-off pattern and shift length

combination is tested for violation. Notice that we need a days-off pattern variable ( ) and a shift

pattern parameter ( ) to do this. For this reason, the feasibility check is a separate subcomponent and

not only part of subcomponent 1 or 2 (see figure 7). Equation (11) makes sure that every decision variable

for which a particular days-off pattern and shift length combination is violated, is set equal to 0.

3.2.3.3. Hybrid method

As the name already suggests, this tour scheduling approach is a combination of the previous two methods.

Figure 8 gives a first glimpse of this combined tour scheduling method:

Figure 8: Hybrid method: Subdivisions of the component ‘Tour Generator’

The subcomponent ‘Step 1’, its feasibility check and the main variable of this subcomponent ( ),

are all similar to those of the set-covering method.

Tour Generator

Step 1

Feasibility Check

Shift scheduling

Feasibility Check

001110111100000000000000

001111011100000000000000

001111101100000000000000

000111011110000000000000

000111101110000000000000

000111110110000000000000

75

The second subcomponent is derived from the shift scheduling subcomponent under the implicit tour

scheduling method, but its content is not exactly the same. Under the implicit modelling approach, a BW

needed to be defined which was the basis for setting up all the different shift patterns. Under this model,

however, shifts are reintroduced just as in the SCM. The difference can also be seen if we investigate the

main variables for this subcomponent (see table 29):

Implicit modelling approach Hybrid modelling approach

Table 29: Comparison of the variables under implicit- and hybrid modelling

The subscripts indicating the band width (BW) and the number of bands per shift length ( ) in the

implicit variable, are replaced by the number of start times ( ) and a set of shift types (K). Table 30

shows the output of the shift scheduling subcomponent of the hybrid tour scheduling approach. The table

has been set up starting from a situation with 3 working shifts and one free shift, a day consisting out of 24

periods, one shift length of 8 periods, three starting periods: 1, 9 and 17 and a BR of 2 periods. Next to the

table, a text area is added showing the link with the variable .

Shift 1 111011110000000000000000

111101110000000000000000

Shift 2 000000001110111100000000

000000001111011100000000

Shift 3 000000000000000011101111

000000000000000011110111

Shift 4 (free shift) 000000000000000000000000

000000000000000000000000

Table 30: Hybrid method: output of the shift scheduling subcomponent

The feasibility check is comparable to the one under the implicit modelling approach, except that one

constraint is added. Namely, going from a BW to shifts, the time between two assignments needs to be

tested again carefully. The respective equations needed for this feasibility check are (12) - (16).

b3,3,1,1,20 = 0 b3,3,1,1,21 = 1 b3,3,1,1,22 = 1 b3,3,1,1,23 = 1 b3,3,1,1,24 = 1

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3.2.4. Cost Generator

Calculating the cost variable is a process of transforming the cost per period per day towards a cost with

the same dimensions of the decision variables of the respective problem formulations. Figure 9 shows a

visual representation of this process:

Figure 9: Cost generator: transformation process

The input, the cost per period per day ( ), needs to be defined by the user. However, a standard cost

pattern is provided. Each cost is depended on the period of the day and whether it is a weekend day or not.

Figure 10 shows the standard cost pattern included in each program.

Figure 10: Default cost per period per day

The transformation process is very straightforward for the set-covering- and hybrid TSP. However, for the

implicit approach, one should take into account that workers are not assigned to tours, but instead to

Cost Generator

Input Output

0

5

10

15

20

25

30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Cost

Period

Day 7

Day 6

Day 1-5

Cost variable Transformation

77

bands. In addition, we will have to calculate the average cost of working a particular band. The formulas for

transforming the costs of figure 10 are provided in table 31 (only FT):

SCM = ∑ ∑ ∑ (

( ( ) ) )

IM

=

∑ ∑ ∑ ∑ ∑

∑ ∑ (

) / BW ]

HM = ∑ ∑ ∑ ∑ ∑ ∑ ∑ (

)

Table 31: Cost Generator: Transformation formulas

The formula of the implicit modelling approach shows how the average cost per band is calculated. Namely,

the total cost for working a particular band is summed and then divided by the BW.

The costs obtained after applying these formulas are now multiplied with their respective decision

variables. This way, we can set the default objective function of each program (see eq. (1) of the

mathematical problem formulations).

3.2.5. Optimization

The optimization of the different TSPs is performed by the state-of-the-art mathematical programming

solver ‘Gurobi Optimizer 5.0.1’. For more information about this commercial optimization tool, I refer the

reader to http://www.gurobi.com/.

The optimization procedure can be divided in five subcomponents (see figure 11):

Figure 11: Subdivisions of the component ‘Optimization’

Optimization

Loading the decision variables

Set the objective function

Add constraints

Optimization Output

http://www.gurobi.com/

78

The first step in the optimization process is to define and load in all the different decision variables. While

defining these variables, we multiply them with their respective objective function variable(s). In the

default setting, these are the cost variables calculated by the cost generator.

The second step is then to choose whether the process has to maximize or minimize the objective function.

Minimization is the default setting in each program.

The third step is to set up the constraints that have a direct link with the decision variables. Three of these

types of constraints return in each of our three programs: the coverage constraint and the two labor ratio

constraints who, respectively, limit the number of PT and low skilled employees. Two additional

equations/constraints need be defined for the implicit- and hybrid tour scheduling solution approach. For

the IM, these are equations (2b) and (11). Eq. (2b) ensures that the number of workers assigned to a

particular working day is equal to the total amount of workers assigned to a band on that same day. Eq.

(11) inserts the maximum- and minimum number of working hour constraint into the model. The two

additional constraints for the HM are equations (14) and (16). Eq. (14) has the same purpose as equation

(11) of the IM. Eq. (16) ensures that the constraint checking the time between two assignments is included

into the model.

After these three steps and all of the calculations done in the demand-, tour- and cost generator, the model

can finally be optimized. Gurobi applies a branch-and-cut algorithm to find a feasible integer solution as

close as possible to the detected lineair lower/upper bound.

If a feasible solution is found, the final step is to generate a useful output. The latter can be used to analyze

and compare with other solutions and even be applied in practice. Analysis and comparison of the obtained

results is what I will describe in the final paragraph of this chapter, the computational results.

3.3. Results

To determine the influence of the tour scheduling categories listed in figure 2, we need to perform a

thorough scenario analysis. But before we can start comparing, a baseline model (BM) needs to be set up.

As said before, the code was written in C++ with Microsoft Visual 2010 Express. The computer used to

perform the optimization was a Toshiba Satellite P300 with an Intel(R) Core(TM)2 Duo CPU P8400 @

2.26GHz processor with 3GB RAM.

79

3.3.1. Baseline model (BM)

3.3.1.1. Setup

To build a model, we need to define all our settings (see figure 4). Appendix B shows the values that I have

chosen for the BM, which are in line with the most common values used in literature and in practice. A

couple of clarifications are appropriate here. First, the number of periods per day is equal to 24, which

suggests that we work under a continuous scheduling horizon. However, shifts start and end on the same

day, so no day overlap is allowed. In other words, although we consider a 24 period day, we study a

discontinuous TSM. Day overlap is a topic that I leave open for further research. Secondly, the BM defines

three working shifts and one free shift. The working shifts start, for FT employees, at period 0, 8 and 16,

respectively. In addition, a night shift is not included into the model, which influences the setup of night

shift related constraints. In detail, the assignment constraint that limits the maximum number of night

shifts is excluded from the model, together with the two work pattern constraints (see table 22).

Furthermore, the type of shift is changed in the second weekend constraint. Thirdly, the multiple and trend

used for the demand generator both equal 0. This means that for each period of the scheduling horizon, a

random number is chosen between the lowest- and highest starting value defined by the user. For the BM,

I have set these values equal to 20 and 25, respectively. The exact same demand pattern is used for all

three models. Finally, the objective function of the BM minimizes the total workforce cost. One reason for

choosing a cost objective function is because of its popularity (see table 5). Another reason is the ability to

make “a trade-off between hiring employees, overtime, casual workers, etc., by assigning a (relative) cost

to all of these factors” (J. Van den Bergh et al., 2012). To make a distinction between FT- and PT personnel,

the respective ratio is set equal to 0.8. This means that the employer only has to pay 0.8*14 euro to a PT

worker if one FT working period is paid 14 euro. The cost pattern used for the BM is the exact same pattern

as shown in figure 10.

3.3.1.2. Results

The comparison of the models is performed by a set of indicators spread across three classes. The

indicators of the first class provide the reader with a detailed overview of the composition of the

workforce: the number of FT- and PT employees, the number of high- and low skilled employees and the

total workforce. The difference in skill class will be only relevant in the respective scenario. The second

class gives an indication about the size of the model. The latter is defined by the number of rows

(constraints), columns (variables) and nonzeros of the respective models. The third and final class contains

four computational results: the integer- and lineair objective bounds, the gap between previous two

indicators and the CPU-time needed to find an optimal solution. Table 32 shows these classes and their

indicators together with the results for the BMs:

80

SCM IM HM

BM BM BM

Workforce indicators

FT employees 93 90 93

PT employees 31 30 31

Total workforce 124 120 124

Size indicators

Rows 337 2.769 905

Columns 844 3.920 1.564

Nonzeros 29.371 24.590 46.007

Computational indicators

Integer objective bound (€) 57.091 56.223 57.075

Lineair objective bound (€) 57.090 56.218 57.075

GAP (%) 0,0018 0,0089 0

CPU-time (sec.) 0,23 10,98 0,14

Table 32: BMs – summary of the results

Interesting to notice in table 32 is the huge similarity between the SCM and HM. This is logical if we look at

both models mathematical formulations. Namely, tours are created on the same way, except that for the

SCM optimization is performed with tours capturing days, shifts and periods, while for the HM it is the task

of the optimization step to find the best combination of these same tour levels. As we will find out later on,

the objective bounds founded for the SCM and HM are indeed always the same. In addition, only the size

indicators and CPU-time will be relevant in comparing the results of these respective models. The same

conclusion was already drawn analysing figure 8. Another finding is the lower objective function of the IM.

This is explained by the introduction of intra-tour start-time flexibility. For the SCM and HM, only 1 starting

period per shift has been scheduled for FT employees and only two for PT employees. The number of

starting periods in the implicit TSM is, however, much larger. This is due to the introduction of bands and

BWs. Table 27 has already shown the consequence of defining BWs on the number of starting times. In

scenario 3 I will affirm this conclusion by modifying the BW for the IM. A last conclusion concerning table 32

is about the number of PT employees. The reader can see that for each model the FT to PT ratio α has been

fully taken into account. This is logical because PT employees are paid less than FT employees, as already

explained in the setup of the BMs.

81

Besides these numerical results, figure 12 gives us the possibility to draw another interesting conclusion

about our BMs. On the left, the graphics show the average weekday coverage pattern for each model. On

the right, you will find the same but now for the weekend.

0

5

10

15

20

25

30

1 3 5 7 9 11 13 15 17 19 21 23

Period

Set-covering Baseline Model: average weekday pattern

FT employees PT employees

Surplus Demand

0

5

10

15

20

25

30

1 3 5 7 9 11 13 15 17 19 21 23

Period

Set-covering Baseline Model: average weekend pattern


Surplus Demand

0

5

10

15

20

25

30

1 3 5 7 9 11 13 15 17 19 21 23

Period

Implicit Baseline Model: average weekday pattern


Surplus Demand

0

5

10

15

20

25

30

1 3 5 7 9 11 13 15 17 19 21 23

Period

Implicit Baseline Model: average weekend pattern


Surplus Demand

82

Figure 12: BMs – coverage patterns

As the reader should have noticed, the distribution of PT employees is different for weekdays and

weekends. More specifically, for the first 5 days of the week we see a smooth distribution of part-timers

during the entire day. But in weekends, the middle of the day is almost completely covered by FT

employees while part-timers do a lot more work in the first and last periods of the day. Most of this

appearance is explained by the cost PT ratio δ and especially the difference in costs for working a weekday

or a weekend day (see figure 10). This, however, does not count for the implicit TSM. Namely, the coverage

pattern for weekdays is almost identical to the one in the weekend. Furthermore, the peak of full-timers in

the middle of a weekend day is much steeper for the explicit tour scheduling methods than for the implicit

one. This is again due to the introduction of intra-tour start-time flexibility.

Next, a couple of scenarios are worked out. For every scenario I will investigate the influence on the

workforce-, size- and computational indicators. It is important for the reader to understand that these

scenarios are independent from each other. More precisely, after every scenario the modification is

dropped again and we start back from the BM for the next scenario. We distinguish the following five

modifications to the BM:

- Scenario 1: Change the objective function, i.e. minimize the workforce size

- Scenario 2: Add break periods into the model, i.e. break placement

- Scenario 3: Change the BW of the IM, i.e. intra-tour start-time flexibility

- Scenario 4: Add skills into the model, i.e. high- and low skilled employees

- Scenario 5: Change the demand pattern and study the relationship with PT employees

0

5

10

15

20

25

30

1 3 5 7 9 11 13 15 17 19 21 23

Period

Hybrid Baseline Model: average weekday pattern


Surplus Demand

0

5

10

15

20

25

30

1 3 5 7 9 11 13 15 17 19 21 23

Period

Hybrid Baseline Model: average weekend pattern


Surplus Demand

83

While studying the results, one will notice that in some cases the optimization process has been stopped

(indicated by one (*) or two (**) stars). One reason for this is the predefined time window. Namely, the

optimization process is always stopped after 3600 seconds. In the scope of the TSPs studied in this thesis,

crossing this time boundary is an indication of a significantly higher computational effort than for other

problems. The other reason is related to the limited amount of memory available to do the analysis. One

will notice that this ‘out-of-memory’ problem only encounters for the SCM and the HM. This already

confirms that explicit tour scheduling approaches require much more memory than IMs (see, for example,

Morris and Showalter 1983, Bechtold and Showalter 1987, Bechtold et al. 1991, Easton and Rossin 1991a,

Brusco and Jacobs 1993, Bechtold and Brusco 1994).

3.3.2. Scenario 1: Objective function modification: minimize the workforce size

The BM needed to minimize the total workforce cost. Interesting would be now to see what the impact is

on the results when we change this objective function. Table 33 lists for each model the results with the

objective of minimizing the total workforce size. The size indicators are not included in this table because

they are identical to those of the BM.

SCM IM HM


FT employees 111 111 111

PT employees 1 1 1

Total Cost (€) 62.840 62.227 62.064


Integer objective bound (€) 112 112 112

Lineair objective bound (€) 112 112 112

GAP (%) 0 0 0

CPU-time (sec.) 0,02 0,08 0,03

Table 33: Scenario 1 – summary of the results

Just like the BMs, the time needed to perform the optimization is negligible. More interesting the see,

however, are the total work schedule costs and the according workforce sizes. Both are illustrated in figure

13:

84

Figure 13: Scenario 1 – Total work schedule cost and -workforce size

Scenario 1 has, as expected, led to a lower workforce size than in the BM, but a higher overall work

schedule cost. The reduction in the workforce size has been accomplished by switching PT workers for FT

workers. This makes sense because part-timers are restricted to 30 working periods during the week, while

this is only the lower bound for full-timers. As a consequence, however, the total work schedule cost has

increased by nearly 10 per cent compared with the BMs. A last point of interest is the average surplus

coverage scheduled per day. Figure 14 compares this average daily surplus for the BM of the set-covering

approach and scenario 1:

Figure 14: Scenario 1 – Average daily surplus coverage

After studying figure 14, we can conclude that the average daily surplus coverage is always higher for

scenario 1 than for the BM. This is contradictive knowing that the objective of scenario 1 is to minimize the

total workforce size while the BM minimizes the overall work schedule cost.

50.00053.00056.00059.00062.00065.00068.000

SCM IM HM

Co

st (

€)

Total Work Schedule Cost

BM Scenario 1

100

105

110

115

120

125

130

SCM IM HM

Emp

loye

es

Total Workforce Size

BM Scenario 1

0

1

2

3

1 2 3 4 5 6 7

Emp

loye

es

Day

Average Surplus coverage

BM

Scenario 1

85

3.3.3. Scenario 2: Break placement

I continue this scenario analysis by adopting breaks into the BMs. How this is done has already been

explained in paragraph 3.2.3. For a visual representation I refer the reader to tables 25, 28 and 30 for the

respective models. The BR is changed from 1 till 4. Table 34 summarizes the results of the break placement

exercise. On the top of the table and below the name of the model, the length of the BR is indicated. To

make the analysis easier, the BM is also added which had a BR equal to 0.

In contrast with the results from scenario 1, break placement does have a great impact on the size of the

problem. Figure 15 gives more insight in the positive relation between the number of rows (constraints)

and columns (variables) and the length of the BR.

Figure 15: Scenario 2 – Rows and columns

As a consequence of this positive relationship, I had to deal with memory and time limit problems. The

huge amount of memory needed for the set-covering- and HM are in line with the findings of tour

scheduling literature (see, for example, Morris and Showalter 1983, Bechtold and Showalter 1987, Bechtold

et al. 1991, Easton and Rossin 1991a, Brusco and Jacobs 1993, Bechtold and Brusco 1994). Moreover, the

higher values of the HM compared to the SCM reflect themselves in a longer pre-solvation time. The latter

is a phenomenon noticed in every optimization exercise. The time limit was reached for the IM adopting a

BR equal to 3 and 4.

0

2.500

5.000

7.500

0 (BM) 1 2 3 4BR

Rows (Constraints)

SCM HM IM

0

5.000

10.000

15.000

20.000

0 (BM) 1 2 3 4BR

Columns (Variables)

SCM HM IM

86


* Solve interrupted: out of memory

** Optimization was stopped; time limit was reached (3600 sec.)

SCM IM HM

0 (BM) 1 2 3* 4* 0 (BM) 1 2 3** 4** 0 (BM) 1 2 3* 4*


FT employees 93 315 126 / / 90 108 97 96 95 93 315 126 / /

PT employees 31 105 42 / / 30 36 32 32 31 31 105 40 / /

Total workforce 124 420 168 / / 120 144 129 128 126 124 420 166 / /

Size indicators

Rows 337 337 337 337 337 2.769 2.871 4.019 5.167 6.315 905 947 1.557 2.167 2.777

Columns 844 802 1.604 2.406 3.208 3.920 3.920 7.840 11.760 15.680 1.564 1.564 3.128 4.692 6.256

Nonzeros 29.371 25.645 51.290 76.935 102.580 24.590 23.936 47.872 71.808 95.744 46.007 43.205 86.410 129.615 172.820


Integer objective bound (€) 57.091 142.307 71.837 61.828 58.769 56.223 61.862 56.257 55.191 54.300 57.075 142.195 71.837 61.772 58.667

Lineair objective bound (€) 57.090 142.307 71.833 61.718 58.597 56.218 61.856 56.252 54.992 54.224 57.075 142.195 71.831 61.683 58.569

GAP (%) 0,0018 0 0,0056 0,1779 0,2927 0,0089 0,0097 0,0089 0,3606 0,1402 0 0 0,0084 0,1441 0,1673

CPU-time (sec.) 0,23 0,17 0,41 916,51 2.158,33 10.98 19.67 3.393,51 3.600 3.600 0,14 0,25 0,87 2793,05 2895,23

87

Another aspect that should be discussed is the influence of incorporating breaks to the total work schedule

cost and the workforce size. Figure 16 illustrates both of them:

Figure 16: Scenario 2 – Total work schedule cost and -workforce size

The enormous increase of the total cost and workforce size for both the SCM and HM adopting a break

length equal to 1 is easy to explain. Namely, only one period during a FT shift can be scheduled as a break

period. In addition, the demand during these particular break periods needs to be covered by PT

employees. This group is, however, limited by the FT to PT ratio. Hence, hiring one extra part-timer leads to

3 (default FT/PT ratio) extra full-timers which explains the drastic increase in both costs and staff size. This

reasoning does not count any more when we adopt a BR larger than 1, cause the program can now switch

between multiple break periods. Therefore, the higher the BR, the more the program can ‘play’ with these

periods and adjust them according to the demand that needs to be covered. This leads in turn to lower

costs and staffing size, but this effect does level off as we can see on figure 16.

3.3.4. Scenario 3: Intra-tour start-time flexibility (IM only)

As said before, one of the scenarios would investigate the impact on the results of changing the BW of the

IM. The BW of the BM was set equal to 1, indicating that employees have to start every work day of their

tour on the same period. Table 35 shows what happens when we increase this BW (the length of the BW is

indicated at the top of the table)

50.000

70.000

90.000

110.000

130.000

150.000

0 (BM) 1 2 3 4

Co

st (

€)

BR


SCM IM HM

100150200250300350400450

0 (BM) 1 2 3 4

Emp

loye

es

BR

Total Workforce Size

SCM IM HM

88

IM

1 (BM) 2 3* 4* 5*


FT employees 90 90 89 89 88

PT employees 30 30 29 29 29

Total workforce 120 120 118 118 117

Size indicators

Rows 2.769 2.649 2.529 2.409 2.289

Columns 3.920 5.040 6.020 6.860 7.560

Nonzeros 24.590 34.612 43.402 50.960 57.286


Integer objective bound (€) 56.223 54.282 50.795 48.480 46.689

Lineair objective bound (€) 56.218 54.277 50.763 48.447 46.643

GAP (%) 0,0089 0,0092 0,0630 0,0681 0,0985

CPU-time (sec.) 10,98 478,47 3.600 3.600 3.600


* Optimization was stopped; time limit was reached (3600 sec.)

The results are in line with our expectations. More specifically, stretching the BW decreases the number of

employees and the total workforce size. In terms of costs, the decrease is even very drastically as visually

shown by figure 17:

Figure 17: Scenario 3 – Total work schedule costs

35.000

40.000

45.000

50.000

55.000

60.000

65.000

1 (BM) 2 3 4 5

Co

st (

€)

BW


89

Increasing the flexibility of the schedule, by for example a longer BR or BW, comes along with a higher

computational effort. The models with BWs equalizing 3 or more all reached the time limit. However, the

gap with the respective lineair objective bounds is always smaller than a tenth of a per cent. Rows and

columns are also influenced by increasing the intra-tour start-time flexibility. While the former is

recognized by a negative correlation with the length of the BW, the second size indicator shows an upward

trend.

Besides costs and computational effort there are many other advantages and disadvantages of increasing

the BW. In this respect I would like to refer the reader back to paragraph 2.3.4 and papers like Isken (2004)

and Showalter & Mabert (1988).

One final point that should be made is about the number of surplus employees. Namely, when one

increases the BW, the number of possibilities to schedule an employee increases tremendously. This gives

the program the occasion to create a work schedule without nearly no surplus coverage. So in practice, if

the employer wants to anticipate sickness or unexpected absenteeism, he or she should add a safety buffer

on top of the demand before optimizing the problem.

3.3.5. Scenario 4: High- and low skilled employees

The jobs that need to be performed in a company do not always require the same high skilled and -paid

type of employee. Therefore, a good savings method is to hire low skilled employees. The adoption of this

type of worker comes along with two parameters in our models. The first parameter is the high-to-low-skill

ratio β which limits the number of low skilled employees per period to a certain proportion of the high

skilled employees working that same particular period. The other parameter, the cost skill ratio γ,

determines the percentage of the paycheck of a high skilled employee given to a low skilled one. In our

BMs, no low skilled employees were employed. Table 36 summarizes the results for β-values between 2

and 5.

90


* Solve interrupted: out of memory ** Optimization was stopped; time limit was reached (3600 sec.)

SCM IM HM

0 (BM) 2 3 4* 5 0 (BM) 2** 3** 4** 5** 0 (BM) 2 3** 4** 5**


High skilled FT employees 93 60 69 / 78 90 58 68 72 76 93 60 69 76 78

High skilled PT employees 31 27 27 / 27 30 26 25 29 27 31 27 27 28 27

Low skilled FT employees 0 33 24 / 15 0 32 23 18 15 0 33 24 17 15

Low skilled PT employees 0 4 4 / 4 0 4 5 1 3 0 4 4 3 4

Total workforce 124 124 124 / 124 120 120 121 120 121 124 124 124 124 124

Size indicators

Rows 337 337 337 337 337 2.769 2.769 2.769 2.769 2.769 905 905 905 905 905

Columns 844 884 884 884 884 3.920 3.920 3.920 3.920 3.920 1.564 1.564 1.564 1.564 1.564

Nonzeros 29.371 38.880 38.880 38.880 38.880 24.590 28.090 28.090 28.090 28.090 46.007 60.632 60.632 60.632 60.632




GAP (%) 0,0018 0,0078 0,0094 0,0185 0,0073 0,0089 0,1807 0,17 0,0206 0,1767 0 0,0058 0,0094 0,0148 0,0092

CPU-time (sec.) 0,23 46,15 70,55 3053,83 51,87 10,98 3.600 3.600 3.600 3.600 0,14 68,13 3.600 3.600 3.600

91

The results are far from surprising. First, we see that the size of the problem increases, just as we have seen

in the previous two scenarios. Secondly, the workforce is scattered across four subclasses: high- and low

skilled FT employees and high- and low skilled PT employees. The number of employees belonging to a

particular subclass is strictly determined by their respective ratios. The last thing we notice is the negative

correlation between the number of low-skilled employees and the total work schedule cost. This is a very

logical pattern given a cost skill ratio γ of 0.7. Figure 18 illustrates for the IM the workforce distribution and

the total work schedule cost in function of the skill ratio:

Figure 18: Scenario 4 – Workforce distribution and total work schedule cost for the IM

3.3.6. Scenario 5: demand pattern modification & the relationship with PT employees

The demand pattern used for the BMs and the previous four scenarios was a very general and low variable

pattern. The multiple and trend were set equal to 0, which ensured that the random numbers were always

located between the lowest- and highest starting value, equalizing 20 and 25, respectively. The mean ideal

staffing level valued 22.5 with an amplitude of ±11,11%. This demand pattern is, however, not

representative for all service environments. Therefore, I have set up a scenario were we investigate the

influence of the demand pattern on our, in the meanwhile well-known, results. The demand patterns vary

on two dimensions: the ‘shape’ of the demand/ideal staffing level (four factor levels) and the variation in

the ideal staffing level across periods (two levels). The former points to the number of daily peaks in the

ideal staffing level. We consider in this scenario one, two, three and numerous daily peaks. “One daily peak

in demand often occurs in retail facilities on weekends. Two daily peaks are often observed in service

environments where demand is related to commuters; for example, drop-off and pick-up demand at dry

cleaners. Three daily demand peaks commonly occur in restaurants. Numerous daily peaks are observed in

0

20

40

60

80

100

0 2 3 4 5

Emp

loye

es

Skill Ratio

Workforce distribution (IM)

High-FT High-PT Low-FT Low-PT

50.000

51.000

52.000

53.000

54.000

55.000

56.000

57.000

0% 20% 25% 33% 50%

Co

st (

€)

Skill ratio (%)


92

service environments where there are multiple components of customer demand, for example, counter

staffing requirements in airport terminals.” (Thompson and Pullman, 2007) The variation in the ideal

staffing levels is in congruence with the amplitudes of these levels. ±20% and 80% are the two amplitudes

studied in this scenario. “The rationale behind this second dimension is that a higher variability should

make it harder to provide the ideal number of staff in every period.” (Thompson and Pullman, 2007)

Actually there is also a third dimension, namely the mean ideal staffing level, but this is fixed on a height of

25 employees. This staffing level holds the middle between a small grocery store and a modestly sized

telemarketing operator (Thompson and Pullman, 2007). The setups considered here are self-made but

strongly consistent with those from a number of earlier studies (see, for example, Thompson and Pullman,

2007; Thompson, 1996; Thompson, 1997).

Changing the demand pattern provides me the opportunity to investigate lots of subscenarios. More

specifically, it should be interesting to see if I can confirm my findings from the previous scenarios. For

example, one should think that longer BRs and BWs come more to their right with high variable demand

patterns. And what would happen if we would minimize the workforce size, instead of the total work

schedule cost as seen in scenario 1. These topics are, however, left open for further research. What I will

study is the relationship between the total work schedule cost and the type of demand pattern. Can we say

that we have to pay more for a, on average, higher variable demand pattern? Is this possible relationship

affected by the “shape” of the ideal staffing pattern? And what about the differences between the SCM, IM

and HM? Furthermore, what is the role of the FT/PT-ratio in all of this? Do part-timers come more to their

right in high variable service environments?

Next, figure 19 illustrates eight demand patterns with each one of them showing a combination of the first

and second dimension. Downwards, I vary dimension 1, the “shape” of the demand pattern, while

dimension 2, the variability of the ideal staffing level, is modified from the left to the right. I assume that

the exact same demand pattern returns every day.

93

Figure 19: Scenario 5 – Demand patterns considered

0

10

20

30

40

1 3 5 7 9 11 13 15 17 19 21 23

Dem

and

Planning Periods

Demand Pattern 1

0

10

20

30

40

1 3 5 7 9 11 13 15 17 19 21 23

Dem

and

Planning Periods

Demand Pattern 5

0

10

20

30

40

1 3 5 7 9 11 13 15 17 19 21 23

Dem

and

Planning Periods

Demand Pattern 2

0

10

20

30

40

1 3 5 7 9 11 13 15 17 19 21 23

Dem

and

Planning Periods

Demand Pattern 6

0

10

20

30

40

1 3 5 7 9 11 13 15 17 19 21 23

Dem

and

Planning Periods

Demand Pattern 3

0

10

20

30

40

1 3 5 7 9 11 13 15 17 19 21 23

Dem

and

Planning Periods

Demand Pattern 7

0

10

20

30

40

1 3 5 7 9 11 13 15 17 19 21 23

Dem

and

Planning Periods

Demand Pattern 4

0

10

20

30

40

1 3 5 7 9 11 13 15 17 19 21 23

Dem

and

Planning Periods

Demand Pattern 8

94

On the following two pages, the results are provided needed to give an answer on the questions posed in

the beginning of this scenario. The models used to do this analysis have the exact same input as for the BM

(see appendix B); only the demand pattern has been changed. Table 38 shows per TSM the results for all

eight demand patterns defined in figure 19. The second table has a similar setup, with that difference that

only FT employees are included into the model. Size indicators are not relevant because they did not

change in comparison with the BMs.

We start our analysis with comparing the total work schedule cost for all eight demand patterns, with

(table 38) and without (table 39) PT employees, see figure 20:

Figure 20: Scenario 5 – Total work schedule cost with and without PT employees

Figure 20 let us conclude that there is indeed a difference in low- and high variable demand patterns, both

with and without PT employees. Namely, the first four, low variable, patterns have about the same

objective function. But when we look at the last four total work schedule costs, we see an immediate

increase. This finding is even clearer when we compare the mean- and standard deviation of the first- and

last four objective functions, both with and without part-timers, see table 37:

In

(1.000)

Table 37: FT + PT Table 38: FT

Low variable patterns High variable patterns Low variable patterns High variable patterns

Mean Std. dev. Mean Std. dev. Mean Std. dev. Mean Std. dev.

SCM 64 2 84 11 72 2 101 10

IM 60 2 70 13 67 3 84 18

HM 64 2 84 11 72 2 101 10

Table 37: Scenario 5 – Mean and standard deviation for low and high variable demand patterns

50.000

62.500

75.000

87.500

100.000

112.500

125.000

1 2 3 4 5 6 7 8

Co

st (

€)

Demand Pattern

Total Work Schedule Cost (FT+PT)

SCM IM HM

50.000

62.500

75.000

87.500

100.000

112.500

125.000

1 2 3 4 5 6 7 8

Co

st (

€)

Demand Pattern

Total Work Schedule Cost (FT)

SCM IM HM

95

Table 38: Scenario 5 – summary of the results (a)

* Solve interrupted: out of memory ** Optimization was stopped; time limit was reached (3600 sec.)

SCM IM HM

1** 2 3 4 1 2 3 4 1** 2 3 4


FT employees 102 102 102 109 94 93 99 102 102 102 102 109

PT employees 34 34 32 36 31 30 31 34 33 29 34 36

Total workforce 136 136 134 145 125 123 130 136 135 131 136 145


Integer objective bound (€) 60.994 63.391 63.902 66.362 58.374 58.644 59.909 62.904 60.996 63.391 63.902 66.360

Lineair objective bound (€) 60.958 63.385 63.902 66.357 58.369 58.639 59.909 62.898 60.934 63.385 63.902 66.356

GAP (%) 0,0590 0,0095 0 0,0075 0,0086 0,0085 0 0,0095 0,1016 0,0095 0 0,0060

CPU-time (sec.) 3.600 2,86 0,34 0,55 60,42 75,67 3,13 192,21 3.600 7,84 0,35 1,22

5 6 7 8* 5 6 7 8 5 6 7 8**


FT employees 117 138 153 / 93 102 150 143 117 138 153 154

PT employees 39 46 51 / 31 34 50 47 39 44 51 51

Total workforce 156 184 204 / 124 136 200 190 156 182 204 205




GAP (%) 0 0,0073 0 0,0309 0,0092 0,0016 0,0097 0,0095 0 0,0012 0 0,0777

CPU-time (sec.) 0,04 2,59 0,06 2.025 354,93 46,32 72,59 26,72 0,06 0,41 0,14 3.600

96

Table 39: Scenario 5 – summary of the results (b)

SCM IM HM

1 2 3 4 1 2 3 4 1 2 3 4


FT employees 125 134 135 134 118 119 132 130 125 134 135 134




GAP (%) 0 0 0 0 0,0096 0 0 0,0086 0 0 0 0

CPU-time (sec.) 0,02 0,02 0 0,02 0,42 7,66 0,05 3,45 0,02 0,02 0,02 0,02

5 6 7 8 5 6 7 8 5 6 7 8


FT employees 159 182 203 200 121 129 198 190 159 182 203 200




GAP (%) 0 0 0 0 0 0 0 0 0 0 0 0

CPU-time (sec.) 0,02 0,06 0,08 0,02 8,64 0,66 4,65 0,31 0 0,06 0,06 0,02

97

The mean values for high variable patterns are significantly higher than demand patterns with low

amplitude. In other words, mean high variable demand comes along with, on average, a more expensive

total work schedule cost.

Another significant finding is the difference in the values of the standard deviations between high- and low

variable demand patterns. This standard deviation says something more about the first dimension, the

‘shape’ of the demand/ideal staffing level. It is clear that the number of peaks does not play an important

role when facing low variable demand patterns (Jacobs and Bechtold, 1993b). However, the standard

deviations for demand patterns with large amplitude are of such a high level that one should take into

account the number of daily peaks when studying these types of demand. Moreover, this relationship

seems to be positive, or at least so for the first three factor levels of dimension 1. Moving from three

towards numerous peaks has a more stagnating relationship. This last finding is even better visually

represented in figure 21 where I have calculated the cost difference per “shape”:

Figure 21: Scenario 5 – Difference in total work schedule cost with and without PT employees

A final word about figure 20 and table 37 has to do with the type of model used to solve the problem. In

line with the conclusions of previous scenarios, the implicit modelling approach is, for each demand

pattern, able to come up with a lower overall cost, compared with the SCM and HM. This has to do with the

increased scheduling flexibility provided to the program because of numerous possible starting periods

(intra-tour start-time flexibility) (cf. supra). However, there are two remarkable results, namely for high

variable demand patterns 5 and 6. Here we see that the IM has found a solution in the same objective

function range as for the low variable demand patterns (see also figure 21, “shape” one and two of the IM).

The underlying reason of this phenomenon is, however, not very straightforward. In addition, the

relationship between the “shape” of the demand pattern and intra-tour start-time flexibility would be an

interesting topic for future research.

-10.000

0

10.000

20.000

30.000

40.000

1 2 3 4

Co

st (

€)

Shape

Absolute Difference in Total Work Schedule Cost per "Shape" (FT+PT)

SCM IM HM

-10.000

0

10.000

20.000

30.000

40.000

1 2 3 4

Co

st (

€)

Shape

Absolute Difference in Total Work Schedule Cost per "Shape" (FT)

SCM IM HM

98

Now I will try to investigate whether there is a difference between tables 38 and 39. In other words, has the

incorporation of part-timers affected the total work schedule cost? Figure 22 illustrates the absolute and

relative difference in cost for each demand pattern. The values are calculated by subtracting the cost for

full-timers with those from FT- and PT employees

Figure 22: Scenario 5 – Absolute and relative cost difference per demand pattern

Based on figure 22, we can easily say that the incorporation of PT employees reduces the total work

schedule cost. In detail, for the SCM and HM, the total cost increases with 17 percentage points if only FT

employees can be scheduled. For the IM this increase values 15 percentage points. One could argue of

course that these percentages strongly depend on the FT/PT ratio α and the cost PT ratio δ. To investigate

this, I have added appendix C and D where I modify these respective ratios for the BM. The demand pattern

that was used for the BM is comparable with demand pattern 4, and therefore also the results. Appendix C

and D learn us that for this type of demand pattern, α and δ do not have such a great influence that they

would scramble the values calculated for figure 22. The relationship between α and δ and dimensions one

and two of the demand patterns is open for further research.

0

10.000

20.000

30.000

1 2 3 4 5 6 7 8

Co

st (

€)

Demand Pattern

Absolute Difference in Total Work Schedule Cost

SCM IM HM

0%

10%

20%

30%

1 2 3 4 5 6 7 8

%

Demand Pattern

Relative difference in Total Work Schedule Cost

SCM IM HM

99

Chapter 4

4. Conclusion

The purpose of this thesis was to model and analyse an integer programming formulation for the TSP based

on an in-depth literature study. The research conducted in this thesis was particularly focused on an

explicit- and implicit set-covering formulation of the TSP and a hybrid form combining both modelling

approaches. In what follows, the most important findings for both the literature study and the

programming part are enumerated.

The art of tour scheduling distinct itself in the width field of personnel planning by combining the

techniques of shift- and days-off scheduling. In addition, if one wants to set up a viable work schedule, he

or she should take into account many rules and restrictions. Since the mid-twentieth century, researches

started to come up with mathematical formulations to capture this complex problem. Until now, two

solutions methods turned out to be most popular: an explicit- and an implicit set-covering approach (there

is also a third derivative model, but this is left open for further research). Set-covering refers to the fact

that every day of the scheduling horizon a certain demand pattern needs to be covered. Explicit and

implicit are two ways of formulating the (decision) variables of the mathematical problem. The former

represents each tour by a separate integer variable while the latter uses a combination of variables

whereby shifts are represented by “starting-time bands” and days-on patterns. Besides these variable

formulations, tour scheduling literature can be classified in four other categories: the objective function(s),

personnel characteristics, shift definitions and constraints.

The objective functions can occur as costs, workforce size and many others like, for example, goal

modelling and balancing objectives. Cost objectives are most often applied, especially in the form of

personnel costs.

Personnel characteristics are introduced to separate the different types of employees. In tour

scheduling, a distinction is regularly made between full- and part-time employees and workers with

different skill levels.

Shift categorization is maybe the most important aspect in terms of classifying existing tour

scheduling literature. Shift overlap, shift start-times, length of the shifts and break placement all fall under

this category and are a good way of incorporating flexibility into the model. The sample of more than 40

100

tour scheduling papers seem to be pretty scattered across all of these categories, except for the last one,

break placement, which only occurs in one third of the papers.

A final way to classify tour scheduling literature is based on the constraints added into the

respective models. Twenty-two constraints were studied and it turned out to be that hard coverage

constraints, constraints which allow overstaffing and constraints that restrict the number of assignments

and consecutive days/shifts are the most popular ones.

Based on the tour scheduling categories identified in literature, it was now task to bring these findings into

practice. In addition, an (explicit) SCM, IM and HM were set up, both mathematical as well as programmed

in C++. The HM combines the techniques of the previous two standard approaches, as said before. Analysis

of the models is performed by ten different indicators, spread over three classes: the workforce

distribution, the size of the problem and computational indicators.

First, a BM was set up, to make comparison possible. This resulted in a couple of interesting figures.

For example, the objective function and the workforce size are identical for the SCM and HM. This is due to

their similarities in the way tours are created. Another finding was the lower objective function of the IM.

This is, in turn, explained by the introduction of intra-tour start-time flexibility. The latter also explains the

more gradual distribution, compared to the SCM and HM, of the workforce during weekdays and

weekends. After studying the BM, five adaptions were configured and analysed thoroughly.

In the first scenario, the cost objective function was changed towards a minimization of the

workforce size. Besides the higher total work schedule cost, I also found out that the average surplus

coverage per day became higher. Remarkable, knowing that the objective of this scenario was to minimize

the workforce size, while the BM was optimized in light of the cost perspective.

The second scenario introduced breaks into the model by help of defining a BR. Just as for the third

and fourth scenario (cf. infra), this modification influenced the size of the problem. More than once,

optimizing the SCM and HM resulted in a memory problem, which is confirmed by existing tour scheduling

literature. Moreover, higher pre-solvation times were found for the HM compared to the SCM. The latter is

a phenomenon that was noticed in every optimization exercise. Break placement also affected the overall

cost of the work schedule and the workforce size. In detail, I found a negative relationship between the

length of the BR and the price paid to employ the entire workforce and the total number of employees.

The third scenario examined, only for the IM, the impact of an increasing BW. It turned out to be

that intra-tour start-time flexibility has a great influence on the total work schedule cost.

Skill levels were added under the fourth scenario. A logical negative relationship was found

between the maximum number of low skilled (and paid) employees and the amount of euros that should

be spend to cover the demand.

101

In a final scenario, an analysis was conducted around demand pattern modifications. The demand

patterns varied over two dimensions; the ‘shape’ of the demand (number of daily peaks) and the variation

in the ideal staffing level across periods. One of the first findings was that mean high variable demand

comes along with, on average, a more expensive total work schedule cost. Another interesting result had to

do with the first dimension, the ‘shape’ of the ideal staffing level. Namely, on the one hand, no significant

relationship was found for low variable demand patterns. But on the other hand, for demand patterns with

high amplitudes, a strong positive trend was identified between the number of peaks and the total work

schedule cost. A third and final study was set up to determine the relationship between full- and part-time

employees and the varying demand patterns. It was immediately clear that the introduction of part-timers

reduces the total work schedule cost. However, the relation between part-time employment and the two

dimensions of the demand patterns was less clear and is left open for further research.

102

Chapter 5

5. Limitations and Future Recommendations

During the last year and a half studying the art of the tour scheduling, I was forced to set some limitations

and refer the reader to past or future research.

A first, very important, limitation was the choice to drop the derivative set-covering TSM. This

model is mostly applied in nurse scheduling, but also in transit operator scheduling, airline crew scheduling

and telecommuting.

The second limitation is related to the Gurobi Optimizer tool. Namely, the latter only solves the

program by a branch-and-price technique. In addition, it should be interesting to see how the results

change if, for example, other column-based solution techniques are applied. Eitzen et al. (2004) has already

done some research about this.

A third group of limitations appeared as a result of defining a BM. Although a scenario analysis was

worked out, numerous other scenarios can be studied. To start, one can analyze the effects of a continuous

TSP instead of the restricted ‘no day-overlap policy’ applied in this thesis. Also, what is the impact on our

results after changing the number of shifts and their respective lengths and start-times? Furthermore, is

there a relationship between the demand pattern dimensions and the BR, the BW and the part-time ratio?

And do we find the same results of introducing low skilled employees if we would also include productivity

levels? One could also change the parameters of the constraints or even add more constraints. Examples

for that last option could be a restriction on the number of shift changes and a prohibition to schedule an

early shift after a weekend off.

A final, more complex, recommendation has to do with the way of modelling. The explicit-, implicit

and hybrid model suggested in this thesis follow the standard tour scheduling rules. One of the options for

the interested reader could be, for example, to change the explicit BR variable to an implicit combination of

backward- and forward pass constraints. This would have the advantage that the number of integer

variables decreases, which could partly solve the memory problems. Bard et al. (2003) enumerates a couple

of papers that investigate how to reduce the number of variables for explicit TSMs. Another option with

respect to break placement and explicit variable reduction is to include breaks after the complete work

schedule has been set up.

x

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xv

Appendix A: Overview Table of Tour Scheduling Characteristics

Appendix A: Overview table of tour scheduling Characteristics

Reference Objective Function

Type of Decision

Variable(s) *

Personnel characteristics Shift definitions

Breaks Labor Contract Skills Overlap

Start times Length

Full time Part time Fixed Definable Fixed Definable

Alfares, 2007 Cost 1 x x x x x

Al-Yakoob and Sherali,

2007

Workforce

Size, Other 3 x x x

Bard, 2004a Cost 2 x x x x x x x x x

Bard, 2004b Cost 2 x x x x x x x x

Bard and Wan, 2008 Other 3 x x x x x x x x x

Bard et al., 2003 Cost 2 x x x x x x x x

Bard et al., 2007 Cost 2 x x x x x x x x

Beaulieu et al., 2000 Other 3 x x x

Berrada et al., 1996 Other 3 x x x x x

Brusco and Jacobs,

1998a Cost 1 x x x x x

Brusco and Jacobs,

1998b Cost 1 x x x x x

Brusco and Jacobs, 2000 Workforce

Size 2 x x x x x

Brusco and Jacobs, 2001 Workforce

Size 1 x x x x x x

* Type 1: SCM, Type 2: IM, Type 3: Derivative model (see paragraph 2.3.2.)

xvi


Type of Decision

Variable(s) *



Start times Length


Brusco and Johns, 1996 Cost 1 x x x x x x x

Burke et al., 2010 Other 3 x x x x x

Chen et al., 2010 Cost 2 x x x x x

Choi et al., 2009; Cost 1 x x x x x

De Grano et al., 2009 Other 3 x x x x x

De Matta and Peters,

2009 Cost 1 x x x x

Eitzen et al., 2004 Other 1 x x x x x

Ertogral and

Bamuqabel, 2008

Cost,

Workforce

Size

1 x x x x x x x x

Felici and Gentile, 2004 Other 1 x x x x

Isken, 2004 Cost 2 x x x x x

Jacobs and Brusco, 1996 Workforce

Size 2 x x x x

Jarrah et al., 1994 Cost 2 x x x x x x x

Kostreva and Jennings,

1991 Other 3 x x x

Lezaun et al., 2006 Other 3 x x x

Lezaun et al., 2007 Other 3 x x x

Lezaun et al., 2010 Other 3 x x x x

Love and Hoey, 1990 Other 1 x x x x x x

Mohan, 2008 Other 3 x x x x


xvii


Type of Decision

Variable(s) *



Start times Length


Moz and Pato, 2004; Other 3 x x x

Ozkarahan, 1989; Cost, Other 1 x x x x x x

Rong, 2010; Cost 2 x x x

Ronnberg and Larsson,

2010; Other 3 x x x x x

Seckiner et al., 2007; Cost 2 x x x x x

Stolletz, 2010; Cost 3 x x x x

Thompson, 1995; Cost, Other 1 x x x x x

Topaloglu and

Ozkarahan (2004); Other 2 x x x x x x

Trilling et al., 2006; Other 3 x x x x x

Valouxis and Housos,

2000; Cost, Other 3 x x x

Valouxis et al., 2012; Other 3 x x x x x x

Warner, 1976b; Other 3 x x x x


xviii

Appendix B: Detailed setup for the Baseline Model

Appendix B: Detailed setup for the Baseline model Settings Notation Value Settings Notation Value

1) General 2) Demand Generator

Days J 7 Lowest value 20

Periods P 24 Highest value 25

Shifts* K 3 Multiple 0

Band width** BW 1 Trend 0

3) Break Placement

Break window BR 0 Min. shift length to schedule a break 6

4) FT personnel

Max. assignments 5 Min. periods between two assignments* S 12

Min. assignments 4 Max. working periods 40

Max. night shifts* / Min. working periods 30

Max. consecutive days 5 Number of shift lengths 1

Min. consecutive days 2 Shift length in periods (shift length 1)

8

Max. consecutive shifts* 5 Number of start times per shift* 1

Min. consecutive shifts* 2 Start time in periods (shift 1, start time 1)*

0

Max. consecutive days off 3 Start time in periods (shift 2, start time 1)*

8

Min. consecutive days off 2 Start time in periods (shift 3, start time 1)*

16

* Not for the IM ** Only for the IM

xix

Settings Notation Value Settings Notation Value

5) PT personnel

Max. assignments 5 Number of shift lengths 4

Min. assignments 2 Shift length in periods (shift length 1) 3

Max. night shifts* / Shift length in periods (shift length 1) 4

Max. consecutive days 5 Shift length in periods (shift length 1)

5

Min. consecutive days 2 Shift length in periods (shift length 1)

6

Max. consecutive shifts* 5 Number of start times per shift* 2

Min. consecutive shifts* 2 Start time in periods (shift 1, start time 1)*

0

Max. consecutive days off 5 Start time in periods (shift 2, start time 1)*

8

Min. consecutive days off 2 Start time in periods (shift 3, start time 1)*

16

Min. periods between two assignments* S 12 Start time in periods (shift 1, start time 2)* 4

Max. working periods 30 Start time in periods (shift 2, start time 2)* 12

Min. working periods 15 Start time in periods (shift 3, start time 2)* 20

FT to PT ratio α 3

6) Skill level

High-to-low-skill ratio β 0

7) Objective function

Minimize total cost Yes Minimize overstaffing No

Minimize total workforce size No Minimize understaffing No

Cost skill ratio γ 0.7 Cost par-time ratio δ 0.8

* Not for the IM

xx

Appendix C: Results of changing the FT/PT ratio of the BM

Appendix C: Results of changing the FT/PT ratio of the BM

* Solve interrupted: out of memory ** Optimization was stopped; time limit was reached (3.600 sec.)

Remark: The default value of the FT/PT ratio α is equal to 3 (see Appendix B).

SCM IM HM

0 2* 3 (BM) 4 5 0 2 3 (BM) 4 5 0 2* 3 (BM) 4 5**


FT employees 113 / 93 97 99 112 82 90 96 98 113 / 93 97 99

PT employees 0 / 31 24 19 0 40 30 24 19 0 / 31 24 19

Total workforce 113 / 124 121 118 112 122 120 120 117 113 / 124 121 118

Size indicators

Rows 337 337 337 337 337 2.769 2.769 2.769 2.769 2.769 373 905 905 905 905

Columns 104 844 844 844 844 3.920 3.920 3.920 3.920 3.920 140 1.564 1.564 1.564 1.564

Nonzeros 5.840 29.371 29.371 29.371 29.371 24.352 24.590 24.590 24.590 24.590 7.928 46.007 46.007 46.007 46.007




GAP (%) 0 0,0267 0,0018 0,0087 0,0086 0 0,0091 0,0089 0,0088 0,0087 0 0,0178 0 0,0087 0,0173

CPU-time (sec.) 0,06 1.274,18 0,23 0,28 448,75 0,03 58,61 10,98 162,15 566,42 0,08 1.455,62 0,14 0,87 3.600

xxi

Appendix D: Results of changing the cost PT ratio of the BM

Appendix C: Results of changing the cost PT ratio of the BM

Remark: The default value of the cost PT ratio δ is equal to 0.8 (see Appendix B).

SCM IM HM

0.6 0.7 0.8

(BM) 0.9 1 0.6 0.7

0.8

(BM) 0.9 1 0.6 0.7

0.8

(BM) 0.9 1


FT employees 93 93 93 93 93 90 90 90 91 93 93 93 93 93 93

PT employees 31 29 31 31 30 30 30 30 30 31 31 29 31 31 31

Total workforce 124 122 124 124 123 120 120 120 121 124 124 122 124 124 124




GAP (%) 0,0091 0 0,0018 0,0034 0,0051 0,0056 0 0,0089 0 0 0,0073 0,0053 0 0 0

CPU-time (sec.) 0,28 0,02 0,23 0,30 0,06 28,78 19,67 10,98 5,69 15,48 1,03 1,22 0,14 0,09 0,16

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