University of Groningen
Master Thesis
Large instance multi-department shift
scheduling
Author:
Rowan Timmermans (s2740192)
Large instance multi-department shift
scheduling
Author:
Rowan Timmermans (s2740192)
Abstract
In service related industries, automated shift scheduling methods can efficiently create schedules that meet varying labour requirements. However, when these requirements become more dynamic, more flexibility in the scheduling criteria is required. In work environments where employees are qualified to work in multiple departments, schedule quality can be improved by allowing them to transfer between departments during a shift. This is referred to as the multi-department extension. Another approach is to increase flexibility in labour rules such as shift duration and break placement, which is referred to as labour flexibility. As flexibility rises, the number of scheduling options rises exponentially. Therefore, in many practical applications, Multi-Department Shift Scheduling (MDSS) problems are difficult to solve in reasonable time.
This thesis focuses on the MDSS problem for instances with high labour flexibility. We propose two techniques that handle the resulting complexity in the model. First, we propose Transfer Shift Selection Algorithms (TSSAs) that limit the solution space by identifying promising shifts in a pre-processing stage. Second, we introduce a hybrid Tabu Search method that iteratively solves sub-problems of two departments to approach the optimal schedule. Compared to existing solution approaches in current literature, these techniques are equipped for and tested on large instances with a very high labour flexibility.
We show that the difficulty in solving such MDSS problems to optimality arises from symmetry in the solution space: there are often many (near-) optimal solutions, which complicates the solving procedure. By using a TSSA, we can deal with a large part of the symmetry problem and improve computational speed by over 90% while still reaching up to 99% of optimality. Furthermore, we show that the Tabu Search approach is an efficient method to reach a quick improvement in a short, fixed time. We conclude that the heuristics proposed in this thesis are efficient and flexible approaches that can be adapted to many di↵erent multi-department shift scheduling applications.
Contents
1 Introduction 4
1.1 Example of the problem . . . 5
1.2 Overview . . . 5
2 Literature overview 6 2.1 Shift scheduling developments . . . 6
2.2 Multi-department context . . . 6
3 Problem formulation 8 4 Transfer Shift Selection Algorithms 10 4.1 Number of possible shifts . . . 10
4.2 Demand TSSA . . . 11
4.2.1 Labour requirement correlation and multi-department shifts . . . 11
4.2.2 Mathematical formulation . . . 12
4.3 Result TSSA . . . 12
4.3.1 Transfer shift value analysis . . . 13
4.3.2 Mathematical formulation . . . 13
5 Tabu Search 15 5.1 Overview of the hybridization scheme . . . 15
5.2 Initialization . . . 15
5.3 Move selection technique . . . 15
5.4 Main operator . . . 16 5.5 Intensification . . . 16 5.6 Diversification . . . 16 5.7 Finalization . . . 16 6 Experiments 18 6.1 Parameter settings . . . 19 6.1.1 DTSSA parameters . . . 19 6.1.2 RTSSA parameters . . . 19
6.1.3 Tabu Search parameters . . . 20
6.2 General model performance . . . 20
7 Experimental insights 22 7.1 TSSA propositions . . . 22
7.2 TSSA performance and symmetry . . . 22
7.3 Tabu Search performance . . . 24
8 Conclusion 26
9 Discussion and further research 27
References 28
1
Introduction
In many service related industries such as restaurants, supermarkets and hotels, employees are scheduled to work in shifts. These shifts are typically created in advance by a manager, who intends to ensure that there are enough employees present on the work floor. The number of required employees in a time period is referred to as the labour requirement, and satisfying this is an important goal in many scheduling problems. Failing to do so results in undercoverage, which can lead to among others customer loss due to long waiting times or employee unhappiness due to stressful work periods. On the other hand, over-satisfying the requirements by planning too many employees, referred to as overcoverage, leads to unnecessary high labour costs. In service related industries, it can be difficult to avoid under- and overcoverage. This is because there can be highly fluctuating labour requirements during a day, which cannot be fully accounted for by the schedule because shifts should adhere to specific labour rules. For example, if there is a 1 hour requirement peak during a stock delivery for a supermarket, we cannot schedule an additional 1 hour shift because an employee has the right to a longer shift duration. However, we could schedule this employee as a stock clerk for 1 hour and then transfer him/her to work as a cashier for the remainder of the shift, in an attempt to avoid under- and overcoverage. This approach is called multi-department shift scheduling, in which a distinction is made between ‘departments’ in a work environment with separate labour requirements. Although this approach could potentially increase scheduling quality, the large number of transfer possibilities can make it a very complex problem to solve.
In order to deal with the complexity of scheduling problems, the demand for automated shift scheduling methods has risen the last decades. The main objective of these methods is to minimize under- and overcoverage by selecting shifts for a daily time window. To do this, the problem is most commonly modeled by a set-covering approach: each possible shift is explicitly defined by a start time, duration, one or more breaks and the employee assigned to it. Then, a few shifts are chosen to construct a schedule. An important advantage of this approach is its adaptability: planners can easily add or remove possible shifts based on their own judgement, allowing them to stay in control of the model’s decisions. Next to labour requirements, the labour rules are an important aspect of scheduling problems, since they influence which shifts are regarded as ‘possible’. When labour rules are not very restrictive, there are a large number of possible shifts and we say that there is high labour flexibility. For an overview of the labour rules considered, we refer to Appendix A.
We make a distinction between the single- and multi-department context. By limiting the scope to one department, single-department scheduling problems ignore the possibility of influences of other departments within a work environment. The multi-department extension broadens the scope by including the option for so-called transfer shifts: employees can transfer from one department to another within a shift. Shift scheduling problems that allow for this extension are referred to as Multi-Department Shift Scheduling (MDSS) problems. By doing so, volatility in labour requirements may be dealt with more efficiently while still adhering to labour rules. Furthermore, in an environment where each employee can work on multiple departments, the employee base can be utilized to its full potential. However, combining the multi-department context with the set-covering approach leads to a substantial increase in model size. Current multi-department scheduling literature is primarily focused on dealing with this through the use of e.g. activity assignment techniques (Lequy et al., 2012), meta-heuristics (Dahmen and Rekik, 2015) and decomposition methods (Attia et al., 2019). These papers have in common that, next to their solution approach, they deal with the model size by limiting labour flexibility: strict labour rules are imposed that limit the scheduling options of the model. However, such an assumption might not always be valid in work environments with a flexible employee base
This thesis explores the MDSS problem and attempts to approach optimal solutions in reasonable time, while allowing for a large degree of labour flexibility. We model the problem as a Mixed-Integer Program (MIP) using a set-covering approach, that minimizes labour costs and under- and overcoverage by constructing a single-day schedule of shifts. We propose two heuristic techniques that deal with the complexity issue of the multi-department context by limiting the solution space. The first technique is based on an idea from Attia et al. (2019), who use a shift selection algorithm that identifies promising transfer shifts before constructing the model. This thesis builds upon that idea by introducing two Transfer Shift Selection Algorithms (TSSAs): the demand and result TSSA. The demand TSSA uses information on labour requirements to identify fluctuations that cannot be dealt with using department shifts. The result TSSA does so by first solving single-department models and using the under- and overcoverage intervals still present to select promising transfer shifts. The second technique is a hybrid heuristic that uses a Tabu Search (TS) method that iteratively improves an optimal single-department solution by exploring neighborhoods with transfer shifts between two departments. A Brand-and-Cut (B&C) method is then applied to efficiently find the best neighboring solution.
performance of the result-based TSSA. However, by using the demand-based TSSA, we can deal with a large part of the symmetry problem and improve computational speed by over 90 % while still reaching up to 99% of optimality. Furthermore, we show that the Tabu Search is an efficient method to reach a quick improvement in a short, fixed time. We conclude that the heuristics proposed in this thesis are efficient and flexible approaches that can be adapted to many di↵erent multi-department shift scheduling applications.
1.1
Example of the problem
To illustrate the potential increase in schedule quality by the multi-department extension, we start with a basic example of a shift scheduling problem. Consider a supermarket consisting of two departments; stock clerks and cashiers. We assume that the majority of the employee base is able to work in both departments. Furthermore, for illustration purposes, we assume that the supermarket is open from 9AM - 5PM and only plans in shifts that cover the entire day. During a day in the supermarket, labour requirements vary; in the morning, a large delivery of the distribution center arrives and 5 stock clerks are required to handle this. Then, from 13.00, more and more people arrive in the supermarket, and the demand for cashiers spikes, whereas the stock clerks are done with the most of their work. Assume that sufficient employees are available to be scheduled. The requirements are illustrated in Figure 1.
Figure 1: Daily labour requirements for stock clerks and cashiers.
If no transfer shifts would be allowed in this occasion, the supermarket manager would either have to plan enough employees for both departments to cover peak hours, or accept a degree of undercoverage in his/her shop. However, a multi-department shift scheduling approach allows several employees to work as a stock clerk until 13.00 and then continue as a cashier. As a result, only 7 employees need to be scheduled to cover all demand instead of 10 in the single-department case. In total, 24 overcovered labour hours can be avoided. This di↵erence is illustrated in Figure 2.
Figure 2: Optimal shifts when minimizing undercoverage, without (top) and with (bottom) transfer shifts.
1.2
Overview
2
Literature overview
In this section, we discuss the positioning of this paper in existing literature and its contribution to it. First, the development of single-department scheduling methods is assessed. Then, the extension of these models to the multi-department context is discussed, together with relevant solution approaches from recent studies.
2.1
Shift scheduling developments
Personnel scheduling methods have been widely researched in literature. Baker (1976) is the first to classify per-sonnel scheduling problems into three types: days-o↵, shift, and tour scheduling problems. Days-o↵ scheduling problems involve the assignment of work and rest days to employees over a multi-day planning horizon. Then, shift scheduling problems consider a single day in this schedule and specify work starting and ending times and possibly the number, duration and position of breaks within worked shifts. Finally, tour scheduling problems aim at simultaneously constructing days-o↵ and shift schedules, taking into account long-term constraints such as labour laws and employee-specific preferences. Although such a complete approach seems convenient to be used by a workforce planner, it can be difficult to use in practical applications. For example, the practise of days-o↵ scheduling is often a personal matter and difficult to capture within a model. We restrict our atten-tion to the daily shift-scheduling problem, and take the results from days-o↵ scheduling, the daily employee availability, as given.
In current literature, shift scheduling models can be divided into two types of approaches: the explicit and implicit approaches. Dantzig (1954) developed the general explicit approach using a set-covering formulation: each possible shift is explicitly defined in terms of start time, duration and break placement. Based on this specification, a binary variable for each shift-employee combination is created, indicating if the shift is assigned to the employee. A major downside of this is that the number of variables quickly increases in the number of shifts and employees, which often results in models that are difficult to solve with a commercial MIP solver. The reason for this difficulty is due to a large degree of symmetry in the solution space. A MIP is symmetric if its variables can be permuted without changing the structure of the problem. In scheduling problems, symmetry in the solution space implies that there are many (near-)optimal schedules. For more information on symmetry in Integer Programming, we refer to Margot (2010). Sherali and Smith (2001) argue that symmetry is one of the most important features of mixed-integer programming models, since symmetry “can hopelessly mire a branch-and-cut solver by burdening it to explore and eliminate such alternative symmetric solutions”.
An implicit approach is a method to decrease model symmetry through a partially implicit model formulation. One of the most common applications of this approach is the implicit break model, originally developed by Bechtold and Jacobs (1990). They argue that a large degree of symmetry is caused by the explicit break definition of each possible shift. In their approach, instead of defining all possible shift-break combinations, each shift is assigned a single break interval. The assignment of each break to the chosen shifts is done in a later stage, the feasibility of which can be ensured by a set of so-called backwards and forwards constraints. They later prove in (Bechtold and Jacobs, 1996) that under certain conditions, this model is equivalent to the explicit approach. Later, Rekik, Cordeau, and Soumis (2010) generalize the concepts and provide the context in which the backwards and forwards constraints can be used. Although the implicit approach is able to significantly decrease model complexity in some contexts, it lacks adaptability with complex constraints. For example, suppose that a workforce planner prefers that employees that start at the same time have simultaneous breaks. This would be a straightforward constraint in an explicit model, but a difficult one when breaks are scheduled implicitly, without having to impose other restrictions. We consider it an important objective to use a model that is adaptable to many practical applications. Hence, we choose not to adopt an implicit break approach.
2.2
Multi-department context
The idea that employee should only be assigned to a specific department has been challenged severely. Ritzman, Krajewski, and Showalter (1976) may be the first to understand the need for disaggregating workforce planning by individual activity, recognizing that an employee could be cross-trained for more than one job. This gave rise to the multi-department (MDSS) and multi-activity shift scheduling (MASS) context. The definition of a department and an activity is usually similar; both include multiple labour requirements and consider shifts that can transfer between them. A typical di↵erence, however, is that multi-department problems usually limit to at most one transfer for each shift, whereas multi-activity problems usually allow multiple transfers between more than two activities.
anonymous shifts are constructed: the employee definition of each variable is dropped. Then, the decision variables are integer variables, indicating how often each shift is chosen for the final schedule. An important downside is that these types of models do not take into account employee-specific labour rules such as weekly contract hours. This makes it a less attractive approach in organizations with an inflexible employee base. However, this approach also can have a commercial advantage: as mentioned before, in many businesses the shift assignment problem still is personal matter between the employer and employees. In those cases, it is preferred that the assignment of shifts is left to the problem owner. Due to the potential reduction in model complexity and the practical relevance of the approach, we adopt this implicit modelling technique.
However, Cˆot´e, Gendron, and Rousseau (2011) and Restrepo, Lozano, and Medaglia (2012) do not consider employee availability in their approach; they assume that enough employees are available to perform the sched-uled shifts. We argue that a shift scheduling model should include a constraint for the number of available employees, and take into account the specific skillset of each employee. For example, in the multi-department context, an employee should be qualified for all departments covered in a shift to be assigned to it in a later stage. Hence, in this thesis, we assume a given heterogeneous set of employees and propose so-called employee capacity constraints that assert that shifts can be assigned in a later stage. To the best of our knowledge, no such constraints have been included before for a heterogeneous workforce on an anonymous MASS/MDSS problem.
The explicit MDSS problem is sometimes referred to as the Employee Scheduling Problem with Inter-Department Transfers (ESP-IDT). Attia et al. (2019) approach a multiple-day ESP-IDT with a three step decomposition-based heuristic. In the first step, they solve a single-department scheduling problem, and use a transfer shift selection algorithm to filter ‘promising’ transfer shifts. Then, in the second step, they decompose the problem into anonymous single-day scheduling problems that include the promising multi-department shifts. From the obtained schedules, they then devise inter-department demand curves, which specify the number of transfers between departments over time. In the third phase, they decompose the initial scheduling problem into single-department problems using these inter-department demand curves and solve each problem separately. The problems considered in the first two steps are very similar to the context of this thesis. The notion of only using a set of promising transfer shifts is very interesting, since it could potentially greatly reduce model size and symmetry by making large cuts in the solution space.
We build upon this idea by introducing two variations of their promising shift selection method, aiming to select transfer shifts more efficiently to approach the optimal solution while maintaining the gain in computation time. Their selection method consists of identifying intervals of under- and overcoverage in the single-department solution and adding transfer shifts at the start and end of those intervals. For example, if an interval of undercoverage starts at a certain time for a department, they select transfer shifts that transfer at that moment from and to that department. In this thesis, we argue that if requirements cannot be covered with single-department shifts in this interval, a transfer shift that transfers away from the undercovered single-department at that interval does not seem likely to be optimal. The first variation proposed in this thesis, the Result Transfer Shift Selection Algorithm (RTSSA), uses this insight to further narrow down the number of selected shifts in an attempt to approach the optimal solution more efficiently.
However, one might say that the aforementioned ideas treat the result, under- and overcoverage, rather than the cause: the labour requirement fluctuations. Furthermore, these ideas do not take inter-department influences into account. The second variation, the DTSSA, follows a more direct approach by looking at inter-department labour requirement fluctuations to select promising transfer shifts. To the best of our knowledge, no version of this shift selection technique has been proposed in previous literature.
3
Problem formulation
Our model considers a fixed one-day time window for which a schedule has to be created. The time window is defined as a set I of separate time buckets of equal length, typically set to 60 to 15 minutes. We consider a work environment with a set of departments D. Furthermore, we assume that for each department d with d2 D, and each time bucket i with i 2 I, the labour requirement rid is known. We denote Rd as the set of
labour requirements for department d2 D.
Considering the time window I and the set of departments D, we generate a set of possible shifts S, based on pre-specified labour rules. We make a distinction between two types of shifts: we have a set of transfer shifts Str, that include a transfer to another department, and a set of internal shifts Sint, that do not. Note that we
have S = Str
[ Sint. Then, each possible shift s is characterized by a unique set of definitions. The labour rules
and shift definitions are summarized in Appendix A.
The problem owner, the manager in charge of scheduling, has to construct a schedule of shifts that mini-mizes the total costs due to labour, transfer, undercoverage and overcoverage. The integer decision value that determines how many of each possible shift s is chosen for the schedule is denoted Xsand has a corresponding
labour cost parameter cl
s. Furthermore, for each transfer shift, we consider a fixed non-negative transfer penalty
ct, which is independent of the pre- and post-transfer department. The under- and overcoverage in each time bucket i are modelled as a slack and surplus variable Yi and Yi+, respectively. We consider a constant under-coverage cost parameter c and overcoverage cost parameter c+. Finally, to link the variables with the labour
requirements, we use a binary parameter sidthat is 1 if shift s covers department d during time bucket i, and
zero else.
As mentioned before, we use some information on employee availability. Consider the supermarket problem in Section 1.1. Suppose that we have 3 employees available that can only work as cashiers, 2 that only can work as stock clerks, and 4 that can work as both. Then, we know that we cannot schedule more than 7 cashier shifts and 6 stock clerk shifts. Additionally, we know that we cannot schedule more than 9 shifts on either the cashier or stock clerk department. To translate this into a mathematical constraint, we assume a given set of employees E, where each employee e has a set of departments De for which it is qualified. To assert that we
can assign each shift to a qualified employee in a later stage, we add a general employee capacity constraint for each subset of departments. First, we denote a subset of departments as D0 ✓ D and the set of all unique non-empty subsets of D asD. Then, we introduce the general employee capacity parameters
CDg0 = e2 E : De\ D0 6= ; , D02 D,
that indicate how many shifts can at most be scheduled that cover one of the departments in D0. Note that this does not yet correctly limit the number of transfer shifts allowed: to assign an employee to a transfer shift, an employee should be qualified for both departments. For example, in the supermarket setting, we should take into account that we can only schedule a maximum of 4 shifts that cover both the cashier and stock clerk department. We denote a department pair as{d, d0}, and the set of all possible pairs as D
2. Then, we introduce
the transfer employee capacity parameters
Cddt 0 = e2 E : d, d0 2 De , {d, d0} 2 D2,
that indicate how many transfer shifts can at most be scheduled that transfer between department d and d0.
Finally, we introduce the binary variable µsd that is 1 if shift s covers department d. Using this variable, we
construct two sets of constraints. First, the general employee capacity constraints limit the number of shifts that cover a department in D0. Secondly, the transfer employee capacity constraints limit the number of transfer
Minimize X s2S clsXs+ X s2Str ctsXs+ X d2D X i2I (c Yid + c+Yid+), (1) subject to: X s2S sidXs+ Yid Y + id = rid, d2 D, i 2 I, (2) X s2S 1 Y d2D0 (1 µsd) ! Xs CDg0, D02 D, (3) X s2Str µsdµsd0Xs Cddt 0, {d, d0} 2 D2, (4) Xs2 N0 8 s 2 S, Yid+, Yid 2 R 8 i 2 I, d 2 D.
The first two aspects of our MIP formulation are based on existing multi-department shift scheduling models. The minimization function (1) combines labour and transfer costs for each shift and under- and overcoverage costs for each department in each time bucket. The first constraint set (2) deals with covering the labour requirement for each department in each time bucket, including slack and surplus variables for under- and overcoverage, respectively. We propose two additional constraints sets that deal with employee availability: constraint set (3) imposes limitations based on general employee capacity, constraint set (4) takes the transfer employee capacity into account.
Table 1: Components of the anonymous MDSS problem. Sets
I Set of time buckets in the time window. D Set of departments in the work environment. Rd Set of labour requirements for department d.
S Set of all possible shifts, given the labour rules. Sint Set of all possible internal shifts.
Str Set of all possible transfer shifts.
E Set of employees.
De Set of departments d for which employee e is qualified.
D Set of all unique non-empty subsets of D. D2 Set of all unique department pairs of D.
Cost and capacity parameters cl
s Labour costs of shift s.
c+ Cost due to overcoverage in a department for one time bucket.
c Cost due to undercoverage in a department for one time bucket. ct Transfer cost for a transfer shift.
CDg0 General employee capacity for departments D0.
Ct
dd0 Transfer employee capacity for department pair {d, d0}.
Binary parameters
sid2 {0, 1} Binary parameter that is 1 if shift s covers department d in bucket i,
and 0 else.
µsd2 {0, 1} Binary parameter that is 1 if shift s covers department d, and 0 else.
Variables
Xs2 N0 Integer decision variable indicating how many times shift s is chosen.
4
Transfer Shift Selection Algorithms
The main complication that comes with the multi-department extension is the substantial increase of the number of possible shifts, and consequently the increase in the size of the solution space. An interesting feature, however, is that the solution of the single-department problem is already a feasible solution of the multi-department one. This raises the question: when do transfer shifts add value to a single-department solution? In this section, we try to narrow down the solution space by answering this question and using this to identify ‘promising’ transfer shifts. More specifically, we would like to construct a subset of transfer shifts ˜Str
⇢ Str that are likely
to improve the case when Str=
;. We refer to the latter as the single-department solution, since there are no transfer shifts and hence little inter-department influence.
Due to the complexity of the shift scheduling problem, we are not able to strictly define the conditions under which a specific transfer shift improves an existing solution, hence we cannot guarantee optimality with a limited number of transfer shifts. However, through the use of reasoning, we have constructed two Transfer Shift Selection Algorithms (TSSAs) that try to approach this theoretically optimal subset of transfer shifts. In this section, we start by illustrating the scaling problem through a theoretical computation of the size of transfer shifts Str as a function of the number of time buckets and departments. Then, we propose two TSSAs
that reduce the set of possible transfer shifts to a subset ˜Str of promising shifts that are likely to improve the single-department solution. Both in reasoning and in functionality, these algorithms make extensive use of the labour rules and shift characteristics defined in Appendix A.
4.1
Number of possible shifts
In this section, we illustrate the complexity issue of using the set-covering approach in the multi-department context. We do this by means of a simplified example with maximum labour flexibility. First, we denote the number of time buckets|I| as n, and the number of departments |D| as m. Consider the following problem:
• We want to make a schedule for a time window of n > 1 time buckets for m > 1 departments. • We consider maximum labour flexibility in terms of shifts, i.e. SD = DD = 1, SD+ = n.
• No breaks are considered.
Observe that for a single department there is one possibility of a shift with length n, two possibilities with length n 1, and so on. Then, we know that the number of possible internal shifts for each department is given by n X k=1 k = n(n + 1) 2 , (1)
which means that the total number of possible internal shifts is given by |Sint
| = mn(n + 1)2 . (2) Furthermore, we know that a shift with length l has l 1 possible transfer times. Hence, if we do not specify the pre- and post-transfer department, there are 1(n 1) possible transfer shifts with length n, 2(n 2) of length n 1, and so on. Then, we know that the number of possible ‘empty’ transfer shifts is given by
n X k=1 k(n k) = n n X k=1 k n X k=1 k2= nn(n + 1) 2 n(n + 1)(2n + 1) 6 = n m|S int | 2n + 13m |Sint |. (3)
Finally, observe that for each transfer shift, we have m(m 1) possible combinations of pre- and post-transfer departments, which means that the number of transfer shifts in this example is given by
|Str | = (m 1)(1 3n 1 3)|S int |, (4)
which, assuming that transfers are not limited by breaks, holds for all possible break configurations. It can be seen that, even without breaks, the number of possible internal shifts already is very large: |Sint
| has a linear relationship with m and a quadratic one with n. This e↵ect is then further multiplied when considering |Str
|. This illustrates the challenge of the multi-department extension in a set-covering approach; in practical applications, we are dealing with up to 10 departments and a time window of over 60 di↵erent time buckets, which results in |Str| ⇡ 3000000 transfer shifts. In this section, we propose two TSSAs that attempt to deal
4.2
Demand TSSA
The Demand-based Transfer Shift Selection Algorithm (DTSSA) is based on an idea of Easton (2011), who show that cross-training employees only adds value when demand streams are negatively correlated. This can also be seen in the supermarket problem in Section 1.1. Around 13.00, there is a clear negative correlation between requirements for stock clerks and cashiers, which is also the transfer time of the optimal transfer shifts. Hence, it might be interesting to look at departments with negatively correlated labour requirements to construct a set of promising transfer shifts. First, we support this claim by illustrating why adding transfer shifts between departments with positive correlation might not make much sense. This is done through the use of several propositions that are supported by numerical tests later on in Section 7.1. Then, we formally define periods with negative correlation as Requirement switches and mathematically introduce the algorithm.
4.2.1 Labour requirement correlation and multi-department shifts
Consider a setting with two departments D = {d, d0}, each with a set of requirements, R
d and Rd0. We
assume a fixed hourly labour cost that is equal in all departments for all time buckets. Recall that we define a department pair as {d, d0}, where {d, d0} 2 D
2. Additionally, for notation purposes, we define a transfer
shift between department d and d0 as a transfer shift with either pre-transfer department d and post-transfer
department d0, or vice versa.
Consider an instance where rid = rid0 for all i 2 I. Observe that in this situation, no transfer shifts add
value to the solution: by transferring from d to d0, you start to cover labour requirements in d0 that also should be covered in d. In Proposition 4.1, we use this observation to make a claim about any pair of departments {d, d0} and any continuous period I0✓ I:
Proposition 4.1 If for a continuous period I0 ✓ I and a pair of departments {d, d0} 2 D
2, we have rid = rid0
for all i2 I0, then transfer shifts between department d and d0 with a transfer in I0 are not considered promising
shifts.
Note that this proposition is still very broad; by only excluding transfer shifts based on equality of requirements, we are likely still left with a large set of transfer shifts. Hence, we start looking at inter-department requirement fluctuations. First, we define a fluctuation parameter f that indicates how many time buckets are considered to identify a requirement fluctuation. Then, for a bucket i and department d, we define pre-bucket requirement rpreid and post-bucket requirement ridpost as
rpreid = ri 1 f,d+· · · + ri 1,d,
ridpost= ri,d+· · · + ri+f,d.
Now suppose that department d has an increase in requirements at bucket i, i.e. ridpre< rpostid . Then, it might seem interesting to add transfer shifts to the model with a transfer from d0 to d at i. However, using similar
rationale as before, if requirements for d0 at bucket i rise just as much, this does not make much sense; we
transfer from a department that just had an increase in labour requirements. We use this insight to define promising shift conditions in Proposition 4.2:
Proposition 4.2 If at a time bucket i2 I and for a department pair {d, d0} 2 D
2, we have ridpost r pre id > r post id0 ridpre0,
then transfer shifts with pre-transfer department d0, post-transfer department d and a transfer at i are considered
promising shifts.
Then, to further reduce the set of promising shifts, we narrow down our definition twice. First, we make Proposition 4.3, in which we state that a transfer shift is only promising if it transfers to (from) a department for which requirements rise (fall) and from (to) a department where requirements do not rise (fall):
Proposition 4.3 If at a time bucket i2 I and for a department pair {d, d0} 2 D
2, we have
ridpre< rpostid and ridpre0 r
post id0 ,
then transfer shifts with pre-transfer department d0, post-transfer department d and a transfer at i are considered
promising shifts. Conversely, if we have
ridpre rpostid and ridpre0 < rpostid0 ,
Secondly, we state in Proposition 4.4 that a transfer shift is promising if it transfers while the requirements drop in the pre-transfer department and rise in the post-transfer department:
Proposition 4.4 If at a time bucket i2 I and for a department pair {d, d0} 2 D
2, we have
ridpre< rpostid and ridpre0 > r
post id0 ,
then transfer shifts with pre-transfer department d0, post-transfer department d and a transfer at i are considered
promising shifts.
4.2.2 Mathematical formulation
In the previous section, we introduced conditions under which a transfer shift is considered ‘promising’. This section uses these conditions to formulate 3 types of requirement switches. First, we start by defining our time buckets i more explicitly; we consider a time window of n time buckets, i.e. I ={i : i = 1, 2, . . . , n}.
The requirements switch types (rs1, rs2, rs3) are respectively based on Proposition 4.2, 4.3, 4.4. More specifically, for any pair of departments {d, d0} 2 D
2and all i = 1 + f, . . . , n f , we define parameters 1
idd0 =
(
1, if ridpost rpreid < ridpost0 r
pre id0 , 0, otherwise, (5) 2 idd0 = 8 > < > :
1, if rpostid < rpreid , ridpost0 r
pre id0, or ridpost ridpre, r post id0 > r pre id0, 0, otherwise, (6) 3 idd0 = ( 1, if ridpost< r pre id , r post id0 > ridpre0, 0, otherwise, (7)
that indicate if a requirement switch of type 1, 2 or 3 from department d to d0 occurs at time bucket i. Then, we define
k
dd0 ={i : iddk 0 = 1}, (8)
as the set of time buckets in which a requirement switch of type k from department d to d0occurs, for k = 1, 2, 3.
Note that the switch types are defined such that 3
dd0 ✓ 2dd0✓ 1dd0. These parameters indicate which transfer
shifts are included in our model. For a type k and each department pair{d, d0} 2 D
2 we then run the following
algorithm:
Algorithm 1: Demand-based TSSA ˜ Str = ; and i = f; while i < n f do i = i + 1; if k idd0 = 1 then
Add all transfer shifts in Str with pre-transfer department d, transfer at i, and post-transfer
department d0 to ˜Str;
end end
Note that rs3 is more strict in selecting transfer shifts than rs2, and rs2 is more strict than rs1. In fact, if we define the set of selected transfer shifts of type k as ˜Str
k , we have ˜S3tr✓ ˜S2tr✓ ˜S1tr.
4.3
Result TSSA
The following shift selection algorithm is more similar to the one proposed by Attia et al. (2019). It is based on the argument that, while transfer shifts might primarily add value to problems with negatively correlated requirements, this still might not be necessary if internal shifts can perfectly cover these requirements. In the supermarket example in Section 1.1, the single-department solution has a clear overcoverage interval for both departments. The optimal transfer shifts, then, transfer to the department at the start of the interval and from the department at the end.
section, we first give a brief intuition on how the under- and overcoverage intervals are translated into promising shift selection criteria. Similar to Section 4.2.1, this is done through the use of propositions that are numerically tested later on in Section 7.1. Then, we mathematically define the RTSSA.
4.3.1 Transfer shift value analysis
Consider a problem similar to the one in Section 4.2.1. Suppose we have a single-department solution with un-dercoverage Y and overcoverage Y+. If the optimal single-department solution has no under- and overcoverage
in any department, we can state that there are no transfer shifts that can improve the current solution; require-ments are perfectly covered, so there are no labour costs to save without incurring undercoverage. Furthermore, assuming non-negative transfer costs, the objective value cannot be decreased by replacing two internal shift by a transfer one. We use this statement to develop a proposition on the added value of transfer shift on a specific time window and department combination:
Proposition 4.5 If for a pair of departments{d, d0} 2 D
2, on a continuous interval I0 ✓ I, we have
Yid+= Yid = Yid+0 = Yid0 = 0 8 i 2 I0,
then transfer shits with a transfer between d and d0 on a bucket i2 I0 are not considered promising shifts.
Now suppose there is undercoverage in an interval [i1, i2] on department d. This implies that internal shifts
cannot cover this interval and a transfer shift might improve the solution. However, it also implies that in the interval before i1 and after i2, requirements can already be covered by internal shifts, hence we do not
necessarily require transfer shifts that cover department d these periods. Based on this, we derive that we consider a transfer shift ‘promising’ if it transfers to d at the start of the undercoverage interval or to another department afterwards.
Conversely, if there is an interval of overcoverage, that implies that internal shifts couldn’t cover neighbouring intervals without having to extend to this one. Using similar reasoning, we derive that we also consider a transfer shift ‘promising’ if it transfers from d at the start of the overcoverage interval or to d afterwards. These statements are summarized in Proposition 4.6 and 4.7:
Proposition 4.6 If for a department d2 D in an interval [i1, i2]⇢ I, we have
Yid > 0 8 i 2 [i1, i2] and Yi1 1,d= Yi2+1,d= 0,
then all transfer shifts with 1. pre-transfer department d and transfer at i2 or 2. post-transfer department d
and transfer at i1 are considered promising shifts.
Proposition 4.7 If for a department d2 D in an interval [i1, i2]⇢ I, we have
Yid+> 0 8 i 2 [i1, i2] and Yi+1 1,d= Yi+2+1,d= 0,
then all transfer shifts with 1. pre-transfer department d and transfer at i1 or 2. post-transfer department d
and transfer at i2 are considered promising shifts.
We use these propositions to formally define the RTSSA in the next section. 4.3.2 Mathematical formulation
We introduce three parameters to influence the number of promising shifts that can be taken into account. First, we introduce parameter ↵, that provides a bu↵er around a transfer bucket used for defining promising shifts. For example, if we have a period of undercoverage at department d starting at bucket i, we add all transfer shifts with post-transfer department d and a transfer in [i ↵, i + ↵] to the set of promising shifts. Secondly, we introduce parameters and that respectively provide a minimum length of the under- and overcoverage interval for it to be taken into consideration. For example, if = 2, we do not consider a single time bucket of overcoverage when adding promising shifts to model. Before running the algorithm, we solve the single-department model, and store the resulting under- and overcoverage values ¯Y , ¯Y+. Then, for each
Algorithm 2: Result-based TSSA ˜ Str =; and i = 0; while i < n do i = i + 1; if ¯Yid > 0 then j = i; while ¯Yjd> 0 and j < n do j = j + 1; end if j i then k = - ↵; while k ↵ do
Add all shifts in Str with a transfer at i + k and post-transfer department d to ˜Str;
Add all shifts in Str with pre-transfer department d and a transfer at j + k to ˜Str; k = k + 1; end end i = j; end if ¯Yid+> 0 then j = i; while ¯Yjd+> 0 and j < n do j = j + 1; end if j i then k = - ↵; while k ↵ do
Add all shifts in Str with pre-transfer department d and a transfer at i + k to ˜Str;
Add all shifts in Str with a transfer at j + k and post-transfer department d to ˜Str;
k = k + 1; end end i = j 1; end end
5
Tabu Search
Although the TSSAs should theoretically decrease the number of variables of the set-covering model, there still is a large potential increase in the number of variables when the number of departments increases. This section tries to deal with this issue by observing that, since we only assume a maximum of two departments within each shift, we can consider a sub-problem involving only two departments as more or less stand-alone. This would allow us to decompose a very large multi-department problem into multiple two-department problems, and iteratively solve these to approach the optimal solution.
To do this, we developed a hybrid heuristic based on Dahmen and Rekik (2015) that combines an adaptation of the Tabu Search (TS) meta-heuristic (Glover and McMillan, 1986) with an exact Brand-and-Cut (B&C) procedure. In this section, we first provide a global overview of all TS ingredients and how the B&C is applied. Then, we further elaborate on each ingredient and provide the full mathematical definition in separate subsections.
5.1
Overview of the hybridization scheme
The TS algorithm begins with solving the single-department scheduling problem by setting Str=; and applying
the B&C procedure. The resulting optimal solution is the initial solution X0 and the initial best solution ¯X
of the heuristic. We begin each iteration t by selecting a department pair{d, d0} for which there is likely an
improvement possible by adding transfer shifts. Then, we define the neighborhood around the previous solution Xt 1as the set of possible internal shifts in department d or d0 and possible transfer shifts between d and d0. Note that this neighborhood is then a restricted version of the MDSS model, where we have a subset S0 ⇢ S
of possible shifts that we would like to use to re-optimize the schedule. We then apply the B&C procedure to explore the neighborhood and find the next best solution Xt. If the solution of Xt is better than ¯X, we
set ¯X = Xt. After a neighborhood is explored, the department pair
{d, d0} is added to the Tabu list; for
the next Nl iterations, this move is excluded from the search. This model uses two additional TS techniques:
intensification and diversification. Intensification is used to further dive into ‘good’ solutions to try to approach a local optimum. In this case, after observing Ni iterations without improvement of the objective function, we
select a move and approach it with an increased neighborhood of transfer shifts. Diversification, on the other hand, tries to escape local optima after Ndnon-improving iterations by visiting solutions that were not explored
much before. We use diversification in this model by splitting transfer shifts into separate internal shifts. The algorithm ends after either a pre-specified time Tmax, or after Nm diversification steps. Algorithm 4 provides
a sketch of the steps followed by the heuristic.
5.2
Initialization
Before starting the iterative method, we first construct a feasible solution by solving the single-department scheduling problem: we set Str = ; and solve the resulting MIP with our B&C procedure. If there is no
under- or overcoverage in the solution X0, then we stop the algorithm; as discussed before, there is no further
improvement possible by including transfer shifts. If there is under- or overcoverage, we define the TS parameters Nl, Ni, Nd, Nm, RTSSA parameters ↵, , and DTSSA parameters f and rs type k. Furthermore, we set
¯
X = X0, construct an empty Tabu List and start the algorithm.
5.3
Move selection technique
In this section, we discuss the technique used to identify which move to take at iteration t, i.e. which department pair is considered for re-optimization. To make this decision, we identify two characteristics of a department pair that indicate if solving a corresponding sub problem might improve the solution. First, to assess if demand is sufficiently negatively correlated, we consider the number of requirement switches of pre-specified type k between both departments, denoted| k
dd0|. Secondly, consider the previous solution Xt 1. Recall that we have
the corresponding under- and overcoverage (Yid+, Yid) for each department d2 D and time bucket i 2 I. Based on this, we define the total under- and overcoverage costs for a department pair{d, d0} as
Cdduo0 = X i2I c (Yid + Yid0) + c+(Yid++ Y + id0). (9)
P (move ={d, d0}) = | kdd0| 2 X {d,d0}2D2\T | k dd0| + C uo dd0 2 X {d,d0}2D2\T Cdduo0 . (10)
Finally, we sample a move from this distribution and use this move for the following iteration.
5.4
Main operator
After choosing a department pair, our goal is to construct a promising neighborhood of possible shifts ˜St dd0.
First, we add all internal shifts of both departments to the neighborhood. Then, we create a restricted set of transfer shifts ˜Str by using the RTSSA for department d and d0, and only add transfer shifts between these
departments to ˜St
dd0. Finally, we add all transfer shifts between d and d0 that were in the previous best solution.
That is: we add all transfer shifts s2 Strfor which we have Xt 1
s > 0 in the previous solution. This set of shifts
allows us to create a restricted MIP to solve the sub-problem. More specifically, we construct the restricted model by solving the original model with S = Sin[ Str and adding the constraint
Xs= Xst 1 8 s 2 S \ ˜Stdd0, (11)
where we consider the previous solution as parameter Xt 1
s for each s 2 S. After solving the model and
obtaining a new solution Xt, we check if this solution is better than ¯X. If so, we set ¯X = Xt. Finally, we add
department pair{d, d0} to the Tabu list and continue the algorithm.
5.5
Intensification
If we do not improve our solution for Niiterations, we start our intensification step. Recall that intensification is
a method to further explore good solutions to approach a local optimum. To do so, we first select a move based on our move selection algorithm and then explore the complete neighborhood of previously selected transfer shifts. That is, we set
˜ Sddt 0 = t 1[ p=1 ˜ Sddp 0, (12)
and solve the restricted model as described for the main operator.
5.6
Diversification
If the solution does not improve for Nd iterations, we start the diversification step. In this step, we randomly
remove a transfer shift between each pair of departments and replace it by two internal shifts. Algorithm 3: Diversification step
Xt= Xt 1for each {d, d0} in D
2 do
s = a random transfer shift between d and d0 such that Xt s> 0;
if no s exists then Go to next {d, d0};
end
s1= a shift in Sinwith department equal to the pre-transfer department of s, such that we have a
minimum di↵erence between the start of s1 and s and the end of s1and the transfer of s;
s2= a shift in Sinwith department equal to the post-transfer department of s, such that we have a
minimum di↵erence between the begin of s2 and the transfer of s and between the end of s2 and
the end of s; Xt s1+ = 1, X t s2+ = 1, X t s = 1; end
5.7
Finalization
After a pre-defined time limit is reaches, or Nm diversification steps have been reached, we stop the TS. The
Algorithm 4: Global overview Tabu search heuristic. t = 0;
¯
X = X0 = Initialization;
while time < Tmax do
t+ = 1;
if non-improving count == Ni then
move = Move selection technique(Tabu list); Xt= Intensification(move, Xt 1) ;
add move to Tabu list;
else if non-improving count == Nd then diversification count += 1;
if diversification count Nmthen
Xt= Diversification(Xt 1); end else break end else
move = Move selection technique(Tabu list); Xt= Main operator(move, Xt 1);
add move to Tabu list; end
if Xtbetter than ¯X then
non-improving count = 0; ¯
X = Xt;
else
non-improving count += 1; if length of Tabu list == Nlthen
Delete oldest move of Tabu list; end
end
6
Experiments
In this section, we assess the experimental setups and results to determine the quality of the proposed solution approaches. To e↵ectively conduct these experiments, we generated multiple instances of a 14-hour work day with a varying number of departments and time buckets, and requirements that can fluctuate each bucket. For the remainder of this section, we refer to an instance as ‘m-|i|’, where m refers to the number of departments, and|i| refers to the length of the time buckets in minutes. Note that moving from |i| = 60 to |i| = 30 means that the number of time buckets doubles. We consider 9 instances with all combinations of number of departments m = 3, 5, 10 and time bucket length |i| = 60, 30, 15. For each instance we set overcoverage costs c+ = 100,
undercoverage costs c = 1000, transfer costs ct= 1 and fixed hourly labour costs of 10.
Recall that an important goal of this thesis is to propose a solution approach that can deal with multi-department scheduling problems that are complex due to flexible labour rules and volatility in labour require-ments. For our instances, we defined a set of labour rules with a high degree of labour flexibility, which is summarized in Table 2. To illustrate the size of the instances we consider, we compare our approach to Attia et al. (2019). For their decomposition heuristic, they consider instances that have 120 di↵erent possible internal shifts on one department. Furthermore, they consider requirement fluctuations on an hourly level. Our instances with |i| = 15, on the other hand, have around 5000 possible internal shifts per department and requirements that can change each 15 minutes.
To assess the e↵ectiveness of a heuristic, we define several performance indicators (PIs). The PIs are based on solution quality, i.e. the objective value, and the computation time. Note that computation time may refer to the full time necessary to construct and solve the model, or only to the solution time of the solver. Since all our approaches have been implemented in an existing commercial framework, we suspect that the model construction duration might be substantial compared to that of more time-efficient techniques. In order to primarily focus on the e↵ectiveness of the TSSAs, we have chosen to not take the model construction time into account for our analysis. For the DTSSA, only the model solving time is taken into account. Additionally, for the RTSSA, solving time of both the single- and the multi-department model is considered. Finally, since the e↵ect of a Tabu Search algorithm is more dependent on algorithm handling times, we take all handling times into account, except for the model construction duration in the main operator and intensification step.
Note that we have an approximate upper and lower bound for our minimization problem in terms of objective value and time. The solution to the single-department problem, i.e. setting Str=;, can be considered an upper
bound in terms of objective value and lower bound in terms of solution time. On the other hand, the solution of the full multi-department model can be considered a lower bound in terms of objective value and upper bound in solution time. We use these definitions to define two PIs that display the e↵ectiveness of a heuristic. First, we denote the objective value and computation time of a solution approach x as OVxand CP Ux, respectively. We
consider the following solution approaches: single-department (sd), multi-department (md) and a heuristic (h) approach. The first PI, OV, shows the decrease in objective value in percentages by the heuristic compared to
the multi-department extension. The second, CP U, shows the percentage of computation time saved compared
to the full multi-department extension. The PIs for a heuristic approach h are computed by
OV h = OVsd OVh OVsd OVmd · 100%, (13) CP U h = CP Umd CP Uh CP Umd CP Usd · 100%. (14)
To assert that the PIs are representative of the quality of the method rather than the instance, we use randomly generated requirements for each instance. First, we generated a base instance for each combination of m and |i| with hourly fluctuations in requirements. Then, for each base instance, we generated n|i| instances with
requirements r0id based on the base requirements rid as follows:
P (r0id= r) = 8 > > > > > > < > > > > > > : 0.125, r = min(r0 id 2, 0), 0.125, r = min(r0 id 1, 0), 0.5, r = r0 id, 0.125, r = r0 id+ 1, 0.125, r = r0id+ 2, 8 i 2 I, d 2 D.
instance. Finally, to di↵erentiate between near-optimal (99.1% - 99.9%) and optimal (100%) solutions, we report the truncated PIs.
In the remainder of this section, we test our solution approaches on the generated instances for various parameter settings and assess the results by means of the PIs. All occasions have been tested using a commercial CBC solver. First, we examine the performance of all three solution approaches for various parameter settings. Then, we use the result to compare the e↵ectiveness of all solution approaches with optimal parameter settings in a general overview.
Table 2: Labour rules used in the generated instances. The values are in buckets. Labour rule |i| = 60 |i| = 30 |i| = 15
Minimum shift duration 6 12 24 Maximum shift duration 8 16 32 Minimum department duration 3 6 12 Break duration 1 2 4 Earliest break end 2 4 8 Latest break end 2 4 8
6.1
Parameter settings
In this section, we assess the parameter settings for each heuristic. To analyze which settings work best, we determined the performance of each setting and compared the PIs. To make sure that the tests were representative for all instance types, we used the middle level of bucket size|i| = 30 and number of departments m = 5.
6.1.1 DTSSA parameters
In Section 4.2 we identified 3 requirement switch types (rs1, rs2, rs3). Note that since rs3 is more strict in selecting shifts than rs2, and rs2 is more strict than rs1, we expect the average computation time to be better and the average solution quality to get worse when we descend in rs type. Furthermore, recall that we identified a fluctuation parameter f , which indicates how many buckets are taken into account to identify a fluctuation in requirements. For this experiment, we consider the values f = (1, 2, 4, 6). The results are summarized in Table 3.
Table 3: Results of the DTSSA for various parameter settings ( OV, CP U).
Switch type f = 1 f = 2 f = 4 f = 6 rs1 (99, 56) (100, 41) (100, 47) (100, 83) rs2 (99, 77) (99, 77) (99, 77) (99, 83) rs3 (86, 98) (99, 93) (99, 81) (99, 91)
Consider the top left entry of the table. Based on this, we can conclude that using the DTSSA with the rs1 indicator and fluctuation parameter f = 1, we approach 99% of the objective value and save 56% of the computation time compared to the full multi-department model. To select the best settings, we look at which attains the highest values for both PIs.
It can be seen that rs3 already filters out a set of transfer shift that is sufficient to approach optimality. Furthermore, fluctuations on an hourly level, i.e. f = 2, seem to be the best indicators for this requirement switch identification type. This combination is used for the remainder of this section.
6.1.2 RTSSA parameters
Table 4: Results of the RTSSA for various parameter settings ( OV, CP U). ( , ) ↵ = 0 ↵ = 1 ↵ = 2 ↵ = 4 (1,1) (99, 55) (100, 3) (100, 0) (100, 0) (1,2) (99, 84) (100, 57) (100, 10) (100, 1) (2,2) (94, 92) (99, 73) (100, 45) (100, 5) (2,4) (70, 94) (87, 88) (100, 77) (100, 70) (4,4) (68, 97) (86, 94) (90, 88) (100, 71)
It can be seen that in our example, ( , ) = (1, 2) is the most e↵ective when combining it with ↵ = 0. An interesting observation is that the ↵ parameter has a large influence on model performance. Hence, a higher ↵ should be chosen in scheduling contexts in which optimality is very important. For the remainder of this section, we set (↵, , ) = (0, 1, 2) for the RTSSA.
6.1.3 Tabu Search parameters
The TS uses a combination of DTSSA, RTSSA and TS-specific parameters. For the move selection procedure, we use fluctuation parameter f = 2 and rs 3, which was proven to be an efficient indication for negatively correlated requirements. Furthermore, for the main operator and the intensification step, to efficiently explore a neighborhood of shifts between two departments, we use settings ↵ = 0, = 1, = 2. Finally, we defined TS-specific parameters (Nl, Ni, Nd, Nm). Since we expect the move selection method to prioritize a select
group of department pairs, we would like to set the Tabu list length Nl to 20%, 50% or even 100% of the
number of department pairs |D2|. Furthermore, we would like to start the intensification round after half of
the Tabu list length in iterations has been reached without improvement. After two unsuccessful intensification rounds, we start the diversification step. Finally, to identify the influence of diversifying, we use 0, 2, 5 or 20 diversification steps. For the ‘5-30’ instance, this leads to the following experiments and corresponding results:
Table 5: Results of the TS for various parameter settings ( OV, CP U).
(Nl, Ni, Nd) Nm= 0 Nm= 2 Nm= 5 Nm= 10
(2, 1, 3) (51, 91) (83, 52) (85, 20) (90, -34) (5, 3, 7) (86, 83) (88, 15) (90, -76) (90, -122) (10, 5, 11) (89, 34) (90, -12) (90, -115) (91, -235)
Note that a negative value of CP U implies that the heuristic is slower than the full multi-department model.
What can be seen is that although diversification steps help to come somewhat closer to optimality, the additional time necessary to get there does not outweigh this benefit. In Section 7.3, we further look into this phenomenon. For the remainder of this section, we set Nm= 0. Note that a main benefit of using the TS algorithm in practise
is that an improved solution can be reached within a strict time limit. Hence, we make a distinction between a quick solution, where we set Nl
⇡ 0.2|D2|, and a good solution, where Nl⇡ 0.5|D2|.
6.2
General model performance
Table 6: Results of all solution approaches for instance ‘30-5’.
Solution approach |S| (x1000) OV CP U (s) OV (%) CP U (%)
Single-department 8 220584 5 0 100 DTSSA 28 108946 41 99.9 93 RTSSA 101 108946 87 99.9 84 TSSA Attia et. al. 163 108835 307 100 30 20% TSSA 55 150562 25 56 95 TS, quick - 153000 53 51 91 TS, good - 120009 91 86 83 Multi-department 305 108835 515 100 0
It can be seen that the TSSAs successfully reduce the number of decision variables. An interesting result is the di↵erence in variables between the DTSSA and the RTSSA: the former requires significantly fewer variables than the latter to reach the same objective value. Moreover, the TSSA by Attia et al. (2019) indeed selects much more shifts and only improves our solution by a very small fraction. Finally, an interesting result is that the random TSSA scores relatively well: with only 20% of the transfer shifts, a 56% improvement of the single-department solution can be attained.
To compare the performance of the approaches for various applications, an overview of all heuristics for all generated instances is provided in Table 7.
Table 7: Results of the heuristic on all instances ( OV, CP U)
m |i| Heuristic 3 60 3 30 3 15 5 60 5 30 5 15 10 60 10 30 10 15 DTSSA (83, 88) (99, 93) (98, 95) (77, 96) (99, 95) (98, 96) (88, 98) (94, 90) (98, 87) RTSSA (80, 76) (99, 84) (90, 85) (94, 95) (99, 81) (93, 79) (92, 89) (95, 84) (90, 65) 20% TSSA (56, 90) (63, 95) (54, 95) (43, 95) (56, 95) (56, 96) (65, 90) (56, 95) (75, 84) TS, quick (99, -112) (90, 55) (90, 34) (73, 83) (51, 91) (53, 87) (64, 88) (60, 95) (55, 92) TS, good (99, -210) (95, 41) (92, 15) (89, 50) (91, 85) (85, 33) (78, 89) (83, 41) (84, 23)
First of all, when the time bucket size lowers, both TSSAs seem to be able to handle the resulting increase in complexity: both PIs remain relatively high when time buckets are 30 or 15 minutes. However, compared to the DTSSA, the RTSSA struggles with solution quality and decreasing computation time when time buckets of 15 minutes are considered. In fact, the DTSSA strictly outperforms the RTSSA on almost all instances. Another interesting result is the performance of the random 20% TSSA: it is able to reach a substantial improvement of the single-department solution in a short time. In Section 7.2, we provide an insight on the performance of the TSSAs.
As expected, the Tabu Search methods do not outperform the full Multi-Department model when only 3 departments are considered. When the number of departments increases, the TS methods reach a solution relatively quickly. However, it can be seen that the solution quality gets worse when more departments are considered. Even the parameter setting that was designed to reach a good solution does not come close to optimality in these cases. In the next section, we further elaborate on this observation.
7
Experimental insights
In this section, we further dive into the results from the previous sections and attempt to provide an insight on why some solutions did or did not work. This is done through the use of graphs and numerical experiments. First, we numerically test the propositions made in Section 4 regarding the situations in which transfer shifts are promising. Then, we analyze the performance of the TSSAs and show why some techniques work better than others. Finally, we provide an insight on why the Tabu Search method does not perform as well as the other heuristics in terms of solution quality.
7.1
TSSA propositions
In Section 4.2 and 4.3, several propositions were made regarding the situation in which a transfer shift is promising. In particular, Propositions 4.2, 4.3, 4.4, 4.6 and 4.7 are directly used to construct the TSSAs. To support these propositions, tests were run to determine if they provide a good indication of the added value of a transfer shift. More specifically, we tested how often a transfer shift is not considered promising according to a proposition and still improves the single-department solution. To do this, we generated instances with 3 departments and 8 time buckets, no breaks, SD = 6, SD+ = 8, DD = 1, and all cost parameters as
in Section 6. For each proposition, we generated 1000 instances for which no promising transfer shifts are identified. For example, for Proposition 4.2, instances were generated where no requirement switch of type 1 occurs. Then, we solved the full MDSS model for these instances and checked how often there are still transfer shifts are selected for the optimal schedule. The results are enlisted in Table 8.
Table 8: Number of instances in which an optimal transfer shift is not identified. Proposition Selection criterion Number of optimal transfer shifts
4.2 Requirement switch 1 0 4.3 Requirement switch 2 12 4.4 Requirement switch 3 28 4.6 Undercoverage intervals 9 4.7 Overcoverage intervals 12
Apart from Proposition 4.2, there always seems to be exceptions in which an optimal promising shift is not identified. This is as expected: as argued in Section 4, we cannot guarantee a perfect identification of promising shifts due to the complexity of the MDSS problem. However, we can see that, apart from these exceptions, the identification techniques of the propositions are e↵ective. For example, Proposition 4.4 states that transfer shifts are only promising at an instance where requirements drop in the pre-transfer department and rise in the post-transfer departments. From Table 8, it can be seen that if no such an occasion occurs, in only 28 out of 1000 cases there is a transfer shift that improves the solution. We conclude that the propositions in Section 4 provide good indications of the added value of a transfer shift to the solution of a MDSS problem.
7.2
TSSA performance and symmetry
Figure 3: The increase of the number of possible shifts due to the number of buckets (left) and departments (right).
First of all, we observe that, as theorized in Section 4.1, the number of shifts indeed increases exponentially in the number of time buckets and departments. Moreover, the DTSSA seems to handle this increase more efficiently, by selecting less than 10% of the number of variables, compared to around 20 30% of the RTSSA. This could explain the di↵erence in computation time between both methods.
However, that does not explain why a 20% random TSSA is significantly faster than the RTSSA while selecting approximately as many variables. This seems to indicate the underlying symmetry issue of the shift selection problems. Since there are so many (near-)optimal solutions, the solver cannot quickly di↵erentiate between good and bad solutions. As argued by Sherali and Smith (2001), this has a large e↵ect on solution time. Since the 20% random TSSA also deletes a large number of good shifts, the solver reaches the optimal restricted solution relatively quick. This also explains why this method still attains a relatively decent solution; even after deleting 80% of all transfer shifts, there are still a large number of good solutions present in the solution space.
This result illustrates the symmetry issue still present in large instance anonymous multi-department shift scheduling models. By allowing for a large degree of labour flexibility, the number of possible shifts becomes very large and so does our model symmetry. For example, there is no di↵erence between scheduling two subsequent shifts of 4 hours or one of 8 hours if both are allowed. To provide an insight on how the solution approaches deal with the degree of symmetry, we solved the ‘3-60’ instance 1000 times for each approach and reported the number of di↵erent optimal solutions. Note that for the RTSSA, one single-department solution was used to create the under- and overcoverage intervals. The results are summarized in Table 10:
Table 9: Number of optimal solutions after 1000 attempts. SD MD DTSSA RTSSA 20% TSSA
12 198 39 165 2
The results support our previous claims on symmetry. First of all, the full multi-department problem shows a large degree of symmetry with 198 optimal solutions after solving the model 1000 times. Secondly, the RTSSA indeed still has a large degree of symmetry compared to the DTSSA. Lastly, a random shift selection algorithm indeed seems to remove almost all symmetry from the solution space.
