Attaining stability in multi-skill workforce scheduling

Attaining stability in multi-skill workforce scheduling

Firat, M., & Hurkens, C. A. J. (2010). Attaining stability in multi-skill workforce scheduling. (BETA publicatie : working papers; Vol. 335). Technische Universiteit Eindhoven.

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Attaining stability in multi-skill workforce scheduling

Murat Firat, Cor Hurkens Beta Working Paper series 335

BETA publicatie WP 335 (working paper)

ISBN 978-90-386-2404-4 ISSN

NUR 982

Attaining stability in multi-skill workforce scheduling

Murat Fırat † Cor Hurkens ‡

Abstract

In this paper, we define a set inequalities that are satisfied by stable multi-skill workforce schedules. In our analysis, a schedule is said to be stable if it does not contain a blocking pair, extending the notion of blocking pair in the Marriage Model of Gale-Shapley. Skill efficiency is chosen as the criterion in the preference structure. The proposed algorithm either constructs a stable multi-skill workforce schedule or decides that no stable schedule exists.

Keywords: stability, marriage problem, multi-skill workforce schedules, stable assignments, instability, blocking pair, integer programming model.

1

Introduction

In this study, we define an set of inequalities that guarantee stability whenever they are satisfied by a multi-skill workforce schedule of technicians and jobs. The stability definition of Fırat et al. (2010) is slightly modified to make it easier and more realistic. Here (also in Fırat et al. (2010)) , the main goal in using the stability concept is to maintain the skill efficiency. The preference structure is defined in the way that the used experts in hard jobs are minimized in job-optimal stable schedules.

A schedule is said to be blocked if a technician-job pair not assigned can improve their individual preferences by being assigned. Fırat et al. (2010) proposed an Integer Linear Pro-gramming (ILP) Model to reassign technicians to jobs using the instability information of the current schedule. The main contribution of this paper is the proposed inequalities that char-acterize stable schedules without requiring the information of a schedule. Satisfaction of these inequalities is the certificate to reach stability and this certificate was posed as an open question by Fırat et al. (2010).

Stability is a property of an assignment satisfying that no two players, not paired to each other, can be better off by being paired. This concept was introduced by David Gale in the beginning of 1960’s. The milestone paper on stability was published by Gale and Shapley (1962) including the polynomial time "deferred acceptance algorithm". After a silence of more than

∗

This research is supported by France Telecom/TUE Research agreement No. ˙46145963.

†

Corresponding author: m.firat@tue.nl. Department of Mathematics and Computer Science, TU Eindhoven, P.O. Box 513, 5600 MB Eindhoven, Netherlands.

‡

c.a.j.hurkens@tue.nl. Department of Mathematics and Computer Science, TU Eindhoven, P.O. Box 513, 5600 MB Eindhoven, Netherlands.

20 years, Gale and Sotomayor (1985) studied the main properties of stable matchings. The authors also showed that stable matchings, if any exist, can be found in polynomial time when preference lists of players may be incomplete. Irving et al. (1987) used graph theoretic methods to find stable marriages in polynomial time. Many researchers studied stability using linear programming and important results were obtained. For example, Vande Vate (1989) showed the integrality of stable matching polytope and Baïou and Balinski (2000) proposed a polynomial time separation algorithm for stable admissions.

The multi-skill workforce scheduling problem is a recently emerging field in scheduling. The problem belongs to the class of multi-skill project scheduling problems (Fırat and Hurkens 2010). Recently, many researchers have worked on several versions of this complex scheduling problem using different techniques like Branch and Bound (Bellenguez and Neron 2007), meta-heuristic methods (Gutjahr et al. 2008, Cordeau et al. 2009), mixed integer linear programming (Hurkens 2009, Fırat and Hurkens 2010), constraint programming (Li and Womer 2009), and genetic algorithms (Valls et al. 2009).

It is not easy to compare different approaches in multi-skill workforce scheduling due to the lack of a benchmark problem set. In the computational challenge ROADEF 2007, France Telecom provided a set of problem instances of a version of multi-skill workforce scheduling (Dutot et al. 2006). This challenge was an opportunity for researchers to work on the same problem and to compare the performance of their solution approaches. (See, for example, Fırat and Hurkens (2010) for more details).

In our multi-skill workforce context, technicians’ expertise in specialization fields are ex-pressed by hierarchical skill levels. Throughout the paper the term skill domain refers to a specialization field. In our scheduling problem, each job requires that the assigned team of technicians has collective capabilities that are above a certain threshold. An instance of our problem is one day of a schedule that is constructed by the combinatorial algorithm of (Fırat and Hurkens 2010). A day schedule includes a group of technicians to perform a certain number of jobs in parallel.

The paper is organized as follows. In section 2 we give the basic definitions and the preference structure of technicians and jobs in the multi-workforce scheduling context. In section 3 the set of linear inequalities characterizing stability is explained. Conclusions and future research directions are discussed in section 4.

2

Basic Definitions

This section explains basic terminology of the multi-skill workforce scheduling.

2.1 Skills

Let D be the set of skill domains. The degree of experience or expertise in a skill domain is interpreted by an hierarchical level. An expert possesses the highest level and a beginner qualifies as the lowest. Let L denote the set of hierarchical skill levels. The skill (l, d) is said to be at level l and belongs to domain d. Skills are specified by matrices in RL×D_.

2.2 Technicians

In the schedules under consideration we are given a set T of technicians. Let t ∈ T and the skills of technician t is specified by St∈ RL×D. Let the skill level of technician t be l ∈ L in skill

domain d. Then S_t(l0,d)= 1 for l0 ≤ l and S_t(l0,d)= 0 otherwise. If we are given St, then the skill

level of t in skill domain d is found by max h

{0} ∪ {l ∈ L|St(l,d) > 0}

i .

Skill value of technician t, denoted by _t, is found by aggregating the skills in all domains at all levels with corresponding weights. If we let W ∈ RL×D _{be the skill weight matrix, then skill}

value of t is calculated by t= hW, Sti.

Let T0⊆ T be a team of technicians. The skill of T0 _{is defined as the skill sum of individual}

technicians in the team and is denoted by ST0 =P

t∈T0S_t. Similarly, _T0 denotes the skill value

of T0 and it is found by _T0 =P

t∈T0t= hW, ST0i.

In a schedule, the job to which technician t is assigned is denoted by J (t) and T (j) denotes the team assigned to job j. It is clear that if in a schedule J (t) = j, then t ∈ T (j).

2.3 Jobs

In the multi-skill workforce scheduling problem instances, a set J = {j₁, j2, . . . } of jobs is

given. Jobs require certain skill qualifications and are processed in parallel. The skills required to perform a task are expressed by a skill requirement matrix. Let task j ∈ J have skill requirement matrix RQ_{j ∈ R}L×D_{. Requirements in RQ}

j are cumulative in the sense that any requirement

in a higher level is carried to lower levels. Therefore we have:

l0 ≤ l ⇒ RQ(l_j0,d) ≥ RQ(l,d)_j , ∀j ∈ J, l, l0 _{∈ L, d ∈ D}

Let RQ∗_{j ∈ R}L×D _{denote the non-cumulative or explicit skill requirement of job j. RQ}∗ j is

obtained from RQj as follows:

RQ∗(l,d)_j = (

RQ(l,d)_j if l = |L|, RQ(l,d)_j − RQ(l+1,d)_{j if 0 < l < |L|.}

For example, let |D| = 4 and |L| = 3, and let the requirement matrix for job j be as follows:

RQj =   1 2 0 0 1 1 0 0 1 0 0 0  =⇒ RQ∗_j =   0 1 0 0 0 1 0 0 1 0 0 0  

In the above example, RQ(1,2)_j = 2 tells us that there must be at least two technicians contributing to the team skill at level 1 in domain 2, namely S_{T (j)}(1,2). One of them must have a skill of at least level 2 (since RQ∗(2,2)_j = 1) and one of at least level 1 (since RQ∗(1,2)_j = 1).

Let T (j) ⊆ T denote the team of technicians assigned to job j. Team T (j) is said to be feasible if it meets the job j’s skill requirements RQ_j and this is expressed by S_{T (j)}≥ RQ_j.

2.4 Contributions to Jobs

As in the study of Fırat et al. (2010), we make the assumption simultaneous skill use which states that technicians can use their skills in more than one skill domain simultaneously while performing a job. The contribution level of technician t to a job j in skill domain d, is defined as the maximum level that is both explicitly required by the job and reached by the technician. It is denoted by CONd_(t,j) and found by the following:

CONd_(t,j)= maxh{0} ∪ {l ∈ L|0 < RQ∗(l,d)j and 0 < S (l,d) t }

i

. (1)

In skill domain d, the highest contribution level that technician t can achieve is found by CONd_{(t,J )} = maxj∈J{CONd_(t,j)}. Each technician orders skill domains lexicographically with

respect to CON_{(t,J )}. Ties due to the same maximum contributions are broken by choosing the domain in which there is less competition and further ties are broken by choosing the domain with minimum index. The skill domain with highest ranking is called favorite domain and it is denoted by d∗_t.

Definition 1 Let t ∈ T and j ∈ J . If, for the skill (l, d), we have RQ∗(l,d)_j > 0 and S_t(l,d) = 0, then (l, d) is called exclusive skill required by j with respect to t. The set of exclusive skills of j with respect to t is denoted by ∆(j, t) ⊂ L × D.

Definition 2 Let µ be an assignment of the multi-skill workforce scheduling (T, J ). In the assignment µ, the skill (l, d) for which we have RQ(l,d)_j = S_{T (j)}(l,d) and RQ(l,d)_j > 0 is called critical and the contributions in this skill are also called critical. Cµ(j) ⊂ L × D denotes the set of

critical skills.

We should note that critical skills and critical contributions depend on the current assign-ment. The following statement is the key observation for the set of inequalities defining stable schedules.

Proposition 3 In assignment µ, let t0 ∈ T (j) and t 6∈ T (j). Technician t can replace t0 _{in T (j)}

if and only if S_t(l,d)0 ≤ S

(l,d)

t for all (l, d) ∈ Cµ(j).

Proof. Firstly, suppose that t can replace t0 in T (j). Let us show that S_t(l,d)0 ≤ S

(l,d)

t is true

for all (l, d) ∈ C_µ(j). The feasibility of the new team implies RQ(l,d)_j − S_{T (j)}(l,d)≤ −S_t(l,d)0 + S

(l,d) t .

Moreover for the critical skills, we know RQ(l,d)_j = S_{T (j)}(l,d). Thus we have S(l,d)_t0 ≤ S

(l,d) t .

Secondly, suppose that S_t(l,d)0 ≤ S

(l,d)

t or 0 ≤ −S (l,d) t0 + S

(l,d)

t for all (l, d) in Cµ(j). Let us

show that the replacement t0by t results in another feasible team. If we consider the critical skill (l, d), we have RQ(l,d)_j = S_{T (j)}(l,d) and hence RQ(l,d)_j ≤ −S_t(l,d)0 + S

(l,d) t + S

(l,d)

T (j) implies the feasibility

of replacing t0 by t. If we consider the uncritical skill (l, d), we have RQ(l,d)_j ≤ S_{T (j)}(l,d) − 1. Considering that −1 ≤ −S_t(l,d)0 + S

(l,d)

t for any skill, we see that RQ (l,d) j ≤ S (l,d) T (j)− S (l,d) t0 + S (l,d) t is

true and uncritical skills are also satisfied. The replacement of t0 by t results in another feasible team._

Note that the replacement in Proposition (3) does not include any comparison between skills of technicians. Therefore incomparable technicians can replace each other depending on the critical contributions. In the next section we show the relation between technician replacement and blocking pairs.

2.5 Preference Structure

We follow the definitions of Fırat et al. (2010). In a matching, two players not paired form a blocking pair if both prefer one another to their currently assigned partners, then the matching is said to be unstable. This is the same definition of a blocking pair in the Marriage Model of Gale-Shapely.

The preference structure in our multi-skill workforce context is defined in a way that skill efficiency is taken care of from both technicians’ and jobs’ point of view. A technician prefers working at the highest possible level in his favorite domain and a job prefers a less overqualified technician group more.

In the multi-skill workforce schedules, it is not possible for jobs to have an ordering of pref-erence, since they need a team rather than an individual. Therefore for a job j, the technicians, not in T (j), are divided into two groups; the ones liked and the ones not liked.

Preference criterion of technicians: Technician t likes job j if and only if CONd∗t

(t,J (t))< CON d∗

t

(t,j) where J (t) is his current job.

Preference criterion of jobs:

Job j likes technician t if and only if, j decreases the total skill value of its team by employing t and releasing one technician. Let the released technician be t0, then we know S_t(l,d)0 ≤ S

(l,d) t for

all (l, d) ∈ Cµ(j) by Proposition (3) and t< t0.

It is important to note that the above preference criterion is different from the one of Fırat et al. (2010). In polynomial time, it is not possible to determine whether a job likes a technician or not, if we use the preference criterion of Fırat et al. (2010). However the above preference criterion of likeness can be checked in polynomial time.

In the classical Gale-Shapley stability if two players from different sets like each other more than their current partners, then they block the current matching. We adapt this idea of blocking pairs to our scheduling context. The pair (t, j) is said to block a schedule if and only if t and j like each other.

The main idea of our contribution. In a schedule, if a technician, say t, prefers a job, say j, to his current job J (t), then in T (j) there must be no technician who can be replaced by t and has higher skill value.

In the next section, we define several properties of multi-skill workforce assignments and we will define the constraints that must be satisfied to attain stability in a schedule.

3

Stability in multi-skill schedules

3.1 Some recent results

Following the terminology of Fırat et al. (2010), nD-nL-nT denotes an instance of multi-skill workforce scheduling. The letters D, L, and T stand for skill domains, skill levels, and team

sizes respectively. If any of these instance property is fixed to certain value, then n is replaced by that value. For example, we give the following theorem for the special case 1D-nL-nT in which the number of skill domains is fixed to one.

Theorem 4 (Fırat et al. (2010)) Stable schedules always exist for feasible 1D-nL-nT instances and they can be constructed in polynomial time.

In the proof of the above theorem, the authors reduced the 1D-nL-nT to the University Admissions Problem. Another result has been obtained on the complexity of finding feasible schedules in nD-1L-2T where jobs have teams of size 2 and technicians are either skilled or not skilled in domains.

Theorem 5 (Fırat et al. (2010)) Constructing a feasible schedule in nD-1L-2T is NP-Hard. The proof of the NP-Hardness is through the reduction of nD-1L-2T to the Three Dimen-sional Matching Problem. We refer to the authors for further details.

3.2 Stable schedules polytope

In this section we define a set of inequalities that must be satisfied for the stability. Let µ be an assignment in multi-skill workforce scheduling (T, J ). Moreover, let xµ_tj = 1 if j ∈ T (j). Otherwise xµ_tj= 0. If technician t likes job j, then y_tjµ = 1. Otherwise y_tjµ = 0. If the requirement of j in skill (l, d) is critical, then β_(j,(l,d))µ = 0. Otherwise β_(j,(l,d))µ = 1. Lastly, if technician t can replace technician t0 in the team T (j), then τ_(t,j,tµ 0₎= 0. Otherwise τ

µ

(t,j,t0₎= 1.

Let us drop the superscript of the assignment and let β_(j,l,d)0 and y0_tj denote the negations of the corresponding terms. Table 1 lists sets, indices, parameters used in the set inequalities defining stable schedules.

Table 1: Sets, indices, parameters and variables of the reassignment model

Sets J (j >, t) whenever t is assigned a job in J (j >, t), he will like j, J (j >, t) ⊂ J

∆(j, t) set of exclusive skills required by j with respect to technician t, ∆(j, t) ⊂ L × D Indices t, t0 technician index, t, t0∈ T

j, j0 job index, j, j0∈ J Parameters δt,t0 1 if t> t0, 0 otherwise

An index is replaced by a set, if the corresponding terms are summed over the elements in that set like x_tJ =P

j∈Jxtj, and S_t∆(j,t)0 β_(j,∆(j,t))0 = P (l,d)∈∆(j,t)S (l,d) t0 β_(j,(l,d))0 . xtJ ≤ 1, ∀t ∈ T (2) STxT j ≥ RQj, ∀j ∈ J (3)

The inequalities (2) and (3) ensure the feasibility of the schedule. Each technician can be assigned to at most one job and the group of technicians assigned to a job must satisfy the skill requirements.

xtJ (j>,t) ≤ ytj, ∀(t, j) ∈ T × J (4)

The technician t likes the job j if and only if he is assigned to one of the jobs that he likes less than j (inequalities (4)).

S_T(l,d)xT j− RQ (l,d) j ≥ β(j,(l,d)), ∀(l, d) : RQ (l,d) j > 0, ∀j ∈ J (5) S_T(l,d)xT j− RQ(l,d)_j ≤ |L||D|β(j,(l,d)), ∀(l, d) : RQ (l,d) j > 0, ∀j ∈ J (6)

Inequalities (5) and (6) determine the critical skills. For the skill (l, d), if we have S_T(l,d)xT j =

RQ(l,d)_j , then β_(j,(l,d)) must be 0 and skill (l, d) is critical. Otherwise β_(j,(l,d)) = 1.

S∆(j,t)_t0 β_(j,∆(j,t))0 ≥ τ_(t,j,t0₎, ∀(t, t0), ∀j ∈ J (7)

S∆(j,t)_t0 β_(j,∆(j,t))0 ≤ |L||D|τ_(t,j,t0₎, ∀(t, t0), ∀j ∈ J (8)

In the above inequalities, S_t∆(j,t)0 β0_(j,∆(j,t)) > 0 means that technician t0 has the critical skill

(l, d) that technician t misses. Therefore τ(t,j,t0₎ = 1 meaning that technician t cannot replace

technician t0 in the team of job j.

xt0_j ≤ y0

tj+ τ(t,j,t0₎+ δ_t,t0, ∀(t, t0), ∀j ∈ J (9)

Theorem 6 In the multi-skill workforce scheduling, an assignment satisfying the inequalities (2)-(9) is stable.

Proof. The inequalities (9) prevent blocking pairs. To show this, suppose that we have an assignment satisfying inequalities (2) through (9) and (t, j) is a blocking pair and j likes t by releasing technician t0 in T (j). It is immediate that τ_(t,j,t0₎ = 0, since t can replace t0 in T (j).

Moreover, we have y_tj = 1 so y0_tj= 0, since J (t) <tj. By preference criterion of jobs, j has less

total skill value when t replaces t0 in its team. This implies t < t0 hence δ_t,t0 = 0. All the

mentioned points make the right side of the inequality (9) is zero and x_t0_j cannot be 1. This

contradicts with t0 ∈ T (j)._

4

Conclusions

This paper we defined a set of inequalities whose satisfaction guarantees stability in multi-skill workforce scheduling. Having defined the polytope of stable schedules, the next questions are (1) How to find technician-optimal and job-optimal stable schedules?(2) What are the prop-erties of these schedules?(3) Is there any relation between stability and robustness of schedules, if yes, which stable schedules are more robust; technician-optimal, job-optimal or any other stable ones?

We will also continue investigating the role of stability in achieving low cost schedules and sensitivity of stability against unexpected changes in jobs or in the technician group.

5

Acknowledgments

The authors would like to thank France Telecom for their cooperation in this project and Alexandre Laugier for his suggestions to start stability idea in multi-skill workforce schedules.

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R.J.I. Basten, M.C. van der Heijden, J.M.J. Schutten

and Critical Link in Global Supply Chain Performance

302 2010

Capacity reservation and utilization for a manufacturer with uncertain capacity and demand

Y. Boulaksil; J.C. Fransoo; T. Tan

300 2009 Spare parts inventory pooling games F.J.P. Karsten; M. Slikker; G.J. van

Houtum

299 2009 Capacity flexibility allocation in an outsourced

supply chain with reservation Y. Boulaksil, M. Grunow, J.C. Fransoo

298 2010 An optimal approach for the joint problem of level

of repair analysis and spare parts stocking

R.J.I. Basten, M.C. van der Heijden, J.M.J. Schutten

297 2009

Responding to the Lehman Wave: Sales

Forecasting and Supply Management during the Credit Crisis

Robert Peels, Maximiliano Udenio, Jan C. Fransoo, Marcel Wolfs, Tom Hendrikx

296 2009

An exact approach for relating recovering surgical patient workload to the master surgical schedule

Peter T. Vanberkel, Richard J. Boucherie, Erwin W. Hans, Johann L. Hurink,

Wineke A.M. van Lent, Wim H. van Harten

295 2009

An iterative method for the simultaneous optimization of repair decisions and spare parts stocks

R.J.I. Basten, M.C. van der Heijden, J.M.J. Schutten

294 2009 Fujaba hits the Wall(-e) Pieter van Gorp, Ruben Jubeh, Bernhard

Grusie, Anne Keller

293 2009 Implementation of a Healthcare Process in Four

Different Workflow Systems

R.S. Mans, W.M.P. van der Aalst, N.C. Russell, P.J.M. Bakker

292 2009 Business Process Model Repositories -

Framework and Survey

Zhiqiang Yan, Remco Dijkman, Paul Grefen

291 2009 Efficient Optimization of the Dual-Index Policy

Using Markov Chains

Joachim Arts, Marcel van Vuuren, Gudrun Kiesmuller

290 2009 Hierarchical Knowledge-Gradient for Sequential

Sampling

Martijn R.K. Mes; Warren B. Powell; Peter I. Frazier

289 2009

Analyzing combined vehicle routing and break scheduling from a distributed decision making perspective

C.M. Meyer; A.L. Kok; H. Kopfer; J.M.J. Schutten

288 2009 Anticipation of lead time performance in Supply

Chain Operations Planning

Michiel Jansen; Ton G. de Kok; Jan C. Fransoo

287 2009 Inventory Models with Lateral Transshipments: A

Review

Colin Paterson; Gudrun Kiesmuller; Ruud Teunter; Kevin Glazebrook

286 2009 Efficiency evaluation for pooling resources in

health care

P.T. Vanberkel; R.J. Boucherie; E.W. Hans; J.L. Hurink; N. Litvak

285 2009 A Survey of Health Care Models that Encompass

Multiple Departments

P.T. Vanberkel; R.J. Boucherie; E.W. Hans; J.L. Hurink; N. Litvak

284 2009 Supporting Process Control in Business

Collaborations

S. Angelov; K. Vidyasankar; J. Vonk; P. Grefen

283 2009 Inventory Control with Partial Batch Ordering O. Alp; W.T. Huh; T. Tan

Structure-Preserving Way

281 2009 The link between product data model and

process model J.J.C.L. Vogelaar; H.A. Reijers

280 2009 Inventory planning for spare parts networks with

delivery time requirements I.C. Reijnen; T. Tan; G.J. van Houtum

279 2009 Co-Evolution of Demand and Supply under

Competition B. Vermeulen; A.G. de Kok

278

277 2010

2009

Toward Meso-level Product-Market Network Indices for Strategic Product Selection and (Re)Design Guidelines over the Product Life-Cycle

An Efficient Method to Construct Minimal Protocol Adaptors

B. Vermeulen, A.G. de Kok

R. Seguel, R. Eshuis, P. Grefen

276 2009 Coordinating Supply Chains: a Bilevel

Programming Approach Ton G. de Kok, Gabriella Muratore

275 2009 Inventory redistribution for fashion products

under demand parameter update G.P. Kiesmuller, S. Minner

274 2009

Comparing Markov chains: Combining

aggregation and precedence relations applied to sets of states

A. Busic, I.M.H. Vliegen, A. Scheller-Wolf

273 2009 Separate tools or tool kits: an exploratory study

of engineers' preferences

I.M.H. Vliegen, P.A.M. Kleingeld, G.J. van Houtum

272 2009

An Exact Solution Procedure for Multi-Item Two-Echelon Spare Parts Inventory Control Problem with Batch Ordering

Engin Topan, Z. Pelin Bayindir, Tarkan Tan

271 2009 Distributed Decision Making in Combined

Vehicle Routing and Break Scheduling

C.M. Meyer, H. Kopfer, A.L. Kok, M. Schutten

270 2009

Dynamic Programming Algorithm for the Vehicle Routing Problem with Time Windows and EC Social Legislation

A.L. Kok, C.M. Meyer, H. Kopfer, J.M.J. Schutten

269 2009 Similarity of Business Process Models: Metics

and Evaluation

Remco Dijkman, Marlon Dumas, Boudewijn van Dongen, Reina Kaarik, Jan Mendling

267 2009 Vehicle routing under time-dependent travel

times: the impact of congestion avoidance A.L. Kok, E.W. Hans, J.M.J. Schutten

266 2009 Restricted dynamic programming: a flexible

framework for solving realistic VRPs

J. Gromicho; J.J. van Hoorn; A.L. Kok; J.M.J. Schutten;