Ariadna Fossas
email: ariadna.fossas@ujf-grenoble.fr

Maxime Nguyen
email: maxime.nguyen@ujf-grenoble.fr

On the automorphism group of the asymptotic pants complex of a planar surface of infinite type

Keywords: mapping class groups, pants complex, infinite type surfaces, group actions, Thompson's groups

We consider a planar surface Σ of infinite type which has the Thompson group T as asymptotic mapping class group. We construct the asymptotic pants complex C of Σ and prove that the group T acts transitively by automorphisms on it. Finally, we establish that the automorphism group of the complex C is an extension of the Thompson group T by Z/2Z.

Introduction

Given a compact surface of genus g with b boundary components, one can build various locally infinite simplicial complexes on which the mapping class group of the surface acts by automorphisms, mainly the curve complex, the arc complex and the pants complex. The automorphism group of each one of these complexes is related to the mapping class group itself. Specifically, Ivanov in [START_REF] Ivanov | Automorphism of complexes of curves and of Teichmüller spaces[END_REF] and subsequently Luo in [START_REF] Luo | Automorphisms of the Complex of Curves[END_REF] proved that the automorphism group of the curve complex is the extended mapping class group for almost all compact surfaces, and Korkmaz extended this result to punctured spheres and punctured tori in [START_REF] Korkmaz | Automorphisms of Complexes of Curves on Punctured Spheres and on Punctured tori[END_REF]. Analogous results were proved for the arc complex by Ivanov [START_REF] Ivanov | Automorphism of complexes of curves and of Teichmüller spaces[END_REF] and Irmak and McCarthy [START_REF] Irmak | Injective simplicial maps of the arc complex[END_REF], for the pants complex by Margalit [START_REF] Margalit | The automorphism group of the pants complex[END_REF], for the Hatcher-Thurston complex by Irmak and Korkmaz [START_REF] Irmak | Automorphisms of the Hatcher-Thurston complex[END_REF], for the arc and curve complex by Korkmaz and Papadopoulos [START_REF] Korkmaz | On the arc and curve complex of a surface[END_REF], and for the complex of non-separating curves by Schmutz-Schaller [START_REF] Schmutz-Schaller | Mapping class groups of hyperbolic surfaces and automorphism groups of graphs[END_REF] and subsequently by Irmak [START_REF] Irmak | Complexes of nonseparating curves and mapping class groups[END_REF].

When dealing with surfaces of infinite type, one issue is to find good analogues for mapping class groups and for the complexes enumerated above. Funar and Kapoudjan defined the notion of asymptotic mapping class group for some surface of genus zero and infinite type in [START_REF] Funar | On a universal mapping class group of genus zero[END_REF], the first one of a series of three papers (see also [START_REF] Funar | The braided Ptolemy-Thompson group is finitely presented[END_REF] and [START_REF] Funar | The braided Ptolemy-Thompson group is asynchronously combable[END_REF]). They constructed several complexes for this type of surfaces and proved that the asymptotic mapping class group acts on them by automorphisms. These complexes can be seen as generalizations of the pants complex to the infinite type case. It is worth to mention that the asymptotic mapping class groups they defined are closely related to Thompson's group T , which is one of the first examples of finitely presented infinite simple groups.

In this paper, we consider the case of a planar infinite surface with non compact boundary, which has the Thompson group T as its asymptotic mapping class group. Then, we construct a locally infinite cellular complex C related to the pants complex of the infinite surface of genus zero of [START_REF] Funar | On a universal mapping class group of genus zero[END_REF]. We prove that the group T acts by automorphisms on the complex and, finally, we establish that the automorphism group of the complex is T ⋊ Z/2Z. We also give a geometric interpretation to this semi-direct product as the extended asymptotic mapping class group, obtaining in this way a complete analogue to Ivanov's theorem for the surface Σ of infinite type.

Our result can also be considered from a different viewpoint. In 1970, Tits proved in [START_REF]Tits. Sur le groupe des automorphismes d'un arbre[END_REF] that most subgroups of automorphism groups of trees generated by vertex stabilizers are simple. This result was extended by Haglund and Paulin in [START_REF] Haglund | Simplicity of groups of endomorphisms of negatively curved spaces. The Epstein birthday schrift[END_REF] to the case of automorphism groups of negatively curved polyhedral complexes which are (in general uncountable) non-linear and virtually simple. On the other hand, Farley proved in [START_REF] Farley | Actions of picture groups on CAT(0) cubical complexes[END_REF] that the Thompson groups F , T and V act properly and isometrically on CAT(0) cubical complexes. In particular, these groups are a-T-menable and satisfy the Baum-Connes conjecture with arbitrary coefficients. This leads to the legitimate question of realizing the simple groups T and V as automorphisms groups of suitable cell complexes. Although the complex which we obtain is not Gromov hyperbolic, it gives a convenient answer to this question for the group T .

Definitions and statement of the results

The surface Σ and its asymptotic mapping class group Let D 2 be the hyperbolic disc and suppose that its boundary ∂D 2 is parametrized by the unit interval (with identified endpoints). Let E denote the triangulation of D 2 given by the family of bi-infinite geodesics representing the standard dyadic intervals, i.e. the family of geodesics I n a joining the points p = a 2 n , q = a+1 2 n on ∂D 2 , where a, n are integers satisfying 0 ≤ a ≤ 2 n -1. Let T 3 be the dual graph of E, which is an infinite (unrooted) trivalent tree. We choose to realize the tree T 3 in D 2 by using geodesic segments obtained from the I n a (with n ≥ 1 and a even) by rotations of angles π 2 n , as shown in figure 1. We also denote by T 3 this realisation.

0 E T 3 1 2 3 4 1 4
Figure 1: The triangulation E of D 2 and its dual tree T 3 . Now, let Σ be a closed δ-neighbourhood of T 3 (see figure 2). Remark that the surface Σ is planar (thus oriented), non compact, contractible and its boundary has infinitely many connected components. Note that we can choose δ small enough to avoid self-intersections of the boundary and such that each intersection I n a ∩ Σ is a single connected arc joining two boundary components of Σ. The triangulation E together with the boundary components of Σ define a tessellation of our surface into hexagons (see figure 2). By abuse of notation we will denote this hexagonal tessellation also by E. Note that the boundary of each hexagon alternates sides contained in different connected components of ∂Σ with sides contained in different arcs of the triangulation E. Let Homeo(Σ) denote the group of orientation-preserving homeomorphisms of Σ.

Definition 2. An element ϕ ∈ Homeo(Σ) is asymptotically rigid if there exists a set of separating arcs {I n1 a1 , . . . , I n k a k } such that:

1. For all 1 ≤ i ≤ k, there exist I mi bi such that ϕ(I ni ai ) ∩ Σ = I mi bi ∩ Σ, and 2. For all 1 ≤ i ≤ k, for all j ∈ N, and for all 0 ≤ l < 2 j the arc ϕ(I ni+j 2 j ai+l ) ∩ Σ is equal to the arc I mi+j 2 j bi+l ∩ Σ.

Remark 1. The set of arcs {I m1 b1 , . . . , I m k b k } from the definition above is a set of separating arcs. Remark 2. Let ϕ be an asymptotically rigid homeomorphism. The first condition of the definition says that there exists a compact subsurface C ⊂ Σ which boundary is composed by segments of ∂Σ and arcs of E such that ϕ sends C to another compact subsurface C ′ of the same type (i.e. which boundary is composed by segments of ∂Σ and arcs of E). The second condition deals with the behaviour of ϕ outside C. Let t 1 , . . . , t k be the connected components of T 3 \(T 3 ∩C) and t ′ 1 , . . . , t ′ k be the connected components of T 3 \ (T 3 ∩ C ′ ). Then the second condition says that each t i is 'naturally' sent to t ′ i .

The group of asymptotically rigid homeomorphisms of Σ will be denoted by Homeo a (Σ).

Definition 3. The asymptotic mapping class group of Σ is the quotient of Homeo a (Σ) by the group of isotopies of Σ.

Remark 3. ([7], section 1.

3) The asymptotic mapping class group of Σ is isomorphic to the Thompson group T , which is the group of orientation preserving piecewise linear homeomorphisms of the circle that are differentiable except at finitely many dyadic rational numbers, such that, on intervals of differentiability, the derivatives are powers of 2, and which send the set of dyadic rational numbers to itself. For a complete introduction to Thompson's groups see [START_REF] Cannon | Introductory notes in Richard Thompson's groups[END_REF].

The complex C

Once the surface Σ defined, we are going to define a 2-dimensional cellular complex C. The vertices of C are the isotopy classes relatively to the boundary of the tessellations of Σ into hexagons which differ from the distinguished tessellation E only in a finite number of hexagons. Note that every hexagon of a tessellation have 3 sides corresponding to different boundary components of Σ and 3 sides corresponding to arcs joining these boundary components.

Let v be a vertex of C, and let a be an arc bounding two hexagons h 1 , h 2 of the tessellation v. Let a ′ be an arc contained in h 1 ∪ h 2 joining two boundary components of Σ such that i([a], [a ′ ]) = 1, where i denote the geometric intersection number and [a] the isotopy class of a relatively to the boundary. Let w be the vertex of C obtained from v by replacing the arc a by a ′ . Then, the vertices v and w are joined by an edge on C. We say that w is obtained from v by flipping the arc a.

Let v be a vertex of C and let a and b be two arcs, each one of which bounding two hexagons of the tessellation v. We say that a and b are adjacent if there exists an hexagon on the tessellation v having both of them on its boundary. There are two possible cases:

1. The two arcs are not adjacent. Then, flipping first a and then b one obtains the same tessellation as the result of first flipping b and then a. Thus, the commutativity of flips defines a 4-cycle on the 1-skeleton of C and we fill it with a squared 2-cell.

2. The two arcs are adjacent. Then, flipping first a and then b one obtains a different tessellation from the result of first flipping b and then a. These two tessellations differ from a flip. We fill the corresponding 5-cycle with a pentagonal 2-cell.

We obtain, in particular, the following analogue to the case of the cellular complexes associated to compact surfaces:

Proposition 4. The asymptotic mapping class group of Σ acts transitively on the complex C by automorphisms.

It is now natural to ask whether or not the automorphism group of the cellular complex C is related to the asymptotic mapping class group of the surface Σ. To answer this question, we construct the following exact sequence:

1 -→ T -→ Aut(C) -→ Z/2Z -→ 1,
and prove that it splits.

Thus, we obtain the following result:

Theorem 1. The automorphism group of the complex C is isomorphic to the semi-direct product

T ⋊ Z/2Z.
Furthermore, the extension is not trivial; we obtain a non trivial action of Z/2Z to Out(T ). Thus, as Brin established in [START_REF] Brin | The chameleon groups of Richards J. Thompson: automorphisms and dynamics[END_REF] that the group of outer automorphisms of T is isomorphic to Z/2Z, we get an isomorphism between Out(T ) and Z/2Z.

Finally, the tools used to prove the main theorem lead us to give a geometric interpretation of this result.

It turns out that one can classify the elements of Aut(C) in two categories: those which preserve (in some natural sense) the orientation of the complex, and those which reverse the orientation. Then, it is natural to define the extended mapping class group of Σ as T ⋊ Z/2Z, which makes theorem 1 a complete analogue to Ivanov's theorem for compact surfaces.

Structure of the paper

This paper is structured as follows. In the second section, we provide a modification of the surface Σ which does not involve changes on the mapping class group. This will allow us to describe the isomorphism between the asymptotic mapping class group and Thompson's group T . In the third section, we briefly discuss the analogy between the complex C and the complexes obtained for compact surfaces and define the action of the asymptotic mapping class group on C. Finally, in the fourth section, we describe the structure of the automorphisms group of the complex C, proving theorem 1.

The first author acknowledges financial support from "La Caixa" and "l'Ambassade de France en Espagne" Post-graduated Fellowship. Both authors would like to thank Javier Aramayona, Francis Bonahon, José Burillo, François Dahmani, Vlad Sergiescu and Juan Souto for useful comments. The authors are particularly indebted to Louis Funar and Frédéric Mouton for enlightening discussions and suggestions.

The surface and its asymptotic mapping class group

In this section we introduce a slightly different surface Σ ′ which allows us to simplify the situation. Both surfaces Σ and Σ ′ have the same asymptotic mapping class group.

Definition 4. The surface Σ ′ is obtained by attaching to the hyperbolic disk D 2 the points of its boundary corresponding to the dyadic rational numbers.

Remark 4. The triangulation E of D 2 is a triangulation of Σ ′ having for vertices all the boundary components of Σ ′ .

Lemma 1. Let ∂ 0 Σ be a connected component of the boundary of Σ, which is a bi-infinite line on D 2 . The adherence of ∂ 0 Σ is ∂ 0 Σ ∪ {q},
where q is a dyadic rational number on ∂D 2 . Thus, there is a natural bijection between the set of dyadic rational numbers ∂Σ ′ and the set of connected components of the boundary of Σ.

Corollary 1. Let ∂ 0 Σ be a connected component of the boundary of Σ, and let I n1 a1 , I n2 a2 be two geodesics of E intersecting ∂ 0 Σ. Then, I n1 a1 and I n2 a2 have a common boundary point (dyadic). Corollary 2. Let {I n1 a1 , . . . , I n k a k } be a set of separating arcs which are cyclically ordered. Then, they are the sides of an ideal hyperbolic k-gon. Corollary 3. One can deform continuously Σ to Σ ′ by an homotopy H : Σ × [0, 1] → D 2 ∪ ∂D 2 such that:

• For t < 1, H(•, t) is an homeomorphism. • For t < 1, H(Σ ∩ I n a , t) ⊂ I n a . • H(•, 1
) is an homeomorphism between the interior of Σ and D 2 .

• Let ∂ 0 Σ be a connected component of the boundary of Σ and let q be the dyadic rational number on its adherence. Then, H(∂ 0 Σ, 1) = q.

One can now define the notion of asymptotically rigid homeomorphism of Σ ′ , and the asymptotic mapping class group by taking Homeo a (Σ ′ ) modulo isotopies of Σ ′ .

Proposition 1. The asymptotic mapping class groups of Σ and Σ ′ are isomorphic.

Furthermore, they are isomorphic to Thompson's group T .

Proof. Let ϕ ∈ Homeo a (Σ). We define ϕ ′ ∈ Homeo a (Σ ′ ) as follows:

ϕ ′ (x) = H (ϕ(y), 1) , if x ∈ D 2 H (ϕ (∂ x Σ) , 1) , if x ∈ ∂Σ ′ ,
where y ∈ Σ such that H(y, 1) = x, and ∂ x Σ is the connected component of ∂Σ having x on its adherence and H is the homotopy from lemma 3. Note that every isotopy of Σ can be extended to an isotopy of D 2 ∪ ∂D 2 . Thus, the application we defined induces an injective morphism for the mapping class groups.

Reciprocally, let [ϕ ′ ] be an element of the asymptotic mapping class group of Σ ′ . Note that we can choose a representative ϕ ′ ∈ Homeo a (Σ ′ ) such that ϕ ′ |Σ ∈ Homeo a (Σ). Thus, we defined an injective morphism from the asymptotic mapping class group of Σ ′ to the asymptotic mapping class group of Σ.

Finally, one can verify that the composition of both is the identity.

Before giving the isomorphism between T and the asymptotic mapping class group of Σ, let us give the analytical definition of Thompson's group T : Definition 5. The Thompson group T is the set of orientation preserving piecewise linear homeomorphisms of the circle S 1 = [0, 1]/ 0∼1 whose points of non-differentiability are dyadic rational numbers, whose derivatives (where defined) are powers of 2, which send the set of dyadic rational numbers to itself.

Let ϕ be a representative of an equivalence classe of the asymptotic mapping class group of Σ ′ . We claim that ϕ acts as an element of the Thompson group T on ∂Σ ′ , which is the set of dyadic rational numbers, and that this action does not depend on the representative. To see this, let {I n1 a1 , . . . , I n k a k } and {I m1 b1 , . . . , I m k b k } be sets of separating arcs associated to ϕ, with ϕ(

I ni ai ) = I mi bi for 1 ≤ i ≤ k. Without loss of generality, we can suppose that a 1 = 0, a k = 2 n k -1 (I 1
a is contained on the compact subsurface bounded by the arcs I n a ) and that both sets are cyclically ordered. Let f be the unique map of the circle satisfying the following conditions:

1. For all 1 ≤ i ≤ k, f a i 2 ni = b i 2 mi , 2. For all 1 ≤ i ≤ k, f a i + 1 2 ni = b i + 1 2 mi , and
3. For all x ∈ a i 2 ni , a i + 1 2 ni , f ′ (x) = 2 ni-mi .
One can see that f ∈ T using the corollary 2 and the fact that ϕ is orientation-preserving.

Reciprocally, given an element f ∈ T it can be proven (adaptation to T of lemma 2.2 from [START_REF] Cannon | Introductory notes in Richard Thompson's groups[END_REF]) that there exists a partition of the unit interval into standard dyadic intervals 0 = x 0 < x 1 < . . . < x k = 1, and an integer 0 ≤ j ≤ k such that:

1. The subintervals of the partition 0 = f (x j ) < f (x j+1 ) < . . . < f (x k ) = f (x 0 ) < . . . < f (x j-1 ) < f (x j ) + 1 = 1 are standard dyadic, and 2. The map f is linear in every subinterval of the partition.

Thus, we can find two ideal k-gons inscribed in E defining an element ϕ in the asymptotic mapping class group of Σ ′ which acts as f on ∂Σ ′ . This ends the proof of the following result (see [START_REF] Funar | The braided Ptolemy-Thompson group is finitely presented[END_REF]):

Proposition 2. The asymptotic mapping class group of Σ is isomorphic to the Thompson group T .

For the purposes of this paper it is useful to think of the Thompson group T as the group generated by two elements: an element α of order 4 and an element β of order 3. The full presentation is the following (for the details see [START_REF] Funar | The braided Ptolemy-Thompson group is finitely presented[END_REF] and [START_REF] Lochak | The universal Ptolemy-Teichmüller groupoid[END_REF]):

T ≃ α, β | α 4 , β 3 , βαβ, α 2 βαβα 2 , βαβ, α 2 β 2 α 2 βαβα 2 βα 2 , (βα) 5 .
The two generators α, β of Thompson's group T can be defined in terms of polygons as follows:

α sends {I 2 0 , I 2 1 , I 2 2 , I 2 3 } respectively to {I 2 1 , I 2 2 , I 2
3 , I 2 0 } and β sends {I 1 0 , I 2 0 , I 2 1 } respectively to {I 2 0 , I 2 1 , I 1 0 }. Remark that the polygons defining an element of T are not unique, although there is a minimal one which satisfies the conditions. Consider this minimal pair of polygons defining an element f ∈ T . Then, if we split a standard dyadic interval of the source polygon in two halves and we also split the standard dyadic interval corresponding to its image in two halves, the element defined by the new pair of polygons is the same. All possible pairs of polygons defining f are obtained doing this expanding operation finitely many times.

The asymptotic pants complex

First of all, we would like to present the relation of the complex C and the different known complexes for the compact case. In some general sense, one can see C as an analogue to the pants complex. For this, take another copy of the initial surface Σ, already tessellated in hexagons by E, and glue the two copies together along their boundary. Now, the arcs of E on the two copies are also glued and they become simple curves which decompose the doubled surface in infinitely many pairs of pants. Hence, the vertices of the complex C can be seen as decompositions in pairs of pants of the double surface, obtained from the initial decomposition by a finite number of elementary moves. An elementary move consists on changing a simple curve γ by another γ ′ , which does not intersect the other curves of the pants decomposition, and with minimal geometric intersection number i([γ], [γ ′ ]) = 2. In our case, we only consider the elementary moves that can be seen on the planar surface Σ. For the details of this construction see [START_REF] Funar | On a universal mapping class group of genus zero[END_REF] and [START_REF] Funar | The braided Ptolemy-Thompson group is finitely presented[END_REF].

The rest of this section is devoted to give another point of view of the complex C using the definition of Σ ′ . For example, we can see the 0-skeleton of C as the set of triangulations of Σ ′ with vertices in ∂Σ ′ which differ from E only on a finite number of triangles. Hence, two vertices u, v of C are joined by an edge if we can obtain v from u by changing the diagonal of a quadrilateral inscribed on the triangulation u. We will that v is obtained by flipping an arc of u.

Remark 5. (Geometric interpretation of the 2-cells) Each 2-cell of the complex can be seen, geometrically, as the object obtained from a triangulation v of the complex C by flipping consecutively two of its arcs. If there exists a triangle in v containing the two arcs on its boundary, then the 2-cell is pentagonal. Otherwise, the 2-cell is squared. Remark 6. Given three vertices u, v, w of C where u, w are different and adjacent to v, there exists a unique 2-cell containing u, v, w. This follows from the precedent remark. Definition 6. We call the 2-dimensional cellular complex C the asymptotic pants complex of the surfaces Σ and Σ ′ .

It is worth to mention that the asymptotic pants complex is closely related to the graph of triangulations of a convex n-gon (see, for example, [START_REF] Bose | Flips in planar graphs[END_REF], [START_REF] Lee | The associahedron and triangulations of the n-gon[END_REF] and [START_REF] Hurtado | Graph of triangulations of a convex polygon and tree of triangulations[END_REF]) and hence, it is also related to the rotation distance in binary trees (see [START_REF] Sleator | Rotation distance, triangulations and hyperbolic geometry[END_REF]). The graph of triangulations of a convex n-gon is a graph whose vertices are the triangulations of the n-gon in n -2 triangles, and where two vertices are joined by an edge if there is a single flip going from one to the other. Thus, the asymptotic pants complex can be seen as a 2-dimensional complex associated to the triangulations graph of an infinite-sided convex polygon. Now, one can derive the connectivity of the complex C from the finite case (see [START_REF] Hurtado | Graph of triangulations of a convex polygon and tree of triangulations[END_REF] for a simple proof). To see this, one can think that every vertex of C has only a finite number of arcs not contained in the triangulation E and so, there exists a convex polygon having all of this arcs on its interior. Furthermore, Lee in [START_REF] Lee | The associahedron and triangulations of the n-gon[END_REF] proved that the complex of triangulations of a convex n-gon is a convex polytope of dimension n -3 (see also [START_REF] Devadoss | Cellular structures determined by polygons and trees[END_REF]). Thus, the complex C is also simply connected.

Proposition 3. The complex C is connected and simply connected.

Finally, we will define the action of T on C. Let v be a vertex of C and let f be an element of T . Consider the set of geodesics G defined as follows: the geodesic joining the dyadic rational numbers p, q is in G if the geodesic joining the dyadic rational numbers f -1 (p) and f -1 (q) belongs to the triangulation v. Remark that the geodesics on G define a triangulation of Σ ′ ; this triangulation will be the image by f of the vertex v and we will denote it by f • v. As a consequence of the definitions of T and C, the triangulation f • v corresponds to a vertex of C. Note that it is possible to find an ideal polygon bounded by the set of separating arcs {I n1 a1 , . . . , I n k a k } which is the source polygon for the element f and which contains all the geodesics that are on the triangulation v and which are not on the base triangulation E.

Let v, w be two vertices joined by an edge in C, and let f be an element of T . It follows from the definition of the action of f on the vertices of C that f • v and f • w are joined by an edge in C. It also follows from the definition of the action on the vertices together with the geometric interpretation of the 2-cells (remark 5) that the action can be extended to the 2-skeleton of C. Thus, we can define a natural map Ψ : T → Aut(C). Proposition 4. The asymptotic mapping class group T of Σ ′ acts transitively on the asymptotic pants complex C by automorphisms. Furthermore, the map Ψ : T → Aut(C) is injective.

Proof. It follows from the paragraph preceding the proposition that T acts on C by automorphisms. We prove the transitivity by showing that given a vertex v of C there exists an element f ∈ T such that f • E = v. Let P be an ideal polygon inscribed on E containing all the edges of the triangulation v which are not on the triangulation E, and having at least the dyadics 0, 1 4 , 1 2 , 3 4 on its boundary. Let a be an interior edge of the triangulation of P in v joining the dyadics p and q. Suppose that p > q. There exists a unique triangle in P having as vertices p, x, q, where x is a dyadic number p > x > q. We associate the triangle of E with vertices 1 2 , 0, 1 4 to this triangle. Analogously, we associate to the triangle with vertices y > p > q (or p > q > y) of P the triangle with vertices 3 4 , 0, 1 2 . Repeating this procedure finitely many times we construct an ideal polygon Q triangulated as E, whose sides are naturally associated to those of the polygon P . Then, the element f ∈ T defined by f (Q) = P satisfies our claim.

For the second assertion it is enough to prove that given an element f ∈ T different from the identity, there exists a vertex v in C such that f • v = v. Let q be a dyadic such that f (q) = q (it exists because f = id). Take a square inscribed in the triangulation E, having q as a vertex and such that it has empty intersection with its image by f . Now, let v be the triangulation obtained from E by flipping the diagonal of this square. Then,

f • v = v.
Proposition 5. The complex C is not Gromov hyperbolic.

The proof uses the following observation: Remark 7. Every flip changes a single arc of the initial triangulation. Therefore, the distance between two vertices v, w ∈ C is bounded below by the number of arcs which are on the triangulation v and are not in the triangulation w.

Proof. Let n be a positive integer. We construct a geodesic triangle u, v, w such that d C (u, v) = d C (v, w) = n, and d C (u, w) = 2n; and we give a point p in the geodesic segment uw such that d C (p, x) ≥ n for all x ∈ uv ∪ vw.

Let s 1 , . . . , s 2n be 2n disjoint squares inscribed in the triangulation E. Let u = E, v be the vertex obtained from E by flipping the diagonals of the squares s 1 , . . . , s n and w be the vertex obtained from v by flipping the diagonals of the squares s n+1 , . . . , s 2n . The segments uv and vw consist on flipping, respectively, the diagonals of s 1 , . . . , s n , and s n+1 , . . . , s 2n , in order. The segment uw consists on flipping first the diagonal of the square s 2n , then the diagonal of s 2n-1 until the diagonal of s 1 , in decreasing order (see figure 3). The point p is the vertex obtained from E by flipping the diagonals of s n+1 , . . . , s 2n . By the remark 7 all these segments are geodesics, and d C (p, x) = n + min{d C (u, x), d C (w, x)}, for all x ∈ uv ∪ vw.

The automorphism group of the complex C

In this section we study the group of automorphisms Aut(C) of the asymptotic pants complex C, and we prove that it is isomorphic to the semi-direct product T ⋊ Z/2Z.

The proof of this result has some similarities with Ivanov's proof (see [START_REF] Ivanov | Automorphism of complexes of curves and of Teichmüller spaces[END_REF]) of the fact that the automorphism group of the arc complex of a compact surface is isomorphic to the extended mapping class group. In particular, he showed that the image of a maximal simplex determines completely an automorphism. In our case, every automorphism is determined by its image of the ball of radius one centred in some vertex v. Then, given an automorphism ϕ of the complex C, we construct an element t ∈ Ψ(T ) ⊂ Aut(C) such that t(E) = ϕ(E). For this step, we need to introduce an auxiliary complex, which is related to the link of v in C.

Construction of the link complex L 2 (v)

Recall that in any triangulation v of Σ ′ , the flips along two different arcs of v do not commute if and only if both arcs belong to the boundary of some triangle of v (remark 5). We construct a sub-complex of C which encodes this information.

Let L 1 (v) be a graph whose set of vertices is the set of vertices of C adjacent to the vertex v. Two vertices are connected by an edge if they lie in the same pentagon in C. Given a vertex of C there is a geometric interpretation of L 1 (v): we associate to every arc a of the triangulation v a vertex which represents the vertex of C obtained from v by flipping the arc a. Then, from the geometric interpretation of the 2-cells of C (remark 5) we see that two vertices of L 1 (v) associated to the arcs a, b are joined by an edge if and only if there is a triangle in v having both a, b on its boundary. Thus, L 1 (v) can be seen as a connected union of triangles, each of them inscribed on a triangle of v, where the intersection between different triangles is either empty or a vertex, and where every vertex belong to two different triangles (see figure 4).

Definition 7. Let v be a vertex of C. The link of v is the 2-dimensional complex L 2 (v) obtained from L 1 (v)
by gluing a 2-cell on each triangle. Furthermore, the link lies on D 2 and every triangle is naturally oriented.

Remark that the link has been constructed using only the combinatorial structure of C in the neighbourhood of a vertex. Note also that C is regular in the sense that all vertices have the same combinatorial structure on their neighbourhoods. Thus, one can derive the following result: Lemma 2. Let v be a vertex of C. Then, every automorphism φ ∈ Aut(C) induces an isomorphism φ * ,v : L 2 (v) → L 2 (w), where w = φ(v). Now, using the transitivity of the action of T on C one obtains: Corollary 4. Let v and w be vertices of C. Then, their links L 2 (v) and L 2 (w) are isomorphic.

Extensions of link isomorphisms

In the previous section we proved that every automorphism ϕ of C induces link isomorphisms between L 2 (v) and L 2 (w), where w = ϕ(v). Now, given two vertices v and w of C and given an isomorphism i : L 2 (v) → L 2 (w) between their links, it is natural to ask under which conditions one can find an automorphism ϕ ∈ Aut(C) such that ϕ * ,v = i. More specifically, which is the main obstruction to the existence of this automorphism? Let ϕ ∈ Aut(C). Let w be the image of v by ϕ. Let (a, b, c) be three vertices of a triangle ∆ on the link L 2 (v), and suppose they are cyclically ordered according to the orientation of the triangle. The vertices ϕ(a), ϕ(b), ϕ(c) are the vertices of a triangle φ(∆) in the link L 2 (w). Now, there are two possible cases:

1. The orientation of ϕ(∆) induces the cyclic order (ϕ(a), ϕ(b), ϕ(c)) on its vertices. Then we say that ϕ is orientation preserving on ∆.

2. The orientation of ϕ(∆) induces the cyclic order (ϕ(a), ϕ(c), ϕ(b)) on its vertices. Then we say that ϕ is orientation reversing on ∆.

Definition 8. Let ϕ ∈ Aut(C) and v be a vertex of C. We say that ϕ is v-orientation preserving (reversing) if it is orientation preserving (reversing) on every triangle of L 2 (v).

Remark 8. Every element f ∈ T induces an automorphism Ψ(f ) ∈ Aut(C) (from proposition 4) which is v-orientation preserving for all vertices v of C.

Lemma 3. Let v and w be vertices of C, and i : L 2 (v) → L 2 (w) be an orientation preserving isomorphism of their links. Then, there exists a unique automorphism ϕ i ∈ Aut(C) such that ϕ * ,v = i. Furthermore, ϕ i ∈ T .

Proof. We first prove the existence by constructing ϕ ∈ T in the following way: let {I d1 c1 , . . . , I d l c l } a set of separating arcs on the triangulation w containing all the edges which differ from E. Let u ′ 1 , . . . , u ′ l be the vertices of L 1 (w) corresponding to the triangulations obtained from w by flipping, respectively, the edges I d1 c1 , . . . , I d l c l . Let {I n1 a1 , . . . , I n k a k } be separating arcs on the triangulation v containing on their interior i -1 (u ′ 1 ), . . . , i -1 (u ′ l ). Let u 1 , . . . , u k be the vertices of L 1 (v) corresponding to the triangulations obtained from v by flipping, respectively, the edges I n1 a1 , . . . , I n k a k . Then, i(u 1 ), . . . , i(u k ) are vertices corresponding to edges which form a set of separating arcs of w, {I m1 b1 , . . . ,

I m k b k }. Define ϕ by ϕ(I nj aj ) = I mj bj for 1 ≤ j ≤ k. It is easy to see that ϕ * ,v = i.
Uniqueness: this proof is based on the remarks 5 and 6. Let ψ ∈ Aut(C) such that ψ * ,v = i. Then, we prove that its action on the 0-skeleton of C must coincide with ϕ.

• ψ(v) = w.

• For every element u on the unit sphere of C centred in v, ψ(u) must be i(u).

• Suppose that ψ is defined in the ball B C (v, n) of C of radius n centred in v for n ≥ 2, and let u be a vertex at distance n + 1 of v. Consider a path p = v, u 1 , . . . , u n-1 , u n , u of length n + 1 joining v and u. The vertices u n-1 , u n , u define a unique 2-cell in C (remark 6), and this 2-cell contains at least a fourth vertex u ′ ∈ B C (v, n). Thus, ψ(u) must be the vertex of the unique 2-cell defined by ψ(u ′ ), ψ(u n-1 ), ψ(u n ) which is adjacent to ψ(u n ) and different from ψ(u n-1 ).

Lemma 4. Let v and w be vertices of C and let i : L 2 (v) → L 2 (w) be an isomorphism of their links such that it is orientation reversing on ∆ 1 = (u 0 , u 1 , u 2 ) and orientation preserving on ∆ 2 = (u 0 , u 3 , u 4 ). Then, there does not exist ϕ ∈ Aut(C) with ϕ * ,v = i.

Proof. Suppose there exists ϕ ∈ Aut(C) such that ϕ The vertices u 1 , v, u 0 define a pentagonal 2-cell of C; let u 5 be the vertex at distance 1 of u 0 and different from v on this cell. Analogously, let u 6 be the vertex at distance 1 of u 0 different from v and lying on the pentagonal 2-cell defined by the path u 4 , v, u 0 . The path u 5 , u 0 , u 6 defines a pentagonal 2-cell of C.

* ,v = i. i(u 1 ) i(u 4 ) w i(u 0 ) ϕ(u 6 ) ϕ(u 5 ) v u 1 u 4 u 6 u 5 u 0 ϕ
On the other hand, i(u 1 ), i(v), i(u 0 ) also define a pentagonal 2-cell of C. Adapting the prove of uniqueness of the lemma 3, ϕ(u 5 ) must be the vertex at distance 1 of i(u 0 ) and different from w on this pentagonal 2-cell. Furthermore, ϕ(u 6 ) must be the vertex at distance 1 of ϕ(u 0 ) different from w and lying on the pentagonal 2-cell defined by the path i(u 4 ), i(v), i(u 0 ). But in this case the path ϕ(u 5 ), ϕ(u 0 ), ϕ(u 6 ) defines a squared 2-cell of C, which contradicts the fact that ϕ ∈ Aut(C). See figure 5 for a picture of the situation.

Remark 9. Let v be a vertex of C and let w be one of its neighbours in C. Let w, u 1 , u 2 and w, u 3 , u 4 denote the triangles of L 2 (v) containing w. Observe that the geometric representation of the link L 2 (w) of w has the same vertices as L 2 (v) and only the triangles mentioned above change to w, u 1 , u 4 and w, u 2 , u 3 . See figures 4 and 6. This provides an alternative prove of lemma 4:

Proof. Suppose there exists ϕ ∈ Aut(C) such that ϕ * ,v = i. Then ϕ * ,u0 must be an isomorphism between the link of u 0 and the link of i(u 0 ). As a consequence of remark 9, u 1 and u 4 are adjacent in the link of u 0 , but i(u 1 ) and i(u 4 ) are not adjacent in the link of i(u 0 ) (see figure 7), which contradicts the fact that ϕ * ,u0 is an isomorphism. Proof. The uniqueness can be proved exactly as in lemma 3. For the existence it suffices to send each vertex to its symmetric with respect to the symmetry of axis I 1 0 .

Proof of theorem 1

In order to simplify the prove of theorem 1 and give a geometric interpretation of it, we introduce the following concept: Proof. As a consequence of lemma 4, given an automorphism ϕ of C, its restriction ϕ * ,E to the link L 2 (E) is either E-orientation preserving or E-orientation reversing. In the first case, ϕ ∈ Ψ(T ) by lemma 3 and thus it is orientation preserving (remark 10). In the second case, consider the automorphism ϕ ′ = ϕ • ϕ R , which is E-orientation preserving and thus orientation preserving. Then, ϕ = ϕ ′ • ϕ R (note that ϕ -1 R = ϕ R ), therefore it is orientation reversing (remark 11). Now, we prove theorem 1:

Proof. We want to construct the following exact sequence:

1 -→ T -→ Aut(C) -→ Z/2Z -→ 1, and prove that it splits.

We denote by Π : Aut(C) → Z/2Z and define it as follows:

ϕ → 0, if ϕ is orientation preserving 1, if ϕ is orientation reversing.

Proposition 6 shows that the map Π is a well defined morphism of groups. As a consequence of lemmas 3 and 5, the map Π is surjective and the isomorphism Z/2Z ≃ ϕ R is a section of Π.

Finally, ker(Π) ≃ T follows from lemma 3.

Therefore, we have an action of Z/2Z into the group of automorphisms of T , where the generator of Z/2Z acts as the automorphism γ R which is defined by γ R (t) = ϕ R • t • ϕ R , for t ∈ T . In particular, the automorphism γ R obtained in T is defined by γ R (α) = α -1 and γ R (β) = α 2 β -1 α 2 . Thus, if we compose γ R with the conjugation by α 2 we obtain the generator of the group of outer automorphisms of T sending α to α -1 and β to β -1 , which is the generator of Out(T ) given by Brin in [START_REF] Brin | The chameleon groups of Richards J. Thompson: automorphisms and dynamics[END_REF].

Remark 12. (Geometric interpretation of theorem 1.) By proposition 6, all automorphisms of C are either orientation preserving or orientation reversing. Thus, if one defines the extended asymptotic mapping class group of Σ as T ⋊ Z/2Z, theorem 1 turns out to be an exact analogue to Ivanov's theorem for the planar surface of infinite type Σ.

ΣFigure 2 :

 2 Figure 2: The surface Σ and its hexagonal tessellation.

Figure 3 :

 3 Figure 3: The points correspond to triangulations obtained from E by flipping the diagonals of s i if there is a one in the i-th coordinate. Discontinuous lines represent distances in C.

Figure 4 :

 4 Figure 4: The link of E.

Figure 5 :

 5 Figure 5: Example of lemma 4.

Figure 6 :

 6 Figure 6: The link of the vertex of C obtained from E by flipping I 1 0 .

Figure 7 :

 7 Figure 7: The map i in ∆ 1 and ∆ 2 . In discontinuous lines the links of u 0 and i(u 0 ).

Definition 9 .Remark 10 .Remark 11 .Proposition 6 .

 910116 Let ϕ ∈ Aut(C). It is orientation preserving (reversing) if it is v-orientation preserving (reversing) for all vertices v of C. As a consequence of lemma 3, the subgroup of orientation preserving automorphisms of C is isomorphic to T . The automorphism ϕ R from lemma 5 is orientation reversing.Using remarks 10 and 11, one obtains the following classification of the automorphisms of C: Every automorphism of C is either orientation preserving or orientation reversing.

Acknowledgements


Matthias Walter
email: matthias.walter@tu-clausthal.de

Thomas Sommer-Dittrich

Jurgen Zimmermann

Jürgen Zimmermann

Evaluating volume flexibility instruments by design-of-experiments methods

Keywords: volume flexibility, design-of-experiments methods, aggregate production planning

Evaluating volume flexibility instruments by design-of-experiments methods

Introduction

Demand fluctuations are a concern for manufacturing firms as they -no matter whether they are certain or random fluctuations -severely impede smooth production and distribution execution. Volume flexibility is one answer to the challenge imposed on a firm by demand variability. Browne et al. (1984, p. 115) define volume flexibility as the capability to operate profitably at different output levels. Similarly, Gerwin (1987, p. 39) relates volume flexibility to the ability to smoothly change the aggregate output of production. Volume flexibility instruments are options of a firm's management to adjust the output level within the planning horizon. Examples for such instruments are additional shifts on Saturdays, hiring as well as firing temporary workers, short time, and changing param- eters of the production system, e.g. altering the cycle time in assembly line production systems.

After having investigated three manufacturing firms in a case study approach and having conducted a field survey [START_REF] Jack | Sources of volume flexibility and their impact on performance[END_REF] concluded 'that volume flexibility has a positive impact on both delivery and financial performance.' Since it is known for long that flexibility is not a free good (cf. Stigler 1939, p. 311) the result of Jack and Raturi poses the following question to a firm: Which volume flexibility instruments should be selected and implemented? This question may be broken down into three research questions:

1. How large is the financial benefit of a volume flexibility instrument? 2. In which way do several volume flexibility instruments interact? 3. How sensitive are volume flexibility instruments to different demand fluctuations?

Although flexibility in general and volume flexibility in particular have been emphasized in academic research for more than 80 years (cf. Kaluza and Blecker 2005, p. 6) a clear and comprehensive theoretical framework is still missing and insights have not been diffused widely into practice because of the 'fuzzy and multidimensional nature of flexibility as a construct ' (de Treville et al. 2007, p. 335). It is the aim of this paper to provide guidance for firms selecting volume flexibility instruments.

Usually, a firm has some volume flexibility instruments at its disposal. Consider a firm that can choose between three different shift models so far. A shift model defines whether only an early shift or even a late or a night shift have to work and specifies at which days of the week and how many hours a shift has to work. The firm thinks about negotiating an additional shift model with the workers' union. Before engaging in the negotiation process and offering compensation the firm must have an good estimate of how much it will save if the fourth shift model can be used. Consequently, the heart of our method is a model that represents the production process and incorporates all the volume flexibility instruments, e.g. the shift models and the hours bank -also known as working time account -when dealing with volume flexibility at the plant level. The model accounts for the relevant cost arising when production is executed given a production plan which specifies a shift model, extra hours, employments and other options for each period. Note that there is no unique model, as a model depends on the considered firm and situation. We present an exemplary model in our case study.

Coming back to the firm considering to opt for a fourth shift model, the valuation of the fourth shift model works in principle as follows. We optimize the production plan with respect to the restrictions of the model two times in separate runs. In the first run we only take into account the three shift models available to the firm so far. In the second run we optimize the production plan under the assumption that the fourth shift model is available. The difference in cost between the two production plans obtained from optimization reflects the value of the additional shift model.

The firm might also wish to raise the limit of unpaid extra hours that can be accumulated in the hours bank in times of high demand. Instead of paying for overtime the hours bank enables the firm to compensate for extra hours by shorter working times in periods of low demand while a constant weekly wage is paid independently of the actual working time. There is a wide spectrum of possible values for the upper limit, e.g. 100, 150 or 200 hours. Furthermore, there may exist an interaction effect between the number of shift models and the limit on accumulated extra hours: The fourth shift model may feature long working hours and may hence only be of advantage when a high limit on accumulated extra hours is chosen. Our method applies design-of-experiment methods to handle International Journal of Production Research 3 the myriad of possible specifications of a single volume flexibility instrument and the multitude of combinations of different instruments. The analysis is framed in a factorial design of experiment that allows us to scrutinize interactions between instruments.

As volume flexibility is one answer amongst others to cope with demand fluctuations we briefly discuss why volume flexibility is favourable in Section 2. In Section 3 we clarify the difference between measurement and valuation of volume flexibility and review related literature. In Section 4 we describe our method which we subsequently apply in Section 5 in a case study conducted with our industry partner, a major European car manufacturer. For an assembly line we present a model that accounts for five volume flexibility instruments which are valued by design-of-experiments-methods. While the case study is based on known demand fluctuations, an extension of our approach to uncertain demand forecasts is outlined in Section 6. A summary and an outlook conclude this paper in Section 7.

The need for volume flexibility

In order to cope with demand variability it is crucial to recall the reasons for and the patterns of this phenomenon. Quite often seasonal fluctuations in demand are a characteristic of a product. Think of scarves that are rather sold in autumn and winter than in hot summer months. A second major cause for variable demand for a product is the life cycle of the product itself, i.e. the typical rise and fall of the demand curve between product launch and market exit. Third, the business cycle causes fluctuations in aggregate demand for a product, and hence affects the number of orders arriving at every single producer. Impacts stemming from the three sources mentioned so far can be forecasted to some extent. However, there is a fourth reason for demand volatility that comprises short-term effects such as price shocks, tax cuts and hikes, devastating test reports or sales promotion efforts of competitors. Those effects do not reveal a typical pattern or occur on a regular basis and are therefore difficult to anticipate.

Bearing these causes in mind a firm may manage both certain and random demand variability in four different ways. At first, a firm might operate at a constant output level and meet excess demand by inventory or with delay when backlogs occur. Secondly, a firm could try to eliminate demand fluctuations by engaging in marketing activities applying pricing and promotion policies. This idea stems from the concept to reduce or even eliminate the need for flexibility by controlling the environment (cf. Simon 1977, pp. 25 et seqq.). However, both strategies have a limited scope and do not fulfil customer needs. Thirdly, a firm producing several products could balance its product portfolio in such a way that the demand fluctuations of the single commodities cause almost constant total capacity requirements. This strategy necessitates product mix flexibility, which is defined as the ability to produce different items at the same time and at changing ratios over time (cf. Gerwin 1987, p. 39, and[START_REF] Gottschalk | Manufacturing flexibility -A concept for visualization and analysis of adaptation strategies in manufacturing[END_REF]. Though, if a firm manages its portfolio primary for the purpose of constant capacity usage it may ignore its strengths and fail to meet customer needs.

The fourth way to encounter demand fluctuations is to use volume flexibility instruments in order to adjust capacity to demand. Establishing volume flexibility instruments basically means to reduce fixed costs. Hence, if the firm adjusts its capacity upwards this action is reversible in short time without loss and if the firm decreases its capacity by say 10% the costs incurred so far will fall by almost 10%. [START_REF] Faißt | Instrumente des Arbeitszeitmanagements[END_REF] analyses volume flexibility instruments that are related to workforce flexibility. [START_REF] Askar | Flexibility planning in automotive plants[END_REF] The shortcomings of the three alternatives mentioned first make volume flexibility a favourable feature. Furthermore, the benefit of volume flexibility rose in recent years. Over the last decades demand became more and more volatile due to faster technological progress and quicker changing customer preferences as a consequence of globally interlinked societies. This development called for a higher degree of volume flexibility and was reflected in the outcomes of negotiations between employers and employees. [START_REF] Ozaki | Negotiating flexibility: The role of the social partners and the State[END_REF] for example summarizes studies which cover countries across the whole world that give impressive evidence for the increasing flexibility in employment and working time. Already in the mid-eighties manufacturing companies made an effort to boost their capacity adjustment capabilities (cf. [START_REF] Van Deelen | Arbeitszeitmodelle bei der BMW AG[END_REF]. Considering European plants of car manufactures [START_REF] Lehndorf | Fabriken mit langem Atem -Der Wandel der betrieblichen Arbeitszeitorganisation in der europäischen Automobilindustrie[END_REF] gives empirical evidence for the increase of volume flexibility. This development is accompanied by a transition from build-to-stock to mass customization or even build-to-order production in some industry lines (cf. [START_REF] Feitzinger | Mass customization at Hewlett-Packard: The power of postponement[END_REF][START_REF] Feitzinger | Mass customization at Hewlett-Packard: The power of postponement[END_REF][START_REF] Sharma | Marketing in the emerging era of build-to-order manufacturing[END_REF]LaPlaca 2005, respectively). [START_REF] Holweg | Succesful build-to-order strategies start with the customer[END_REF] stress that a build-to-order strategy will fail without volume flexibility. [START_REF] Salvador | Mix flexibility and volume flexibility in a build-to-order environment -synergies and trade-offs[END_REF] also claim that, besides mix flexibility, volume flexibility is a prerequisite for a successful build-to-order strategy. 1Altogether, the developments expose firms to the situation to choose and establish appropriate volume flexibility instruments. The choice should be based on a valuation of volume flexibility instruments. This task is not easy as 'flexibility is a complex, multidimensional, and hard-to-capture concept' (Sethi and Sethi 1990, p. 289). We shed light on the issues related to flexibility, its measurement and valuation in the next section.

Flexibility: measurement and valuation issues

In this section we clarify the difference between measurement and valuation of flexibility after some general notations on flexibility.

Due to the complex nature of flexibility a large body of related literature exists. Flexibility has been approached by disciplines as economics, decision science, strategical management, and manufacturing management (cf. Upton 1995, p. 207). The latter discipline comes close to our perspective. For surveys within the field of manufacturing management we refer to [START_REF] Gupta | Flexibility of manufacturing systems: Concepts and measurements[END_REF], [START_REF] Sethi | Flexibility in manufacturing: A survey[END_REF][START_REF] Sethi | Flexibility in manufacturing: A survey[END_REF][START_REF] De Toni | Manufacturing flexibility: A literature review[END_REF][START_REF] De Toni | Manufacturing flexibility: A literature review[END_REF]. Even within one discipline there is lack of an univocal nomenclature and a unitary model of flexibility (cf. Sethi and Sethi 1990, p. 289). Thus, it is no surprise that [START_REF] Slack | The flexibility of manufacturing systems[END_REF] who interviewed managers found out that a comprehensive understanding of flexibility is scarce. However, de Groote (1994) provided a framework which could integrate previous models of flexibility and which is applicable in our case, too, as we will demonstrate later on. [START_REF] Slack | The flexibility of manufacturing systems[END_REF] distinguishes two dimensions of flexibility, namely response and range flexibility. Response flexibility accounts for the time (and cost) needed to adjust to change, e.g. to adjust the output level. Range flexibility describes the range of states a system can adopt or -with a slightly different meaning -the variety of available alternatives for the adaptaption of the system to changes. Koste and Malhotra (1999, p. 78) International Journal of Production Research 5 system can adopt but also the difference between the states. This difference has to be considered when flexibility is measured. In regard to valuing flexibility, this difference as well as the number of possible states may have no impact. Defining flexibility as the capability to adapt to change, measuring flexibility means to quantify this capability in terms of time, cost, and range. Valuing flexibility, in contrast, means to quantify the financial benefit of this capability, either ex post given historic data or ex ante given a set of realistic scenarios for the development of the firm's environment. To illustrate the distinction we consider a firm that exhibits some volume flexibility instruments. Measuring may certify the firm a high degree of volume flexibility when the instruments enable the firm to quickly adjust output levels over a wide range incurring only moderate costs. Nevertheless, this volume flexibility is not of any value to the firm if it operates in a stable environment where demand is almost constant. In the following we will have a closer look at measures and ways of valuation of volume flexibility.

There are different proposals for measuring volume flexibility (cf. [START_REF] Suarez | An empirical study of manufacturing flexibility in printed circuit board assembly[END_REF]. These proposals include the response as well as the range dimension and often refer to both dimensions (cf. Gerwin 1993, Koste and[START_REF] Koste | A theoretical framework for analyzing the dimensions of manufacturing flexibility[END_REF]. [START_REF] Stigler | Production and distribution in the short run[END_REF] introduced an ordinal measure to compare the volume flexibility of production systems: the flatter the average cost curve of a production system the more volume flexible the system is. Other measures were proposed by [START_REF] Azzone | Comparing manufacturing systems with different flexibility: A new approach[END_REF] and [START_REF] Son | Economic measure of productivity, quality and flexibility in advanced manufacturing systems[END_REF], for example. [START_REF] Gupta | On measurement and valuation of manufacturing flexibility[END_REF] pointed out the difficulties in measuring flexibility and recommended the value of flexibility as a surrogate measure. A high value of flexibility for a firm indicates that its current flexibility is low and vice versa. Recently, [START_REF] Olivella | An entropy-based measurement of working time flexibility[END_REF] assessed working time flexibility, a component of volume flexibility, using three measures, one of them relating to the range dimension and another relating to the response dimension. As it seems there is no mutual consent in the literature. None of the measures proposed in literature including the synthetic ones leads the way. For our method we will take the response as well as the range dimension into account. [START_REF] Gottschalk | Manufacturing flexibility -A concept for visualization and analysis of adaptation strategies in manufacturing[END_REF] propose the ratio of relative change in output to time needed for the change as a key figure to measure response flexibility on a time basis. A similar definition of a key figure can be given with respect to cost. The general definition of those key figures indicates the need for a detailed context-sensitive analysis. Such an analysis is a preliminary step of our method. Though, it is not our goal to get a precise measure. But the analysis helps us to incorporate the relevant aspects of an instrument into our model aimed at valuation.

In regard to valuing flexibility [START_REF] Kulatilaka | The strategic value of flexibility: Reducing the ability to compromise[END_REF] distinguish the strategic value from the option value of flexibility. Whereas the former embodies power in negotiations the latter represents the benefit of adaptation to changing conditions which is the value we are interested in. [START_REF] Kulatilaka | Valuing the flexibility of flexible manufacturing systems[END_REF] presented a stochastic dynamic programming model to maximize the expected net present value of uncertain cash flows. He applied the model to a system offering the flexibility instruments to postpone the initial investment, switch between two operating modes, and abandon the investment. [START_REF] Wild | Die Flexibilität von Betriebsvereinbarungen zur Arbeitszeit[END_REF] evaluates in-plant agreements between employers and employees. The agreements differ from one another with respect to the characteristics of volume flexibility instruments, e.g. the number of shift models and the daily working times. He applies a stochastic dynamic programming model to solve a multi-objective optimization problem. One objective -termed volume flexibility -is to minimize the daily deviation of actual personnel capacity from required capacity to fulfil demand, another one is to minimize [START_REF] Khouja | An aggregate production planning framework for the evaluation of volume flexibility[END_REF] takes on the measure of [START_REF] Stigler | Production and distribution in the short run[END_REF] to determine the optimal investment in volume flexibility. He assumes a quadratic average cost curve. Different degrees of volume flexibility are represented by a single parameter C 1 . The costs accrued for creating a degree of volume flexibility described by C 1 are given by g(C 1 ). An optimal C 1 , i.e. an optimal degree of volume flexibility is determined using Monte Carlo simulation. Though, the method falls short of linking the variable C 1 and the cost function g(C 1 ) to volume flexibility instruments.

Recently [START_REF] Baykasoglu | Quantifying machine flexibility[END_REF] proposed an abstract generic measure of flexibility that incorporates volume flexibility besides other types of flexibility related to a system. Since the measure accounts for uncertainty in the environment of the system it actually is a valuation. Applying this way of valuation asks for heavy aggregation which does not seem suitable for instruments that are repeatedly used over time.

A methodical way to value flexibility instruments under complex conditions was presented by [START_REF] Barad | Flexibility in logistic systems -modeling and performance evaluation[END_REF] who analysed a military logistics problem that comprised combat units located at different positions. They expressed the value of flexibility as the probability that no shortage occurred at combat units during a replenishment cycle. Increasing the number of linkages between combat units enabling shipments between them was one instrument that supported flexibility. For each instrument Barad and Even Sapir considered a low and a high level, e.g. zero and three linkages between combat units. They calculated the shortage probability for all the combinations of levels of the instruments. Though only a small number of levels was investigated, they could draw conclusions in regard to the effect of instruments and their interactions because the combinations were chosen systematically using design-of-experiments methods. In the next section we will refine this method and explain the power of systematically varying the availability of instruments for an efficient analysis. An additional contribution of our paper is that we combine this method with optimization techniques.

Methodology

In this section we outline our method that determines the impact of different volume flexibility instruments and their interactions. We apply design-of-experiments methods that are widely used in the fields of science and engineering when the impact of several input variables on a response variable has to be estimated. Design-of-experiments methods are characterized by purposefully altering the input variables to gain maximum insight with a small number of experiments (cf. [START_REF] Montgomery | Design and analysis of experiments[END_REF][START_REF] Box | Statistics for experimenters: design, innovation, and discovery[END_REF][START_REF] Kleppmann | Taschenbuch Versuchsplanung: Produkte und Prozesse optimieren[END_REF], for an application in logistics cf. [START_REF] Barad | Flexibility in logistic systems -modeling and performance evaluation[END_REF], for an application to in-store sales promotion cf. [START_REF] Bell | A Placket-Burmann experiment to increase supermarket sales of a national magazine[END_REF].

Our input variables, also called factors, are the volume flexibility instruments introduced in Section 1, e.g. extra hours. For each factor only a small set of levels is considered, e.g. 100 (t 1 ) and 200 (t 2 ) hours for the upper limit of the hours bank. The response variable is the minimum total cost (π) incurred by the production schedule which exploits the factor levels. Coming back to the framework of de Groote (1994) mentioned in Section 3 and using his notation the benefit of a higher upper limit for the hours bank can be expressed as the difference π(t 2 , e)π(t 1 , e) where e denotes a particular environment, in our case it represents a particular demand. Each environment e is characterized by its As a rule it is not possible to conduct an empiric analysis to value several volume flexibility instruments because time series data of a production system represent only a few degrees of volume flexibility and exhibit many variables not controlled for. Therefore, we conduct an experimental analysis that allows us to systematically vary factor levels for an arbitrary set of instruments. Since the time horizon has to span at least one year due to seasonal effects and the damage potential of improper factor levels is tremendous, experiments cannot be carried out on real production systems. Hence, we carry them out on a mathematical model that incorporates the relevant characteristics of the production system.

Design-of-experiments methods are typically used when a model of the considered system does not exist. They allow to identify the relevant factors affecting a dependent stochastic response variable. The identified factors are included in a preliminary model. In contrast, our method requires a complete model of the system including potentially relevant factors. Furthermore, such a complete model may lack stochastic response variables and may hence be deterministic, i.e. the numerical value of the response variable can be calculated given values for the factors. Nevertheless, design-of-experiments methods are still necessary to identify relevant factors because the complexity of the model inhibits a thorough understanding of the consequences of changes in factor levels. These changes are model parameter changes that are usually examined applying sensitivity analysis. And that is just what design-of-experiments methods provide by purposefully altering the input variables. Hence, a model and the experiments are the crucial features of our method that comprises three steps.

Step 1: Preliminary analysis A preliminary analysis is required to model the production system including the volume flexibility instruments properly. At the outset volume flexibility instruments deemed to be suitable in the broadest sense ought to be selected. This preliminary selection of instruments should not be too restrictive as it is for the experiments to find out the really suitable ones. The analysis of instruments must comprise the response aspect as well as the range aspect. In regard to response flexibility costs are crucial. Note that 'fixed' costs are incurred for creating and maintaining volume flexibility and 'variable' costs arise when a volume flexibility instrument is utilized (cf. Eversheim and Schaefer 1980, pp. 246 et seq.). The range flexibility offered by a volume flexibility instrument can be expressed in product units and number of output levels and the step size. It has to be compared with the expected range of product demand and the expected levels of product demand, respectively. The analysis can enable us to judge whether a volume flexibility instrument should be taken into account or not. Though, an instrument should only be excluded if it is dominated by other instruments. This, however, is not easy to see due to the multitude of dimensions of flexibility and the complexity of single dimensions (e.g. of range flexibility). Hence, the main benefit of an detailed analysis is the basis it provides for the model needed for valuation.

Step 2: Model building and algorithm design

Next, a mathematical model has to be developed which captures the features of all the volume flexibility instruments. The decision variables of the model will fix the production schedule and determine the number of hires and fires etc., i.e. the decision variables will describe the usage of the volume flexibility instruments. The objective function has to consider all relevant costs that arise due to the use of instruments, e.g. paid extra hours, shift allowances and costs for changing the production speed. Furthermore, the objective function has to account for relevant costs incurred for excess capacity or loss of demand.

As real life production systems can differ considerably from one another, a model has to match the special features of the considered production system. The model type and the problem size will affect the choice of an algorithm. In case of a mixed integer program a branch-and-bound procedure or a heuristic method could be applied. In case of non-linear restrictions dynamic programming can be adequate. The task of the algorithm is to determine a feasible production schedule that minimizes the objective function.

Step 3: Applying design-of-experiments methods First, we consider the 'simple case' where we obtain a linear optimization problem P and instrument levels are represented by right-hand sides b i of restrictions i ∈ {1, . . . , m} (e.g. extra hours ≤ 100h). In such cases solving the dual problem P may provide sufficient information. The optimal dual value corresponding to a restriction of P, the so called shadow price, reflects the savings in production costs when the restrictions' right-hand side is increased by one unit (e.g. from 100h to 101h). For well-behaving linear problems the total effect of increasing two right-hand side values b i is equal to the sum of the corresponding dual values because the second derivative of the dual objective function with respect to any combination of two b i is equal to zero, i.e. interaction effects do not occur. For ill-behaving problems though, an interaction effect may exist for two instruments whose dual values are equal to zero. Furthermore, in regard to Lagrangian dual problems this approach does not work for a wide range of problems such as many mixed integer non-linear programs, where a duality gap exists.

For those cases we suggest a (fractional) factorial experiment. For this experiment we consider each instrument at two levels. The low level refers to the less flexible specification of an instrument. This specification can even represent the lack of the instrument. The high level refers to the more flexible state.

Experimental runs are conducted for different combinations of factor levels to determine the corresponding system response. In our case the experiment is conducted as a series of optimization runs. The minimum total cost of production obtained represents the response that is used to estimate factor effects. If the response is a stochastic variable due to stochastic input data such as an absenteeism rate, for each level combination several runs should be conducted to prevent erroneous conclusions. If the optimization problem is deterministic or the solution of replicated runs is too time-consuming only one response per level combination will be determined.

For the experiment we use the following notation: Let k be the number of factors and for each factor i = 1, . . . , k a low level (-) and a high level (+) are specified.

An one-factor-at-a-time experiment starts with a baseline run where all k factors are at their low level. Then k runs are executed in which the factors one by one are set to their high level while the other k -1 factors stick to the low level. The main drawback of an one-factor-at-a-time experiment is that interactions between instruments cannot be captured. To remedy this shortcoming factorial designs can be applied. In full factorial designs runs for every level combination are executed. In fractional factorial designs only a fraction of the complete set of runs is conducted where the runs are systematically chosen to yield as much information as possible. Full and fractional factorial designs facilitate a multidimensional sensitivity analysis.

As an example we consider an experiment with k = 3 factors A, B, and C, each at two levels. A full factorial design for these factors is called a 2 k = 2 3 design that comprises eight runs for a single replicate as depicted in Table 1.

[Table 1 about here.] The main effect of factor i is defined as the average difference in the response of runs with i at its high level and with i at its low level. When the run label represents the corresponding response value, the effect of factor A, denoted by A, is calculated as follows:

A = a + ab + ac + abc 4 - (1) + b + c + bc 4
If significant interaction occurs the experimenter cannot rely on the main effect as it strongly depends on the level of other factors. To calculate an interaction effect consider factor A and B. Their interaction effect, denoted by AB , measures the difference between the effect of A when B is at its high level and the effect of A when B is at its low level. Note that AB also measures the difference between the effect of B when A is at its high level and the effect of B when A is at its low level.

AB = (abc-bc)+(ab-b) 2 -(ac-c)+(a-(1)) 2 2 = (abc-ac)+(ab-a) 2 -(bc-c)+(b-(1)) 2 2
The interaction effect ABC describes the average difference between the interaction effect AB when C is at its high level and the effect AB when C is at its low level. The signs of the responses required to calculate interaction effects are shown in the corresponding columns of Table 1.

The relation between response value y and level combinations (x A , x B , x C ) can also be described by a multiple linear regression model, where x i = -1 refers to the low level of factor i and x i = 1 to the high level of factor i:

y = β 0 + β 1 x A + β 2 x B + β 3 x C + β 4 x A x B + β 5 x A x C + β 6 x B x C + β 7 x A x B x C + ǫ
For discrete factors x i is restricted to the values 1 and -1. For continuous factors it is bounded by these values assuming a linear relationship between the factor and the dependent variable y within the boundaries. The β-values have to be estimated from the observations y. β 0 can best be estimated by the average response of all runs, β 1 , . . . , β 7 are best estimated by the half of the corresponding effect, e.g. β 4 = 1 2 AB . In case of a stochastic response and replicated runs, ǫ represents the random error, i.e. the residual between the observed response and the response determined by the model. The residuals are also not equal to zero when high order interactions have been excluded from the model. This exclusion is common as interactions of more than two factors are very rare.

To identify significant effects we check if the hypothesis β 1 = . . . = β 7 = 0 can be rejected. To accomplish this we use analysis of variance techniques where the total sum of squares, i.e. the total squared deviation of single responses from the mean response value, is partitioned into deviations that derive from effects and random deviations. A test value is defined by the ratio of the mean sum of squares related to effects to the mean sum of squares related to random error. The test value is compared to a quantile of the F distribution. If the mean deviation caused by an effect exceeds the mean random variability in responses to a certain degree, i.e. if the test value is greater than the quantile of the F distribution it can be concluded that this effect is significant.

As the number of runs required for a full factorial design grows exponentially in the number of factors it may be efficient to use fractional factorial designs. Having outlined our methodology we will apply it to volume flexibility instruments considered for a real life production system in the following section.

Case study

Our case study stems from the production system of a major European car manufacturer who assembles cars to fulfil orders and forecasted demand. We consider a paced assembly line with identical unit cycle time at every station. The model of the production system presented below is described in detail by [START_REF] Askar | Optimierte Flexibilitätsnutzung in Automobilwerken[END_REF]. It is used for aggregate planning, i.e. for simultaneous planning of the workforce level and production volume in every period when demand is known or estimated in advance (cf. Nahmias 2005, pp. 108 et seqq.). Hence, we focus on the value of volume flexibility instruments with respect to predictable changes in demand. Afterwards we outline how the approach can be extended to the case of uncertain demand forecasts.

Preliminary analysis

To model the assembly process and to select volume flexibility instruments that may decrease production costs we have to identify the parameters that determine the output level and the related costs. Main parameters are the cycle time and the operating time. The latter is related to the workforce size as working time regulations have to be obeyed. We establish mathematical relations between the parameters and the output level on the one hand and between the parameters and the costs on the other hand. Each parameter is constrained by technical conditions or contracts that constitute the degree of volume flexibility (cf. Figure 1).

[Figure 1 about here.] For the planning horizon of two years our industry partner considered two demand scenarios that reveal typical seasonal patterns of a car market (cf. Figure 2). The management identified five instruments A, . . . , E promising a cheaper production plan: total workforce is increased from 10% to 15%.

[Figure 2 about here.]

Instruments A and B enable the car manufacturer to reach more output levels. Implementing both options might be redundant. Instruments C and D allow the firm to choose shift models with high working times for a longer time period without hiring additional workers to facilitate shifts off. Instrument E is supposed to decrease costs in two ways: The substitution effect of the instrument allows to replace some permanent workers by less paid temporary workers, the adaptation effect allows to better adjust workforce size, especially in downturns because permanent workers are usually not laid off.

To assess these five instruments a model is required that determines production costs depending on the availability of the instruments. We present such a model in the next section.

Optimization model

The notation used in the optimization model is summarised in Table 2. We consider the production of a car on a single model assembly line. The demand for all periods t ∈ T is given by D t . The quantity produced in period t is denoted by P t , it may differ from D t to allow for smoothing. However, the cumulated deviation is limited by an upper bound UB t and a lower bound LB t (cf. Constraints (3)). The production volume is limited by the available capacity which depends on the unit cycle time ct t ∈ CT and the operating time TO t,smt that takes into account calendar information and the shift model sm t ∈ SM selected for period t (Constr. ( 4)). A shift model sm defines which shift groups SG ∈ SG(sm) have to work. The set of shift groups comprises the early shift, the late shift, and the night shift. Additionally, a shift model specifies at which days and how many hours a shift group has to work and the required share of floaters, denoted by ShF sm,SG , who replace workers during their breaks.

The required workforce level essentially depends on the number of stations necessary to assemble a car. Let TA t denote the assembly time of a representative car, the required net workforce level W net t,SG for shift group SG of shift model sm t can be calculated from Constr. (5). Additional workers are required to compensate for workers who are on sick leave and vacation. When the share of absent workers is expressed by ShA t , we obtain the total staff demand per period W total t from Constr. ( 6). The required number of workers in period t has to be met by permanent staff WP t and temporary staff WT t (Constr. ( 7)). Let the hirings and dismissals in period t be denoted by ∆WP t for permanent workers and ∆WT t for temporary workers. These decision variables allow to formulate flow Constr. ( 8) to link workforce levels of consecutive periods. For temporary workers the corresponding flow equations hold. Initial values for t = 0 are known. Due to limited training capacity the number of employments per period must not exceed the level WCap t (Constr. ( 9)). The share of temporary workers is limited to ensure a certain quality level and to comply with in-plant agreements reached with the staff association. Let ShT t denote the maximum share of temporary workers. Then Constr. (10) have to hold for every period.

The actual total working time TW act t that is performed in period t is calculated from the actual working times of each shift group SG, denoted by TW act t,SG , taking into account the number of workers needed on the shop floor (Constr. ( 11)). Let TH contr t be the number of hours a worker is supposed to work per period according to his labor contract or a collective agreement. The total sum of contracted hours TW contr t in period t is calculated 12)). In general, the actual working time does not equal the contracted hours. The difference is recorded on an hours bank where overtime and shifts off are averaged over the workforce. The hours bank balance is denoted by THB t . Workers who are laid off at the beginning of period t are paid for their overtime. Consequently, their overtime is cancelled from the hours bank (Constr. ( 13)). Note that Constr. ( 13) are non-linear constraints. The recorded extra time must not exceed an upper limit THB UL t and after a certain period of time the hours bank has to be balanced, i.e. accumulated extra hours have to be compensated for by time off. Reducing overtime can be managed by hiring additional workers that allow shifts off for a part of the workforce. The upper limit as well as the regular balancing can be expressed by Constr. ( 14).

The costs C t incurred in period t can be partitioned in variable production costs including costs for smoothing, C prod 15)). Apart from variable unit production costs c v t we consider costs for earliness, c e t , and tardiness, c t t , i.e. costs for storing and shortage costs (Constr. ( 16)). When the wages are calculated we account for wage differences between permanent and temporary workers who earn WageP t and WageT t , respectively (Constr. ( 17)). To derive the costs for shift allowances we distinguish several shift types ST ∈ ST (sm t ), namely early, late, night, and Saturday early shift. For each shift type ST the actual working hours TW t,ST are determined. Again, we discriminate between permanent and temporary workers. The shift allowances AP ST and AT ST for permanent workers and temporary workers, respectively, depend on the shift type ST (Constr. ( 18)). If the workforce level is reduced and extra hours were accumulated on the hours bank, the permanent and temporary workers who leave the company receive WageP

Minimize y

(1) subject to International Journal of Production Research 13

y = t∈T C t (1 + i) -t (2) t τ =1 D τ -LB t ≤ t τ =1 P τ ≤ t τ =1 D τ + UB t t ∈ T (3)
P t ≤ TO t,smt ct t t ∈ T (4) W net t,SG = TA t ct t (1 + ShF smt,SG ) SG ∈ SG(sm t ), t ∈ T (5)
W total t = SG∈SG(smt ) W net t,SG (1 -ShA t ) t ∈ T (6) WP t + WT t ≥ W total t t ∈ T (7) WP t = WP t-1 + ∆WP t , WT t = WT t-1 + ∆WT t t ∈ T (8)
∆WP + t + ∆WT + t ≤ WCap t t ∈ T (9) WT t ≤ WP t ShT t 1 -ShT t t ∈ T (10)
TW act t = SG∈SG(smt ) W net t,SG TW act t,SG t ∈ T (11)
TW contr t = TH contr t (WP t + WT t ) (1 -ShA t ) t ∈ T ( 12
)
THB t = TW act t -TW contr t WP t + WT t + THB t-1 min {WP t-1 + WT t-1 , WP t + WT t } WP t + WT t t ∈ T ( 13
)
THB t ≤ THB UL t t ∈ T (14)
C t = C prod t + C wage t + C allow t + C ovti t + C change t t ∈ T (15)
C prod t = c v t P t + c e t t τ =1 P τ - t τ =1 D τ + + c t t t τ =1 D τ - t τ =1 P τ + t ∈ T (16)
C wage t = WageP t WP t + WageT t WT t t ∈ T (17)
C allow t = ST ∈ST (smt ) TW act t,ST W net t,SG(ST ) WP t AP ST + WT t AT ST WP t + WT t t ∈ T (18)
C ovti t = THB + t-1 (WP t-1 -WP t ) + WageP ovti t + (WT t-1 -WT t ) + WageT ovti t t ∈ T (19)
C change t = C change,ct t + C change,sm t + C change,SG t + C change,W t t ∈ T (20) sm t ∈ SM, ct t ∈ CT , ∆WP t ∈ Z, ∆WT t ∈ Z t ∈ T (21)
[Table 2 about here.]

Theorem 5.1 : Problem (1) -( 21) is NP-hard.

Proof : By restriction we demonstrate that the outlined problem contains a weightconstrained shortest path problem (WCSPP) which is known to be NP-hard (cf. Garey and Johnson 1979, p. 214). Let the states of the production system reachable in every period correspond to the set V of nodes of a directed graph G(V, A, l, w). The states are characterized by the components sm t , ct t , WP t , and WT t with t ∈ T = {1, . . . , T }. Additionally, V contains a single source node s associated to t = 0. Arcs a ∈ A connect nodes of period t with nodes of period t + 1 for t = 0, . . . , T -1. Hence, the graph is free of cycles. The arcs are associated with lengths l and weights w. The lengths l correspond to costs for wages, allowances etc. The weights w correspond to the change in accumulated overtime between any two states and can be negative. Let THB UL be a constant upper bound on the total weight. Finding a feasible production plan with minimum total costs for our problem requires to determine a shortest path from s to each of the nodes reachable in period T with a total weight not greater than THB UL in every period.

Hassin (1992) stated pseudo-polynomial algorithms based on dynamic programming to solve the WCSPP. To solve our problem we apply the dynamic programming approach described by [START_REF] Askar | Optimierte Flexibilitätsnutzung in Automobilwerken[END_REF] where the time periods t are associated with the stages. The state space is constituted by the variables sm t , ct t , WP t , WT t and augmented by additional state variables THB t and t τ =1 (P τ -D τ ) required to record the accumulated overtime and the accumulated deviation of production from demand (cf. Bertsekas 2000, pp. 34 et seqq.). Further state variables assure that a certain number of periods elapses between consecutive changes of shift models and cycle times. The algorithm is not exact because discretization is used to reduce the number of states and decisions: The workforce level is discretized, for recorded extra hours and for the cumulated deviation of production from demand intervals are distinguished. The step size of discretization determines the trade-off between solution quality and solution time. Since our problem has not to be solved weekly a 'near-exact' algorithm that emphasizes solution quality seems suitable.

Design-of-Experiments analysis

With the experimental designs outlined in Section 4 we investigate the five instruments deemed suitable for the car manufacturer. Table 3 summarizes the characteristics of the instruments.

[Table 3

about here.]

For the full factorial design 2 5 = 32 runs had to be conducted for each demand scenario. The design and the results of the runs for scenario 1 are depicted in Table 4.

[Table 4 about here.] Since we consider a deterministic case a single replicate of the runs is exhaustive. However, a single replicate does not provide a random error which is required for an analysis of variance. To circumvent this problem, effects of high-order interactions can be pooled as an estimate of error because main effects and low-order interactions usually have a dominant impact on a system response. Table 5 shows the analysis of variance for a model that only accounts for main effects and 2-factor interactions. The analysis indicates that the main effects A, B, D, and E as well as the interactions AB and BD are significant, while instrument C does not exhibit an effect on costs.

[Table 5 about here.] Though, adopting this approach runs the risk of missing significant high-order interactions. Therefore, in the first step of our analysis we use a normal probability plot to International Journal of Production Research graphically identify the major effects.1 For this plot effects are ordered according to magnitude. Then each effect is plotted against its cumulative percentage. Negligible effects are supposed to be normally distributed with mean zero. Hence, they will fall along a straight line in the plot, whereas important effects have non-zero means and stand out from the straight line. Figure 3 depicts the normal probability plot for scenario 1 that does not exhibit significant high-order interactions.

[Figure 3 about here.]

In the next step the important effects A, B, D, E, AB, and BD that stand out from the straight line in Figure 3 are considered in the regression model whereas negligible ones are pooled as an estimate of error. The estimated coefficients of the important effects are stated in Table 6.

[Table 6 about here.]

The factors A, B, D and E for themselves and interaction BD decrease costs, their coefficient estimate is negative. Interaction AB exhibits a small positive coefficient estimate as it marginally increases costs. The largest savings by far can be realized when instrument B, an additional cycle time, is available. On average, exploiting this option yields a saving of 7.2 million ¤ (twice the coefficient estimate). Dominant interaction effects would favor different levels for the corresponding factors. However, interactions do not play a major role here. The joint availability of instruments A and B exhibits low redundancy. Altogether factors A and B decrease costs. Hence, the high level can be chosen for both factors. Interaction effect BD is also small: Cycle time changes are slightly supported by a longer balancing period as higher production speed allows the use of a 2-shift model with long working hours instead of a 3-shift model with less working time for an individual worker. Similar results are obtained for demand scenario 2. In both cases instruments B, E, and A facilitate large cost savings. If more than one scenario for future demand is likely decision makers can for example weight the results of each scenario or apply a minimax decision rule to minimize the maximum possible cost given a negotiation strategy or a budget for investing in additional flexibility instruments.

Extension to uncertain demand forecasts

So far, we have assumed for each scenario that the corresponding demand is known. However, for a long planning horizon demand is usually a random variable and uncertainty about demand is increasing in time. Hence demand forecasts are prone to errors. A manufacturing firm may not revise its production schedule each day where production deviates from demand. Yet, when a forecast is updated the production plan can be adjusted to new information using volume flexibility instruments. The quality of this adjustment depends on the available instruments and on their features. If the option to change the cycle time is available this option offers more flexibility in case the cycle time can be changed at short notice than if prior notice has to be given well in advance.

To value volume flexibility instruments in such a setting our approach can be adapted as follows. An experimental run for a combination of factor levels can be conducted as a simulation run into which the optimization procedure of Section 5.2 is embedded. The simulation starts with a production plan that results from an optimization subject to the initial demand forecast. We randomly determine the number of forecast updates N < |T | and the periods t f u n , n = 1, 2, . . . , N , where a forecast update is obtained. For each period t f u n a reoptimization is executed taking into account that some variables are fixed for a certain number of subsequent periods since notice has to be given in advance. The revised production plan is executed until period t f u n+1 . For each simulation run the total costs including costs for deviation of production volumes from the forecasted demand values are recorded as the response value y. Replicates for each level combination are easily obtained as stochastic simulation is used. Utilising the same random seed for each level combination and classifying stochastic forecast values as a random error, design analysis is straight forward.

Since we consider instruments that are used on a tactical planning level we propose to evaluate instruments with respect to demand forecast changes. However, short-term volume flexibility should be measured with respect to changes in actual demand.

Summary

In this work we put forward arguments why volume flexibility is of great advantage for many firms which are confronted with demand variability. Manager must select volume flexibility instruments for implementation which accrues costs because investments in technical equipment or compromises with unions are required. To support their decisions we propose a three-step method to value flexibility instruments. The method is based on a preliminary analysis and an optimization model of the considered production system and applies design-of-experiments techniques to identify the most useful instruments in an efficient way. A full or fractional factorial design allows to perceive interaction effects between instruments that may render a joint implementation of instruments either very valuable or redundant.

In a case study we applied the method to an assembly line of a major European car manufacturer. We assumed that demand changes were known in advance. Out of five instruments we found one very valuable instrument and one of no value. Furthermore, serious interactions did not occur in that case. We repeated the analysis for a slight variation of demand. The results were robust to this second demand scenario that represented another likely future market development.

Our method is also suited for unexpected changes in demand that can be integrated into our approach by simulation in combination with reoptimization (rescheduling). A stochastic input would provide replicates for each factor level combination and would favor fractional over full factorial designs. Comparing the relative value of a volume flexibility instrument for different firms would be an avenue for future research because our work suggests that the relative value is strongly related to firm specific parameters.

F

List of Figures

  Evaluating volume flexibility instruments by design-ofexperiments methods

  for fix period wages, shift allowances and extra hours. His model can be used to value volume flexibility instruments by setting an aspiration level for the deviation objective and minimizing labour costs.

A

  Additional cycle time: Instead of only one unit cycle time, instrument A offers the opportunity to switch to a smaller cycle time, i.e. to a higher production speed. B Additional shift models: In the inflexible state two 2-shift models and one 3-shift model are available. In the flexible state two additional 2-shift models and two additional 3-shift models are available offering distinguished operating and working times. C Allowing more extra hours: Instrument C increases the maximum number of accumulated extra hours from 100 to 200 hours. D Longer balancing period: In the flexible state the period within which the hours bank has to be balanced is expanded from one year to two years. E Allowing more temporary workers: The maximum share of temporary workers in the

  the whole workforce except for those workers who are on vacation or on sick leave (Constr. (

  g. for night and Saturday shifts, costs for compensating for overtime, C ovti t , and costs for changes C change (Constr. (

  ovti

  every extra hour(Constr. (19)). Finally, we consider costs that arise whenever a variable changes from period to period. These change costs comprise costs for altering the unit cycle time, C change,ct t , the shift model, C change,sm t , the number of shift groups, C change,SG t , and a change in the workforce level, C change,W t (Constr. (20)). The decision variables are constrained in Constr. (21). The overall aim stated in the objective function (1) is to minimize the total discounted cost of production, y, for the planning horizon. Denoting the cost of capital by i the discounted costs y are defined in Constr. (2).

Figure 2 .Figure 3 .

 23 Figure 2. Investigated demand scenarios (solid line: scenario 1, dotted line: scenario 2)

Figure

  Figure 1.Objects of the preliminary analysis

  Figure 1.Objects of the preliminary analysis

Table 2 .

 2 Glossary of symbols (alphabetically ordered in each subsection)

	o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y

Table 3 .

 3 Selected instruments for the analysis

	Instrument Description	Inflexible state (-) Flexible state (+)
	A	Number of shift models	two 2-shift models, four 2-shift models
			one 3-shift model	three 3-shift models
	B	Number of cycle times	one	two
	C	Upper limit on the maximum	100 hours	200 hours
		number of accumulated extra hours
	D	Length of balancing period	one year	two years
		of the hours bank
	E	Maximum share of temporary workers 10%	15%

Table 4 .

 4 Design and results of the 2 5 full factorial experiment

				Factor levels		Response
		A	B	C	D	E
		Shift	Cycle Extra Balancing Temporary	Costs
	Run models times hours	period	workers	in ¤
	1	-	-	-	-	-	160,819,583
	2	+	-	-	-	-	158,208,635
	3	-	+	-	-	-	153,411,104
	4	+	+	-	-	-	151,484,794
	5	-	-	+	-	-	160,819,583
	6	+	-	+	-	-	158,208,635
	7	-	+	+	-	-	153,411,104
	8	+	+	+	-	-	151,484,794
	9	-	-	-	+	-	160,803,175
	10	+	-	-	+	-	157,195,206
	11	-	+	-	+	-	152,382,054
	12	+	+	-	+	-	150,475,420
	13	-	-	+	+	-	159,452,827
	14	+	-	+	+	-	157,094,095
	15	-	+	+	+	-	151,549,638
	16	+	+	+	+	-	150,324,720
	17	-	-	-	-	+	156,890,731
	18	+	-	-	-	+	153,918,834
	19	-	+	-	-	+	150,243,582
	20	+	+	-	-	+	148,113,986
	21	-	-	+	-	+	156,890,731
	22	+	-	+	-	+	153,918,834
	23	-	+	+	-	+	150,243,582
	24	+	+	+	-	+	148,113,986
	25	-	-	-	+	+	157,283,229
	26	+	-	-	+	+	153,733,287
	27	-	+	-	+	+	148,104,869
	28	+	+	-	+	+	147,150,140
	29	-	-	+	+	+	157,109,899
	30	+	-	+	+	+	153,833,385
	31	-	+	+	+	+	148,091,044
	32	+	+	+	+	+	147,150,140

Table 5 .

 5 Analysis of variance for scenario 1 (high-order interactions pooled as an estimate of error) = 99.58%, adjusted R 2 = 99.18%, * and bold p-value indicate significance at a level of α =5%

	Degrees of

Table 6 .

 6 Regression analysis for the important effects = 99.38%, adjusted R 2 = 99.23%, bold p-value indicates significance at a level of α =5%

	Effect	Coefficient estimate Standard error	t-value	p-value
	Average response	153, 684, 863	66, 396 2314.65	< 0.001
	A: Shift models	-1, 159, 308	66, 396	-17.46	< 0.001
	B: Cycle times	-3, 576, 429	66, 396	-53.86	< 0.001
	D: Balancing period	-451, 543	66, 396	-6.80	< 0.001
	E: Temporary workers	-1, 760, 472	66, 396	-26.51	< 0.001
	AB	338, 120	66, 396	5.09	< 0.001
	BD	-253, 389	66, 396	-3.82	0.001
	R 2

A noteworthy aspect of the qualitative empiric research of[START_REF] Salvador | Mix flexibility and volume flexibility in a build-to-order environment -synergies and trade-offs[END_REF] is that the authors reveal side effects of steps intended to increase only mix or only volume flexibility. Those side effects can be conflicting or synergistic, i.e. promote one type of flexibility at the expense of the other or support both types.

http://mc.manuscriptcentral.com/tprs Email: ijpr@lboro.ac.ukInternational Journal of Production Research

Half-normal plots can also be used for this purpose and are recommended (cf. Montgomery

2005, p. 227). For reasons of demonstration we choose a normal plot.

Acknowledgements

The authors are grateful to two anonymous referees for their helpful comments and suggestions.