B Herlemont

O Ogievetsky

Petr Kulish

Rings of h-deformed differential operators

We describe the center of the ring Diff h (n) of h-deformed differential operators of type A. We establish an isomorphism between certain localizations of Diff h (n) and the Weyl algebra W n extended by n indeterminates.

Introduction

The ring Diff h (n) of h-deformed differential operators of type A appears in the theory of reduction algebras. A reduction algebra R A g provides a tool to study decompositions of representations of an associative algebra A with respect to its subalgebra in the situation when this subalgebra is the universal enveloping algebra of a reductive Lie algebra g [M, AST]. We refer to [T, Zh] for the general theory and uses of reduction algebras.

Decompositions of tensor products of representations of a reductive Lie algebra g is a particular case of a restriction problem, associated to the diagonal embedding of U(g) into U(g) ⊗ U(g). The corresponding reduction algebra, denoted D(g), is called "diagonal reduction algebra" [START_REF] Khoroshkin | Diagonal reduction algebras of gl type[END_REF]. A description of the diagonal reduction algebra D(gl n ) in terms of generators and (ordering) defining relations was given in [START_REF] Khoroshkin | Diagonal reduction algebras of gl type[END_REF][START_REF] Khoroshkin | Structure constants of diagonal reduction algebras of gl type[END_REF].

The diagonal reduction algebra D(gl n ) admits an analogue of the "oscillator realization", in the rings Diff h (n, N ), N = 1, 2, 3, . . . , of h-deformed differential operators, see [START_REF] Khoroshkin | Diagonal reduction algebra and reflection equation[END_REF]. The ring Diff h (n, N ) can be obtained by the reduction of the ring of differential operators in nN variables (that is, of the Weyl algebra W nN = W ⊗N n ) with respect to the natural action of gl n . Similarly to the ring of q-differential operators [WZ], the algebra Diff h (n, N ) can be described in the R-matrix formalism. The R-matrix, needed here, is a solution of the so-called dynamical Yang-Baxter equation (we refer to [F, GV, ES] for different aspects of the dynamical Yang-Baxter equation and its solutions).

The ring Diff h (n, N ) is formed by N copies of the ring Diff h (n) = Diff h (n, 1). The aim of the present article is to investigate the structure of the ring Diff h (n). Our first result is the description of the center of Diff h (n): it is a ring of polynomials in n generators.

As follows from the results of [START_REF] Khoroshkin | Rings of fractions of reduction algebras[END_REF], the ring Diff h (n) is a noetherian Ore domain. It is therefore natural to investigate its field of fractions and test the validity of the Gelfand-Kirillov-like conjecture [GK]. For the ring of q-differential operators, the isomorphism (up to a certain localization and completion) with the Weyl algebra was given in [O]. The second result of the present article consists in a construction of an isomorphism between certain localizations of Diff h (n) and the Weyl algebra W n extended by n indeterminates. In particular, our isomorphism implies the isomorphism of the corresponding fields of fractions.

According to the general theory of reduction algebras, the ring Diff h (n) admits the action of Zhelobenko operators [START_REF] Zhelobenko | Representations of reductive Lie algebras[END_REF][START_REF] Khoroshkin | Mickelsson algebras and Zhelobenko operators[END_REF] by automorphisms. The Zhelobenko operators generate the action of the braid group B n . However the Weyl algebra W n admits the action of the symmetric group S n by automorphisms. As a by-product of our construction we define the action of the symmetric group by automorphisms on the ring Diff h (n). Moreover these formulas can be generalized to produce the action of the symmetric group by automorphisms on the rings Diff h (n, N ) for any N , and on the diagonal reduction algebra D(gl n ). We conjecture that the general reduction algebra R A g admits an action, by automorphisms, of the Weyl group of g. Section 2 contains the definition of the ring of h-deformed differential operators and some of their properties used in the sequel. In Section 3 we present a family of n quadratic central elements. We then describe an n-parametric family of "highest weight" representations of Diff h (n) and calculate values of the quadratic central elements in these representations. In Section 4 we introduce the necessary localizations of Diff h (n) and of the Weyl algebra, check the Ore conditions and establish the above mentioned isomorphism of the localized rings. In Section 5 we prove the completeness of the family of central elements constructed in Section 3. Then we describe the action of the symmetric group on Diff h (n, N ) and D(gl n ) as well as the action of the braid group, generated by Zhelobenko operators, on a localization of the Weyl algebra. Also, we present a 2n-parametric family of representations of the algebra Diff h (n) implied by our construction.

Notation

Throughout the paper, k denotes the ground ring of characteristic zero.

The symbol s i stands for the transposition (i, i + 1).

We denote by U(h) the free commutative k-algebra in generators hi , i = 1, . . . , n. Set hij = hihj ∈ h. We define Ū(h) to be the ring of fractions of U(h) with respect to the multiplicative set of denominators, generated by the elements (

h ij + k) -1 , k ∈ Z. Let ψ i := k:k>i hik , ψ i := k:k<i hik and χ i := ψ i ψ i , i = 1, . . . , n .
(1) Let ε j , j = 1, . . . , n, be the elementary translations of the generators of U(h), ε j : hi → hi + δ j i . For an element p ∈ Ū(h) we denote ε j (p) by p[ε j ].

Definition and properties of rings of h-deformed differential operators

Let R = { Rkl ij } n i,j,k,l=1 be a matrix of elements of Ū(h), with nonzero entries

Rij ij = 1 hij , i = j , and Rij ji =      h2 ij -1 h2 ij , i < j, 1 , i ≥ j .
(2)

The matrix R is the standard solution of the dynamical Yang-Baxter equation

a,b,u Rij ab Rbk ur [-ε a ] Rau mn = a,b,u Rjk ab [-ε i ] Ria mu Rub nr [-ε m ] (3) of type A.
The ring Diff h (n) of h-deformed differential operators of type A is a Ū(h)-bimodule with the generators x j and ∂j , j = 1, . . . , n. The ring Diff h (n) is free as a one-sided Ū(h)-module; the left and right Ū(h)-module structures are related by hi

x j = x j ( hi + δ j i ) , hi ∂j = ∂j ( hi -δ j i ) . (4)
The defining relations for the generators x j and ∂j , j = 1, . . . , n, read (see [START_REF] Khoroshkin | Diagonal reduction algebra and reflection equation[END_REF])

x i x j = k,l Rij kl x k x l , ∂i ∂j = k,l Rlk ji ∂k ∂l , x i ∂j = k,l Rki lj [ε k ] ∂k x l -δ i j , (5)
or, in components,

x i x j = hij + 1 hij x j x i , i < j , (6)
∂i ∂j = hij -1 hij ∂j ∂i , i < j , (7)
x i ∂j =        ∂j x i , i < j , hij ( hij -2) ( hij -1) 2 ∂j x i , i > j , (8)
x i ∂i = j 1 1 -hij ∂j x j -1 . (9)
The ring Diff h (n) admits Zhelobenko automorphisms qi , i = 1, . . . , n -1, given by (see [START_REF] Khoroshkin | Diagonal reduction algebra and reflection equation[END_REF])

qi (x i ) = -x i+1 hi,i+1 hi,i+1 -1 , qi (x i+1 ) = x i , qi (x j ) = x j , j = i, i + 1 , qi ( ∂i ) = - hi,i+1 -1 hi,i+1 ∂i+1 , qi ( ∂i+1 ) = ∂i , qi ( ∂j ) = ∂j , j = i, i + 1 , qi ( hj ) = hs i (j) . (10)
The operators qi , i = 1, . . . , n -1, generate the action of the braid group, see [START_REF] Zhelobenko | Representations of reductive Lie algebras[END_REF][START_REF] Khoroshkin | Mickelsson algebras and Zhelobenko operators[END_REF].

The ring Diff h (n) admits an involutive anti-automorphism , defined by

( hi ) = hi , ( ∂i ) = ϕ i x i , (x i ) = ∂i ϕ -1 i , where ϕ i := ψ i ψ i [-ε i ] = k:k>i hik hik -1 , i = 1, . . . , n, (11)
The proof reduces to the formula

ϕ i [-j ] ϕ i = h2 ij -1 h2 ij for 1 ≤ i < j ≤ n .
The construction of central elements in the next Section uses the elements

Γ i := ∂i x i for i = 1, . . . , n . (12)
We collect some properties of these elements.

Lemma 1. We have

(i) Γ i x j = hij + 1 hij x j Γ i and Γ i ∂j = hij -1 hij ∂j Γ i for i = j, i, j = 1, . . . , n.
(ii) qi (Γ j ) = Γ s i (j) for i = 1, . . . , n -1 and j = 1, . . . , n.

(iii) Γ i Γ j = Γ j Γ i for i, j = 1, . . . , n.
Proof. Formulas (i) and (ii) are obtained by a direct calculation; (iii) follows from (i).

We will use the following technical Lemma.

Lemma 2. Let A be an associative algebra. Assume that elements hi , Zi , Zi ∈ A, i = 1, . . . , n, satisfy hi hj = hj hi , hi Zj = Zj ( hi + δ j i ) , hi Zj = Zj ( hi -δ j i ) , i, j = 1, . . . , n .

Let hij := hihj and ψi :=

k:k>i hik , ψ i := k:k<i hik , i = 1, . . . , n .
Assume that the elements hij are invertible. Then (i) the elements Zi satisfy

Zi Zj = hij + 1 hij Zj Zi for i < j , i, j = 1, . . . , n
if and only if any of the two families

{ Z•i } n i=1 or { Z •i } n i=1 where Z•i := ψ i Zi , Z •i := Zi ψ i (13) is commutative; (ii) the elements Zi satisfy Zi Zj = hij -1 hij Zj Zi for i < j , i, j = 1, . . . , n if and only if any of the two families { Z• i } n i=1 or { Z • i } n i=1 where Z• i := ψ i Zi , Z • i := Zi ψ i ( 14
)
is commutative.

Proof. A direct calculation.

3 Quadratic central elements

Let e k := i 1 <•••<i k hi 1 . . . hi k , k = 0, .
. . , n, be the elementary symmetric functions in the variables h1 , . . . , hn . Set

c k := j ∂e k ∂ hj Γ j -e k ,
where Γ j , j = 1, . . . , n, are the elements defined in (12).

It follows from Lemma 1 that qj (c k ) = c k for all j = 1, . . . , n -1 and k = 1, . . . , n.

Proposition 3. The elements c k , k = 1, . . . , n, belong to the center of the ring Diff h (n).

Proof. We shall use the generating functions

e(t) := n k=0 e k t k = i (1 + hi t) and c(t) := n k=1 c k t k = u(t)e(t) + 1 with u(t) := t i 1 1 + hi t Γ i -1 .
The expression u(t) is introduced for convenience; the denominator 1 + hi t, which is not defined in the ring Diff h (n), vanishes in the combination u(t)e(t).

We shall check that the polynomial c(t) is central. We have

x j e(t) = 1 + ( hj -1)t 1 + hj t e(t)x j . ( 15
)
Next, it follows from Lemma 1 that

x j u(t) = k:k =j t 1 + hk t hkj hkj + 1 Γ k + t 1 + ( hj -1)t k 1 1 -hjk Γ k -1 -1 x j .
The coefficient of Γ k in this expression is equal to

1+ hj
1+( hj -1)t for both k = j and k = j. Therefore,

x j u(t) = 1 + hj t 1 + ( hj -1)t u(t)x j . (16)
Combining ( 15) and ( 16) we find that c(t) commutes with x j , j = 1, . . . , n. For ∂j one can either make a parallel calculation or use the anti-automorphism (11).

Lemma 4. (i) The matrix V , defined by V k j :=

∂e j ∂ hk , is invertible. Its inverse is (V -1 ) j i = (-1) j-1 hn-j i χ i ,
where the elements χ i are defined in (1).

(ii) We have

χ j Γ j = hn j -hn j c(-h-1 j ) . ( 17
)
Proof. (i) See, e.g. [OP], Proposition 4.

(ii) Rewrite the equality

c k = j V j k Γ j -e k in the form Γ j = k (V -1 ) k j (c k + e k ) = 1 χ j k (-1) k-1 hn-k j (c k + e k ) = - hn j χ j (c(-h-1 j ) + e(-h-1 j ) -1)
.

Since e(-h-1 j ) = 0, we obtain (17).

Highest weight representations. The ring Diff h (n) admits an n-parametric family of "highest weight" representations. To define them, let D n be an Ū(h)-subring of Diff h (n) generated by { ∂i } n i=1 . Let λ := {λ 1 , . . . , λ n } be a sequence of length n of complex numbers such that λ i -λ j / ∈ Z for all i, j = 1, . . . , n, i = j. Denote by M λ the one-dimensional k-vector space with the basis vector | . Under the specified conditions on λ the formulas Proof. It is sufficient to calculate the value of c k on the highest weight vector | . In terms of generating functions we have to check that

e(t)u(t) : | → - i (1 + (λ i -1)t) | . It follows from [KO5], section 3.3, that Γ j | = χ j [ε j ] χ j | , j = 1, . . . , n .
Therefore we have to check that

e(t) t i 1 1 + hi t χ j [ε j ] χ j -1 : | → - i (1 + (λ i -1)t) | . (18)
We use another formula from [START_REF] Khoroshkin | Diagonal reduction algebra and reflection equation[END_REF] (Note 3 after the proof of Proposition 4.3 in Section 4.2)

l h0 -hl -1 h0 -hl + j 1 h0 -hj χ j [-ε j ] χ j = 1 ,
where h0 is an indeterminate. After the replacements h0 → t -1 and hj → -hj , j = 1, . . . , n, this formula becomes e(t)[-ε] e(t)

+ t j 1 1 + hj t χ j [ε j ] χ j = 1 , where ε = ε 1 + • • • + ε n , which implies (18).

Isomorphism between rings of fractions

It follows from the results of [START_REF] Khoroshkin | Rings of fractions of reduction algebras[END_REF] that the ring Diff h (n) has no zero divisors. Let S x be the multiplicative set generated by x j , j = 1, . . . , n. The set S x satisfies both left and right Ore conditions (see, e.g., [A] for definitions): say, for the left Ore conditions we have to check only that for any x k and a monomial m = ∂i 1 . . . ∂i A x j 1 . . . x j B there exist s ∈ S x and m ∈ Diff h (n) such that sm = mx k . The structure of the commutation relations (6-9) shows that one can choose s = (x k ) ν with sufficiently large ν. Denote by S -1

x Diff h (n) the localization of the ring Diff h (n) with respect to the set S x .

Let W n be the Weyl algebra, the algebra with the generators X j , D j , j = 1, . . . , n, and the defining relations

X i X j = X j X i , D i D j = D j D i , D i X j = δ j i + X j D i , i, j = 1, . . . , n .
Let T be the multiplicative set generated by X j D j -X k D k + , 1 ≤ j < k ≤ n, ∈ Z, and X j , j = 1, . . . , n. The set T satisfies left and right Ore conditions (see [START_REF] Khoroshkin | Rings of fractions of reduction algebras[END_REF], Appendix). Denote by T -1 W n the localization of W n relative to the set T.

Let a 1 , . . . , a n be a family of commuting variables. We shall use the following notation:

H j := D j X j , H jk := H j -H k , Ψ j := k:k<j H jk , Ψ j := k:k>j H jk , C(t) := n k=1 a k t k , Υ i := H n i 1 -C(-H -1 i ) .
The polynomial C has degree n so the element Υ i is a polynomial in H i , i = 1, . . . , n.

Theorem 6. The ring S -1

x Diff h (n) is isomorphic to the ring k[a 1 , . . . , a n ] ⊗ T -1 W n .
Proof. The knowledge of the central elements (Proposition 3) allows to exhibit a generating set of the ring S -1

x Diff h (n) in which the required isomorphism is quite transparent. In the localized ring S -1

x Diff h (n) we can use the set of generators

{ hi , x i , Γ i } n i=1 instead of { hi , x i , ∂i } n i=1 . By Lemma 4 (ii), { hi , x i , c i } n i=1 is also a generating set. Finally, B D := { hi , x •i , c i } n i=1
, where x •i := x i ψ i , i = 1, . . . , n, is a generating set of the localized ring S -1

x Diff h (n) as well. It follows from Lemma 2 that the family {x •i } n i=1 is commutative. The complete set of the defining relations for the generators from the set B D reads hi hj = hj hi , hi x

•j = x •j ( hi + δ j i ) , x •i x •j = x •j x •i , i, j = 1, . . . , n , c i are central , i = 1, . . . , n . (19)
In the localized ring k[a 1 , . . . , a n ] ⊗ T -1 W n we can pass to the set of generators B W := {H i , X i , a i } n i=1 with the defining relations

H i H j = H j H i , H i X j = X j (H i + δ j i ) , X i X j = X j X i , i, j = 1, . . . , n , a i are central , i = 1, . . . , n . ( 20
)
The comparison of ( 19) and ( 20) shows that we have the isomorphism

µ : k[a 1 , . . . , a n ] ⊗ T -1 W n → S -1 x Diff h (n)
given on our generating sets B D and B W by

µ : X i → x •i , H i → hi , a i → c i , i = 1, . . . , n . ( 21
)
The proof is completed.

We shall now rewrite the formulas for the isomorphism µ in terms of the original generators of the rings S -1

x Diff h (n) and k[a 1 , . . . , a n ] ⊗ T -1 W n .
Lemma 7. We have µ :

X i → x i ψ i , D i → (ψ i ) -1 hi (x i ) -1 , a i → c i , i = 1, . . . , n . (22)
and

µ -1 : hi → H i , x i → X i 1 Ψ i , ∂i → Υ i Ψ i (X i ) -1 , i = 1, . . . , n . (23)
Proof. We shall comment only on the last formula in (23). Lemma 4 part (ii) implies that µ -1 (χ i Γ i ) = Υ i and the formula for µ -1 ( ∂i ) follows since ∂i = Γ i (x i ) -1 .

Comments

We shall now establish several corollaries of our construction.

1. We can now give the description of the center of the ring Diff h (n).

Lemma 8. The center of the ring Diff h (n) is formed by polynomials in the elements {c i } n i=1 . Proof. This is a direct consequence of the defining relations (19) for the generating set B D . Indeed, any central element ζ must have h-weight zero, so it belongs to a subring generated by c i and hi , i = 1, . . . , n. Interpret ζ as a rational function in hi , i = 1, . . . , n. Since ζ commutes with x •i , i = 1, . . . , n, this rational function is periodic, with period 1, with respect to any hi , i = 1, . . . , n. Therefore ζ belongs to the subring generated by c i , i = 1, . . . , n, as stated.

Another proof consists in using the isomorphism µ and the triviality of the center of the Weyl algebra.

The symmetric group S n acts by automorphisms on the algebra W

n , π(X j ) = X π(j) , π(D j ) = D π(j) for π ∈ S n .
The isomorphism µ translates this action to the action of S n on the ring S -1

x Diff h (n). It turns out that the subring Diff h (n) is preserved by this action. We present the formulas for the action of the generators s i of S n .

s i (x i ) = -x i+1 hi,i+1 , s i (x i+1 ) = x i 1 hi,i+1 , s i (x j ) = x j for j = i, i + 1 , s i ( ∂i ) = - 1 hi,i+1 ∂i+1 , s i ( ∂i+1 ) = hi,i+1 ∂i , s i ( ∂j ) = ∂j for j = i, i + 1 , s i ( hj ) = hs i (j) . (24)
3a. For the R-matrix description of the diagonal reduction algebra D(gl n ) in [START_REF] Khoroshkin | Diagonal reduction algebra and reflection equation[END_REF] we used the ring Diff h (n, N ) formed by N copies of the ring Diff h (n). We do not know an analogue of the isomorphism µ for the ring Diff h (n, N ). However a straightforward analogue of the formulas (24) provides an action of S n by automorphisms on the ring Diff h (n, N ).

We recall that the ring Diff h (n, N ) is a Ū(h)-bimodule with the generators x j,α and ∂j,α , j = 1, . . . , n, α = 1, . . . , N . The ring Diff h (n, N ) is free as a one-sided Ū(h)-module; the left and right Ū(h)-module structures are related by hi x j,α = x j,α ( hi + δ j i ) , hi ∂j,α = ∂j,α ( hi -δ j i ) .

The defining relations for the generators x j,α and ∂j,α , j = 1, . . . , n, α = 1, . . . , N , read

x i,α x j,β = k,l Rij kl x k,β x l,α , ∂i,α ∂j,β = k,l Rlk ji ∂k,β ∂l,α , x i,α ∂j,β = k,l Rki lj [ε k ] ∂k,β x l,α -δ α β δ i j , (26)
or, in components,

x i,α x j,β = 1 hij x i,β x j,α + h2 ij -1 h2 ij x j,β x i,α , x j,α x i,β = - 1 hij x j,β x i,α + x i,β x j,α , 1 ≤ i < j ≤ n , (27) ∂i,α ∂j,β = - 1 hij ∂i,β ∂j,α + h2 ij -1 h2 ij ∂j,β ∂i,α , ∂j,α ∂i,β = 1 hij ∂j,β ∂i,α + ∂i,β ∂j,α , 1 ≤ i < j ≤ n , (28)
x i,α ∂j,β = ∂j,β x i,α , x j,α ∂i,β = hij ( hij + 2)

( hij + 1) 2 ∂i,β x j,α , 1 ≤ i < j ≤ n , (29)
x i,α ∂i,β = n k=1 1 1 -hik ∂k,β x k,α -δ α β , 1 ≤ i ≤ n. ( 30
)
Lemma 9. The maps s i , i = 1, . . . , n -1, defined on the generators of Diff h (n, N ) by

s i (x i,α ) = -x i+1,α hi,i+1 , s i (x i+1,α ) = x i,α 1 hi,i+1 , s i (x j,α ) = x j,α for j = i, i + 1 , s i ( ∂i,α ) = - 1 hi,i+1
∂i+1,α , s i ( ∂i+1,α ) = hi,i+1 ∂i,α , s i ( ∂j,α ) = ∂j,α for j = i, i + 1 ,

s i ( hj ) = hs i (j) , (31)
extend to automorphisms of the ring Diff h (n, N ). Moreover, these automorphisms satisfy the Artin relations and therefore give the action of the symmetric group S n by automorphisms.

Proof. After the formulas (31) are written down, the verification is a direct calculation.

3b. The operators s i := s i , where is the anti-automorphism (11), generate the action of the symmetric group S n by automorphisms as well. The action of the automorphism s i , i = 1, . . . , n -1, involves only the element hi,i+1 (as the action of the automorphism s i ) and is given by

s i (x i,α ) = - 1 hi,i+1 x i+1,α , s i (x i+1,α ) = hi,i+1 x i,α , s i (x j,α ) = x j,α for j = i, i + 1 , s i ( ∂i,α ) = -∂i+1,α hi,i+1 , s i ( ∂i+1,α ) = ∂i,α 1 hi,i+1 , s i ( ∂j,α ) = ∂j,α for j = i, i + 1 , s i ( hj ) = hs i (j) . (32)
4. The diagonal reduction algebra D(gl n ) is a Ū(h)-bimodule with the generators L j i , i, j = 1, . . . , n. The defining relations of D(gl n ) are given by the reflection equation, see [START_REF] Khoroshkin | Diagonal reduction algebra and reflection equation[END_REF] R12

L 1 R12 L 1 -L 1 R12 L 1 R12 = R12 L 1 -L 1 R12 , where L = {L j i } n i,j=1
is the matrix of generators (we refer to [C, S, RS, KS, IO, IOP, IMO1, IMO2] for various aspects and applications of the reflection equation).

For each N there is a homomorphism ([KO5], Section 4.1)

τ N : D(gl n ) → Diff h (n, N ) defined by τ N (L j i ) = α x j,α ∂i,α .
Moreover τ N is an embedding for N ≥ n.

The formulas (31) show that the image of τ N is preserved by the automorphisms s i .

The element s i (τ N (L j k )) can be written by the same formula for all N . Since τ N is injective for N ≥ n we conclude that the formulas (31) induce the action of the symmetric group S n on the diagonal reduction algebra D(gl n ) by automorphisms.

The resulting formulas for the action of the automorphisms s i , i = 1, . . . , n -1, on the generators L k j , j, k = 1, . . . , n, read

s i (L i j ) = -L i+1 j hi,i+1 , s i (L i+1 j ) = L i j 1 hi,i+1 , j = i, i + 1 , s i (L j i ) = - 1 hi,i+1
L j i+1 , s i (L j i+1 ) = hi,i+1 L i j , j = i, i + 1 ,

s i (L i i ) = L i+1 i+1 , s i (L i i+1 ) = -L i+1 i ( hi,i+1 -1) 2 , s i (L i+1 i ) = -L i i+1 1 ( hi,i+1 + 1) 2 , s i (L i+1 i+1 ) = L i i ,
s i (L k j ) = L k j , k = i, i + 1 and j = i, i + 1 .

5.

The isomorphism µ can be also used to translate the action (10) of the braid group by Zhelobenko operators to the action of the braid group by automorphisms on the ring k[a 1 , . . . , a n ] ⊗ T -1 W n . It turns out that this action preserves the subring T -1 W n . Moreover, let T 0 be the multiplicative set of T generated by X j D j -X k D k + , 1 ≤ j < k ≤ n, ∈ Z. Then the action of the operators qi , i = 1, . . . , n -1, preserves the subring T -1 0 W n . We present the formulas for the action of the operators qi , i = 1, . . . , n -1:

qi (X i ) = 1 H i,i+1
X i+1 , qi (X i+1 ) = X i H i,i+1 , qi (X j ) = X j for j = i, i + 1 ,

qi (D i ) = D i+1 H i,i+1 , qi (D i+1 ) = 1 H i,i+1
D i , qi (D j ) = D j for j = i, i + 1 .

6. The isomorphism (23) allows to construct a 2n-parametric family of Diff h (n)-modules different from the highest weight representations. Let γ := {γ 1 , . . . , γ n } be a sequence of length n of complex numbers such that γ i -γ j / ∈ Z for all i, j = 1, . . . , n, i = j. Let V γ be the vector space with the basis v j := (X 1 ) j 1 +γ 1 (X 2 ) j 2 +γ 2 . . . (X n ) jn+γn , where j := {j 1 , . . . , j n } , j 1 , . . . , j n ∈ Z .

Under the conditions on γ, V γ is naturally a T -1 W n -module. Define the action of the elements a k on the space V γ by a k : v j → A k v j where A := {A 1 , . . . , A n } is another sequence of length n of complex numbers. Then V γ becomes an k[a 1 , . . . , a n ] ⊗ T -1 W n -module and therefore Diff h (n)module which we denote by V γ, A . The central operator c k acts on V γ, A by scalar multiplication on A k .

  hi : | → λ i | , ∂i : | → 0 , i = 1, . . . , n , define the D n -module structure on M λ . We shall call the induced representation Ind Diff h (n) Dn M λ the "highest weight representation" of highest weight λ. Lemma 5. The central operator c k , k = 1, . . . , n, acts on the module Ind Diff h (n) Dn M λ by scalar multiplication on -e k | hi →λ i -1 , the evaluation of the symmetric function -e k on the shifted vector {λ 1 -1, . . . , λ n -1}.

Acknowledgments. The work of O. O. was supported by the Program of Competitive Growth of Kazan Federal University and by the grant RFBR 17-01-00585.


Prof. Dr. habil Dmitry Ivanov
email: divanov@hwr-berlin.de

Alexander Pavlov

Alexandre Dolgui

Dmitry Pavlov

Boris Sokolov

Disruption-driven supply chain (re)-planning and

Keywords: multi-stage supply chain, supply chain dynamics, supply chain planning, disruptions, recovery, structure dynamics, ripple effect, linear programming, service level, maximal flow, multi-period models

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Supply chain (SC) planning is a referenced research problem to successfully meet customer needs while improving operations performance efficiency. SC planning (SCP) decides on directing commodity flows in the SC to efficiently balance supply and demand at the aggregate level in the medium-term planning horizon subject to a given facility location structure, capacities, customer demand, and order quantities ((Tayur et al. 1999[START_REF] Amiri | Designing a distribution network in a supply chain system: Formulation and efficient solution procedure[END_REF][START_REF] Hinojosa | Dynamic supply chain design with inventory[END_REF][START_REF] Manzini | Strategic design and operational management optimization of a multistage physical distribution system[END_REF], Mula et al. 2010[START_REF] Dolgui | Supply Chains Engineering: Useful Methods and Techniques[END_REF][START_REF] Chopra | Supply Chain Management -Strategy, Planning and Operation[END_REF], Constantino et al. 2012).

In SCP, it is mandatory to take into account uncertainty and risks in order to provide practically relevant problem statements and decision-oriented solutions of computational models [START_REF] Wu | Decision making in enterprise risk management: A review and introduction to special issue[END_REF]. Knemeyer et al. (2009), [START_REF] Simchi-Levi | From superstorms to factory fires: Managing unpredictable supply chain disruptions[END_REF], [START_REF] Ambulkar | Firm's resilience to supply chain disruptions: Scale development and empirical examination[END_REF] and Eckstein et al. (2015) underline crucial role of disruption events and recovery policies in the SC planning. Recent literature suggests considering recurrent or operational risks and disruptive risks [START_REF] Chopra | The importance of decoupling recurrent and disruption risks in a supply chain[END_REF].

In 2000-14, SC disruptions (e.g., because of both natural and man-made disasters, such as on 11 March 2011 in Japan, floods in Thailand in 2011, fire in the Phillips Semiconductor plant in New Mexico, etc.) occurred in greater frequency and intensity, and thus with greater consequences (Chopra andSodhi 2014, Simchi-Levi et al. 2014). Such disruptive risks represent a new challenge for SC managers who face the ripple effect (Ivanov et al. 2014a) subject to structural disruptions in the SC. The ripple effect describes the impact of a disruption on SC performance and the disruption-based scope of changes in the SC structures.

Recent studies extensively considered SCP or SCP within the SC design (SCD) decisions taking into account disruption risks (e.g., [START_REF] Dolgui | A Model for Supply Planning under Lead Time Uncertainty[END_REF][START_REF] Xia | Real-time disruption management in a two-stage production and inventory system[END_REF][START_REF] Snyder | Reliability Models for Facility Location: The Expected Failure Cost Case[END_REF][START_REF] Xiao | Supply chain disruption management and evolutionarily stable strategies of retailers in the quantity-setting duopoly situation with homogeneous goods[END_REF][START_REF] Dolgui | Supply planning under uncertainties in MRP environments: a state of the art[END_REF][START_REF] Wilson | The impact of transportation disruptions on supply chain performance[END_REF][START_REF] Chauhan | A Continuous Model for Supply Planning of Assembly Systems with Stochastic Component Procurement Times[END_REF][START_REF] Cui | Reliable facility location design under the risk of disruptions[END_REF][START_REF] Klibi | The design of robust value-creating supply chain networks: A critical review[END_REF][START_REF] Chen | Joint inventory-location problem under the risk of probabilistic facility disruptions[END_REF][START_REF] Peng | Reliable logistics networks design with facility disruptions[END_REF][START_REF] Kouvelis | Contingency strategies in managing supply systems with uncertain leadtimes[END_REF][START_REF] Schmitt | A quantitative analysis of disruption risk in a multi-echelon supply chain[END_REF][START_REF] Baghalian | Robust supply chain network design with service level against disruptions and demand uncertainties: A real-life case[END_REF][START_REF] Li | Reliable facility location design under disruptions[END_REF][START_REF] Lim | Facility location decisions with random disruptions and imperfect estimation[END_REF], Ivanov et al. 2013, 2014[START_REF] Dolgui | A state of the art on supply planning and inventory control under lead time uncertainty[END_REF][START_REF] Sawik | Integrated selection of suppliers and scheduling of customer orders in the presence of supply chain disruption risks[END_REF][START_REF] Paul | Real time disruption management for a two-stage batch production-inventory system with reliability considerations[END_REF][START_REF] Aloulou | A bibliography of non-deterministic lot-sizing models[END_REF].

The scope of the disruption rippling and its performance impact depends both on robustness reserves (e.g., redundancies like inventory or capacity buffers) and speed and scale of recovery actions [START_REF] Hendricks | Association between supply chain glitches and operating performance[END_REF][START_REF] Sheffi | A supply chain view of the resilient enterprise[END_REF][START_REF] Tomlin | On the Value of Mitigation and Contingency Strategies for Managing Supply Chain Disruption Risks[END_REF][START_REF] Das | Integrating effective flexibility actions into a strategic supply chain planning model[END_REF][START_REF] Bode | Understanding responses to supply chain disruptions: Insights from information processing and resource dependence perspectives[END_REF], Ivanov and Sokolov 2013[START_REF] Kim | Guilt by Association: Strategic Failure Prevention and Recovery Capacity Investments[END_REF]. The existing studies considered both disruptions without recovery actions and disruptions with recovery actions. We focus this study on the second group. The recovery options comprise facility fortification (e.g., back-up suppliers, warehouses, depots and transportation channels), inventory and capacity expansion. Contingency plans or backup planning (e.g., alternative suppliers or shipping routes) need to be developed (Knemeyer et al. 2009[START_REF] Cui | Reliable facility location design under the risk of disruptions[END_REF][START_REF] Yang | Using a dual-sourcing option in the presence of asymmetric information about supplier reliability: Competition vs diversification[END_REF][START_REF] Benyoucef | Supply chain network design with unreliable suppliers: A lagrangian relaxation-based approach[END_REF][START_REF] Li | Reliable facility location design under disruptions[END_REF]. The recovery must happen quickly to expedite stabilization and adaptation in order to ensure SC continuity and avoid long-term impact. In implementing such recovery policies, companies need a tool supported by collaboration and SC visibility solutions for assessing the impact of disruption on the SC as well as the effects from redirecting material flows [START_REF] Sheffi | A supply chain view of the resilient enterprise[END_REF][START_REF] Simchi-Levi | From superstorms to factory fires: Managing unpredictable supply chain disruptions[END_REF][START_REF] Chopra | Reducing the risk of supply chain disruptions[END_REF], Gedik et al. 2015).

For example, Toyota extends its SC subject to multiple-sourcing and building new facilities on the supply side. Amazon prefers holding fast-moving items in distribution centres while slowmoving items tend to be stored centrally. Apple localized production in China but the distribu-tion network is global. Such SC segmentation also helps to reduce disruption risk implications [START_REF] Chopra | Reducing the risk of supply chain disruptions[END_REF]. Samsung tends to have at least two suppliers even if the second one provides only 20% of the volume [START_REF] Sodhi | An analysis of sources of risk in the consumer electronics industry[END_REF]. Online fashion retailer ASOS was badly affected by a fire in the UK warehouse in 2005 when operations were almost stopped for one month. They developed a contingency policy for such disruptions. Among others, an additional warehouse was established in Asia. This helped the company to recover within two days in June 2014 when new fire occurred in the UK warehouse.

Despite a wealth of literature on SCP with disruption considerations, most of the existing studies consider the recovery policies under the assumption that the disrupted facilities or transportation channels do not return into the SC operation during the planning horizon. There are only a few studies that incorporate SC plan reconfiguration into the performance impact assessment. To the best of our knowledge there is no published research that considers only temporary absence of some SC elements taking into account the duration of disruptions with the capacity recovery and the costs of this recovery.

In putting these aspects into a research focus, the objective of this research is to incorporate disruptions duration and capacity recovery into SCP decisions and performance impact assessment. The proposed method has been developed in a practical way with the aim to optimize the SCP and to develop a model for a multi-stage SC with structure dynamics considerations.

The remainder of this paper is organized as follows. Section 2 analyses recent literature. Section 3 considers a case-study and proposed methodology. In Section 4, the mathematical model is presented. Section 5 is devoted to the experimental calculation of optimal SC plans. In Section 6, different actions for increasing the SC resilience are analysed. Managerial insights are presented in Section 7. The paper concludes by summarizing the most important findings and outlining future research needs.

Literature review

Different approaches to protect and control SCs in regard to disruptions and to coping with uncertainty have been developed in recent years [START_REF] Dolgui | A Model for Supply Planning under Lead Time Uncertainty[END_REF][START_REF] Santoso | A stochastic programming approach for supply chain network design under uncertainty[END_REF][START_REF] Dotoli | Design and optimization of integrated E-supply chain for agile and environmentally conscious manufacturing[END_REF][START_REF] Chopra | The importance of decoupling recurrent and disruption risks in a supply chain[END_REF], Schütz et al. 2009[START_REF] Chauhan | A Continuous Model for Supply Planning of Assembly Systems with Stochastic Component Procurement Times[END_REF][START_REF] Klibi | The design of robust value-creating supply chain networks: A critical review[END_REF][START_REF] Georgiadis | Optimal design of supply chain networks under uncertain transient demand variations[END_REF][START_REF] Peng | Reliable logistics networks design with facility disruptions[END_REF][START_REF] Lim | A facility reliability problem: Formulation, properties and algorithm[END_REF], Constantino et al. 2012[START_REF] Baghalian | Robust supply chain network design with service level against disruptions and demand uncertainties: A real-life case[END_REF]. The surveys on specific aspects of these problems are presented in [START_REF] Dolgui | Supply planning under uncertainties in MRP environments: a state of the art[END_REF][START_REF] Dolgui | A state of the art on supply planning and inventory control under lead time uncertainty[END_REF], Alloulou et al. 2014).

In this Section, we analyse how the consideration of disruptions with recovery policies has been done in literature so far.

Mixed-integer programming

Mixed-integer programming (MIP) with application to reliable SCD and SCP has been a broad research avenue over the past ten years. The reliable location model was first introduced by [START_REF] Snyder | Reliability Models for Facility Location: The Expected Failure Cost Case[END_REF]. The UFL model aims at finding optimal SC design with assignments of customers to locations with the objective to minimize the sum of fixed and transportation costs in the SC. The relevant development of MIP models can be considered regarding the facility fortification. [START_REF] Lim | A facility reliability problem: Formulation, properties and algorithm[END_REF] incorporated a totally reliable bаck-up supplier that can be used if a primary supplier is destroyed. The related costs are incorporated into the objective function but the fortification budget remains incapacitated. [START_REF] Li | Reliable facility location design under disruptions[END_REF] extended this model by intro-ducing limits on the fortification budget in a single-product case with eight distributors and up to 150 customers.

In addition, inventory considerations have been included. [START_REF] Chen | Joint inventory-location problem under the risk of probabilistic facility disruptions[END_REF] presented a joint inventory-location model under the risk of probabilistic facility disruptions. [START_REF] Benyoucef | Supply chain network design with unreliable suppliers: A lagrangian relaxation-based approach[END_REF] considers SC design with unreliable suppliers. [START_REF] Rafiei | Reliable multi period multi product supply chain design with facility disruption[END_REF] developed a comprehensive model for a problem statement with multiple products and many periods. They considered the levels of inventory, back-ordering, available machine capacity and labour levels for each source, transportation capacity at each transhipment node and available warehouse space at each destination. The problem also considered the facility fortification by taking into account the back-up supplier with reserved capacity and a back-up transhipment node that satisfied demands at higher prices without disruption facility. The solution to the model is based on a priority-based genetic algorithm. [START_REF] Gedik | Vulnerability Assessment and Re-Routing of Freight Trains under Disruptions: A Coal Supply Chain Network Application[END_REF] model disruptions and train re-routing actions in a coal supply chain network and assess impacts of disruptions in terms of transportation and delay costs using a two-stage mixed integer programming (MIP) model. An "interdictor" chooses a limited amount of elements to attack first on a given network, and then an "operator" dispatches trains through the residual network. The MIP model explicitly incorporates discrete unit flows of trains on the rail network with time-variant capacities. A K-th shortest path algorithm is used to enumerate all routes between points. The authors consider a real coal rail transportation network and generate scenarios to provide tactical and operational level vulnerability assessment analysis with incorporation of rerouting decisions, travel and delay costs analysis, and the frequency of interdictions of facilities for the dynamic rail system.

Stochastic programming

Stochastic programming models are often scenario-based where parameters are represented by a set of discrete scenarios with a given probability of occurrence. In standard stochastic programming models [START_REF] Santoso | A stochastic programming approach for supply chain network design under uncertainty[END_REF][START_REF] Goh | A stochastic model for risk management in global chain networks[END_REF], demand is considered as an uncertain parameter. In [START_REF] Azaron | A multi-objective stochastic programming approach for supply chain design considering risk[END_REF], facility disruptions and capacity expansion costs are also considered to be uncertain. [START_REF] Sawik | Integrated selection of suppliers and scheduling of customer orders in the presence of supply chain disruption risks[END_REF] developed a stochastic programming model to integrated supplier selection, order quantity allocation and customer order scheduling in the presence of SC disruption risks.

In the study by [START_REF] Madadi | Supply Network Design under Risk of Supply Disruptions and Tainted Materials Delivery[END_REF], a problem of supply network design under risk of supply disruptions is considered. Tainted materials delivery disruptions are modeled as events which occur randomly and may have a random length. A mixed-integer stochastic model is proposed and solved by a meta-heuristic algorithm. [START_REF] Torabi | Resilient supplier selection and order allocation under operational and disruption risks[END_REF] propose a bi-objective mixed twostage stochastic programming model for supplier selection and order allocation problem under operational and disruption risks. The model considers several proactive strategies such as suppliers' business continuity plans, fortification of suppliers and contracting with backup suppliers. The computational results demonstrate the significant impact of considering disruptive events on the selected supply base. [START_REF] Hishamuddin | A recovery model for a two-echelon serial supply chain with consideration of transportation disruption[END_REF] presented a recovery model for a two-echelon serial SC with consideration of transportation disruption. Their model is capable to determine the optimal ordering and production quantities during the recovery period to minimize total costs. [START_REF] Shao | Supply disruption and reactive strategies in an assemble-to-order supply chain with time-sensitive demand[END_REF] analyse an assemble-to-order system with a backup source to offer on-time delivery and compensation policy to compensate customers for waiting in each period during a disruption. The findings suggest that the backup sourcing strategy is preferred at the beginning of the supply disruption, while the compensation strategy is preferred as time elapses. [START_REF] Hu | Managing risk of supply disruptions: Incentives for capacity restoration[END_REF] analyse incentive mechanisms to motivate a supplier's investment in the capacity restoration. They consider the cases when the incentive is committed to ex-ante (prior to disruption) as well as when it is committed to ex-post (after disruption). The analysis indicates if the buyer offers incentives, both the buyer and supplier (weakly) prefer the ex-ante commitment over the ex-post one. [START_REF] Lewis | Managing inventory in global supply chains facing port-of-entry disruption risks[END_REF] analyse the disruption risks at ports of entry with the help of closure likelihood and duration which are modelled using a completely observed, exogenous Markov chain. They developed a periodic review inventory control model that indicates for studied scenarios that operating margins may decrease 10% for reasonably long port-of-entry closures or eliminated completely without contingency plans, and that expected holding and penalty costs may increase 20% for anticipated increases in port-of-entry utilization. Gupta et al. (2015) study from game-theoretical perspective the implications of the contingent sourcing strategy under competition and in the presence of a possible supply disruption. The time of the occurrence of the supply disruption is uncertain and exogenous, but the procurement time of components is in the control of the firms. The results imply that supply disruption and procurement times jointly impact the firms' buying decisions, optimal order quantities and their expected profits. Subsequently, this study considers the impact endogenizing equilibrium sourcing strategies of asymmetric and symmetric firms, and of capacity reservation to mitigate disruption.

Inventory management

2.4 Simulation, system science and control theory [START_REF] Schmitt | A quantitative analysis of disruption risk in a multi-echelon supply chain[END_REF] presented a quantitative estimation of the disruption risk at production and supply capacities in a multi-echelon SC using discrete-event simulation. They also consider dual sourcing as a contingency action. The disruption risk is actioned by "weeks of recovery" as the amplification of the disruption. [START_REF] Carvalho | Supply chain redesign for resilience using simulation[END_REF] analysed impacts of transportation disruptions on lead-time and overall costs in an automotive SC using ARENA-based simulation. [START_REF] Unnikrishnan | Online freight network assignment model with transportation disruptions and recourse. Transportation research record[END_REF] developed a scenario-based model with an adaptive routing policy. [START_REF] Vahdani | A hybrid multi-stage predictive model for supply chain network collapse recovery analysis: A practical framework for effective supply chain network continuity management[END_REF] applied fuzzy program evaluation and review technique to calculate the completion time of SC operations in the case of a severe disruption. [START_REF] Xu | Predicted supply chain resilience based on structural evolution against random supply disruptions[END_REF] used AnyLogic software and modelled SC as an agent system to study the disruption at suppliers and recovery policies on the SC service level. [START_REF] Paul | Real time disruption management for a two-stage batch production-inventory system with reliability considerations[END_REF] analysed series of disruptions over time and presented an inventory control-based model to develop optimal recovery policies for real time disruption management for a two-stage batch production-inventory system with reliability considerations. They consider multiple disruptions and cases where new disruption may or may not affect the recovery plan of earlier disruptions.

Stochastic maximal flow models or minimal cost flow models [START_REF] Lin | A simple algorithm for reliability evaluation of a stochastic-flow network with node failure[END_REF][START_REF] Chou | Process flexibility revisited: The graph expander and its applications[END_REF]) and fuzzy models (Selim et al. 2008, Constantino et al. 2011) have also been applied. [START_REF] Chou | Process flexibility revisited: The graph expander and its applications[END_REF] propose an algorithm for determining the system reliability with respect to the maximum flow of a network achieving a given demand. The nodes can fail randomly and the demand is assumed deterministic; no cost/profit functions are considered. In the study by [START_REF] Lin | A simple algorithm for reliability evaluation of a stochastic-flow network with node failure[END_REF], the goal is to determine the production quantity of a particular product for a given random demand.

The authors use a simplified supply chain that consists of plants and retailers and represent it as a deterministic bipartite graph. For solution, a heuristic scheme for determining the assignment policy is proposed. The resulting expander graphs are interesting due to the spectral properties, that is, they do not degrade by increasing the number of nodes. Constantino et al. (2012) presented a hierarchical approach to the strategic supply chain design addressing supply planning and allowing the improvement of the manufacturing supply chain agility in terms of ability in reconfiguration to meet performance, and considering the supplier capacity constraints. The approach employs digraph modeling and integer LP to optimal supply chain design. The authors avoid stochastic models by aggregating deterministic product flows within the integer LP model.

Summarizing, the literature mainly considers three basic types of disruptive risks that should be considered by SC managers: production, supply and, transportation disruptions. Recent studies show that efficiency paradigms of lean processes, single sourcing, etc. have failed in disruption situations. Disruptions in a global SC, especially in its supply base, does immediately affect the entire SC. With the increased specialization and geographical concentration of manufacturing, disruptions in one or several nodes affect almost all the nodes and links in the SC [START_REF] Kim | Guilt by Association: Strategic Failure Prevention and Recovery Capacity Investments[END_REF].

Recent literature discussed different recovery strategies:  Back-up suppliers,  Back-up depots and transportation channels/modes  Inventory and capacity buffers  Facility fortification  Capacity expansion Finally, recent empirical literature on SC disruptions that indicates significantly more efforts into pre-disruption stage as compared with recovery activities [START_REF] Bode | Understanding responses to supply chain disruptions: Insights from information processing and resource dependence perspectives[END_REF], Blackhurts et al. 2011). The rationale behind this situation may be that investments in protection are tangible and directly decrease net profits while recovery costs are anticipated.

Reaction to disruptive events can be performed depending on the severity of disruptions:

 Parametrical adaptation  Structure adaptation Parametrical adaptation represents the simplest case where stabilization and recovery are possible through tuning of some critical parameters like lead-time or inventory. Structure adaptation considers back-up supplier on contingency transportation plans.

Operations research along with system dynamics and control theory contain a number of useful methods that can be used for analysis and for mitigating the ripple effect. Different methods are suited to different problems. No single technique is likely to prove a panacea in this field. MIP formulations with facility fortification consider product shift to back-up suppliers if primary suppliers are disrupted. Simulation techniques consider "what-if" scenarios which can be used by SC managers in the case of disruption occurrence to quickly estimate the recovery policies and impacts on operational and financial performance. [START_REF] Chopra | Reducing the risk of supply chain disruptions[END_REF] and Simchi-Levi et al. ( 2014) point out that one has to concentrate mainly on mitigation strategies and identification of the impact of disruption on financial and operational performance regardless of what caused the disruption. In addition, a general short-coming of existing studies, as pointed out by [START_REF] Cui | Reliable facility location design under the risk of disruptions[END_REF] and [START_REF] Li | Reliable facility location design under disruptions[END_REF] is that the dynamics of SC execution is not considered. The disruptions are mostly considered as static events, without taking into account their duration, stabilization/recovery policies. Other possibilities for modelling real flows are system dynamics [START_REF] Villegas | Supply chain dynamics: Analysis of inventory vs order oscillations trade off[END_REF] and control theory [START_REF] Ivanov | Applicability of optimal control theory for adaptive supply chain planning and scheduling[END_REF]Sokolov 2013).

While mathematical and stochastic optimization has its place at the SC design and planning stages without recovery considerations, they fail to throw much light on the dynamic behaviour of the SC. The impact of SC design and SCP on SC performance at the execution and recovery stage can be enhanced by using models based on the dynamics of the execution processes.

Summarizing, investment in SC protection can help to avoid many problems with disruptive events. However, it is impossible to avoid disruption completely. Simchi-Levi et al. ( 2014) underline that focus should be directed to the recovery policies regardless of what caused the disruption. Therefore, adaptation is needed to change SC plans, schedules or inventory policies in order to achieve the desired output performance (Eckstein et al. 2015).

The contribution of this study is a multi-objective problem formulation for SC reconfiguration model (trade-off "service level vs. costs") that includes performance impact assessment with SC re-planning in the case of structure dynamics (trade-off "efficiency vs. resilience") taking into account temporary unavailability of some SC elements and their recovery in time. The SC comprises Tier 3 suppliers (the triangles), Tier 2 suppliers (the circles), Tier 1 suppliers (the rectangles), assembly plants and markets. We assume that the SC displays the following characteristics: (i) SC performance depends on its ability to execute despite of perturbations; (ii) some SC elements may become unavailable due to disruptions, (iii) some SC elements recover in time and (iv) the SC experiences performance degradation if some of its elements fail.

The problem statement captures the following elements: The objective is to find the aggregate product flows to be moved from suppliers through the intermediate stages to the markets subject to maximizing the service level and minimizing the total cost under (i) constrained capacities and processing rates and (ii) SC structure dynamics for a multi-period case. It is to compute the performance impact of disruptions with consideration of the material flow reconfiguration taking into account disruption and recovery time and costs.

 Multi-stage, multi-product, multi-period

Methodology

The research comprises three stages:

 To develop a mathematical model  To use this model for computing the SC plan and economic performance at the existing SC subject to disruption and recovery of SC elements  To investigate the impact of different resilience actions on SC performance and distribution (re)planning

The preliminary analysis of the multi-stage SCP problem with uncertainty in structure dynamics has shown that it could be modelled as a stochastic maximal flow model, i.e., as maximal flow models or minimal cost flow models. However, in some aspects these methods are rather restrictive. First, existing studies have not explicitly considered possible structure dynamics and its impact on the flows and cost. Second, deviations or failures in the network structures and opera-tions are possible, but not unrealistic or describable with some probabilistic assumptions. In addition, the SC structural states do not change permanently, but rather at some intervals. Therefore, intervals of structural constancy can be considered.

Another possible way to model the considered problem could be the LP/MIP-based implementation. However, structural changes in the SC and recovery in different periods do not allow direct application of LP/MIP models if disruption duration and recovery needs to be included into the analysis.

We propose to apply a hybrid approach and distribute static and dynamic parameters between a LP and a dynamic model. The dynamic model is necessary in order to describe the structure dynamics. The dynamic model contains piecewise functions in the right parts of differential equations in order to describe the facility unavailability time and recovery period in the SC can also be used to add the property of partial availability and gradual recovery of capacities. This can also be done with the help of piece-wise functions which will describe the capacities in the SC.

Since the commodity flows will be described in continuous time in the dynamic model, the disruption recovery time can be described more accurately with the help of a piece-wise function that allows considering disruption durations and gradual capacity recovery. The LP model is used for solving transhipment problem within intervals of structural constancy since modelling the material flow balance in the dynamic model is not efficient. Assumed that the network elements does not change their capacities within some subintervals (structure constancy intervals), for these intervals, a multi-objective LP model with transit nodes and two side constraints is formulated. Exclusion of demand constraint allows transit from the classical LP to maximal flow problem, formulated as an LP model. The graph of structural reliability (Ivanov et al. 2014) can be used to model optimistic and pessimistic scenarios. These scenarios are used for computational experiments with the developed model.

In line with the existing research on reliable SC design, we investigate also different resilience actions such as back-up suppliers and capacity reservation regarding their impact on economic performance under disruptions.

Mathematical model

The model described in this Section is a generalized form of the previously developed models in (Ivanov et al. 2013(Ivanov et al. , 2014)). The differences to those models are as follows:

 BOM constraints are included in Eqs (3), ( 4), ( 13) and ( 14 is the set of node numbers for the nodes transmitting products to at time interval

) (t X  } ), ( {  N i t A xi  } , ), ( { ) (   N j i t e t E xij   } , ), ( { ) (    N j i t w t W ij   j i  j i   ik N  i A  k
is the set of node numbers for the nodes receiving products from at time interval is the set of possible SC plans is the number of a time interval (i.e., interval of structural constancy) in the planning horizon T=(t 0 ; t f ] is the number of an item in the SC.

 is a number of an execution scenario is the number of the delivering node in the SC, is the number of the receiving node in the SC, Elements of the set which describe transportation, processing, and warehouse operations are as follows:

is the total ordered quantity from all suppliers is the maximal warehouse capacity of the node is the maximal transportation channel intensity for the product between and is the maximal inbound processing intensity for the product in is the maximal outbound processing intensity for the product from is the transportation costs intensity for the product from to is the inventory costs for the product at the node is the processing costs intensity for the product at the node is the return (utilization) cost for product at the node is the fixed cost of the node and channel from to , are the fixed costs of the node and channel from to respectively at time interval number k , are the recovery costs of the node and channel from to respectively at time interval number k Assume for each pair <A i A j > that manufacturing and transportation capacities may be disrupted and the availability of the connection between two stages in the SC (i.e., tier 1 and tier 2) can be described by a given preset matrix time function of time-spatial constraints . We have , if the channel between A i and A j is available and not disrupted within the given period of time, and , otherwise. In addition, we introduce as a variable that denotes the importance of the product and which is a variable that denotes the urgency of the product .

Decision variables are defined as follows:

is the amount of product transmitted from to and received at at time interval number k; is the product amount relating to the node and to be stored at warehouse at time interval number k,

 ik N  i A  k } {      } ,..., 2 , 1 {  L k  } ,..., 2 , 1 { p P    } ,..., 2 , 1 {  n i     N i } ,..., 2 , 1 {  n j     N j } , ), ( { ) (    N j i t w t W ij   k I  ) (t V i  l A  ) (t ij    i A  j A  ) (t i    l A  ) (t i    i A  ) (t c ij   i A  j A  ) (t h i   i A  ) (t i    i A  ) (t r i   i A  ( ) ij t   i A  i A  j A  ik f ijk f i A  i A  j A  ik b ijk b i A  i A  j A  ) (t ij  1 ) (  t ij  0 ) (  t ij        k ij x    i A  j A  j A  k j y    j A 
is the product amount relating to the node and to be delivered at time interval k is the product amount relating to the node and to be returned (as caused by the missing capacity of SC nodes and channels) at time interval number k is the binary control variable of transportation for product in the scenario if a transportation channel from to is selected, 0 -otherwise is the 0-1 control variable of delivery to the customer ; if a product is delivered to the customer from , 0 -otherwise.

Dynamic model

The following dynamic model describes the dynamics of material flows in the SC ( 1)-( 11):

(1)

(2)

(3) (4) please split this expression into two ( 5)

, ( 6
) (7) (8) (9) (10) (11)
The objective functions (1)-( 3) describe minimization of return flows, throughput maximization and inventory minimization, respectively. Eq. ( 4) is the cost minimization objective. Eq. ( 5) de-

k j g    j A  k j z    j A  } 1 , 0 { ) (  t u ij    Sc 1 ) (  t u ij  xi A xj A } 1 , 0 { ) (  t i   i A  1 ) (  t i    xi A ; ) ( 0 1 1 1 dt t z J f t t p n i i           0 2 1 1 ( ) ( ) ; f t n p i i i t J t t dt                  ; ) ( 0 1 1 3 dt t y J f t t p n i i           0 4 1 1 1 1 1 1 1 1 1 1 1 ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )) f t n n n n n n n p p p p ij ij i i i i i i ij ij i j i i i i j t J c t u t h t y t r t z t t t t e t dt                                                              }. ,..., 2 , 1 { ), ( ) ( ) ( ) (           n N i t z t y t x t x i i i i            1 ( ) ( ) ( ) ( ) ( ) ( ) ( ), n i i i i ji ji ji j x t t t t e t t u t i N                                 1 ( ) ( ) ( ) ( ) ( ) ( ), , n i i i ij ij ij j x t t t e t t u t i N                             ], , ( , ) ( 0 0 1 f i p i t t t N i V t y             0 ( ) {0,1}, ( ) , ( ) {0,1} , , , ( , ], ij i i f u t t t i j N P t t t                   ]. , ( , , 0 ) ( , 0 ) ( 0 f i i t t t P N i t z t z              , 0 ) ( ) ( 0 0      N i t z t y i i    
scribes the dynamics of the product subject to the node and ensures that the sum of the outgoing flow, inventory and return flow should equal the incoming flow.

We denote as actual inbound delivery (i.e., the received product flow) of the product to , as actual outbound delivery of the product from , as volume of stocked products in the warehouse of , and as unprocessed and returned volume of the product from .

The intensity of the inbound deliveries of into is described in Eq. ( 6) that shows that the incoming flow may not exceed maximal inbound processing capacity and the capacity of the selected transportation channel.

Analogously, the intensity of the outbound deliveries of from is described in Eq. ( 7). are considered as parameters, functions are unknown, and controls uniquely define and . Therefore, the pairs can be considered as states of the SC represented through the dynamic system. Piecewise functions are contained in the right parts of Eqs ( 6) and ( 7) in order to describe the structure dynamics and disruption durations.

Constraints are described as Eqs ( 8)-( 11). Eq. ( 8) reflects the maximal warehouse capacity. Eq. ( 9) ensures that control variables take integer values. Eq. ( 10) is non-negativity constraint for the return flows. Eq. ( 11) shows the starting conditions, e.g., at the beginning of the planning interval, no inventory and return flows exist. Control variables are constrained by ( 8)-( 10) and ( 11) are evaluated subject to (5)-( 7).

Since each scenario is characterized by a number of structural states during certain intervals of structural constancy . The structure and the parameters of SC undergo changes at discrete time points . These points divide the planning interval into subintervals , . The SC structure does not vary at each k-sub-interval and is in the state within this interval. The assumption on the intervals of structural constancy allows transit from the dynamic to static models.

Static model

Functions can be integrated and transformed to variables

. Then, the model ( 5)-( 7) can be rewritten as balance equations ( 12)-( 13) and capacity constraints ( 14) as follows:

(12) ,( ],,{(

 i A  ) (t x i      i A  ) (t x i      i A  ) (t y i    i A  ) (t z i   i A   i A   i A  ( ), ( ), ( ), ( ) i ij i i t t t t               ( ), ( ), ( ), ( ), ( ) i i ij i i y t z t u t t t               ( ), ( ), ( ) ij i i u t t t           ) (t z i  ) (t y i  ) ( ), ( t z t y i i     ) (t e ij  ) ( , ) ( t t u i ij       Sc } { ,.., , 2 1  k i i i St ) ,..., , ( 1 0 k t t t ] , ( 0 k t t  L ]} , ( ],..., , ( ],...,
1 1 2 1 1 0 f L L k k t t t t t t t t t T       ] , ( 1 k k k t t T    k i i i St ,.., , 2 1 ( ), ( ) , ( ) ij i i u t t t           1 1 1 ( ) , ( ) , ( ) k k k k k k t t t ij k ij i k i i k i t t t u u t dt t dt t dt                              ( 1) , ik ik i k i k ji k ji k i k i k i k i k ij k ij k i k i k j N j N I y u u y z                                                       (13)
; ; ;

; .

(

) 14
The meaning of Eqs. ( 12)-( 14) is identical to their dynamic form ( 5)-( 11).

The objectives ( 1)-( 4) can be rewritten as follows:

(15)

(16) (17) (18)
respectively.

The search for optimal SC plan is performed under preference relations (e.g., weights) based on the following criteria:

We use the successive concessions method for multi-objective decision-making.

Since

, then causes maximization of . On the other hand, reduces stocked quantities. By setting as the objective with highest priority, the model will calculate the solution for maximal flow and push inventory to the final customer (as requested by service level strategy), even if this may result in higher transportation costs. So the rational precedence relation in the considered settings would

( 1) ( ) ( ) ( ) . ik ik ij k ji k i k i k i k i k i k i k j N j N x x y y g g z I                                   ) ( 0 1      k k k ij k ij t t x      i p k i V y        1 0 1 0 ( ) i k i k k k g t t             1 0 ( ) i k i k k k g t t             0  k i z   ; 1 1 1 1       p n i L k k i z J         2 1 1 1 ; n L p i k i k J g                ; 1 1 1 3       p n i L k k i p y J        ( 1) ( 1) 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1, 1 1 1, ik ik ik i k i k n n L n L n L p p p ij ij k i i k i i k i j k i k i k n L L L n p i i k ik ijk ik i k k i n k i i n j N N N N N J c x h y g r z f f b                                                                                       ( 1) 2 2 1 ik i k ik ik L L n ijk k k i j N j N N N b                                * min ) ( 1     J max ) ( 2     J min ) ( 3     J min ) ( 4     J 3 2 1    J J I J             min ) ( 1 J ) ( ) ( 3 2       J J           max ) ( 2 J          max ) ( 2 J 1 2 3 5 6 8 9
be . This sequence is therefore the basic rule for using the successive conces- sions method.

Since the set is convex and polyhedral, and objective functions are linear, the following scalar objective can be built based on the additive convolution:

, ( 19
)
where , are priority coefficients for and , are priority coefficients between the group and . With the suggested convolution, the Pareto set points can be gained with the help of varying the coefficients α (one possible solution is presented in Eg. ( 19)). The coordination of the planning results of static and dynamic models is carried out through the variables , , and (aggregate amount of products) of the static models and the corresponding variables , , and from the dynamic model, where vector u is optimal control vector, i.e., the optimal SC plan.

Experimental results

This section describes an example of the practical application and managerial insights. Since the developed model is an LP model with transit nodes and two-side constraints, it can be solved with the simplex method with consecutive plan improvement techniques implemented in any LP solver. The model includes k m equations and

           k m f k l l 3 1
unknown variables, where m is the number of nodes in the SC; k is the number of time intervals; l f is the number of variables characterizing amounts of received (transmitted) products at the l-interval. Since the modified simplex method is used in the computational procedure, we avoid extensive numerical experiments. However, we note that the computational complexity increases subject to the number of different types of products and intensity-depended costs of production and transportation.

Computing the SC plans and economic performance for the existing SC

Consider a part of an SC in automotive industry that is composed of two Tier 2 suppliers (nodes #1 and #2), one Tier 1supplier (node #3), two assembly plants (nodes #5 and #6), and two markets (nodes #8 and #9) (see Fig. 2).

Fig. 2. Current structure of the supply chain

For computational experiments, the following data set has been used:

 Consider six periods 4 2 1    J J J   } {      max ) ( ) ( ) ( ) ( 4 4 1 1 3 2 2 3                     J J J J рез 1 2 1     0 , 2 1    2 1 ,   J J 1 4 3     0 , 4 3    } , { 2 1   J J 4  J ijk x jl y jl g ) (t x ijk ) (t y jk ) (t g jk
 Demand is known from Sales & Operations Planning and set up at 250-240-230-240-250-240 units in market #8 and 220-210-200-210-220-210 units in market #9 in each period respectively.  The supplier #1 is the main Tier 2 supplier that delivers 400 units in each period  The supplier #2 is the secondary Tier 2 supplier that delivers 100 units in each period  Without loss of generality we consider the BOM factor as 1:1 for all stages  The processing intensities are as follows: node #3 -550 units, nodes #5 and 6 -300 units respectively, channel 13-500 units, channel 23=150 units, channel 35=300 units, channel 36=250 units, channel 58=280 units, channel 69=240 units  The warehouse capacities are as follows: node #1=150 units, node #2=70 units, node #3=250 units, nodes #5 and 6 = 100 units, nodes #8 and 9 = 50 units  The price for each final product in each market is $65  Upstream SC costs is $25 per unit (i.e., the costs on the upstream SC stages prior to the Tier 2 stage)  Transportation costs is $4 for each arc  Processing costs is $2 per unit  Inventory holding costs is $2 per unit in each period  Return flow costs is $15 per unit  Total fixed cost is composed of a firm part (proportionally to the number of SC elements) and the operating part (proportionally to the design capacity): $200 for each node and channel and $5 for each unit of processing or transportation capacity  Recovery costs is $1 for each unit of disrupted capacity in each period For analysis, the management considers two scenarios (Table 1). Disrupted elements in the optimistic scenario -6 6,1 58 -23

Disrupted elements in the pessimistic scenario 1 1 4 4,2 58 -In Fig. 3, results of optimal planning subject to the given data set and highest priority of the service level component in the goal function (Eqs. ( 1)-( 3) and ( 15)-( 17)) for the optimistic and pessimistic scenarios are presented. In Table 3, the performance impact with consideration of the reconfiguration plan is presented.

Running the developed planning model on the assumption of the highest priority of the service level component in the goal function, the optimal solution for the optimistic scenario leads to delivering 2,140 units which is equivalent to a service level of 78.7%, subject to the estimated demand of 2,720 units in six periods. For the pessimistic scenario, optimal solution delivers 1,280 units, i.e. the service level is of 47.1%.

Table 3. Performance impact with consideration of the reconfiguration plan

In the optimistic scenario, the SC still remains profitable, in the pessimistic scenario the losses of $13,390 arise. In comparison to the ideal case (no disruptions), in both optimistic and pessimistic scenarios the service level decreases to 78.7% and 47.1%, respectively. Moreover, the profits would decrease to $20,500 and $-13,390 as compared to $42,720, in the ideal case.

Analysis of different resilience actions

In both cases considered in Sect. 5.1, the profits and service level (i.e., sales volume) significantly decrease. This is the starting point for the SC managers to develop actions to increase the SC resilience. The central trade-off in this analysis is the question of efficiency vs resilience. In the next step, the developed model is applied to analysis of different actions for a possible resilience increase (measured through the service level and sales volumes) subject to this trade-off.

The management considers some structural and parametrical resilience increase actions (Fig. 4):

 The supplier #2 is used as a back-up supplier, i.e., it delivers under normal conditions 100 units in each period and can extend the quantity to 400 units if necessary  Back-up Tier 1 suppler #4 that can be used in the case of disruptions in node #3  Back-up assembly capacity #7 that can be used in the case of disruptions in nodes #5 and #6  Alternative transportation channels  Increase in warehouse storage and processing capacity  Increase in transportation channel capacity  Increase in order quantities from the Tier 2 suppliers #1 and #2.

Each of these actions could increase the resilience and have positive impact on the service level. However, the costs would also increase. The task is to identify experimentally the action (or a combination of actions) that ensures the best balance "efficiency vs service level" through the SC (re)planning. For experiments, the following assumptions have been made regarding processing and transportation intensities of new elements: new transportation channels -120% of the processing intensity of the outgoing node; node #4 -250 units in each period, node #7 -150 units in each period. In Table 3, the planning results are summarized. It can be observed that structural changes significantly impact the performance and resilience.

The most significant contribution to resilience increase is the action #7 where 100% service level can be achieved despite the disruptions. Obviously, this is one of the most expensive actions as well. In this particular case, action #7 results in a negative profit. The highest profit can be achieved using actions #3 and #4 which can be considered for SC re-design. In the optimistic scenario, for action #3, the service level increases by 8.9%, and the profit rises by 76.2% as In Fig. 6, SC costs for seven structural resilience actions are presented. In Fig. 7, the resilience and efficiency views of different structural actions are depicted. In Fig. 7, performance impact of structural resilience actions for optimistic and pessimistic scenarios is presented.

Fig. 7. Impact of different resilience policies on SC resilience and efficiency

It can be observed that structural changes differently impact the performance and resilience in the optimistic and pessimistic scenarios. The most significant contributions to the resilience increase in both scenarios are the actions #6 and #7. However, the profits are negative in these cases. The highest profit can be achieved using actions #3 and #4 which can be considered for SC re-design. Even in the pessimistic scenario, actions #3 and #4 achieves profits.

However, since structural changes cannot be implemented in short-term period, the management also considers the following parametrical measures: The modelling results are summarized in Table 5 and Fig. 8.

Table 5 Performance impact of different resilience policies for parametrical actions

The impact of different resilience policies on SC economic performance for parametrical actions in graphical form is reported in Fig. 8. It can be observed from Table 5 and Figure 8 that parametrical actions help to improve the SC resilience measured in service level and sales volumes. The most significant contribution to resilience increase is the action #6 in both scenarios. Obviously, this is the most expensive action. In this particular case, action #6 results in negative profits in both scenarios. Actions #1 and #2 can be considered for SC re-planning. In the optimistic scenario, for action #1, the service level increases by 0.7% and the profit declines by 0.9%. For action #2, the service level increases by 1.02% and the profit declines by 6.4%.

Managerial Insights

Disruption risks may result into ripple effect and structure dynamics in the SC. It is to notice that the scope of the rippling and its performance impact depend both on robustness reserves (e.g., redundancies like inventory or capacity buffers) and speed and scale of recovery actions (Knemeyer et al. 2009, Ivanov and Sokolov 2013[START_REF] Hu | Managing risk of supply disruptions: Incentives for capacity restoration[END_REF][START_REF] Kim | Guilt by Association: Strategic Failure Prevention and Recovery Capacity Investments[END_REF], Pettit et al. 2013). In many practical settings, companies need analysis tools to estimate both the SC efficiency and SC resilience. For SC resilience, the impacts of recovery actions subject to different disruptions and performance indicators need to be estimated.

The results of this study contribute to support decisions in these practical problems. The developed model can help the SC risk managers to identify how the existing SCD is resilient for different disruption scenarios. The model also considers recovery strategies (i.e., reconfiguration) that can be used by SC managers and translated into the SCD and SCP changes.

With the use of the developed approach, SC managers can compare different possible SCDs regarding their resilience (measured through the sales volumes or service level) using the proposed model. Since the computation includes the recovery actions, the developed model can help to identify opportunities to reduce disruption and recovery costs by SC re-design and re-planning.

The proposed model analyses effective ways to recover and re-allocate resources and flows after a disruption. It also considers disruption durations and operative reactions to disruptions. Such a model can be used by SC risk specialists to analyse the performance impact of different resilience and recovery actions and adjust mitigation and recovery policies with regard to critical SCD elements and SCP parameters.

Conclusions

SC planning models known in literature have been extensively considered in the light of severe disruptions. Taking into account the gap between practical needs and the literature, we developed a multi-objective formulation for the SC reconfiguration model (trade-off "service level vs. costs") that offers performance impact assessment with SC re-planning in the case of structure dynamics (trade-off "efficiency vs. resilience") taking into account temporary unavailability of some SC elements and their recovery in time. Therefore, this study extends the existing models to multi-stage SC (re)planning problem by an explicit connection of performance impact assessment and SC plan reconfiguration with consideration of the duration of disruptions and the costs of recovery.

The approach is based on a hybrid model containing elements of system dynamics and linear programming. The SC structure dynamics and recovery is considered in the dynamic model while a linear programming model is used for the flow balance. We pass from a standard linear programming model to a maximal flow problem by excluding demand constraints.

The results have some major implications. First, it suggests a method to compare different actions for supply chain resilience regarding the performance impact. Second, since the commodity flows are described in continuous time in the dynamic models, the disruption recovery time can be described more accurately. Third, this offers the possibility to include a service level and sales volumes as resilience indicators into the performance consideration.

Subsequently, through numerical computations, we obtain additional managerial insights. The developed model can provide the SC risk, sales, production and purchasing managers with the analytical tool of how to identify the SC resilience and the impact of recovery strategies in order to compare different possible SCDs regarding their resilience and efficiency. This extends the existing theoretical knowledge and practical applications which can be used by SC managers to apply robust methods to SC design and planning in the case of disruption occurrence to estimate the recovery policies and impacts on operational and financial performance.

Finally, some extensions of this model in future can be considered. First, the possibility of addressing decision components of different time horizons and levels of detail arises from a combination of a static LP and dynamic OPC models. This combination can result in a hybrid multiperiod production-routing model that can be investigated in detail in the future research.

Second, additional restrictions, e.g., total recovery budget with time value or other optimization objectives, e.g., minimizing delivery times, can be included in a future analysis. On-line adaptation can be an additional possible future research direction. Here different adaptation options (e.g., flow re-direction, capacity adjustment, and structure adjustment) and their costs can be compared. In addition, a comparison between investments in robustness vs. costs of adaptation can be made.

Some limitations of the proposed approach belong to its centralized planning focus. However, for the considered case-study the proposed model has been successfully validated and practically tested on different examples. In future, an extrapolation of other case-studies with different numbers of intervals of structural constancy and other SC structures may reveal some additional insights in the application of the proposed model. Additional research into dynamic and inverse models is needed. Cumulative impacts of different structural and parametrical actions can be investigated.

  3. Problem statement and methodology 3.1 Problem statement We investigate a multi-stage SC. Consider a referenced automotive SC design (SCD) structure subject to the study by Simchi-Levi et al. (2014) (Fig. 1).

Fig

  Fig. 1. SCD structure

  SCP  Network structure is dynamic, nodes and arc may become unavailable for different duration  Bill-of-materials (BOM) and different processing intensities  Reconfiguration of material flows in the case of a disruption and computation of the performance impact of the disruption taking into account optimal reconfiguration and recovery costs  Upstream supply control in the case of disruption subject to the total SC costs minimization  Different pro-active actions: structural and parametrical  Sales volumes and service level as resilience indicators  Sourcing, transportation, processing, return, fixed, inventory, and recovery costs are used for efficiency assessment and are assumed to be linear functions from the quantities.  Multiple objective views: logistics (costs minimization), customer (service level), system (supply chain design and disruptions)  Time duration of a disruption modelled as continuous time function in dynamic model  Impact of disruptions on the economic performance which is simulated using a hybrid staticdynamic optimization model  The transportation, processing, and inventory volumes are constrained by maximal capacities. The demands in markets may change in each period  Inventory from previous periods may be used in the following periods  If the processing and warehouse capacity is exceeded by the delivered quantity, unprocessed and non-stored goods are sent back subject to additional return costs (i.e., penalties)

  )  Production processing intensities are included in the model at different stages  Capacity degradation/recovery dynamics is described in Eqs (3) and (4)  Recovery costs is included into the objective function (12) 4.1 Set-theoretical formulation Let's introduce some notations and definitions: = is the set of nodes in the SC is the set of arcs in the SC is the set of operational characteristics for the transportation (if ) or processing at warehouse (if )

Fig. 3 .

 3 Fig. 3. Planning results for the initial SCThe production and shipment quantities are marked green, red and blue, according to the type of product in BOM. The yellow triangles show the storage capacities and their actual utilization. The grey rectangles depict the manufacturing design and utilized processing capacity respectively. The numbers on the arc represent the design transportation capacities and their actual utilization. The red nodes and channels are disrupted. The yellow arrows at nodes #8 and #9 depict the delivered volume of goods. The grey arrows depict the return flows. The results are presented in Table2.

Fig. 4 .

 4 Fig. 4. Possible extensions to SC structure

  Fig. 5. Planning results for the re-designed SC (action #4)

Fig. 6

 6 Fig. 6 Impact of different resilience policies on SC efficiency Analogously, pessimistic scenario can be investigated.

Fig. 8 .

 8 Fig. 8. Performance impact of different resilience policies for parametrical actions

Table 1

 1 Optimistic and pessimistic scenarios

	Period

Table 2

 2 SC reconfiguration plan for the optimistic scenario

	Node / Arc					Inventory / Shipment quantity	Manufacturing / sales quantity
	k=1	k=2	k=3	k=4	k=5	k=6	k=1 k=2 k=3 k=4 k=5 k=6 k=1 k=2 k=3 k=4 k=5 k=6
	1	1	-	1	1	1					150 400 400	400 400 400
	2	2	2	2	2	2		10			70	90	100 100 100 100 70
	3	3	3	3	3	3		250 80	250	200 490 490 110 250 500 300
	5	5	5	5	5	5				100 100	250 240	250 240
	6	-	-	6	6	6				20	50	240	280 210 220 210
	8	8	8	8	8	8			50		250 230 230 50	250 240
	9	9	9	9	9	9	20				220	210 220 210
	13 13 -	13 13 13 400 400		400 400 250
	23 23 23 23 23 -	90	90	110 100 100 140
	35 35 35 35 35 35 250 240 280 100 250 160
	36 -	-	36 36 36 240			230 250 240
	58 58 58 -	58 58 250 240 280		250 210
	69 -	-	69 69 69 240			210 220

Table 4

 4 Performance impact of different resilience policies in the optimistic scenario

		Node #2	Node #4 and	Node #7 and	Channels	Actions	Actions	Actions
		and	channels	channels 78	68 and	1+2	3+5	4+6
	Indicator	channel	45 and	and 79	59
		23	46
		Action 1	Action 2	Action 3	Action 4	Action 5	Action 6	Action 7
	Revenue	139100	139100	151450	160550	139100	160550	176800
	Transporta-
	tion costs	28280	26600	24320	29640	28880	33400	32800
	Inventory
	holding costs	2680	3260	1300	920	3600	4080	3940
	Return costs	22350	150	450	450	20100	14100	10350
	Fixed costs	13100	15200	12350	12600	16300	18400	18800
	Upstream
	costs	110000	65000	65000	65000	110000	110000	110000
	Recovery
	costs	2510	3310	2510	2990	3310	3310	3310
	Processing
	costs	8560	8560	9320	9880	8560	9880	10880
	Total costs	187480	122080	115250	121480	190750	193170	190080
	Profit	-48380	17020	36200	39070	-51650	-32620	-13280