Gayane Vardoyan

Saikat Guha

Philippe Nain

Don Towsley

On the Capacity Region of Bipartite and Tripartite Entanglement Switching and Key Distribution

On the Capacity Region of Bipartite and Tripartite

Entanglement Switching and Key Distribution Gayane Vardoyan * , Saikat Guha † , Philippe Nain ‡ , and Don Towsley * * College of Information and Computer Sciences, University of Massachusetts, Amherst, {gvardoyan, towsley}@cs.umass.edu † College of Optical Sciences, The University of Arizona, saikat@optics.arizona.edu ‡ Inria, France, philippe.nain@inria.fr

We study a quantum switch serving a set of users. The function of the switch is to convert bipartite entanglement generated over individual links connecting each user to the switch, into bipartite or tripartite entangled states among (pairs or groups of) users at the highest possible rates at a fixed ratio. Such entanglement can then be converted to quantum-secure shared secret bits among pairs or triples of users using E91-like Quantum Key Distribution (QKD) protocols. The switch can store a certain number of qubits in a quantum memory for a certain length of time, and can make two-qubit Bell-basis measurements or three-qubit GHZ-basis projective measurements on qubits held in the memory. We model a set of randomized switching policies. Discovering that some are better than others, we present analytical results for the case where the switch stores one qubit per user at a given time step, and find that the best policies outperform a time division multiplexing (TDM) policy for sharing the switch between bipartite and tripartite entanglement generation. This performance improvement decreases as the number of users grows. The model is easily augmented to study the capacity region in the presence of qubit decoherence, obtaining similar results. Moreover, decoherence appears to have little effect on capacity. We also study a smaller class of policies when the switch can store two qubits per user. The full manuscript can be found at https://arxiv.org/abs/1901.06786.

Multi-qubit entangled states are fundamental building blocks for quantum computation, sensing, and security. Consequently there is a need for a quantum network that can generate such entanglement on demand between pairs and groups of users [START_REF] Pant | Routing Entanglement in the Quantum Internet[END_REF]- [START_REF] Van Meter | Quantum networking[END_REF]. In this work, we study the performance of the simplest multi-user network, a star-topology quantum switch connecting k users, where each user is connected to the switch via a separate optical link as shown in Figure 1. Bipartite, two-qubit maximally-entangled states, i.e., Bell pairs (or EPR states) are generated at a constant rate across each link, with the qubits getting stored at local quantum memories at each end of the links. As these link entanglements start appearing, the switch uses two-qubit Bell-state measurement (BSM) between pairs of locally-held qubits and three-qubit Greenberger-Horne-Zeilinger (GHZ) basis measurements between triples of locally-held qubits to provide two-qubit and three-qubit entanglements to pairs and triples of users, respectively [START_REF] Nielsen | Quantum Computation and Quantum Information[END_REF]. The capacity of such a switch to provide these two types of entanglements to the users depends on the switching mechanism, the number of quantum memories and their decoherence rates, and the number of links.

Using the Ekert-91 (E91) protocol for Quantum Key Distribution (QKD) [START_REF] Ekert | Quantum Cryptography Based on Bell's Theorem[END_REF], pairs of users can convert shared Bell states into shared secret bits whose security derives from quantum mechanics. Similarly, a variant of the E91 protocol can be employed to convert shared GHZ states between three users into tripartite quantum-secured shared secret bits. If QKD is the ultimate goal of the quantum network, the end users do not need to have quantum memories. They can keep making projective measurements chosen randomly from a set of mutually-unbiased measurement bases as specified in the E91 protocol, and extract shared keys in post-processing using reconciliation, error correction and privacy amplification. Since shared key generation is a strictly weaker task than entanglement generation, an entanglement distribution rate is always also an achievable rate for QKD. Therefore, hereon in this paper, we will focus exclusively on entanglement generation rates. The creation of an end-to-end entanglement requires two steps. First two-qubit Bell states are generated pairwise between a qubit stored locally at the switch and a qubit owned by a user. Once such link-level two-qubit entangled states have been created, the switch performs joint (entangling) measurements (over j ≥ 2 locally-held qubits that are entangled with qubits held by j distinct users), which, if successful, produces a j-qubit maximally-entangled state between the corresponding j users. After the measurement, if the result of the measurement is communicated to the end nodes, they can use conditional local unitaries on their qubits to convert the shared entanglement into a generic pre-determined jqubit GHZ state of their choice. Link-level entanglement generation, as well as entangling measurements, when realized with practical systems, are inherently probabilistic [START_REF] Guha | Rate-loss Analysis of an Efficient Quantum Repeater Architecture[END_REF]. We assume that only two-user (two-qubit) and three-user (threequbit) entanglements are created, i.e., 2-qubit BSMs and 3qubit GHZ basis measurements are done at the switch. For simplicity, we assume that these j = 2 or 3 qubit measurements at the switch take negligible time and always succeed.

Each link attempts two-qubit entanglements in each time slot of length τ seconds, and with probability p, establishes one entangled pair successfully. For simplicity, we model the time to successfully create a link entanglement as an exponential random variable with mean 1/µ = τ /p. We assume that each link can store B = 1, 2, . . . qubits. We also assume that qubits at the switch can decohere and model decoherence time as an exponential random variable with mean 1/α. We assume a step-function decoherence model where the twoqubit entanglement goes from a maximally-entangled qubit pair (one ebit) to zero entanglement. In this work, we only consider B = 1, 2. Last, when a qubit is stored at the switch, with its entangled pair stored at a user, we refer to this as a stored link entanglement.

We assume that all possible bipartite and tripartite user entanglements are of interest and consider two classes of probabilistic policies, one for B = 1 and the second for B = 2, that provide the flexibility to generate both types of entanglements with arbitrary rates. Policies in both classes incorporate the oldest link entanglement first (OLEF) rule whereby when a link entanglement is created it is always matched up with stored link entanglements when possible rather than be stored. This has the nice consequence, when coupled with the assumption that links are homogeneous but statistically independent, that the system can be modeled by a continuous time Markov chain (CTMC) where the state simply tracks the number of stored entanglements for two users. Moreover, the OLEF rule ensures an efficient use of resources (i.e., link-level entanglements) in the presence of decoherence.

In this work, we employ a set of time division multiplexing (TDM) policies as a baseline. When operating according to a TDM policy, the switch performs BSMs γ fraction of the time and tripartite GHZ basis measurements 1 -γ fraction of the time, for γ ∈ [0, 1]. The goal is to determine whether there exist policies that yield higher switch capacity than TDM policies. To this end, we consider a class of randomized policies. When properly configured, these randomized policies outperform TDM. However the relative difference between the two randomized policies and TDM goes to zero as the number of links k → ∞. We also observe that increasing the number of memories from one to two increases capacity but that the advantage diminishes as k gets large. We also explore the effect that decoherence-the locally stored qubits at each end of the link being subject to a noise process that reduces the entanglement between the two qubits-has on capacity. In the cases of B = 1 with and without decoherence, we have simple closed form expressions for capacity whereas for the case of B = 2, our results are numerical. Both types of systems are modeled using CTMCs and capacities of bipartite and tripartite end-to-end entanglements are computed using the stationary distributions. An example of the capacity regions for B = 1, 2 is shown in Figure 2. In the figure, C 2 represents the bipartite capacity and C 3 the tripartite capacity. Note that the entanglement generation rate simply scales the capacities by µ, so we have set it equal to 1 for all examples shown. For the case of B = 1, we study the entire capacity region of the switch under a set of randomized policies that enable the switch to make more diverse decisions than the set of TDM policies. While we find many such policies that outperform TDM, one in particular stands out: a policy that yields the most efficient operation in terms of optimizing both bipartite and tripartite entanglement capacities. This policy is labeled in Figure 2a and can be stated as follows: (i) Suppose there is currently only one stored link entanglement. Let l 1 be the link that is associated with this entanglement. When another link (not l 1 ) generates an entanglement, the switch should store it (as opposed to using the two linklevel entanglements in a BSM). Note that if l 1 is the link that generates a new entanglement, the older (stored) one is dropped by the switch since B = 1, and the newer one is stored for future use. (ii) Suppose that there are currently two stored link-level entanglements (note that since B = 1, they must belong to two different links). Let l 1 and l 2 be the two links associated with these entanglements. If another link, l 3 generates an entanglement, the switch must always use the three distinct link-level entanglements to serve a tripartite end-to-end entanglement between the three users. However, if either l 1 or l 2 generates another entanglement, the switch must always perform a BSM using the two stored entanglements, and the new entanglement is stored for future use. Note that the latter rule directs the switch to not waste an entanglement whenever it is possible to use it in a measurement.

For B = 1, we analytically show that this policy is optimal. Further, we derive tight upper bounds on the achievable capacities given the set of randomized policies considered in this work. Note that the set of TDM policies form a line connecting the maximum achievable bipartite capacity, C * 2 , and the maximum achievable tripartite capacity, C * 3 . C * 2 and C * 3 are obtained when the switch performs only BSMs or 3qubit GHZ basis measurements, respectively. We call this the TDM line. Let the point ( Ĉ3 , Ĉ2 ) represent the tripartite and bipartite capacity produced by the optimal policy described above. We show that the upper bounds on the achievable capacity form a triangular region above the TDM line, such that the vertex of the triangle is the point ( Ĉ3 , Ĉ2 ), i.e., the point farthest away from the TDM line. Denote the area of the triangular capacity region above the TDM line as A , and the area below the TDM line as A T DM . Then the total area of the achievable capacity region is A T = A + A T DM . We show that as k → ∞, A /A T → 0. This result is proof that as the number of links grows, the advantages of using the alternate, randomized policies as opposed to TDM diminish.

For the case B = 1, the set of policies that we explored was exhaustive; i.e., we consider every policy that can be modeled by the CTMC. For the case B = 2, we opt for a less exhaustive search, but the goal is the same: determine whether there exist policies that perform better than TDM. Our analysis of this system is numerical, but the results closely resemble those of the B = 1 case: there is a region above the TDM line that represents a set of policies which outperform TDM. Also, as k increases, we observe that the ratio of the region above the TDM line to the achievable capacity region decreases. Moreover, we compare the capacity regions obtained from the B = 1 and B = 2 systems and discover while there is something to be gained from the extra buffer space for a small number of users, the advantage becomes less apparent for a larger number of users. For B = 1, the analysis of the capacity region is very similar to the original system without decoherence. As one expects, the effect of decoherence is that the achievable capacity region shrinks as α increases. Interestingly, we observe that this effect also diminishes as the number of links k grows. Examples of this phenomenon are shown for B = 1 and B = 2 in Figures 3 and4, respectively. For all examples, we use an entanglement generation rate of µ = 1; for qualitative results, we only need to concern ourselves with the value of α relative to µ. In typical scenarios, α is at least one order of magnitude smaller than µ.

To summarize, in this work, we explore a set of policies for a quantum switch that can store up to two qubits per link and whose objective is to perform bipartite and tripartite joint measurements to distribute two and three qubit entanglement to pairs and triples of users. We present analytical results for the case where the per-link buffer has size one. By comparing against TDM policies, we discover that better policies in terms of achievable bipartite and tripartite capacities exist, but that as the number of links grows, the advantage of using such policies diminishes. We also compare the capacity regions for systems with different per-link buffer sizes and observe that systems with fewer links benefit more from the extra storage space than systems with a larger number of links. Finally, we model decoherence for both types of systems and present analytical results for the case with per-link buffer size one. Observations and analysis show that as the number of links increases, the effects of decoherence become less apparent on systems.

Fig. 1 :

 1 Fig. 1: A quantum switch serving k users in a star topology.

= 5 Fig. 2 :

 52 Fig. 2: Comparison of capacity regions for systems of buffer sizes one and two with varying number of links k, and entanglement generation rate µ = 1.

  (a) k = 3 (b) k = 10

Fig. 3 :

 3 Fig. 3: Capacity region for a system of buffer size one and varying number of links k, decoherence rates α, and entanglement generation rate µ = 1. The solid lines are the upper boundaries of the capacity region, and the dashed are TDM lines.

Fig. 4 :

 4 Fig.4: Capacity region for a system of buffer size two and varying number of links k, decoherence rates α, and entanglement generation rate µ = 1. The solid lines are the upper boundaries of the capacity region, and the dashed are TDM lines.


Georgios Psaradakis

Limiting case of an isoperimetric inequality with radial densities and applications

Keywords: Mathematics Subject Classification: 49Q20, 26D10, 46E35, 35J75 Isoperimetric inequality, Radial density, Norm inequality with weights, Pólya-Szegö inequality

. Further applications include an L p version with the best constant of the functional analogue of this isoperimetric inequality and also a weighted Pólya-Szegö inequality.

Introduction 1.Overview, notation and central results

The Gagliardo-Nirenberg inequality in its sharp form states that

R n |∇f | dx ≥ nω 1/n n R n |f | n/(n-1) dx 1-1/n ∀ f ∈ C 1 c (R n ), (1.1)
Here and throughout the whole paper, ω n is the volume of a unit ball in R n , n ∈ N \ {1}, and | • | denotes both absolute value of numbers and length of vectors. Also, whenever G ⊆ R n is open, then C 1 c (G) stands for the set of continuously differentiable scalar functions with compact support in G and C 1 c (G; R n ) for all n-vector functions with C 1 c (G) components. Inequality (1.1) was proved by Maz'ya in [START_REF] Maz'ja | Classes of domains and imbedding theorems for function spaces[END_REF] and independently by Federer and Fleming in [START_REF] Federer | Normal and integral currents[END_REF]. It was shown in both papers that it is equivalent to the classical isoperimetric inequality in R n . Nowadays this equivalence is formulated as follows (see [START_REF] Evans | Measure theory and fine properties of functions[END_REF]Section 5.6.2]): (1.1) is equivalent to

P(E) ≥ nω 1/n n L n (E) 1-1/n , (1.2)
valid for any L n -measurable set E ⊂ R n with finite perimeter P(E); that is

P(E) := sup E div ϕ dx : ϕ ∈ C 1 c (R n ; R n ), |ϕ| ≤ 1 in R n < ∞,
Recall that equality holds in (1.

2

) if E = B R (x 0 ) := x ∈ R n : |x -x 0 | < R , any R > 0, any x 0 ∈ R n .
Maz'ya and Shaposhnikova in [START_REF] Maz | A collection of sharp dilation invariant integral inequalities for differentiable functions[END_REF]Corollary 5.1]-(see also [25, Corollary in Section 4.8]), have obtained the best constant in the following scale invariant weighted generalization of (1.1), the weights being powers of the distance to the origin: if 0 ≤ a < n -1 and an/(n -1) ≤ b ≤ a + 1, then

R n |∇f | |x| a dx ≥ C n,a,b R n |f | (n-b)/(n-1-a) |x| b dx (n-1-a)/(n-b) ∀ f ∈ C 1 c (R n ), (1.3)
where

C n,a,b := nω n (n -b) (n-1-a)/(1+a-b) (1+a-b)/(n-b) .
The case a = 0 has been established earlier in [START_REF] Maz'ja | On certain integral inequalities for functions of many variables[END_REF]. Without the sharp constant, inequality (1.3) goes back to Il'in; see [START_REF] Il'in | Some integral inequalities and their applications to the theory of differentiable functions of many variables[END_REF]Theorem 1.4,pg 367]. It is a subcase of the Caffarelli-Kohn-Nirenberg interpolation inequality; see [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF] and also the exhaustive work of Rabier [START_REF] Rabier | Embeddings of weighted Sobolev spaces and generalized Caffarelli-Kohn-Nirenberg inequalities[END_REF]. As with (1.1), the sharp estimate (1.3) has the isoperimetric counterpart

P(E; |x| -a ) ≥ C n,a,b L n (E; |x| -b ) (n-1-a)/(n-b) , (1.4)
valid for any L n -measurable set E ⊂ R n satisfying P(E; |x| -a ) < ∞. Here we have set (see [START_REF] Baldi | Weighted BV functions[END_REF])

P(E; |x| -α ) := sup E div ϕ dx : ϕ ∈ C 1 c (R n ; R n ), |ϕ(x)| ≤ |x| -α in R n , L n (E; |x| -b ) := E |x| -b dx. Taking a = b = 0 in (1.4) we recover (1.2). If a = 0 then -if b = a + 1, equality in (1.4) (resp. (1.3) formulated in BV (R n )) holds if E = B R (0) (resp. f = χ B R (0)
), that is to say the ball has to be centered at the origin. This is an effect of having weighted both sides by powers of the distance to the origin. The obvious changes go through if the distance is taken from a point x 0 ∈ R n .

-if b = a + 1, the same as above is true for (1.4) but equality in (1.3) is additionally achieved for any nonnegative, radially decreasing function f . Section 3 of this paper is devoted to an extension of this last case (see the first application in Section 1.2).

Several mathematicians have shown interest on different aspects of isoperimetric inequalities involving various kinds of weights; see for instance [START_REF] Brock | A weighted isoperimetric inequality in an orthant[END_REF], [START_REF] Cabré | Sobolev and isoperimetric inequalities with monomial weights[END_REF], [START_REF] Cabré | Sharp isoperimetric inequalities via the ABP method[END_REF], [START_REF] Chiacchio | Isoperimetric inequalities for the first Neumann eigenvalue in Gauss space[END_REF], [START_REF] Croce | An isoperimetric inequality for a nonlinear eigenvalue problem[END_REF], [START_REF] Csato | An isoperimetric problem with a density and the Hardy-Sobolev inequality in R 2[END_REF], [START_REF] Figalli | On the isoperimetric problem for radial log-convex densities[END_REF], [START_REF] Fusco | On the isoperimetric profile for a mixed Euclidean-log-convex measure[END_REF], [START_REF] Greco | An embedding theorem in Lorentz-Zygmund spaces[END_REF], [START_REF] Morgan | Existence of isoperimetric regions in R n with density[END_REF] and [START_REF] Pratelli | On the isoperimetric problem with double density[END_REF]. The work [START_REF] Alvino | Some isoperimetric inequalities on R n with respect to weights |x| α[END_REF] contains an extended list of the relevant references. Moreover, it gives new isoperimetric inequalities of the type (1.3) and (1.4) in the case 0 > a > b-1 (see Theorem 1.1-(iii) & (iv) there) and applications to Caffarelli-Kohn-Nirenberg inequalities. Let us mention that the preceded work [START_REF] Csato | An isoperimetric problem with a density and the Hardy-Sobolev inequality in R 2[END_REF] falls within this particular range for the parameters (n = 2, b = 0 and 0 > a > -1) and uses different methods. However, it seems that the above mentioned paper [START_REF] Maz | A collection of sharp dilation invariant integral inequalities for differentiable functions[END_REF] by Maz'ya and Shaposhnikova, where (1.4) is established, was unnoticed in the corresponding to radial weights recent literature (see for instance the reference given in [START_REF] Alvino | Some isoperimetric inequalities on R n with respect to weights |x| α[END_REF] for inequality (1.4), which is stated as Theorem 1.1-(ii) there).

The aim of this work is to investigate the end point case a = n -1 in (1.4) and (1.3) (this amounts to l + N = 0 in the notation of [START_REF] Alvino | Some isoperimetric inequalities on R n with respect to weights |x| α[END_REF]). Note the parameter assumptions for (1.3) to be valid force b to equal n in this case, so the right hand side of (1.3) is infinite unless f is supported away from the origin. We provide sharp substitutes for (1.4) and (1.3) by logarithmically correcting the weights in both of their sides. More precisely, let

X(t) := (1 -log t) -1 , t ∈ (0, 1],
and observe lim t→0 + X(t) = 0. Standard calculus shows that given R > 0, for any δ ∈ (0, R] one has

B δ (0) |x| -n X 1+θ |x| R dx = nω n δ 0 r -1 X 1+θ r R dr < ∞ if and only if θ > 0.
Keeping this in mind we read the basic result of the paper Theorem A. Suppose Ω is a bounded domain in R n , n ∈ N \ {1}, containing the origin and set R Ω := sup x∈Ω |x|. For all γ ∈ (0, n -1] and any

f ∈ C 1 c (Ω), it holds that Ω |∇f | |x| n-1 X γ |x| R Ω dx ≥ C n,γ Ω |f | n/(n-1) |x| n X 1+γn/(n-1) |x| R Ω dx 1-1/n , (1.5)
with the best constant C n,γ := nω

1/n n γ/(n -1) 1-1/n .
Remark 1.1. Clearly, (1.5) fails for γ = 0. On the other hand, the restriction γ ≤ n -1 is not essential. Note for instance that 0 ≤ X(t) ≤ 1 for all t ∈ [0, 1], so the right hand side will further decrease upon increasing the exponent on X. Of course the constant ceases to be optimal. We don't search for the best constant in case γ > n -1 here. Our aim is to demonstrate that taking γ > 0 smaller and smaller, the weight X 1+γn/(n-1) (|x|/R Ω ) relaxes less the singularity at 0 of the weight |x| -n , but the estimate fails for γ = 0. A similar open ended condition for the parameter indicating the power on a logarithmic correction has appeared recently in the Leray-Trudinger estimate of [START_REF] Psaradakis | A Leray-Trudinger inequality[END_REF]. See [START_REF] Adimurthi; Filippas | On the best constant of Hardy-Sobolev inequalities[END_REF], [START_REF] Filippas | Optimizing improved Hardy inequalities[END_REF], [START_REF] Gkikas | Optimal non-homogeneous improvements for the series expansion of Hardy's inequality[END_REF] and [START_REF] Psaradakis | An optimal Hardy-Morrey inequality[END_REF] for examples where the range of γ has to be in a closed interval for the purposes there.

Our isoperimetric inequality with radial densities will be a direct consequence of Theorem A: with

L n E; |x| -n X 1+γn/(n-1) (|x|/R Ω ) := E |x| -n X 1+γn/(n-1) (|x|/R Ω ) dx, whenever E ⊆ Ω is L n -measurable, we have Corollary A. Suppose Ω is a bounded domain in R n , n ∈ N \ {1}
, containing the origin and set R Ω := sup x∈Ω |x|. For all γ ∈ (0, n -1] there holds

P E; |x| 1-n X γ (|x|/R Ω ) ≥ C n,γ L n E; |x| -n X 1+γn/(n-1) (|x|/R Ω ) 1-1/n , (1.6
)

for any L n -measurable set E ⊆ Ω satisfying P(E; |x| 1-n X γ (|x|/R Ω )) := sup E div ϕ dx : ϕ ∈ C 1 c (Ω; R n ), |ϕ(x)| ≤ |x| 1-n X γ (|x|/R Ω ) in Ω < ∞.
Moreover, equality holds if E is a ball centered at the origin.

Applications

The choice b = a + 1 in (1.3) leads to the following L 1 weighted Hardy inequality

R n |∇f | |x| s-1 dx ≥ (n -s) R n |f | |x| s dx ∀ f ∈ C 1 c (R n ), (1.7)
whenever s ∈ [1, n) (here we have switched from "b" to "s" to be consistent with the notation in [START_REF] Psaradakis | L 1 Hardy inequalities with weights[END_REF]). Actually, it takes only an integration by parts to see that for any s ∈ R \ {n} there holds

R n |∇f | |x| s-1 dx ≥ |n -s| R n |f | |x| s dx, (1.8) valid for all f ∈ C 1 c (R n ) if s < n, or all f ∈ C 1 c (R n \ {0}) if s > n.
Moreover, it is easy to check when s < n, that equality holds for any nonnegative, radially decreasing function. In contrast, if s > n it is known (see [START_REF] Psaradakis | L 1 Hardy inequalities with weights[END_REF]Section 2.1]) that the constant appearing in (1.8) is again optimal but the inequality itself can be improved. More precisely, let Ω be any domain in R n containing the origin. For s ≥ n set

I[f ] := Ω |∇f | |x| s-1 dx -(s -n) Ω |f | |x| s dx, f ∈ C 1 c (Ω \ {0}).
(1.9)

We will establish the following improvement for (1.8).

Theorem B Let Ω be a bounded domain in R n containing the origin, and set R Ω := sup x∈Ω |x|.

Then for all γ > 0, s ≥ n, and any f ∈ C 1 c (Ω \ {0}), it holds that

I[f ] ≥ γ R s-n Ω Ω |f | |x| n X 1+γ |x| R Ω dx (1.10) + C n,γ R s-n Ω Ω |f | n/(n-1) |x| n X 1+γn/(n-1) |x| R Ω dx 1-1/n
, where the second term on the right fails to appear when γ = 0.

Remark 1.2. For the first term on the right, we already know (see [START_REF] Psaradakis | L 1 Hardy inequalities with weights[END_REF]Remark 2.6]) that it fails to appear when γ = 0. Hence the above inequality includes the optimal homogeneous weighted norm improvement and the optimal critical Sobolev weighted norm improvement at the same time.

Other applications include a p-version of Theorem A with p ∈ (1, n) (see Corollary 4.1 & Theorem 4.3 for the best constant) and also a weighted Pólya-Szegö inequality (see Theorem 5.4).

Throughout the rest of this paper, Ω denotes a bounded domain in R n , n ∈ N \ {1}, containing the origin and R Ω := sup x∈Ω |x|. Furhermore, L n stands for the Lebesgue measure in R n and σ for the n -

1-dimensional Hausdorff measure in R n . B r (x) is the open ball in R n having radius r > 0 and centre at x ∈ R n ; ∂B r (x) is its boundary. In particular B n := B 1 (0) and S n-1 := ∂B 1 (0). Also, ω n := L n B 1 (x) and so σ ∂B 1 (x) = nω n .

Proof of Theorem A and Corollary A

The proof is based in the ideas of [START_REF] Catrina | On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence) and symmetry of extremal functions[END_REF] and [START_REF] Horiuchi | Best constant in weighted Sobolev inequality with weights being powers of distance from the origin[END_REF], as applied in [START_REF] Alvino | Some isoperimetric inequalities on R n with respect to weights |x| α[END_REF] and [START_REF] Adimurthi; Filippas | On the best constant of Hardy-Sobolev inequalities[END_REF]. We start noting that since

0 ∈ Ω, given f ∈ C 1 c (Ω) we have f ∈ C 1 c (B R Ω (0)
) and hence it suffices to establish (1.5) with Ω = B R Ω (0). Moreover, being invariant under scaling, it is enough to establish it for R Ω = 1.

Consider the transformation B n x → (t, θ) ∈ [1, ∞] × S n-1 given by t := |x| 1-n , θ := |x| -1 x.
Then

t x i = -(n -1)t n/(n-1) θ i , θ x i = t 1/(n-1
) (e i -θ i θ), and writing g(t, θ) for f (x) we have

f x i = g t t x i + ∇ θ g • θ x i = -(n -1)t n/(n-1) g t θ i + t 1/(n-1) g θ i .
Altogether,

|∇f | = (n -1)t n/(n-1) g 2 t + (n -1)t -2 |∇ θ g| 2 1/2 .
Also, the absolute value of the determinant of the Jacobian matrix of this transformation is

J(t, θ) = (n -1) -1 t -1-n/(n-1) , therefore, with A := g ∈ C 1 ([1, ∞) × S n-1 ) \ {0} : g(1, θ) = 0 , we have nω 1/n n = inf f ∈C 1 c (B n ) f ≡0 B n |∇f | dx B n |f | n/(n-1) dx 1-1/n = (n -1) 1-1/n inf g∈A ∞ 1 S n-1 t -1 g 2 t + (n -1)t -2 |∇ θ g| 2 1/2 dσ(θ)dt ∞ 1 S n-1 t -1-n/(n-1) |g| n/(n-1) dσ(θ)dt 1-1/n . (2.1)
Next we define

C := inf f ∈C 1 c (B n ) f ≡0 B n |x| 1-n |∇f |X γ (|x|) dx B n |x| -n |f | n/(n-1) X 1+γn/(n-1) (|x|) dx 1-1/n . Consider this time the transformation B n x → (τ, θ) ∈ [1, ∞] × S n-1 given by τ := X -γ (|x|) = (1 -log |x|) γ , θ := |x| -1 x.
Then τ x i = -γτ 1-1/γ e τ 1/γ -1 θ i , θ x i = e τ 1/γ -1 (e i -θ i θ), and writing h(τ, θ) for f (x) we get

f x i = h τ τ x i + ∇ θ h • θ x i = e τ 1/γ -1 -γτ 1-1/γ h τ θ i + h θ i . These imply |∇f | = e τ 1/γ -1 γ 2 τ 2(1-1/γ) h 2 τ + |∇ θ h| 2 1/2 .
The absolute value of the determinant of the Jacobian matrix of the transformation is

J(τ, θ) = 1 γ τ 1/γ-1 e -n(τ 1/γ -1) ,
and taking into account that the transformation says |x| = e 1-τ 1/γ and X γ (|x|) = τ -1 , an elementary computation implies

C = γ 1-1/n inf h∈A ∞ 1 S n-1 τ -1 h 2 τ + γτ 1-1/γ -2 |∇ θ h| 2 1/2 dσ(θ)dτ ∞ 1 S n-1 τ -1-n/(n-1) |h| n/(n-1) dσ(θ)dτ 1-1/n . (2.2)
To compare the two infima in (2.1) and (2.2) we observe that since γ ∈ (0, n -1] and τ ≥ 1, we know

γτ 1-1/γ ≤ (n -1)τ.
Hence we may combine these equations to conclude

C ≥ nω 1/n n γ n -1 1-1/n .
The rest is a routine procedure. Supposing that R ∈ (0, R Ω ) we choose E = B R (0) and f = χ B R (0) , the characteristic function of B R (0). Translating for the moment |∇f | dx as the variation measure of f , it is known in this case that both

Ω |∇f | |x| n-1 X γ |x| R Ω dx and P E; |x| 1-n X γ (|x|/R Ω ) , are equal to ∂B R (0) |x| 1-n X γ (|x|/R Ω ) dσ = R 1-n X γ (R/R Ω )σ ∂B R (0) = nω n X γ (R/R Ω ).
On the other hand

Ω |f | n/(n-1) |x| n X 1+γn/(n-1) |x| R Ω dx = B R (0) 1 |x| n X 1+γn/(n-1) |x| R Ω dx = nω n R 0 r -1 X 1+γn/(n-1) r R Ω dr.
Noting that d dr X γn/(n-1) (r/R Ω ) = n/(n -1) r -1 X 1+γn/(n-1) (r/R Ω ), we conclude

Ω |f | n/(n-1) |x| n X 1+γn/(n-1) |x| R Ω dx 1-1/n = ω n n -1 γ 1-1/n X γ (R/R Ω ) = nω n X γ (R/R Ω ) C n,γ
, as required. This together with a standard mollification of the BV (Ω) function χ B R (0) shows that the constant C n,γ in (1.5) is the best possible. At the same time we have shown that equality holds in (1.6) whenever E is a ball centered at the origin.

Proof of Theorem B

By pushing further an argument of [START_REF] Psaradakis | L 1 Hardy inequalities with weights[END_REF] we deduce here the following improvement for (1.8) Proposition 3.1. For all s ≥ n, γ ≥ 0 and any f ∈ C 1 c (Ω \ {0}), it holds that

I[f ] ≥ γ R s-n Ω |f | |x| n X 1+γ |x| R Ω dx + 1 R s-n Ω |∇f | |x| n-1 X γ |x| R Ω dx. Proof. Let ϕ ∈ C 1 (Ω \ {0}; R n ).
Integrating by parts we easily get

Ω |∇f ||ϕ| dx ≥ Ω |f |divϕ dx, (3.1)
for all f ∈ C 1 c (Ω \ {0}). Choosing ϕ(x) = -1 - |x| R Ω s-n X γ |x| R Ω |x| -s x, x ∈ Ω \ {0}, there holds |ϕ(x)| = 1 -(|x|/R Ω ) s-n X γ (|x|/R Ω ) |x| 1-s for x ∈ Ω \ {0}, hence Ω |∇f ||ϕ| dx = Ω |∇f | |x| s-1 dx - 1 R s-n Ω |∇f | |x| n-1 X γ |x| R Ω dx. (3.2)
On the other hand,

div(v) = (s -n)|x| -s + γR n-s Ω |x| -n X 1+γ |x|/R Ω , x ∈ Ω \ {0}, thus Ω |f | div ϕ dx = (s -n) Ω |f | |x| s dx + γ R s-n Ω Ω |f | |x| n X 1+γ |x| R Ω dx. (3.3)
The proposition now follows by inserting (3.2) and (3.3) in (3.1).

Proof of Theorem B. The inequality of Theorem B readily follows from Theorem A and Proposition 3.1. In order to establish the optimality assertion, consider the function

f δ (x) := χ Bη\B δ (x); x ∈ R n , (3.4)
where, for any r > 0, by B r we denote (until the end of this proof), the open ball of radius r centered at the origin. Here 0 < δ < η < R Ω and η is fixed. It is easily seen that the distributional gradient of f δ is given by

∇f δ = ν ∂B δ δ ∂B δ -ν ∂Bη δ ∂Bη ,
where, for any r > 0, ν ∂Br stands for the outward pointing unit normal vector field along ∂B r , and by δ ∂Br we denote the Dirac measure on ∂B r . Also the variation measure of f δ is

|∇f δ | dx = δ ∂B δ + δ ∂Bη .
Using (3.4) we showed in [29, Remark 2.4 & Remark 2.6] that the constant s -n in the inequality

I[f ] ≥ 0 for all f ∈ C 1 c (Ω \ {0}) (see (1.9) for the definition of I[f ]
) is optimal and that if γ = 0, the first term on the right of (1.10) fails to appear. In the same fashion we have

I[f δ ] Ω |f δ | n/(n-1) |x| -n X(|x|/R Ω ) dx 1-1/n = δ 1-s σ(∂B δ ) + η 1-s σ(∂B η ) -(s -n)nω n η δ r n-1-s dr nω n η δ r -1 X(r/R Ω ) dr 1-1/n = (nω n ) 1/n 2η n-s log X(η/R Ω ) -log X(δ/R Ω ) 1-1/n = o δ (1).
A standard mollification of the BV (Ω) function f δ applies to see that the above computation holds in the limit. This shows that if γ = 0, the second term on the right of (1.10) fails to appear.

Limiting case in the Caffarelli-Kohn-Nirenberg inequality

Performing a weighted variant of a classical argument of [START_REF] Federer | Normal and integral currents[END_REF] (see also [13, Section 4.5.1, pg 140]) we obtain next a substitute for the end point case (a = n -p) of the parameters in the following p-version of (1.3):

if 1 ≤ p < n, 0 ≤ a < n -p and an/(n -p) ≤ b ≤ a + p, then R n |∇f | p |x| a dx ≥ C n,p,a,b R n |f | (n-b)p/(n-p-a) |x| b dx (n-p-a)/(n-b) ∀ f ∈ C 1 c (R n ). (4.1)
This was also established by Il'in in [START_REF] Il'in | Some integral inequalities and their applications to the theory of differentiable functions of many variables[END_REF] and reproved later in [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF] as a particular case of a multiplicative embedding inequality with weights; the Cafarelli-Kohn-Nirenberg inequality.

Ω |∇f | p |x| n-p X α |x| R Ω dx (4.2) ≥ C p n,α,p Ω |f | np/(n-p) |x| n X 1+(α+p-1)n/(n-p) |x| R Ω dx 1-p/n
, where C n,α,p = (n -p)nω

1/n n (α + p -1)/(n -p) 1-1/n / p(n -1) .
Proof. For p = 1 this is (1.5). Let 1 < p < n. We replace |f | by |f | θ in (1.5), where θ > 1 will be selected below. With β := 1 + γn/(n -1), we find

C n,γ Ω |f | θn/(n-1) |x| n X β |x| R Ω dx 1-1/n ≤ θ Ω |f | θ-1 |∇f | |x| n-1 X γ |x| R Ω dx = θ Ω |f | θ-1 |x| n(p-1)/p X β(p-1)/p |x| R Ω |∇f | |x| (n-p)/p X γ-β(p-1)/p |x| R Ω dx ≤ θ Ω |f | (θ-1)p/(p-1) |x| n X β |x| R Ω dx 1-1/p Ω |∇f | p |x| n-p X γp-β(p-1) |x| R Ω dx 1/p .
Choose θ so that θn/(n -1) = (θ -1)p/(p -1). Then θn/(n -1) = np/(n -p). Thus

C n,γ θ Ω |f | np/(n-p) |x| n X β |x| R Ω dx 1/p-1/n ≤ Ω |∇f | p |x| n-p X α |x| R Ω dx 1/p
, where we have set α := γp -β(p -1). Since β = 1 + γn/(n -1), we have

γ = (α + p -1)(n -1)/(n -p).
From this, condition 0 < γ ≤ n -1 is translated to 1 -p < α ≤ n + 1 -2p and also

β = 1 + γn/(n -1) = 1 + (α + p -1)n/(n -p),
as required.

Remark 4.2. The above estimate was known only for α = 0 when p = 2 (see [START_REF] Filippas | Optimizing improved Hardy inequalities[END_REF]Lemma 3.2]) and for α = 2 -p when p ∈ (1, 2) (see [START_REF] Gkikas | Optimal non-homogeneous improvements for the series expansion of Hardy's inequality[END_REF]Section 3]).

The best constant in (4.2) for α = 0 when p = 2 was found in [START_REF] Adimurthi; Filippas | On the best constant of Hardy-Sobolev inequalities[END_REF], while for α = 2 -p when p ∈ (1, 2) and n ≥ 3 can be extracted from the computations performed in [19, Section 3]. We can obtain the best constant for the whole range of the parameters n, α, p introduced in the above corollary, by arguing as in the proof of Theorem A.

Theorem 4.3. If 1 ≤ p < n, then for all α ∈ (1 -p, n + 1 -2p], the best constant in (4.2) is given by S n,α,p = α + p -1 n -p 1-1/n S n,p ,
where S n,p is the best constant in the Sobolev inequality (see [START_REF] Aubin | Problèmes isopérimétriques et espaces de Sobolev[END_REF] & [START_REF] Talenti | Best constant in Sobolev inequality[END_REF]).

Proof. To avoid repetition with the proof of Theorem A, we present only the basic tasks towards the proof of (4.2) and leave their verification to the reader. As before, we may assume

f ∈ C 1 c B n . Consider the transformation B n x → (t, θ) ∈ [1, ∞] × S n-1 given by t := |x| p-n , θ := |x| -1 x.
Then, working as in the proof of Theorem A, we obtain

S p n,p = inf f ∈C 1 c (B n ) f ≡0 B n |∇f | p dx B n |f | np/(n-p) dx 1-p/n = (n -p) p(n-1)/n inf g∈A ∞ 1 S n-1 t p-2 g 2 t + (n -p)t -2 |∇ θ g| 2 p/2 dσ(θ)dt ∞ 1 S n-1 t -1-n/(n-p) |g| np/(n-p) dσ(θ)dt 1-p/n , (4.3)
where A is as in the proof of Theorem A.

Next we define

S p n,α,p := inf f ∈C 1 c (B n ) f ≡0 B n |x| p-n |∇f | p X α (|x|) dx B n |x| -n |f | np/(n-p) X 1+(α+p-1)n/(n-p) (|x|) dx 1-p/n . Consider this time the transformation B n x → (τ, θ) ∈ [1, ∞] × S n-1 given by τ := X -α-p+1 (|x|) = (1 -log |x|) α+p-1 , θ := |x| -1 x.
It is not difficult to see that this gives

S p n,α,p (α + p -1) p(1-1/n) = inf h∈A ∞ 1 S n-1 τ p-2 h 2 τ + (α + p -1)τ 1-1/(α+p-1) -2 |∇ θ h| 2 p/2 dσ(θ)dτ ∞ 1 S n-1 t -1-n/(n-p) |h| np/(n-p) dσ(θ)dt 1-p/n . (4.4)
To compare the two infima in (4.3) and (4.4) we observe that since α ∈ (1 -p, n + 1 -2p] and τ ≥ 1, we know

(α + p -1)τ 1-1/(α+p-1) ≤ (n -p)τ.
Hence we may combine these equations to conclude

S n,α,p ≥ α + p -1 n -p 1-1/n S n,p .
It remains to show that the reverse inequality is also true. This is a consequence of the fact that the infimum in (4.3) is attained by radially symmetric functions (see [START_REF] Aubin | Problèmes isopérimétriques et espaces de Sobolev[END_REF] & [START_REF] Talenti | Best constant in Sobolev inequality[END_REF]), which in turn means the value of the infimum in (4.3) is the same if we consider admissible functions depending on t only. With this in mind we read S n,α,p ≤ S (radial) n,α,p =

α + p -1 n -p 1-1/n S (radial) n,p = α + p -1 n -p 1-1/n S n,p ,
and the proof is complete.

Remark 4.4. The analogous remark of Remark 1.1 applies to the above theorem.

A Pólya-Szegö inequality

In this section, whenever x ∈ Ω \ {0} and γ ∈ (0, n -1], we write for convenience

v(x) := |x| -n X 1+γn/(n-1) (|x|/R Ω ), w(x) := |x| 1-n X γ (|x|/R Ω ).
Definition 5.1. For any L n -measurable E ⊆ Ω we define successively:

(i) E ⊂ R n to be the ball centered at the origin and satisfying

L n (E; v) = L n (E ; v), (5.1)
(ii) χ E : R n → {0, 1} to be the characteristic function of E ; that is χ E := χ E .

(iii) the v-weighted rearrangement of a Borel measurable f : Ω → R given by

f (x) := ∞ 0 χ {|f |>t} (x) dt.
Remark 5.2. Since Ω contains the origin, given E ⊆ Ω, we know E ⊆ B R Ω (0). Hence the radius of E never exceeds R Ω . As a consequence, the argument of X on the right hand side of (5.1) is well defined.

Remark 5.3. The function f is nonnegative, measurable, radial and radially non-increasing. Moreover, {|f | > t} = {f > t} for all t ≥ 0, which together with (5.1) implies the weighted equimeasurability formula

L n {|f | > t}; v = L n {f > t}; v for all t ≥ 0. (5.2)
The two statements of Corollary A together with (5.1) imply

P E; w ≥ C n,γ L n (E; v) 1-1/n = C n,γ L n (E ; v) 1-1/n = P E ; w .
In particular, if E ⊆ Ω is a sufficiently smooth domain this reads

∂E w dσ ≥ ∂E w dσ. (5.3)
With this at hand, performing a weighted variant of a standard argument (we follow the presentation of [START_REF] Fusco | Geometrical aspects of symmetrization[END_REF]Theorem 3.1] for this), we establish here the following Pólya-Szegö inequality with radial density.

Theorem 5.4. Let f ∈ C 1 c (Ω), p ≥ 1 and γ ∈ (0, n -1]. Then Ω |∇f | p |x| n-p X ϑ |x| R Ω dx ≥ Ω |∇f | p |x| n-p X ϑ |x| R Ω dx, (5.4)
where ϑ is given by ϑ := γp -1 + γn/(n -1) (p -1).

Proof. We first recall some facts from basic geometric measure theory. In particular we are going to use the coarea formula; [25, Theorem 1.

whenever g ∈ L 1 (Ω) is nonnegative (simply choose Φ = gχ {|u|>s} /(|∇u|+ε), ε > 0, in (5.5), and then take the limit as ε → 0 using the monotone convergence theorem). We performed the above analysis in order to show that the assumption essinf|∇u| > 0 for (5. To prove the theorem we take u = f and Φ As a final result we will apply Theorem 5.4 to produce an embedding inequality in weighted Lorentz spaces (see [START_REF] Maz'ja | Sobolev Spaces (2nd revised and augmented edition[END_REF]Section 4.8]). This is an improvement of the embedding implied by Corollary 4.1 with α = ϑ. Recall that the weighted Lorentz L(P, Q; v), P > 1, Q ≥ 1, space is defined through the seminorm Proof. By the fact that d dr X γn/(n-1) (r/R Ω ) = n/(n -1) r -1 X 1+γn/(n-1) (r/R Ω ), we find first

= |∇f | p-1 |x| p-n X ϑ (|x|/R Ω ) in (5.5) to obtain Ω |∇f | p |x| n-p X ϑ |x| R Ω dx = ∞ 0 {|f |=t} |∇f | p-1 |x| n-p X ϑ (|x|/R Ω ) dσ(x)dt ≥ ∞ 0 {|f |=t} w dσ p {|f |=t} 1 |∇f | v dσ 1-p dt ≥ ∞ 0 {f =t} w dσ p - d dt L n {|f | > t}; v 1
L n B |x| (0); v = n -1 γ ω n X γn/(n-1) |x| R Ω , x ∈ Ω.

  Φ is a Borel measurable nonnegative function in Ω and u : Ω → R is Lipschitz. Rademacher's theorem asserts u is differentiable L n -a.e. in Ω and setting N u := {x ∈ Ω : ∇u(x) = 0}, we may take Φ = χ Nu in (5.5) to obtain 0 = Nu |∇u| dx = ∞ 0 σ {|u| = t} ∩ N u dt. Hence σ {|u| = t} ∩ N u = 0 for L 1 -a.e. t ≥ 0. This implies (5.5) can take the form {|u|>s} g dx = ∞ s {|u|=t} g |∇u| dσdt for L 1 -a.e. s ≥ 0,

6 )

 6 to hold true (see [13, Proposition 3, Section 3.4.4]) is redundant. Observe finally that (5.6) implies d dt L n {|u| > t}; g = -{|u|=t} g |∇u| dσ for L 1 -a.e. t ≥ 0.(5.7)

7 )

 7 -p dt where we have applied first Hölder's inequality to reach the middle line, and then (5.3) together with (5.7) for g = v to reach the last line. Note that the level sets of f are smooth enough by virtue of Sard's lemma. Hence, using (5.2)Ω |∇f | p |x| n-p X ϑ |x|R Ω dx ≥ for g = v but with u = f this time. On the other hand, taking u = f andΦ = |∇f | p-1 |x| p-n X ϑ (|x|/R Ω ) in (5.5) Ω |∇f | p |x| n-p X ϑ |x| R Ω dx = ∞ 0 {f =t} |∇f | p-1 |x| n-p X ϑ |x| R Ω dσ(x)dt= |∇f | is constant on the level sets of f .

Corollary 5 . 5 .

 55 [f ] L(P,Q;v) :=L n (Ω;v) 0 f * (s) Q d(s Q/P ) n ({|f |>t};v) (s) dt.Under the assumptions of Theorem 5.4, there holdsΩ |∇f | p |x| n-p X ϑ |x| R Ω dx ≥ ω p/(n-p),p;v) .

  Hence, noting f (x) = f * L n B |x| (0); v , we computeΩ |∇f | p |x| n-p X ϑ |x| R Ω dx = Ω [f * ] L n B |x| (0); v ∇ x L n B |x| (0); n) ds,where R is the radius of the ball Ω . The one dimensional weighted Hardy inequality (see[START_REF] Maz'ja | Sobolev Spaces (2nd revised and augmented edition[END_REF] Section 1.3.1]) applies to deduceΩ |∇f | p |x| n-p X ϑ |x| R p s -p/n ds,which is the desired estimate.

  The best constant in (4.1) when p > 1 goes back at least to[START_REF] Horiuchi | Best constant in weighted Sobolev inequality with weights being powers of distance from the origin[END_REF] Lemma 3.3]. It can also be found in[START_REF] Maz | A collection of sharp dilation invariant integral inequalities for differentiable functions[END_REF] Corollary 5.1] as a particular case of their more general inequality involving Lorenz spaces; see[START_REF] Maz | A collection of sharp dilation invariant integral inequalities for differentiable functions[END_REF] Theorem 5.2].

Corollary 4.1. If 1 ≤ p < n, then for all α ∈ (1 -p, n + 1 -2p] and any f ∈ C 1 c (Ω), it holds that

Acknowledgments The author is grateful to the anonymous referees for their critical comments and also for spotting a couple of inconsistencies. The author would like to thank Giuseppina di Blasio and Giovanni Pisante for their hospitality and useful advice. Part of this work was done while visiting the Department of Mathematics and Physics of University of Campania "Luigi Vanvitelli" with the aid of the INdAM/GNAMPA Visiting Professor program (Bando 30/11/2018).


Göknur Sirin
email: goknur.sirin@ecp.fr

Torgeir Welo
email: torgeir.welo@ntnu.no

Bernard Yannou
email: bernard.yannou@ecp.fr

Eric Landel
email: eric.landel@renault.com

VALUE CREATION IN COLLABORATIVE ANALYSIS MODEL DEVELOPMENT PROCESSES

Keywords: Modeling and Simulation, Lean Systems Engineering, Collaborative Product Development, Model Reuse, Pull and Push Concept

Integration and coordination of engineering analysis model is a vast development field in the context of complex product development. Engineers' siloed way of working in combination with lack of efficiency in current model development process may cause inconsistency based on model interfaces, human errors, miscommunication between teams and misinterpretations. In lean terms, this may create multiple wastes, including waiting, overproduction leading to excess inventory, unnecessary processing and may be the most harmful: defects (e.g., incorrect models) with rework consequences. Hence, product manufacturing companies must establish effective processes to add value throughout the multidisciplinary distributed modeling environment. The goal of this paper is to propose a pull-control model development process, providing model architecture integration and coherent control in early design phase. This paper proposes also an appropriate reuse strategy; this allows for utilizing plug-and-play type modular product models managed through a single-source of authority concept. A pull-control development process helps prevent potential rework arising from inconsistencies related to definitions, know-how and stakeholders communication at an early stage of the design process. Also, the proposed black box models reuse strategy helps reduce human-related error such as lack of domain knowledge, experience and misinterpretations. The proposed method is used to identify and visualize potential improvement in terms of increased model transparency and reuse when transforming from the present to the suggested future modeling strategy. The research has been conducted by synthesizing findings from a literature review, in combination with observations and analysis of current analysis model development practices within the automotive OEM Renault in France.

INTRODUCTION

A. Background Nowadays analysis models are becoming critical tools for automotive vehicle development. These models can be used to better understand the impact of different operating conditions on vehicle attributes and performances. To accurately represent vehicle behavior in a given field situation, these analysis models are created by integrating different models of the subsystems within a vehicle, such as chassis, engine, transmission, etc. These subsystem models are referred to as domain models. However, vehicle modeling is a complex endeavor which includes multiple disciplines, coordination between multiple stakeholders (i.e. Internal and external model suppliers, model architect and system architect), management of large amounts of data and information throughout the lifecycle of the modeling process [START_REF] Sirin | Creating a Domain Ontology to Support the Numerical Models Exchange between Suppliers and Users in a Complex System Design[END_REF]. Today, one of the major challenges is to manage the creation of system level models by integrating associated domain level model and providers, while delivering a viable system level model in less time. Since the quality of the complex multidisciplinary system model is strongly dependent on input from multiple sources, domain experts from different disciplines must establish a common understanding for collaborating and transforming knowledge across the different disciplines [START_REF] Sirin | Creating a Domain Ontology to Support the Numerical Models Exchange between Suppliers and Users in a Complex System Design[END_REF]. Hence, ensuring confidence in simulation results (i.e., model credibility) is particularly challenging for the design of multidisciplinary systems in which domain models within different disciplines are tightly coupled. In addition there are many factors that may cause inconsistencies, such as model interfaces, human error, miscommunication between teams and misunderstood assumptions. These inconsistencies are all sources to uncertainty and its propagation in multidisciplinary modeling environment is more complicated than in a single disciplinary domain. The effect of the uncertainties in one domain model may propagate to another through interrelated variables, and the final system output will suffer from the accumulated effect of the individual uncertainties. Thus, the information flow in modeling practice is one of the key aspects of its uncertainty, which may imply risk that the product attributes will not ultimately meet user needs [START_REF] Oberkampf | A New Methodology for the Estimation of Total Uncertainty in Computational Simulation[END_REF] In a vehicle development environment, the modeling process involves several companies (suppliers) and several design teams evolving through large-scale partnership [START_REF] Vosgien | Lean Approach to Integrate Collaborative Product Development Processes and Digital Engineering Systems[END_REF]. A large amount of data is then created by the different partners (company internal and external suppliers) involved in designing a new product using communication and IT technologies to access and exchange product design information in a context of simultaneous and concurrent systems engineering [START_REF] Vosgien | Lean Approach to Integrate Collaborative Product Development Processes and Digital Engineering Systems[END_REF]. Consistency between the models in the design process within a socio-technical system is not only governed by the technology tools available but also to a large degree by human factors, such as confidence in model quality as well as company cultural factors such as procedures for appropriate model reuse and more formal communication. On the contrary, for example, lack of a well-defined model reuse strategy can be a source for inaccuracy, leading to uncertainty and time delays as data is lost or is not consistently modified in the translation from one stakeholder to another. Here the term model reuse refers to modifying existing domain models to fulfill purposes for which they were not originally made. Model reuse is an important overacting objective since it can reduce development time and decrease costs. However, model reuse requires well defined documentation along with towering engineering knowledge and experience about the system [START_REF] Malak | Validating Behavioral Models for Reuse[END_REF]. On the other hand, if the engineers lack critical parts of domain knowledge, model validation can be as time consuming and costly as adequate models have to be developed from scratch [START_REF] Malak | Validating Behavioral Models for Reuse[END_REF]. The viewpoint defended in this paper is that unless a Modeling and Simulation (M&S) practitioner understands the model's contextual dependencies accurately and unambiguously, model reuse will continue to be an ineffective trial-and-error effort [START_REF] Yilmaz | On the Need for Contextualized Introspective Models to Improve Reuse and Composability of Defense Simulations[END_REF]. To this end, the basic motivation for building an adequate system level model is to encourage using domain models as a black box. To be able to integrate the domain level models to a full-vehicle system model, we need to prepare a detailed virtual prototype of the system in the early design phase. This need requires integrating multiple views (functional, structural and physical) and leveraging a flow-correctness check mechanism to the current model development procedure. It helps also to synchronize between actual needs of the downstream process and what the upstream process is delivering (see Fig. 1 right side image). Today, however, most of the high level model development and integration activities do not integrate this kind of early detailed model design and correctness check. In this manner, the system level model integration activity is conducted by finding, modifying and integrating existing models (physical prototype) without knowing which sub models are meaningful to connect together and, more specifically, how they can be connected to each other and with which interface requirements (see Fig. 1 left side image). It is typically a Push way of working.

B. Objectives and Research Questions

The solution proposed to overcome some of the problems indicated above is a detailed Model Design phase integration and a correctness check in early model building process (see Fig. 1 right side image). Thus, definition of project's scope with different views (i.e., functional, structural and physical) in the design phase vastly influences the model development and its overall performance. Understanding the complexity of design in functional, structural and physical contexts at an early stage is important in defining appropriate end facility of the project [START_REF] Kohler | Long-term design, management and finance for the built environment[END_REF].

Today, many companies try to use the V-cycle system engineering process for product development as proposed by Frosberg and adapted by the National Council on Systems Engineering [START_REF]Systems Engineering Fundamentals, DOD Supplementary text[END_REF]. However, companies' model building process is typically functioning as a push system rather than a pull system (see Fig. 1 left side image); i.e., a push system is one where an upstream operation transmits work to a subsequent downstream operation without being requested as a need for further processing. The push concept, also called 'over-the-walldesign/modeling', is only efficient when looked at from a local, silo-structure's view point. In a push-type model development process, testing and integration of hardware products are mainly done in later process stages using physical prototypes. This may create multiple wastes, including waiting (i.e., the model is not available when needed or it is sitting waiting for somebody to process it further), over-production (i.e., the model is not needed), which may lead to excess inventory (i.e., data/models not utilized), unnecessary processing (i.e., sending files/models not requested or recreating existing models), and may be the most harmful of them all, namely, defects (i.e., incorrect models), which is subsequently used as basis for design decisions. The common denominator is lack of understanding of needs of intermediate 'customers', causing rework [START_REF] Hagel | From Push to Pull-Emerging Models for Mobilizing Resources[END_REF]. An alternative model development process must be more effective and efficient than the existing. The aim of this work is to maximize the value creation throughout engineering analysis model process, while increasing the confidence in the model. In this connection, a suitable measure may be to apply principles from Lean Thinking to the Systems Engineering (SE), combining these two concepts into a common modeling strategy, henceforth called Lean Systems Engineering (LeanSE) [START_REF]The Guide to Lean Enablers for Managing Engineering Programs[END_REF][START_REF] Rebentisch | Lean Systems Engineering: Research Initiatives in Support of a New Paradigm[END_REF]. Hence, LeanSE may be a suitable strategy for achieving improved modeling practices in systems engineering. Adding a Detailed Model Design stage and Correctness Check in the current model development process improves the traditional V-cycle because this allows problems to be identified early. As shown on the right hand side of Fig. 1, this may reduce rework in the more expensive implementation and physical prototype validation phase, which is the main driver for product development cost [START_REF] Weck | Feasibility of a 5x Speedup in System Development due to Meta Design[END_REF].

To maximize the value in collaborative modeling environment depends on advances in many areas. Among the key issues here are: Addressing these two issues will help identify wastes, inefficiencies, and non-value added activities, which should be eliminated come up with at a more desirable "future state" as an intermediate stage towards the possible "ideal state" serving as a longer-term goal.

The reminder of this paper is organized as follows. In Section 2, we provide an investigation and literature review about Complexity management and Value creation. In section 3, we introduce our methodology to optimize the current model development activities by proposing a Pull Process with a detailed Model Design Phase and Correctness Control and a Model Reuse Strategy. The conclusion and future research are in Section 4.

DETAILED INVESTIGATION AND LITERATURE REVIEW 2.1 Managing Complexity with Different Views and View Points

The applicability of any model depends on the accuracy and reliability of its output. Yet, because all models are imperfect abstractions of reality and precise input data are rarely available, all output values are subject to inaccuracies. Accuracy is the closeness of the agreement between the measured value and true value. High accuracy implies a low error. Precision is the closeness of the agreement under the same conditions, and reflects the repeatability of the result [START_REF] Van Der Velden | Probabilistic Certificate of Correctness for Cyber Physical Systems[END_REF]. Moreover, input data errors and modelling uncertainties are not independent of each other as they can interact in various ways.

Sources of uncertainty may be categorized in model inputs, numerical approximations, or in the form of mathematical models. Input uncertainty arises when the requirements that define a design problem are inaccurate. Model inputs include both parameters used in the model of the system and data from the surroundings (see Fig. 3). Model input data includes parameters such as geometry, constitutive model, and boundary conditions, and can come from a range of sources including experimental measurement, theory, other supporting simulations, or even expert opinion [START_REF] Oberkampf | A New Methodology for the Estimation of Total Uncertainty in Computational Simulation[END_REF]. Data from the surrounding includes boundary conditions and additional knowledge (see Fig. 3). Additionally, in a design process there are different stakeholders such as model and system architects, model providers, who have different needs and viewpoints. In Model Based System Engineering, Views and Viewpoints can be used to model the perspectives of different stakeholders and their interests. A viewpoint describes a particular perspective of interest to a set of stakeholders, while a view is a stereotyped package that is said to conform to a particular viewpoint [START_REF] Friedenthal | A practical guide to sysml: The systems modeling language[END_REF]. In a traditional document-based approach, each stakeholder works from their own domain-specific tools and documents that they need to perform their tasks. For a model-based approach, this necessity remains. Ideally, each stakeholder would only use models which have been customized to their "view" of the system, in lieu of any documents. Practically speaking, however, this is not yet possible. Documents and presentations are still an integral part of the engineering design process. To support this functionality for a SysML-driven MBSE approach, many tools have been developed that allow users to generate and modify documents linked to a SysML model [START_REF] Mcginnis | A Simple Example of SysML-Driven Simulation[END_REF].

Figure 2. Views and View Points

As shown in Fig. 2, the proposed Detailed Model Design phase contains three views: Functional view identifies the boundaries of the system, the requirements and defines what the system has to accomplish for the users. Structural and behavioral views illustrate the system as a white box and define how the system will work to fulfill expectations. Physical view indicates how the system will be developed and built, along with interfaces and domain model specifications with model providers. These three views provide the basis for creating a virtual prototype in the early design phase. After establishing these three views, the next step is to send a request to domain model suppliers. The request has to contain the industrial criteria and integration strategy, including what is acceptable for each supplier. The process that the authors explained in the section 3 covers especially Structural and Behavioral views creation.

2.2

Lean Thinking and Value Creation In Lean SE, value is defined simply as mission assurance (the delivery of a flawless complex system, with flawless technical performance during the product or mission life cycle), satisfying the customer and all other stakeholders, which implies completion with minimal waste, minimal cost, and the shortest possible schedule [START_REF] Womack | Lean Thinking. Banish Waste and Create Wealth in Your Corporation[END_REF].

Figure 3. Value Creation in Modeling Process

As illustrated in Fig. 3, the aim of this work is to maximize the value of the analysis model. This means to increase the confidence in the analysis model by decreasing time to development and cost. To be able to evaluate added value, addition has to be paid to the current Model Development Process. However, identifying value in the model development process is not straightforward. The complexity of the process, distance from the final customer, shifting market conditions, and uncertainties of technical performance, cost, and schedule, all make a simple definition of value based on customer (model user) needs unworkable for process improvement [START_REF] Bauch | Lean Product Development: Making waste transparent[END_REF]. Overall, creating value through Product Development Process (PDP) activities includes creating information and knowledge about the product definition while reducing risk and uncertainties in the project. The activities require internal and external inputs supported by resources (human, material, IT resources) to provide added-value outputs. In this paper, without going into details in Lean philosophy, we would like to inspire from lean thinking to be achieved through the identification, monitoring, analysis and continuous improvement of value chains in the Modeling and Simulation activities. The purpose is to create a continuous flow of material and information to deliver the desired customer value with the least possible waste of resources and minimized delays. In "Lean Thinking" [START_REF] Womack | Lean Thinking. Banish Waste and Create Wealth in Your Corporation[END_REF], the authors define an iterative approach, inspired by the Deming's "quality wheel", sequenced in 5 basic steps:

Defining what makes or creates value for customers 2. Identifying the value stream 3. Promote the flow of the stream by ensuring that the stages of value creation are optimized 4. Pull the flow downstream 5. Striving for perfection to achieve excellence

We will try to improve the current model development process based on these 5 basic steps.

Non-Value Added Activities in Product Development

Process Today, supplying analysis models especially from external providers may be challenging. For example, when supplying a model from the requirement elicitation phase to model integration tests, the probability of failing is very high since, among others, there is limited common vocabulary between the two stakeholders. The source of the problem is also based on wrong or insufficient knowledge transmission from automotive manufacturer to model suppliers. Due to lack of common understanding and transparency, the result may be that the provided model does not fully conform to the requirements (Fig. 4). As a final result the mentioned activities take a lot of time, sometimes as much as 1-6 months, of several problem solving meetings and integration tests. Based on our observation in Renault, the final product performance is mostly related to the following three key points:

• Knowledge generation and appropriate model reuse

The knowledge encapsulated in each analysis model must be standard and coherent to be used by another party. The knowledge must be captured for reuse in future projects. To reduce the possible risk caused by inappropriate model reuse, therefore, one needs to define a robust model reuse strategy. Here documentation alone is insufficient; equally important is model providers' towering knowledge and experience. The model reuse strategy is discussed in Section 3. • Common understanding and Team communication: Knowledge sharing is one of the key points in the model development process. Providing a common vocabulary for the M&S users can help communicate fact-based decision in a maker's assessment of the credibility of M&S results. The ability for users to select from a list of options is an immensely important capability. Because creating full-vehicle simulation models is a multidisciplinary process, it is important that the same strategies are used across different teams of domain experts. By limiting large groups of users to the same vocabulary and set of options whenever possible, inconsistencies arising from miscommunication or misinformation can be reduced significantly. Company. Since these 5 categories are strongly linked to the project targets [START_REF] Oehmen | Waste in Lean Product Development[END_REF]. Process overview: However, we can argue that, due to the collaboration process in product development, the delivery delays in products have considerable financial impacts for all collaborators in this process. Some studies have shown also that delays in development were factors that had the most negative impact on expected profits, and that exceeding the development budget was the factor with the least impact [START_REF] Womack | Lean Thinking. Banish Waste and Create Wealth in Your Corporation[END_REF][START_REF] Bauch | Lean Product Development: Making waste transparent[END_REF]. In addition, managing several projects simultaneously is not a trivial issue, especially for companies developing different complex products like automobile engine. One of the key issues is the resource allocation and finding the balance between single project optimum and overall organizational benefits. In addition to this observation, we would like also to highlight that poor understanding of client's needs, human error and lack of expertise are the also the key issues in complex model development process. This is the reason why, section 2 and 3 introduce Pull-type model building process with a Detailed Model Design stage and Early Correctness Check and the most reliable Model Reuse Strategy.

METHODOLOGY

PULL-TYPE MODEL BUILDING PROCESS

This section aims to demonstrate the flow and pull concept for downstream activities (see Figure1 right side). The "Flow" principle enables the value creation process to flow smoothly and continuously without waste, such as unintended stops, waiting, rework, or backflow. On the other hand; "Pull" promotes the culture of tailoring tasks and their outputs to meet the legitimate needs of internal or external customer, while eliminating wasteful activities. In addition, the flow principle contains several measures related to the practices intended to boost the flow. These include frequent clarification of requirements as well as frequent opportunities for decision-making, using effective communications and coordination practices [START_REF] Hagel | From Push to Pull-Emerging Models for Mobilizing Resources[END_REF][START_REF]The Guide to Lean Enablers for Managing Engineering Programs[END_REF]. The major aim of this detailed design phase is to promote the communication and engineering data exchange between different design actors especially between external model providers and model architect(s).

Actors:

The [START_REF] Kato | Development of a Process for Continuous Creation of Lean Value in Product Development Organizations[END_REF] Copyright © 2014 by ASME and simulation knowledge. They have also a deep understanding of both the system-level requirements for the vehicle model, as well as how their models must interface with other domain models.

Activities Overview:

Each activity addresses different questions. Par example; analysis application plan of System Architect addresses; detailed description of set of analyses to be run such as what will the analysis be used for, which vehicle architecture are we considering? Etc…Model Requirements contain vehicle and domain model requirements. In vehicle model requirements, we try to know which simulation environment will be used, which interfaces are needed etc. In domain model requirements activity; our aim is to know what assumptions should be made, are there assumptions or restrictions that will affect other domains etc. Domain and Vehicle model specifications addresses the questions such as how should the model be created, how should it be verified and validated? How should the domain models be integrated and how should the integrated models be verified and validated? Based on output from the previous steps, the System and Model Architect can draft a set of requirements for the vehicle system model. This includes specifying the computer and operating system that the model should be run on, which solvers and file formats to use, which language(s) the model should be written in, etc. These requirements are used to drive the development of the individual domain subsystem models, whose requirements are defined at a later stage. System architecture selection and characterization is extremely useful in complex, multidisciplinary vehicle system analysis. Architectures provide a holistic view of a system and allow different stakeholders to work together with a common basis in the same vehicle system definition [START_REF] Branscomb | Supporting Multidisciplinary Vehicle Analysis Using a Vehicle Reference Architecture Model in SysML[END_REF]. There are many different architecture exist for capturing vehicle architectures; for example, if the goal of a system design study is to examine different vehicle architectures, such as a traditional internal combustion engine version or a full-hybrid version. It is of the utmost importance that this information is presented explicitly; otherwise it could result in major inconsistencies between the models create by different domain engineers. In particular, communication may be challenging between model architect and domain model suppliers since they in most cases are not collocated. Defining lists of model attributes for model specification and most vital attributes to reduce interoperability problems is very important. Domain engineers must specify both the control attributes that they need and the attributes that they plan to provide from their models. This complete list of model attributes is then reviewed by model architect, who negotiate with each domain team to develop a consistent set of signals for the entire vehicle. These system and model architects must negotiate with both the domain engineers providing signals and those receiving signals, so that all of the analysis models are compatible. Because this is an iterative process, it may require several rounds of negotiations with all of the different teams before a common vehicle-wide set of control signals can be agreed upon. Using a formal check list and a correctness control is critical for early virtual prototype validation.

Correctness Control

After having established the domain level signals (interfaces and domain level model specifications), the model architects integrate these sub models into a full virtual prototype. The model architect can evaluate alternative architectures against different model accuracy constraints. S/he has to detect any potential problem before the IVVQ (Integration, Verification, Validation, and Qualification) phase. The virtual prototype contains various correctness checks for interoperability problems such as domain models software names, versions, models' min/max values, units, the direction of acausal connections, models' accuracy levels, etc. (see Figures 6 and7) [START_REF] Van Der Velden | Probabilistic Certificate of Correctness for Cyber Physical Systems[END_REF]. scavenging"), and they invoke software components they have written or purchased previously. However, multidisciplinary analysis model reuse is much more risky than software code reuse because the former requires engineer's deep domain knowledge and system experience [START_REF] Malak | Validating Behavioral Models for Reuse[END_REF]. Reuse has different meaning for different modelers. In one case, reuse could be limited to only recomposing existing models from a library, without modification of components. In other contexts, reuse can involve both reuse without modification, as well as modifying an existing component if it meets modeling needs and/or reuse will speed development.

Model Refactoring

The way that engineering teams exchange modeling and simulation data is often siloed and highly inefficient. One of the primary consequences of silo structure is model refactoring. The term refactoring refers to developing an existing model from scratch. Change control and version management tend to be manual or more or less absent, and models and simulation data are often stored on local drives or network shares. Different engineering teams in the same organization may be solving the same problem, or even one that has already been solved, and they lack effective ways to achieve good solutions. Refactoring is the source of wastes like inventory, over-processing and over-production. As a result, companies try to find alternative solutions to refactoring for saving time and money. One of the alternatives to model refactoring is establishing a modeling reuse strategy. If correctly developed, reusing existing models provides a high potential in reducing modeling efforts [START_REF] Yilmaz | On the Need for Contextualized Introspective Models to Improve Reuse and Composability of Defense Simulations[END_REF].

Model Reuse

One reuse strategy is to consider standard sub-models as a black box, whose functionality may be related to input, output, control variables and decision parameters without any knowledge of its internal workings. The opposite is a white box object or system where its internal components or logic is available for inspection and modification, such as an application code. The model reuse concept may also refer to the case of modifying an existing component or full product models to fulfill purposes for which they were not originally made. Progress in model reuse requires significant Copyright © 2014 by ASME developments in several areas such as 1) understanding what information is needed to support reuse and how this should be represented, 2) developing mechanisms, automated and manual to collect and record this information, 3) understanding how to design for reuse, 4) developing analysis and search tools to locate appropriate existing components, 5) documentation of Validation and Verification (V&V) steps, which cannot be fully fulfilled [START_REF] Mohamed | Component Based Modeling and Simulation of Value Stream Mapping for Lean Production Systems[END_REF].

Model Reuse (via modification): White Box Model

Reusing of the entire simulation model which is complex and particularly challenging for model validation. Model reuse through modification requires well-defined documentation and towering engineer knowledge about the system and the conditions of interest. The diversity of objectives for different simulation uses makes creation of models that can satisfy all intended simulation needs infeasible. On the other hand, if the user lacks critical domain knowledge, model validation can be as time-consuming and costly as developing a similar model from scratch (Model Refactoring) [START_REF] Malak | Validating Behavioral Models for Reuse[END_REF]. Possible risks and consequences of reusing models include errors-prone decision, loss of intellectual properties and difficulty to modify a model when requirements change. Thus, the knowledge encapsulated in each numerical model needs to be coherent for it to be used for different purposes. In most cases, however, model developers and users do not have the same level of understanding of the model, which may cause them to use different naming conventions, model organizations, numbers of ports, and other conventions. Not only is it a wasted time for subgroups to duplicate each other's work, it can also introduce errors. Unless an M&S practitioner understands the model's contextual dependencies accurately and unambiguously, model reuse (via modification) will continue to be an ineffective trial-and-error effort. Therefore, the main motivation for model plug-ins is to introduce the 'single source of authority' concept as a part of the model reuse strategy [START_REF] Moffett | The Source of Authority for Commercial Access Control[END_REF]. There exist technologies that may lower the cost of reusing a model or allow reuse of models in previously impractical situations. However, improper reuse of numerical models can undermine these gains as the consequences of a bad decision made with an invalid model can easily outweigh the benefits of reusing the model. The open question about numerical model reuse is not only whether engineers reuse existing models properly, but how valuable they can make the practice throughout the modeling lifecycle.

Single source of authority or right from me

In the M&S reuse activity, there are likely to be at least two groups of people involved, sometimes many more (e.g., external or internal model suppliers). The basic idea is therefore to give the full modification right to the model developer (provider). It means that only the model developer is eligible to modify his/her model. Hence, the model provider is suggested to be the single source of authority in the model reuse strategy deployment [START_REF] Moffett | The Source of Authority for Commercial Access Control[END_REF]. In the literature, the term "Single Source of Authority" is used for access control to a computer system to reduce the likelihood of data being overwritten, or for other security issues. Model providers must be confident that their models are used appropriately and model user can plug-and-play the provided model. In other words, the provided model must comply with the needs of the model users [START_REF] Moffett | The Source of Authority for Commercial Access Control[END_REF]. The term single source of authority is consistent with a term commonly used in the lean community: "Right-from-me" [START_REF] Konstructions -Bakelit | KB Lean Enterprise[END_REF]. Right from me, means that we should get it right the first time in all process stages from preparation of tender documents, and model to right-time delivery. In the development of fit-forpurpose instructions model, it is necessary to prevent mistakes to the greatest possible extent. When we discover any nonconformityabnormal situation, all employees have a duty to act, correct or halt the process. Everything from mistakes in drawings to faults in the equipmentthe people from the previous manufacturing stage or supplier who have caused the non-conformity must be informed immediately (real time).

Model Reuse: Black Box Model

Black box model refers here to reuse of an existing model for the same purpose for which it was originally constructed without any modifying. Model users should be able to assemble the existing model in a plug-in manner, thus minimizing the time, cost and expertise required to construct comprehensive models within the context of their organization. This is possible when a model is used on a routine basis to support tactical decision making within known and defined limits. It is not possible, however, to be sure that reuse is viable when a model is used for a purpose different from for which it is built or is used in combination with other models, possibly based on different sets of assumptions. If a model is to be reused for a purpose other than that for which is it's constructed, it is vital to establish a new credibility assessment process against which the model's validity may be assessed in its new environment of use. Assuming that the characteristics of a reused model to transfer from one provider to another, like its credibility will transfer from one application to another, are simply not justified (Fig. 4) [START_REF] Yilmaz | On the Need for Contextualized Introspective Models to Improve Reuse and Composability of Defense Simulations[END_REF][START_REF] Mohamed | Component Based Modeling and Simulation of Value Stream Mapping for Lean Production Systems[END_REF][START_REF] Moffett | The Source of Authority for Commercial Access Control[END_REF].

  (a) Creating a pull-type model building process (Preliminary Model Design / Virtual Prototype) based on the actual needs of immediate model users; (b) Defining a model sharing and reuse strategy as an enabler to reduce model recreation and errors associated with new model construction and;

Figure 4 .

 4 Figure 4. Principal actors and communication in complex model development strategy • Trust in provided model quality:Among inhibitors to consistency management, humans who create or provide models can be a primary cause when not conforming to the common language, or laws of nature, or introducing changes in a model. To this end, model provider or developer develop fit-for-purpose instructions model. A simulation model and its results have to be credible for the decision-makers to accept them as "correct." Note here that a credible model is not necessarily valid. The following items may increase credibility of a model:• The decision-maker's (model user) and model suppliers' understanding and agreement with the model's assumptions, • Demonstration that the model has been verified and validated, • Reputation of the model developer (provider), • Importance of detailed model design and correctness check in early development phase. As shown in Table1, the authors define 5 categories of waste in PDP adapted by Oppenheim: Waiting, Inventory, Over production, Defects and Transportation. The examples and causes are developed based on our observations in the

Figure 5 .

 5 Figure 5. Proposed Pull Type Model Design Process

Figure 6 Figure 7

 67 Figure 6 Accuracy control between ports

Figure 8

 8 Figure 8 Model Reuse Strategy, General View

Figure 9

 9 Figure 9 Model Reuse in Engineering PracticeIn the literature, the term "Single Source of Authority" is used for access control to a computer system to reduce the likelihood of data being overwritten, or for other security issues. Model providers must be confident that their models are used appropriately and model user can plug-and-play the provided model. In other words, the provided model must comply with the needs of the model users[START_REF] Moffett | The Source of Authority for Commercial Access Control[END_REF]. The term single source of authority is consistent with a term commonly used in the lean community: "Right-from-me"[START_REF] Konstructions -Bakelit | KB Lean Enterprise[END_REF]. Right from me, means that we should get it right the first time in all process stages from preparation of tender documents, and model to right-time delivery. In the development of fit-forpurpose instructions model, it is necessary to prevent mistakes to the greatest possible extent. When we discover any nonconformityabnormal situation, all employees have a duty to act, correct or halt the process. Everything from mistakes in drawings to faults in the equipmentthe people from the previous manufacturing stage or supplier who have caused the non-conformity must be informed immediately (real time).Model Reuse: Black Box ModelBlack box model refers here to reuse of an existing model for the same purpose for which it was originally constructed without any modifying. Model users should be able to assemble the existing model in a plug-in manner, thus minimizing the time, cost and expertise required to construct comprehensive models within the context of their organization. This is possible when a model is used on a routine basis to support tactical decision making within known and defined limits. It is not possible, however, to be sure that reuse is viable when a model is used for a purpose different from for which it is built or is used in combination with other models, possibly based on different sets of assumptions. If a model is to be reused for a purpose other than that for which is it's constructed, it is vital to establish a new credibility assessment process against which the model's validity may be assessed in its new environment of use. Assuming that the characteristics of a reused model to transfer from one provider to another, like its credibility will transfer from one application to another, are simply not justified (Fig.4)[START_REF] Yilmaz | On the Need for Contextualized Introspective Models to Improve Reuse and Composability of Defense Simulations[END_REF][START_REF] Mohamed | Component Based Modeling and Simulation of Value Stream Mapping for Lean Production Systems[END_REF][START_REF] Moffett | The Source of Authority for Commercial Access Control[END_REF].

and Model Architect and

	Type of Waste		Examples	Causes
	Waiting		•People waiting for	• Lack of access and poorly
	Unavailable	information	designed or executed process
	information, Late	•Participant	to provide information
	delivery of		Availability	• How much time will each
	information, Lack of		participant be able to
	resource,			provide the modeling effort?
	Scheduling			•What time frames are
				available for inter-team
				coordination?
	Inventory		• Too much	•Poor understanding of user
	Complicated	information	needs
	retrieval;		• Multiple/redundant	•Tendency for everybody to
	Outdated, Obsolete	sources	maintain Lack of "version
	information	• Outdated/obsolete	control"
			information	• Lack of disciplined system
				for updating new and
				purging old information
				• Inadequate archiving
				standards or practices
	Over Production		•Poor understanding of each
	Creation of		• Pushing, not	participant's needs
	unnecessary data and	pulling	• Send all information to
	information,	model	everyone, rather than to meet
	Information over-	• Lack of reuse	specific needs
	dissemination	strategy	• Tendency to "over-design"
				• More detail than necessary
				in early design
	Defects		•Errors in data	• Human error, lack of
	Erroneous model	reporting/entries	expertise
	information	and	•Errors in	• Poorly designed input
	reports. Lack of	information provided	templates
	reviews,	tests,	to customers	• Lack of disciplined
	verifications	and	•Information does	reviews, tests, verification
	validation, Lack of	not	and validation
	interpretation	and	make sense to user	• Lack of model
	responsibility			specification and early
				correctness control
	Transportatio	•Delay	• Lack of clear information
	n		•Refactoring	flow paths, failure of process
	Unnecessary		to produce information
	movement			needed
	of information		• Incompatible information
	between			types(drawings vs. digital
	people,			descriptions)
	organizations,		• Incompatible software	Model Providers.
	or systems, Information and Software		systems or tools • Lack of availability, knowledge, or training in	System Architects are the sponsor of model development activity. He or she defines the projects' expected time, cost
	incompatibility		conversion and linking	and decision parameters issues. Domain Model Providers are
				systems	the domain experts who build models with theirs specific
				• Software/hardware incompatibilities	domain knowledge. Model Design activity contains formal architecture design with domain models' interfaces
					definitions, vehicle level and domain level model
					specifications including an early interfaces consistency control
					between specified interfaces. Model Design phase gives a
					structural and semi-behavioral view about the system to be
					modeled. Thus, this transversal view from Functional to
					Physical View should be managed by a new actor of the

auteurs propose a high-level overview of Detailed Model Design process as illustrated in Figure5. Note that there are three actors shown here: System collaboration named "Model Architect". The authors would like to highlight that, in today's design environment, there is no or not clearly defined role of the Model Architect. In this proposed detailed model design process, we suppose that each Model Architect has a multidisciplinary vision of a product,

Table 1 .

 1 Waste in PDP adapted by Oppenheim

Copyright © 2014 by ASME

CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK

The authors are discussed the context of reuse and a pull-type model building process (Preliminary Model Design / Virtual Prototype). The fact that we would like to use the suppliers' model as a black box fashion, it requires to check the correctness of the suppliers' models to detect any potential problem before the IVVQ (Integration, Verification, Validation, and Qualification) phase. The pull type process presented in this paper is an improvement over the traditional V-cycle because correctness check at early design phase happens and it may reduce rework in the more expensive implementation and physical prototype validation phase, which is the main driver for product development cost. This paper introduces also some important and novel terms such as single source of authority and model plug and play to highlight the traceability and intellectual properties of a simulation model. This paper does not give a concrete result in terms qualitative and quantitative improvement of current model development activity but this work proposes a robust methodology for possible improvement. The proposed methodology is supposed to be test by an industrial case study for understand its capacity. Thus the future work includes the validation protocol of proposed concepts in terms of value addition to company's current situation. Also, authors want to estimate and compare the task durations, estimated probabilities of rework and resource limitations once the overall structures of the 'as-is' and 'to-be' process models will be agreed.