Mohammmed Bachir

INTRODUCTION TO THE PROBABILISTIC 1-LIPSCHITZ MAPS

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Introduction

We introduce the notion of probabilistic 1-Lipschitz map defined on a probabilistic metric space and give an analogous to the result of Mac Shane (see [START_REF] Shane | Extension of range of functions[END_REF]) on the extenstion of real-valued Lipschitz functions from a subset of a metric space to the whole space.

A distribution function is a function F : [-∞, +∞] -→ [0, 1] which is nondecreasing and left-continuous with F (-∞)) = 0; F (+∞) = 1. The set of all distribution functions will be denoted by ∆. The subset of ∆ consisting on distributions F such that F (0) = 0 will be denoted by ∆ + . For F, G ∈ ∆ + , the relation

F ≤ G is meant by F (t) ≤ G(t), for all t ∈ R. For all a ∈ R, the distribution H a is defined as follow H a (t) = 0, if t ≤ a 1, if t > a For a = +∞, H ∞ (t) = 0, if t ∈ [-∞, +∞[ 1, if t = +∞
It is well known that (∆, ≤) and (∆ + , ≤) are complet lattice with the minimal element H ∞ and the maximal element H 0 (see [START_REF] Hadžić | Pap Fixed Point Theory in Probabilistic Metric Space[END_REF]). Thus, for any nonempty set I and any familly (F i ) i∈I of distributions in ∆ (resp. in ∆ + ), the function F = sup i∈I F i is also an element of ∆ (resp. of ∆ + ).

Axioms 1.1. In this work, we assume that ∆ + is equipped with a law ⋆ (a triangual function) satisfying the following axioms:

(i) F ⋆ L ∈ ∆ + for all F, L ∈ ∆ + . (ii) F ⋆ L = L ⋆ F for all F, L ∈ ∆ + . (iii) F ⋆ (L ⋆ K) = (F ⋆ L) ⋆ K, for all F, L, K ∈ ∆ + . (iv) F ⋆ H 0 = F for all F ∈ ∆ + . (v) F ≤ L =⇒ F ⋆ K ≤ L ⋆ K for all F, L, K ∈ ∆ + .
(vi) Let I be a set, (F i ) i∈I a familly of distributions in ∆ + and L ∈ ∆ + . Then,

sup i∈I (F i ⋆ L) = (sup i∈I (F i )) ⋆ L. Definition 1.2. We say that ⋆ is continuous at (F, L) ∈ ∆ + ×∆ + if lim n→+∞ (F n ⋆ L n )(t) = (F ⋆L)(t) for all t ∈ R point of continuity of F ⋆L, whenever lim n→+∞ F n (t) = F (t) for all t ∈ R point of continuity of F and lim n→+∞ L n (t) = L(t) for all t ∈ R point of continuity of L. Example 1.3. (see [3], [1]) Let T : [0, 1] × [0, 1] -→ [0, 1]
be a left-continuous t-norm, then the operation ⋆ defined for all F, L ∈ ∆ + and for all t ∈ R by

(F ⋆ L)(t) := sup s+u=t T (F (s), L(u)) (1.1)
is continuous at each point (F, L) ∈ ∆ + × ∆ + and satisfies the axiomes (i)-(vi).

In all this work, we assume that ∆ + is equipped with a continuous law ⋆ satisfying the axioms (i)-(vi). Definition 1.4. Let G be a set and let D : G × G -→ (∆ + , ⋆, ≤) be a map. We say that (G, D, ⋆) is a probabilistic metric space if the following axioms (i)-(iii) hold:

(i) D(p, q) = H 0 iff p = q.

(ii) D(p, q) = D(q, p) for all p, q ∈ G (iii) D(p, q) ⋆ D(q, r) ≤ D(p, r) for all p, q, r ∈ G

Probabilistic 1-Lipschitz Maps

We define probabilistic continuous functions.

Definition 2.1. Let (G, D, ⋆) be a probabilistic metric space and f : G -→ ∆ be a function. We say that f is continuous at

x ∈ G if lim n f (x n )(t) = f (x)(t) for all t ∈ R point of continuity of f (x) and all sequence (x n ) n ⊂ G such that lim n D(x n , x) = H 0 (i.e. such that lim n D(x n , x)(t) = H 0 (t) for all t ∈ R).
Now, we introduce the notion of probabilistic 1-Lipschitz map.

Definition 2.2. Let (G, D, ⋆) be a probabilistic metric space and f : G -→ ∆ be a map. We say that f is a probabilistic 1-Lipschitz map if, for all x, y ∈ G we have:

D(x, y) ⋆ f (y) ≤ f (x).
Proposition 2.3. Every probabilistic 1-Lipschitz map is continuous.

Proof. Let f be a probabilistic 1-Lipschitz map and (x n ) n ⊂ G be a sequence such that lim n D(x n , x) = H 0 . On one hand, since (∆, ≤) is a complete lattice, the law ⋆ is continuous and

D(x n , x) ⋆ f (x) ≤ f (x n ), then f (x) = H 0 ⋆ f (x) = lim inf n D(x n , x) ⋆ f (x) ≤ lim inf n f (x n ) . Thus, f (x) ≤ lim inf n f (x n ). On the other hand, since D(x n , x) ⋆ f (x n ) ≤ f (x), it follows that lim sup n f (x n ) = H 0 ⋆ lim sup n f (x n ) = lim sup n (D(x n , x) ⋆ f (x n ) ≤ f (x). Thus, lim n f (x n ) = f (x) and so f is continuous. By Lip 1 ⋆ (G, ∆) (resp. Lip 1 ⋆ (G, ∆ +
)), we denotes the space of all probabilistic 1-Lipschitz maps (resp. all ∆ + -valued 1-Lipschitz maps). For all x ∈ G, by δ x : G -→ ∆ + we denote the map δ x : y → D(y, x) and by δ, we denote the operator δ : x → δ x .

Proposition 2.4. Let (G, D, ⋆) be a probabilistic metric space. Then, we have that

δ a ∈ Lip 1 ⋆ (G, ∆ + ) for each a ∈ G and the map δ : G -→ Lip 1 ⋆ (G, ∆ + ) is injective. Proof. The fact that δ a ∈ Lip 1 ⋆ (G, ∆ + ) for each a ∈ G follows from the property: D(x, y) ⋆ D(y, a) ≤ D(x, a) for all a, x, y ∈ G. Now, let a, b ∈ G be such that δ a = δ b . It follows that δ a (x) = δ b (x) for all x ∈ G. In particular, for x = b we have that D(a, b) = δ a (b) = δ b (b) = H 0 , which implies that a = b. Let f ∈ Lip 1 ⋆ (G, ∆ + ) and F ∈ ∆ + , by f, F : G -→ ∆ + ,
we denote the map defined by f, F (x) := f (x) ⋆ F for all x ∈ G. We easly obtain the following propostion.

Proposition 2.5. Let (G, D, ⋆) be a probabilistic metric space. Then, for all f ∈ Lip 1 ⋆ (G, ∆ + ) and all F ∈ ∆ + , we have that f, F ∈ Lip 1 ⋆ (G, ∆ + ).

Recall the Lipschitz extention result of Mac Shane in [START_REF] Shane | Extension of range of functions[END_REF]: if (X, d) is a metric space, A a nonempty subset of X and f : A -→ R is k-Lipschitz map, then there exist a k-Lipschitz map f : X -→ R such that f|A = f . We give bellow an analogous of this result for probabilistic 1-Lipschitz maps.

Theorem 2.6. Let (G, D, ⋆) be a probabilistic metric space and A be a nonempty subset of G. Let f : A -→ ∆ be a probabilistic 1-Lipschitz map. Then, there exists a probabilistic 1-Lipschitz map f : G -→ ∆ such that f|A = f . Proof. We define f : G -→ ∆ as follows: for all x ∈ G, f (x) := sup a∈A D(a, x) ⋆ f (a).

We first prove that f (x) = f (x) for all x ∈ A. Indeed, let x ∈ A. On one hand we have f

(x) = H 0 ⋆ f (x) = D(x, x) ⋆ f (x) ≤ sup a∈A D(a, x) ⋆ f (a) = f (x).
On the other hand, since f is probabilistic 1-Lipschitz on A and x ∈ A, then D(a, x)⋆f (a) ≤ f (x) for all a ∈ A. It follows that f (x) := sup a∈A D(a, x) ⋆ f (a) ≤ f (x). Thus, f (x) = f (x) for all x ∈ A. Now, we show that f is probabilistic 1-Lipschitz on G. Indeed, Let x, y ∈ G. For all a ∈ A we have that D(a, x) ⋆ D(x, y) ≤ D(a, y). So, D(a, x) ⋆ f (a) ⋆ D(x, y) ≤ D(a, y) ⋆ f (a). By taking the supremum over a ∈ A and using axiom (vi) we get f (x) ⋆ D(x, y) ≤ f (y). Hence, f is probabilistic 1-Lipschitz map on G that coincides with f on A.


C Holmqvist

D Feinberg

A Zazunov

Emergence of a negative charging energy in a metallic dot capacitively coupled to a superconducting island

Keywords: numbers: 73, 23, Hk, 74, 78, Na

We consider the hybrid setup formed by a metallic dot, capacitively coupled to a superconducting island S connected to a bulk superconductor by a Josephson junction. Charge fluctuations in S act as a dynamical gate and overscreen the electronic repulsion in the metallic dot, producing an attractive interaction between two additional electrons. As the offset charge of the metallic dot is increased, the dot charging curve shows positive steps (+2e) followed by negative ones (-e) signaling the occurrence of a negative differential capacitance. A proposal for experimental detection is given, and potential applications in nanoelectronics are mentioned.

At low temperatures, the electronic transport through pointlike metallic nanostructures (quantum dots) is dominated by the electronic Coulomb repulsion between additional electrons. When a small-capacitance dot is weakly coupled to a normal metallic reservoir, the average number of charges in the dot, n N , increases one by one with the gate voltage V gN , leading to conductance peaks [1]. This Coulomb blockade phenomenon has recently enabled an individual control of charge or spin, for instance in view of quantum information protocols [2,3] . We address here the possibility of inverting the sign of the charging energy. Indeed, creating a negative effective charging energy in a normal (non-superconducting) metallic dot would induce attractive correlations, triggering for instance pair tunneling from/to a normal reservoir [4] or a charge Kondo effect [5,6], or giving rise to super-Poissonian shot noise [6,7]. Going beyond the single dot case, attractive correlations between electrons in spatially separated dots could help implementing quantum information protocols involving two-qubit gates, when the qubits are carried by the charge (spin) of the last added electron. Historically, attractive interactions in the solid state are known as valence-skipping states [8], and negative-U centers [9]. Another possible mechanism for electronic attraction is mediated by optical phonons, binding two electrons as a bipolaron in confined geometries for strong electron-phonon coupling and lattice polarizability [10]. Due to the low polarizability and small effective carrier mass, bipolarons are unlikely to form in a clean GaAs/AlGaAs two-dimensional electron gas (2DEG), although they might do so in presence of a few donor impurities [11], or with more polarizable multilayer materials [12]. Molecular junctions are also promising for achieving a negative charging energy [4,13].

In the present Letter, we propose an alternative mechanism, pointing out that the repulsive charging energy in a metallic dot (N) connected to a normal reservoir (Fig. 1) can be turned into an attractive one when N is capacitively coupled to a superconducting island (S). The latter is connected to a superconducting reservoir by a Josephson junction (JJ) and operates in the Cooper pair box regime, e.g. it fluctuates between two pair number states [14,15]. Here we assume that electron tun-neling between S and N is negligible; therefore no proximity effect occurs in the N dot. We instead focus on the charging properties of the N grain as its gate voltage is varied. The S island acts as an effective dynamical gate, whose effects turn out to be nonlinear. As the main result of this Letter, the Coulomb charging energy in N can be overscreened by the neighboring pair fluctuations in S, and an effective local attraction appears between electrons added into N. As a corollary, certain charge states are "skipped" as the N gate voltage is varied. The resulting charging curve becomes non-monotonous, displaying positive steps (+2e) followed by negative ones (-e). A related screening effect was proposed by Averin and Bruder for controlling the coupling between two superconducting charge qubits [16]. Notice that if N were coupled to both drain and source reservoirs, our set-up would be similar to a Cooper pair box coupled to a single-electron transistor (SET). The latter has been studied in great detail as a read-out device for a superconducting (charge) qubit embodied in the S island [17]. In this case, the capacitive coupling between N and S must be small, in order to minimize the decoherence due to backaction of the normal part of the device onto the superconducting one, whereas in our proposal, the coupling is very strong. The JJ connecting the S island to the reservoir has a Josephson energy E J and capacitance C J , and a gate imposes an offset ν S = C gS V gS /e, with C gS ≪ C J . Similarly, the N island is connected to a normal reservoir by a tunnel junction, with single-electron tunneling rate Γ and capacitance C N , and experiences a gate offset ν N = C gN V gN /e, with C gN ≪ C N . Most importantly, the islands N and S are coupled by a large capacitance C 0 > C N , C J . We assume the superconducting gap in S to be larger than the charging energy, such that only even charge number states n S occur in S. At low temperatures, quasiparticle tunneling in S can be neglected. Defining

C ΣS = C J + C 0 + C gS and C ΣN = C N + C 0 + C gN ,
E C = E CN [(n N -ν N ) 2 + b(n S -ν S ) 2 + 2r √ b(n N -ν N )(n S -ν S )] (1)
with

E CN = e 2 /[2C ΣN (1 -r 2 )].
The asymmetry parameter b and the coupling parameter r < 1 are not independent, and r < min (b, 1/ √ b). Eq. ( 1) determines the charge stability diagram of the isolated NS system in the (ν N , ν S ) plane. First, for a value ν S imposing an integer number of pairs in S, say ν S = 2, the charging number n N increases monotonously with ν N . Next, consider a case where n S fluctuates, for instance ν S = 1. For small r, as shown in Fig. 2a, n N is again a monotonous function of ν N : the sequence of charge states (n N , n S ) as ν N increases reads (0, 0), (0, 2), (1, 0), (1, 2), (2, 0), (2, 2), . . . (notice the oscillation of n S ). The result is very different if r is large. In fig. 2(b), for ν S = 1, n N increases with ν N but in a non-monotonous way, the charge state sequence being (1, 0), (0, 2), (2, 0), (1, 2), (3, 0), (2, 2), etc. The corresponding charging staircases are plotted in the insets.

One sees that the transition from (n N , 2) to (n N + 2, 0) at ν N = n N +1 "skips" the charge state n N +1 in the grain. This signals a negative effective charging energy in N which overcomes the Coulomb repulsion. After increasing by two units, n N decreases by one unit, yielding a negative differential capacitance (NDCA) C dif f = C gN (dn N /dν N ) at half-integer values of ν N . Strikingly, the total number of steps, positive or negative, is doubled with respect to the usual case. Both charge skipping and NDCA occur above the dotted line indicated in Fig. 4 displaying a (b, r) diagram. From the above charging energy, an effective attractive potential U < 0 can be estimated for ν S = 1 as

U = E C (0, 2) + E C (2, 0) - 2E C (1, 2) = e 2 (1 -2r √ b)/[C ΣN (1 -r 2 )].
The necessary condition for the occurrence of a negative charging energy is thus 2r √ b > 1.

To further analyze this possibility in an open NS system, let us consider the full Hamiltonian: where k (q) denotes electron states in the normal reservoir R (grain N), and the Coulomb interaction E C is given by eq. ( 1). The total charge in N is expressed as n N = qσ c † qN,σ c qN,σ . Assuming constant densities of states in N and R, the singleelectron transition rate from R to N is given by Γ

H = E C + kσ ε k c † kR,σ c kR,σ + qσ ε q c † qN,σ c qN,σ + [ kqσ T k,q c † kR,σ c qN,σ - E J 2 |n S + 2 n S | + H.c.] , (2)
(+1) = [δE (+1) C /e 2 R N ] [exp (δE (+1) C
/k B T )-1] -1 within the golden rule approximation, where R N is the tunnel resistance.

Considering first the case of small E J ≪ E CS = e 2 /[2C ΣS (1 -r 2 )], we perform a T-matrix calculation of the transition rates from (0, 2) to (2, 0) (close to ν N = 1) and from (2, 0) to (1, 2) (close to ν N = 1.5). For the first transition, we take into account three possible configuration paths involving higher-energy states: (0, 2) → (1, 2) → (2, 2) → (2, 0), (0, 2) → (1, 2) → (1, 0) → (2, 0), and (0, 2) → (0, 0) → (1, 0) → (2, 0). For the second transition, only one excited state is involved: (2, 0) → (1, 0) → (1, 2) and (2, 0) → (2, 2) → (1, 2). The shape of each step is calculated at finite temperature by solving the master equation governing the dynamics of the probabilities p(0, 2), p(2, 0) for the positive step and p(2, 0), p(1, 2) for the negative one. The master equation reads ṗ(a) = Γ b→a p(b)-Γ a→b p(a) with p(b) = 1 -p(a) for the states a, b involved in the transition. Here the probabilities of other states are neglected, which is justified close to ν N = 1 or ν N = 1.5 and if the steps are sufficiently narrow. The resulting steps are shown in Fig. 3.

As a result, a positive step +2e (where the charge number n N = 1 is skipped) and a consecutive negative step -e are stabilized. Notice that contrary to the usual case where all transitions between n and n ± 1 are real and obey the same master equation [19], here the rates are of higher order and the virtual states involved in one transition become real states (with first order rate) for the next one. A full treatment of all processes is beyond the scope of this Letter.

Let us now turn to the case of a large Josephson energy, E J > E CS . The adiabatic assumption [16], setting the phase difference to φ across the JJ, allows to solve the Hamiltonian (2), neglecting the normal electron tunneling term. The resulting adiabatic Hamiltonian H ad = E C -E J cos φ describes a Cooper pair box with an effective gate voltage, which is an adiabatic function of n N . In the tight-binding limit E J /E CN ≫ b, assuming that the junction dynamics is confined to the lowest Bloch band, one obtains the sum of the N dot charging energy and the adiabatic Bloch band energy :

H ad = E CN (1 -r 2 )(n N -ν N ) 2 -∆ 0 cos[π(ν S - r √ b (n N -ν N ))] , (3)
where the bandwidth is given by [14]

∆ 0 = 16 2 π b E CN E J 2bE CN 3/4 e - √ 8EJ /bECN . ( 4
)
The second term in H ad represents a nonlinear screening potential acting on the charge in N. The offset ν S controls the phase of the cosine term, and an appropriate choice (for instance ν S ≈ 1) achieves a negative curvature of H ad , viewed as an effective charging energy E ef f CN for the gauged charge in N, n N -ν N . The required condition reads (π 2 /2)r 2 ∆ 0 /[b(1r 2 )] > 1, yielding the lines in Fig. 4. Clearly, a large E J puts a stronger constraint on the coupling capacitance C 0 , and requires larger values of r than for small E J . The shape of the charge skipping and negative steps is then calculated, using a master equation based on charge states n N = 0, 2 or n N = 2, 1, respectively. The adiabatic transition rates are given by

Γ ad = [δE ef f CN /e 2 R N ] [exp (δE ef f CN /k B T ) -1] -1 .
The corresponding steps (Fig. 3) are less pronounced than in the small E J case.

Searching the optimum regime must account for the fact that for too large r values, the system behaves like one single island and its energy no longer depends on the location of the charge. The Coulomb blockade requires temperatures much smaller than the energy difference between two charge states. An optimal r is close to 0.75 (with b = 1) for small Josephson energies, and the requirement for Coulomb blockade reads k B T < E CN /4. The value r = 0.8 was used in the steps calculations. A temperature of T ∼ 30 mK and a typical charging energy of E CN ∼ 0.1 meV were used, yielding C N = C J ∼ 2 fF and C 0 = 4C N = 8 fF. The tunnel resistance R N is chosen as R N /R K = 10 with R K = h/e 2 ≈ 25.8 kΩ. This yields a bare single-electron tunneling rate of Γ ∼ 10 9 s -1 . In presence of the S island, the effective tunneling rates are 10 7 s -1 (n N decreasing from 2 to 1) and 5 • 10 3 s -1 (n N increasing from 0 to 2), respectively. With the above parameters, the maximum attraction |U | is of the order of 50µeV.

Notice that such a negative charging energy cannot render the dot superconducting, because it concerns the energy required to add one or two electrons on the dot, rather than a true attractive potential felt by all electrons near the Fermi level. The effective "negative-U" potential manifests itself only when the dot is weakly coupled to a reservoir such that its charge can fluctuate. This is similar to a single "-U " center weakly hybridized with a normal bulk metal [9]. Here, pair fluctuations with the reservoir are assumed to be incoherent. On the contrary, a very small tunneling term T N S between N and S would allow for a true phase coherence between states n N , n N + 2, thus a proximity effect in a quite unusual regime, where T N S < |U |.

Let us briefly discuss the issue of phase coherence in the Cooper pair box. Charge skipping only requires pair tunneling between the superconducting reservoir and the S island in order to screen the repulsive interaction in the normal grain. No phase coherence is needed, as shown by our calculation in the small E J case. Moreover, even in the large E J case, charge fluctuations in N should strongly react back upon S and reduce the phase coherence. A full treatment goes beyond the adiabatic approximation [16]. One can anticipate that large fluctuations in the phase φ renormalize E J downwards, making the small-E J case generic.

We now propose a scheme for detecting the nonmonotonous charging of the N grain. SETs, or point contacts [20] provide very sensitive detection of the local change in the electrostatic potential. In double-dot setups with weak mutual coupling, the potential variations in each dot can be measured by a different neighboring point contact [21]. In the present case, placing a point contact close to N does not measure δn N , but instead δV [18]. Notice that the tunneling rates calculated above are much reduced compared to the bare singleelectron rate Γ. Therefore the use of point contacts permits, non only a time-averaged [21], but even a time-resolved and directional [22] detection of the charge variations in N and S. On the other hand, cross-correlation shot noise measurements, as in ref. [23], would require higher currents. In practice, a possible setup inspired by ref. [21] is proposed in Fig. 1. It involves a superconducting strip with a Cooper pair box, coupled laterally to an InGaAs/AlGaAs 2DEG, suitably tuning the Schottky barrier present at the interface between the superconductor and the 2DEG. Notice that the geometry of Fig. 1 could be modified, including drain and source such as to allow transport through N [4].

N = (C -1 ) N N (eδn N ) + (C -1 ) N S (eδn S ) = e [δn N + r √ b δn S ]/[C ΣN (1 -r 2 )]. If 2r √ b > 1,
In conclusion, we have proposed a mechanism inducing a controllable negative charging energy, thus attractive correlations, in one or several metallic dots. We believe that such an effect would be useful in view of more complex nanoelectronics devices. The authors are grateful to T. Martin, M. Fogelström, and G. Johansson for useful discussions, and S. Andergassen for careful reading of the manuscript. D. F. and A. Z. were partially supported by the contract AC NANO NR0114, and C. H. was supported by the Swedish Research Council (VR) under grant 621-2006-3072.

FIG. 1 :

 1 FIG.1: Schematic view of a normal grain (N) coupled with a strong capacitive coupling (controlled by 3, 9) to a Cooper pair box composed of a Josephson junction connecting superconducting reservoir 2 and island, and gate 10. Electrons tunnel between N and its reservoir (defined by 7, 8). Detection is made by sweeping the gate voltage 4 and measuring the island voltages using quantum point contacts for both N (5,6,7) and S(1,11,12).

  and introducing the parameters b = C ΣN /C ΣS and r = C 0 / √ C ΣN C ΣS , the total charging energy of the NS system can be written as[18]

FIG. 2 :

 2 FIG. 2: Charge stability (or honeycomb) diagram for b = 1, r = 0.2 (a) and r = 0.8 (b). The insets show the charging curves for N, taken along the dotted line. In case (b) the charging staircase (inset) exhibits charge skipping effects.

FIG. 3 :

 3 FIG. 3: Charging staircase : r = 0.8, b = 1, ECS = ECN , kBT /ECN = 3 • 10 -2 , RN /RK = 10; (full line) EJ /ECN = 0.5 ; (dotted line) EJ /ECN = 2.

  doubling of the number of steps can be detected, but not the nonmonotonous charging curve. To access the latter, it is suitable to measure δV S = e [r √ b δn N + b δn S ]/[C ΣN (1 -r 2 )] as well, with a second point contact close to S, and reconstruct δn N = C ΣN [δV N -rδV S / √ b]/e. The parameters C ΣN , r, b can easily be measured from the stability diagram obtained in the normal state in the presence of a very weak tunneling between N and S


Audrey Abi Akle

Stéphanie Minel

Audrey Abi

Akle Estia

Bernard Yannou

Information Visualization for Selection in Design by Shopping

Keywords: Visual Design, Computer Aided Design (CAD), Decision Making

published or not. The documents may come

Information visualization for selection in Design by

Shopping

Introduction

In engineering design, once the design has been formalized, a necessary design task is to select from amongst candidate designs or parametric values [START_REF] Otto | The method of imprecision compared to utility theory for design selection problems[END_REF]. This can be done in the preliminary concept phase, where the designer has to choose one solution over other alternatives, or in the detail design, where the designer has to choose particular values to use in a design model. The main challenge when designing complex systems (such as cars or aircraft) lies in resolving the inherent trade-offs that exist between the overall system and subsystems, and between conflicting and competing objectives.

When exploring the design space, the design is selected after evaluating the elements present in order to identify optimal solutions by reducing the design space to an area of performance. In this method, visualization techniques are used as decision support tools. Graphs are useful to quickly visualize feasible solutions as opposed to impractical solutions, as well as those violating engineering constraints or client requirements. Three different situations can be represented with more or less data:  Representing the single vector of design parameters featuring the product solution, (𝑋): this refers to the feasible design space,  Representing the single vector of solution performances for feasible solutions, (𝑌): this refers to the feasible performance space,  Representing two sets of design parameters and corresponding performances for feasible solutions (respecting constraints and requirements),( 𝑋 𝑌 ): this refers to the feasible design and performance space.

Graphical supports are effective for design parameters [START_REF] Simpson | Impact of response delay and training on user performance with text-based and graphical user interfaces for engineering design[END_REF], engineering optimization [START_REF] Barron | Graphical user interfaces for engineering design: impact of response delay and training on user performance[END_REF]) and conceptual design [START_REF] Yannou | Faster generation of feasible design points[END_REF]. There is already much research in visual design and it has been shown that fast graphical design interfaces impact user performance in terms of design efficiency, We know that exploring the design space is the embodiment of a paradigm where designers shop for the best solution. It is called Design by Shopping, a term coined by [START_REF] Balling | Design by shopping: A new paradigm?[END_REF]. Indeed, [START_REF] Balling | Design by shopping: A new paradigm?[END_REF] noted that the traditional optimization-based design process to "formulate the design problem, obtain analysis models and execute an optimization algorithm" leaves designers unsatisfied. Designers, like consumers, want to "shop" to gain an insight into trades, feasible and impractical solutions, and to learn about their alternatives before making decisions. Design by Shopping firstly allows designers to explore the design space and secondly to optimize and choose an optimal solution from a set of possible designs, and then develop realistic expectations with regard to what is possible.

So, in Design by Shopping, the selection phase should be supported by graphs that allow designers to observe all the design alternatives, therefore enabling them to distinguish between right and wrong responses. The graph should allow differentiation between design parameters and performance variables. With graphs, the designer needs to compare candidates according to criteria. In this way, the graph should help the designer find a so-called "optimal" solution. In this sense, it should not create confusion or increase the designer's workload.

We have thus identified three graphs which are useful for representing multidimensional sets of data (>3 variables) and with a limited number of alternatives (<50): Combined Table (CT), Parallel Coordinate Plot (PCP) and Radar Chart (RC). We carried out experiments with 30 participants and designed three scenarios to mimic the design activity. The framework was simplified in order to focus our study on the selection phase (13 design parameters and 5 performance variables where it is necessary to achieve trade-offs between conflicting objectives). We identified the graph best suited to the selection phase in Design by Shopping: the Parallel Coordinates Plot (PCP).

In this paper, we describe the different approaches to design selection in the first section, followed by a description of the tools available for exploring the design space. We then characterize the Design by Shopping paradigm and efficient graphs for the selection phase, before presenting the experimental design and the measurements used in the experiment that lead to our study's results. Finally, we discuss the results and conclude.

As already noted by [START_REF] Yannou | Set-based design by simulation of usage scenario coverage[END_REF], design optimization is a well-established field of research with two primary approaches: unique objective function optimization and multiple objective optimizations. In our case, we are concerned with the second approach in which design solutions are computed and/or a desired compromise (on the Pareto surface for instance) is found either automatically or interactively.

In the "automatic" approach, a finite set of "best" solutions is directly proposed to the designer. There are many methods, such as the weighted-sum of preferences, indifference points or compensation of the weight (see [START_REF] Scott | Using indifference points in engineering decisions[END_REF], 2005). In this case, we also consider the "semi-automated" approach where the direct specification of importance weights is an ad-hoc process, for instance. This approach involves the designer at the end of the process.

In the "interactive" approach we distinguish two close paradigms: Uncertainty Reduction and Design by Shopping. In the Uncertainty Reduction paradigm, the designer iteratively reduces intervals on variables allowing design knowledge to be gained, thereby reducing uncertainty through the process of convergence towards an optimal solution [START_REF] Malak | Multi-attribute utility analysis in set-based conceptual design[END_REF]. It is worth mentioning the use of Constraint Programming as a relevant method for reducing uncertainty in the preliminary design stage (cf. [START_REF] Yannou | Truss dimensioning with an uncertainty reduction paradigm[END_REF][START_REF] Yannou | Dimensioning a product in preliminary design through different exploration techniques[END_REF]. With this paradigm, the designer is part of the process and the design selection is derived from the process. In the Design by Shopping paradigm, the designer explores feasible design solutions and interactions between variables. A design point can be selected and consequently the design space is reduced by discovering knowledge and gaining insights during the process [START_REF] Zhang | Supporting knowledge exploration and discovery in multi-dimensional data with interactive multiscale visualisation[END_REF]. With this paradigm, the designers are part of the process and they are direct in the simulation loop.

We also focus on the work of Miller et al. which attempts to capture the key attributes from the decision-making literature tailored to the engineering domain [START_REF] Miller | Preference Construction, Sequential Decision Making, and Trade Space Exploration[END_REF]. More precisely, they compare three models: the widely known normative decision-making model, the marketing model of [START_REF] Shocker | Consideration set influences on consumer decision-making and choice: Issues, models, and suggestions[END_REF], and the trade space exploration model developed by [START_REF] Simpson | Visual steering commands and test problems to support research in trade space exploration[END_REF].

The first one concerns the traditional optimization process that is to develop a model, defines constraints/objectives, chooses an algorithm, optimizes and checks satisfaction (see [START_REF] Miller | Preference Construction, Sequential Decision Making, and Trade Space Exploration[END_REF]). This process (and by extension, selection using this process) follows the "automatic" approach already mentioned. The second model comes from the domain of marketing and was defined by [START_REF] Shocker | Consideration set influences on consumer decision-making and choice: Issues, models, and suggestions[END_REF] where "consumers" sequentially reduce the space of considered choices through a number of discrete sets. There are four sets: the universal set corresponding to all possible choices, the awareness set corresponding to solutions for which the decision-maker becomes aware, the consideration set corresponding to the potential solutions and the choice set defined as the final consideration set prior to a final choice. Finally, the last model described by [START_REF] Miller | Preference Construction, Sequential Decision Making, and Trade Space Exploration[END_REF] is the Trade Space Exploration which is an embodiment of the Design by Shopping paradigm. The authors point out that this model combines the first two; it describes the "arrows" between sets defined in the marketing model. Indeed, the visualization tools are used to assist Decision-Makers in constructing their preferences while the space is proceeded upon to reduce the universal sets into a sequence of consideration sets to a choice set and finally a choice.

Design by Shopping was coined by [START_REF] Balling | Design by shopping: A new paradigm?[END_REF]. He noted that the traditional optimization-based design process of "formulate the design problem, obtain analysis models and execute an optimization algorithm" leaves designers unsatisfied with their results because the problem is usually improperly formulated: "the objectives and constraints used in optimization were not what the owners and stakeholders really wanted. In many cases, people do not know what they really want until they see some designs" [START_REF] Zhang | Supporting trade space exploration of multi-dimensional data with interactive multi-scale nested clustering and aggregation[END_REF]. Design by Shopping places emphasis on revealing the range of options available without presuming that all the selection criteria have previously been elicited [START_REF] Feather | Application of a broad-spectrum quantitative requirements model to early-lifecycle decision making[END_REF].

Designers, like consumers, want to "shop" to gain an insight into trades, feasible and impractical solutions, and to learn about their alternatives before making decisions. Design by Shopping initially allows designers to explore the design space, before choosing an optimal solution from a set of possible designs and finally developing realistic expectations of what is possible.

According to [START_REF] Ferringer | Many-objective reconfiguration of operational satellite constellations with the large-cluster epsilon non-dominated sorting genetic algorithm-II[END_REF], "the ultimate reduction of the solution space to a single Decision-Maker preferred design proceeds using this trade-off information in a Design by Shopping paradigm". Rothrock & Narayanan (2011) noted similar findings reported in other fields. For example, [START_REF] Wilson | Thinking too much: introspection can reduce the quality of preferences and decisions[END_REF] showed that people do worse at some decision tasks when asked to analyze the reasons for their preferences or evaluate all the attributes of their choices. [START_REF] Shanteau | Competence in experts: The role of task characteristics[END_REF] observed that when people are dissatisfied with the results of a rational decision-making process, they often change their rating to achieve their desired results. This paradigm introduces greater "control" for decision-makers by "allowing them to first form their preference based on the visualization of the entire design space, and then choose an optimal design." [START_REF] Rangavajhala | Decision Making in Design Under uncertainty with multiobjective Robust Design Optimization 205[END_REF]. This paradigm is as much about eliminating the wrong answers as it is about finding the right answer. Design by Shopping is classified as an a posteriori articulation of preferences used to solve multi-objective optimization problems. It replaces Human directly in the simulation loop.

In Design by Shopping, therefore, the selection phase should be supported by graphs that allow designers to observe all the design alternatives, as well as the wrong and right responses. The graph should allow differentiation between design parameters and performance variables. The designer should be able to compare candidates according to specific criteria. The graph should be a help to the designer, a means to select the socalled "optimal" solution. In this sense, it should not create confusion or an additional workload for the designer.

Interactive design visualization

It has already been shown that fast graphic design interfaces impact user performances in terms of design efficiency, design effectiveness and the design search process [START_REF] Ligetti | Assessing the impact of graphical design interfaces on design efficiency and effectiveness[END_REF] The Rave tool (Daskilewicz & German, 2012) is a computational framework designed specifically as a research platform for design decision-support methods. It provides a "blank canvas" on which we can arrange graphs, tables, images, text, and user interface controls (automatically linked together). In order to visualize design points, this tool proposes several graphs as line plot, scatter plot, 3D scatter plot, scatter plot matrix, parallel coordinates plot, bar charts and density scatter plot (Fig. 5). Rave also provides continuous visualization. These graphs are useful for representing multidimensional sets of data with an unlimited number of alternatives (design points). Clearly, these graphs are adapted to this method especially when a designer generates a sample of design points, but the design space exploration activity consists of three phases: discovery, narrowing and selection; and during the selection phase, designers face a reduced design space with a limited number of design points (compared to the first two phases). As already noted, exploring the design space is the embodiment of a paradigm where designers shop for the best solution. In this context, it should identify the specific characteristics of this paradigm especially for the selection phase.

Graphs For The Selection Phase

In our context of achieving trade-offs and selecting an optimal solution, several graphs (design space representations) are available to us. As already mentioned, this is a case-representation of multidimensional sets of data with limited numbers of alternatives (design points). Based on the works of [START_REF] Miettinen | Survey of methods to visualize alternatives in multiple criteria decision making problems[END_REF] [START_REF] Boy | Suggested Interactivity: Seeking Perceived Affordances for Information Visualization[END_REF]. In our implementation, each row is a design point (or alternative) and each column is a variable (or component). The interaction with the graph takes place on the labels at the top of each column. By clicking on these labels, the values in the columns are sorted from highest to lowest. In addition to the values in the table, we added a heatmap. Theoretically, heatmaps are matrices in which the cells are colored according to their values. We have already shown in (Abi Akle et al., 2013) that the most efficient visualization for a decision-maker would be a monochromatic heatmap in red. In each column, the highest value has a bright red cell. The colors used range from red for the column's highest value to white for the lowest. In addition, on mouse over, the entire row of values (corresponding to an alternative or possible solution) is underlined and the alternative's title appears at the top left of the interface. This representation is particularly useful for identifying patterns such as correlations [START_REF] Cook | Exploring gene expression data, using plots[END_REF]. The Parallel Coordinates Plot is defined as a graph displaying multiple criteria without drastically increasing the complexity of the display [START_REF] Inselberg | Parallel Coordinates: Visual Multidimensional Geometry and Its Applications[END_REF]. In PCP, variable values are displayed on separate axes laid out in parallel. The design points (or alternatives) are depicted as profile lines that connect points on the respective axes. According to [START_REF] Gettinger | A comparison of representations for discrete multi-criteria decision problems[END_REF], this representation can be readily interpreted and provides a good overview. Furthermore, patterns such as positive, negative and non-trivial (multiple) correlations may be identified at a glance. In our implementation, the interaction is directly applied on the graph. To filter the values, simply place the mouse on one of the axes and, with a click, drag the cursor to obtain a desired range of the criteria like a window, outside which value vectors are excluded. In addition, the mouse, when moved above a line, is used to display the title of the design point (or alternative) at the top left of the interface (Fig 7). As with PCP, the design points are depicted as profile lines that connect points on the respective axes. In our implementation, interactions are achieved by the mouse first on the points (on the axes) displaying the variable value and second on lines displaying the title of the design point (alternative). Filtering is done using a form. We chose to use a form so as to not overload the graph. In the field of health care [START_REF] Saary | Radar plots: a useful way for presenting multivariate health care data[END_REF], radar charts are seen as a form of powerful graph to effectively convey meaning of multivariate data. To answer our research questions, we conducted a controlled experiment that adopted a within-subject approach.

Each participant performed the experiment on three graphs tested consecutively. To diversify the participant selection process, we proposed an experiment based on three scenarios leading to the selection of one alternative where a car has to be chosen. The three scenarios have been imagined to mimic the design activity in a simplified framework in order to focus on the selection phase of the design process. Scenarios are designed with personas method (see for example [START_REF] Long | Real or imaginary: The effectiveness of using personas in product design[END_REF])) as brief marketing in which a profile target is described. For each of the three scenarios, trade-offs between conflicting objectives have to be achieved in order to select an alternative, ie. choose an ideal in a car purchasing situation according to the scenario.

The Three Scenarios

We designed the Power Scenario, Size Scenario and Journey Scenario. The three scenarios are defined as follows:

 "Size" Scenario: The target user has a family of three children and a spouse. They work all day long in a big city where it is difficult to park. That is why you should select a car that is as small as possible in order to be able to park easily (width and length) but with sufficient internal volume for the target user and his/her family (height, trunk capacity and trunk (max-min) capacity)

 "Power" Scenario: Looking for the most powerful car. The maximum speed of the vehicle must be as high as possible with the greatest acceleration (time to accelerate from 0 to 100km/h). The car should also have a high horsepower and significant motor torque. Finally, to ensure the car is as powerful as possible, you choose a car that is light (weight of the car).

 "Journey" Scenario: The target user is someone who travels almost every weekend with his/her car. You therefore look for a car with a big petrol tank and minimum consumption levels, because the target user doesn't want to stop often on the way to refuel, which would make the trip too expensive. Comfort is also an important factor: the car must be large (width and large trunk) and finally, a car with good handling should be selected (maximum rim diameter).

The data used in each graphical representation is the same: 40 cars with 18 variables (13 design parameters and 5 performance variables) for each problem scenario (italic text in the scenario descriptions). The 18 variables are as follows: price, engine size, number of cylinders, horsepower, power (rpm), motor torque, car weight, width, length, height, tank, trunk (max. and min.) capacity, rim diameter, maximum speed, acceleration (time to reach 100km/h), fuel consumption and CO² emissions. The cars' names have been erased and replaced by a number (ranging from 1 to 40). Data is gathered from real cars. In order to obtain a wide range of data, we have chosen cars from different categories including large 4x4 and small city cars. In addition, all participants are aware that each criterion has the same weighting.

Procedure

The experiment is divided into three main steps (see Fig 9). The first step begins with a presentation of the experiment's objectives and a preliminary questionnaire in order to obtain subject profiles and their pre-existing knowledge of the graphs tested. The second step is the main part of our experiment and consists of testing the 3 graphical supports after a short training course. In this part, we ask the participant to read the scenario description, to "play" a role (as described in the scenario) and to use the graphic tool to deduce a single solution.

We call the chosen solution a "trade-off" because the scenarios are designed with five conflicting objectives. It is therefore necessary to make a trade-off. The training course is about finding the needed item (car) as well as interacting with graphs. The third step is a questionnaire with the aim of collecting subjective measurements (participant effort, confidence, etc.).

Fig 9. Illustration of the procedure

The total time for a complete session is about 1h10 (ie. 35h of experiments over 4 weeks). It is worth noting that for step 2, in order to avoid the effect of rank and order induced by the graph's sequence, we use the counterbalance method. In our case, this means different graphic sequences (eg. PCP -RC -CT, PCP -CT -RC, CT -PCP -RC, etc.) and different scenario sequences. In addition, this method allows scenarios to be alternated for testing graphics and balancing the effect of participants' learning.

Participants

Subjects were recruited from various teams of scientists in our research building such as engineering design, mechanical, electronic, or computer engineering. All subjects were voluntary participants. The 30 subjects carried out the experiment on the three graphic supports (PCP, CT, RC). Thus, we obtained 30*3=90 test results (the experiment respects a within-subject approach). The mean age of the subject was 30.9 years (SD=7.3). The gender distribution was 53.3% women and 46.7% men. We observed that only 3.3% of participants had already used a PCP, 66.7% a Radar Chart and 33.3% a combined table.

Measurements

Variables were either measured during the tests with the CamStudio (http://camstudio.org/ ) tool or collected through the final questionnaire (n°2). 11

Description Of Variables

The influence of the graph on trade-off assessment is measured with two variables: the time spent on decisionmaking -selecting an optimal solution -and the quality index for the selected design point or selected car (the quality index calculation is described in the next section). Also, we noted the number of actions performed to finally choose a car and the time spent between actions in order to identify the influence of the graph on the path. For us, the path refers to the user engagement as defined by Boy and colleagues: "By engagement, we specifically mean a user's investment in the exploration of a visualization." (Boy et al., 2015). "Action" refers to all interactions with the graphic tool. It uses the "brush", "filter" or "sorting" functions. Every action is an indication of the designer's mental activity (data mining). These four variables are collected during the scenario phase.

We used a questionnaire at the end of the experiment to obtain "subjective" measurements: Influence of the graph on the cognitive load and Influence of the graph on the level of confidence. To measure cognitive load, we chose the DALI method (Pauzié, 2008) which is the method which provides the greatest breakdown of the cognitive load through its questionnaire compared to NASA TLX or SWAT (visual effort, for example). We also adapted DALI to our case (DALI is designed for driving so we adapted the questions for the design selection).

Answers to questions are given on a scale of 0 to 100. Overall, performance metrics are developed when simulations are run. In our case, there are no simulations, either from subjects or from us. We aim to automatically calculate an index to determine which graph allows designers to make the best decisions. In the following section, we describe the construction of a quality index linked to a design scenario from the Pareto dominance and a basic index.

Pareto Dominance

Considering a multi-objective problem such as:

min(F(x) = (f1(x), f2(x), …, fn(x)), n≥2

Where :

 x ∈ X
 X is the decision space (or design parameter space)

 Y is the objective space (or performance space)

 And Y = F(X)

We define dominance as : y dominates z if

∀ i ∈ [1 … n], fi(y) ≤ fi(z) and ∃ k ∈ [1 … n] | fk(y) < fk(z).
And the Pareto Dominance is defined by: One solution

x p ∈ X is Pareto-Optimal if ∄ x ∈ X | x dominates x p
The representation of the set of Pareto-Optimal solutions in the objective space is called Pareto front. Finally, the performance evaluation of the dominated solutions is realized by calculating the Euclidian distance between the dominated solution and the Pareto front. In our case, we calculate the minimum distance between the dominated solution and non-dominated solutions d (Equation 1).

𝑑 = min 𝑥 ( √∑ (𝑥 𝑖 -𝑥 𝑖 𝑝 ) 2 𝑛 𝑖=1 ) (1)
 ∀ i ∈ [1 … n],
where n is the number of performance variables.

 ∀ x ∈ X dominated solution and x p ∈ X non-dominated solution.

Then, for each scenario we apply the Pareto dominance principle and calculate the distance with the Pareto front for all the dominated solutions. Obviously, the distance allows us to compare the dominated solutions, but we need an additional index in order to compare the Pareto-Optimal solutions.

Basic Index

For each scenario, a basic index ij is calculated (Equation 2), after which a global basic index i is calculated (Equation 3). The basic index is based on the Compromise Programming (CP) method, which appears to us the best fit to our case. CP assumes that any decision maker seeks a solution as close as possible to the ideal point. This is equivalent to minimizing the difference (i.e. distance) between utility at the ideal point and utility at a frontier point on the criteria map, a meaningful statement as minimizing distances to the utopia is the ethos of compromise programming.

𝑖 𝑗 = 1 𝑛 ∑ | 𝑓 𝑘 (𝑃)-𝑓 𝑘 (𝐴) 𝑓 𝑘 (𝐼)-𝑓 𝑘 (𝐴) | 𝑛 𝑘=1 , k=1 … n (2) 𝑖 = | 𝑖 𝑗 -𝑖 𝑗 𝑚𝑖𝑛 𝑖 𝑗 𝑚𝑎𝑥 -𝑖 𝑗 𝑚𝑖𝑛 | (3)
Where fk is a performance variable (n=5 for each scenario). fk(P) is the value of design point P for the variable fk. And: (I for the Ideal and A for the Anti-Ideal). We then calculate the global basic index i with equation ( 3) so as to get a value below or equal to 1, 1 being the best car considering the 5 variables of performance with an identical weighting. With this method, the three cases are comparable (cf. Fig11 a. and b.).

 If fk

Our Quality Index

Our quality index distinguishes non-dominated from dominated solutions. It allows non-dominated solutions to be favored and dominated solutions to be "penalized".

The quality index for the non-dominated solutions is the basic index rescaled from 0 to 1. Hence, a quality index is equal to 1 for the best and 0 for the worst solution in the set of non-dominated.

The quality index for dominated solutions corresponds to the distance to the Pareto front multiplied by (-1). Thus, the dominated solutions receive a negative quality index. The closest solution to the Pareto front receives a quality index close to 0 and the solution furthest from the Pareto Front receives the most "negative" quality index.

Results

We First of all, we check if there are any "scenario effects" and/or "learning effects" according to the sequence order through the experiment.

Scenario and learning effects

We have a sample of N = 30 subjects and we are in a within approach (3 groups/ scenarios /sequences). Also, all measured variables during the test are quantitative. Therefore, we apply the ANOVA-within statistical test. The results are shown in Table 1. The results of these tests concerning the scenario effect are not significant. So there is no "scenario effect" on the 8 variables. The results of the tests concerning the learning effect, however, show a significant difference for the time to completion (see Fig. 12). The results of the t-tests show that the time to completion is longer in sequence 1 than in sequences 2 and 3. This means for the rest of the analysis that we should split data between sequence 1 and sequence 2 and 3 when comparing time to completion with the 3 graphs.

Effect of Graphs

Influence of the Graph on the selected solution (TIME & QUALITY)

In order to compare the response time between the three graphs during the sequence 1, we apply the ANOVAbetween statistical test with a sample of N=30 subjects. The results are not significant. We obtain F(2,27)=0.99 and p=0.39. For sequences 2 and 3, we apply several t-tests because we are in a within approach (2 groups i.e. 2 graphs using in the sequences 2 and 3) and we have a sample of N=10 subjects: [RC]vs[CT]: t(9)=2.01 p< 0.08;

[PCP]vs[CT]: t(9)=0.19 p< 0.85 and [PCP]vs[RC]: t(9)=0.68 p< 0.51 (see Fig. 13).

The results of these t-tests are not significant. There are therefore no significant differences between the three graphs for time completion. Then, we obtain F(2,58)=0.03 and p=0.98 for the quality index. The results for trade-offs are not significant.

There are therefore no significant differences between the three graphs for the selected solution.

Indeed, we observe good results (i.e. high quality index values) for the three graphs: i=0.78 with PCP, i=0.77 with RC and i=0.77 with CT (see Fig 14).

Fig 14.

Mean and Standard Error for the variable QUALITY

Influence of the graph on the path (NBR_ACTION & TIME_ACTION)

As in the previous section, we have a sample of N = 30 subjects and it is a within approach (3 groups). Also, all measured variables during the test are quantitative. Therefore, we apply an ANOVA within statistical test (and the T-test for the post-hoc analysis).

The results for the performance of data mining are significant because we obtain F(2,58) =14.1 and p<0.01 for the number of actions carried out and F(2,58)=10.7 and p<0.01 for the average time between actions. So we perform a post-hoc analysis: t-test pairwise comparison (see Table 2). The results of these tests show that the PCP is the most efficient for data mining with an action number equal to 9.67 and an average time between actions of 26.19 seconds compared to 3.97 action numbers for RC and 3.60 for CT and 72.96 seconds between actions for RC and 50.29 seconds for CT (see Fig 15).

Table 2. Results for the t-test for the number of actions and the time between actions

T-test pairwise comparison for the number of actions for the time between actions

Graph's influence on the designer

In this analytical stage, we use Friedman (and Wilcoxon) for statistical analysis (N = 30) because the variables are ordinal (qualitative) and we consider a significance level α=5%.

Cognitive load (ATTENTION, VISUAL & STRESS)

The cognitive measurement is divided into three indicators: effort of attention, visual effort and perceived stress. The results for the three indicators are significant. We obtain csqr=8.87, df=2and p<0.01 for the effort of attention, csqr=8.15, df=2 and p<0.02 for the visual effort and csqr=19.27, df=2 and p<0.01 for the perceived stress. So we perform a post-hoc analysis: Wilcoxon pairwise comparison (see Table 3).

Confidence (CERTAINTY & PREFERENCE)

The confidence measurement consists of two indicators: the level of certainty and preference (for path and to select one solution).

We apply the Friedman test and the results for the level of certainty are not significant. We obtain csqr=1.22, df=2 and p=0.54. We therefore cannot conclude that there is a significant difference between the three graphs for this indicator (PCP=55.61/100; RC=50.44/100 and CT=63.83/100).

Finally, the graph preference for the design process' selection phase is tested with two modalities: the preference of one graph for path; and selecting one solution. Here, we use the Friedman test for statistical analysis because the answers are ranks and we have 3 within groups.

The results for the preference are significant. We obtain csqr=16.47, df=2 and p<0.01 for data mining and csqr=9.8, df=2 and p<0.01 for decision-making. There is a difference between the three graphs for these two modalities. So, we perform a post-hoc analysis: Wilcoxon signed-rank test pairwise comparison (as N=30>20, we consider the W-value). For N=30 at p>0.05, W-value=137 (Table 4). The results of these tests show a significant difference for RC which is the least preferred graph with a rank=2.6 for data mining (compared to 1.6 for PCP and 1.8 for CT) and rank=2.4 to select one solution (compared to 1.6 for PCP and 1.9 for CT).

Correlations

The last part of the analysis is devoted to finding correlations between TIME and QUALITY, TIME and CERTAINTY and TIME and cognitive load (ATTENTION, VISUAL & STRESS). Firstly, we realize the measurements on all data, then per sequence and per graph. We use the Pearson product-moment correlation coefficient (R) and the coefficient of determination (R²). Moreover, we use the value of t associated with the calculated value of R, along with the corresponding p-value to assess the significance of any particular instance of R.

TIME vs QUALITY

The results obtained show that there is no correlation between the time to completion and the quality of the selected solution.

TIME vs CERTAINTY

We obtain significant results for the correlation between TIME and CERTAINTY (see Table 5). The measurement across all the data gives us a negative correlation with a 3.35% variability in TIME linked with variability in CERTAINTY. Unfortunately, this link is very low. However, the measurement of the data linked to the CT graph gives us a negative correlation with 40% of variability in TIME linked with variability in CERTAINTY (see Fig 16). So, there is a link (a trend) between time and the confidence of the subject when using the Combined Table : the longer the particpants take, the lower their level of confidence. The results are unexpected. When using Radar chart, more the participants take time less they have effort of attention whereas more the participants take time more they have effort of attention when using the Combined Table.

TIME vs VISUAL

We obtain one significant result for the correlation between TIME (mean= 210.97) and VISUAL (mean= 53) with N=30, R²=0.17, t=2.36 and p=0.01. The measurement on the data linked to the CT graph gives us a positive correlation with 16.6% of variability in TIME linked with variability in VISUAL (see Fig 17). So, there is a link (low) between time and the visual effort when using the Combined Table : More the participants take time more they have visual effort.

TIME vs STRESS

Once again, we obtain one significant result for the correlation between TIME (mean= 211.00) and STRESS (mean= 41.13) with N=30, R²=0.23, t=2.88 and p<0.01. The measurement on the data linked to the CT graph gives us a positive correlation with 22.9% of variability in TIME linked with variability in STRESS (see Fig 17). So, there is a link between time and the perceived stress when using the Combined Table : More the participants take time more they have stress.

Synthesis of results

To sum up our results, we have a greater performance for the "path" (ie. data mining) with the graph Parallel Coordinates Plot (PCP). There is no distinction between the three graphs for the selected solution (both quality index and time indicators). Furthermore, the designer is less satisfied than with the graph Radar Chart (RC). The results are summarized in Table 7.

The correlation analysis enables us firstly to note that, overall, the selection of a good design is not related to the time that participants spent on the graphs. Moreover, we observe, on the use of CT, there is a link, a trend between the time to completion and cognitive workload and time and trust. It seems that time is linked to the cognitive effort by the participants. Conversely, participants take more time and more confidence decreases. More broadly, it suggests to us that the participants experienced an important cognitive workload are those who remained longer on the graph. Reasonably, we find that the duration of the exercise is linked to the difficulty perceived by the subject, for the CT. The results are reversed for the RC. Participants who spent few time on the RC have felt a strong effort of attention. The results for the RC are difficult to interpret, perhaps should we promote designers to spend more time on the RC in order to reduce the effort of attention. At last, in line with the cognitive workload results, participants who spent time on the graph CT have little confidence in their selected solution.

Finally, we should therefore conclude that the Parallel Coordinates Plot (PCP) is the best graphic support to help designers select one optimal solution whilst exploring the design space.

Conclusion & Discussion

In conclusion, our experiment enables us to make certain recommendations regarding the choice of graph for the selection phase in Design by Shopping. More specifically, we deliver a directive for the choice of a graph representing design points to efficiently select an optimal solution in a set of feasible solutions defined by their design and performance value vectors.

Indeed, our results show that it is more efficient to use the Parallel Coordinate Plot, which performs better in our experiment. It is worth noting that our recommendations apply design multi-criteria decisions or, in other words, trade-offs between conflicting objectives to obtain an optimal solution. Finally, we propose using two graphs on the same screen. The Parallel Coordinate Plot is particularly useful for data mining and selecting an alternative, while the combined table specifically provides detailed information (eg. specific values). Nevertheless, both graphs must match (linked views) ie. if a filter is applied to one graph, the data must be automatically filtered on the second graph.

Eikenes and Morrison (2010) argue and show that the interface is not just a flat layer between the user and computer, but is rather a complex, mediating, cultural artefact. Thus, appropriate distinctions between designer profiles, such as expert designer or novice designer, can help improve results. As shown by Wolf et al. [START_REF] Wolf | A preliminary study of novice and expert users' decision-making procedures during visual trade space exploration[END_REF], there are procedural differences in how novices and experts use multi-dimensional data visualization and exploration tools when solving an engineering design problem. In the same way, increasing the training phase could substantially enhance the difference between the three graphs for the design selection phase because we observe a poor use of the PCP (only 3.3% of participants have already used a PCP). In fact, Wolf et al. [START_REF] Wolf | The Importance of Training for Interactive Trade Space Exploration: A Study of Novice and Expert Users[END_REF] have already shown the importance of training for the entire process of exploring the trade space. In addition, we believe that an indicator after the training phase could provide valuable information about the result, thus enabling the participant to measure the graph's practical level. This would, for instance, allow us to identify a lack of practice at the origin of non-optimal decisions, for instance.

Since knowledge of the three graphs is very heterogeneous (3.3% of participants have already used a PCP, 66.7% a Radar Chart and 33.3% a combined table), we cannot impose a time limit, because the results would reflect a lack of use of graphical tools. Thus, greater training which measures practice would allow us to add a time limit and reveal richer results.

Obviously, our experiment focuses on the selection phase; it would be advisable to develop this approach by identifying graphs adapted to designers for the entire process. Firstly, this will test whether the PCP is the best graph for the selection phase included in the process of exploring the design space, ie. following the generation of design points -reduction of space design loop. Secondly, it will enable us to verify that the PCP is a graph that can support all of the phases of the process in an effective manner. Finally, it may be useful to envisage the use of graphical tools in a collaborative framework such as that used in [START_REF] Canbaz | Constraint programming simulation of a distributed set-based design framework with control indicators[END_REF][START_REF] Canbaz | Preventing and resolving design conflicts for a collaborative convergence in distributed set-based design[END_REF]).

Fig 1 .

 1 Fig 1. From exploring the Design Space to the research question

Fig 2 .

 2 Fig 2. Screenshot of ARL Trade Space Visualizer: a) Glyph Plot, b) Parallel Coordinates, d) Scatter Matrix (Stump et al., 2004) The VIDEO tool (Kollat & Reed, 2007) is used to separately visualize on two graphs the objective space (ie. performance variables) and decision space (ie. design parameters) with the 3-D Scatter Plot using color and size glyphs (Fig 3).

Fig 3 .

 3 Fig 3. Screenshot of VIDEO system (Kollat & Reed, 2007) The LIVE tool (Yan et al., 2011, 2012) is, on the one hand, used to visualize in two graphs, the input variables (design parameters) on one side (Fig 4 b.) and the output variables (performance variables) on the other (Fig 4 a.). On the other hand, a tree structure is displayed which represents the classification of input variable combinations (Fig 4 d.). Both spaces (input/output) are projected in a 2D Scatter Plot in which color and shape glyphs (cluster creation) can be added. The tree structure is presented in a Treemap[START_REF] Shneiderman | Tree visualization with tree-maps: 2-d space-filling approach[END_REF] in which each node corresponds to an input attribute with a splitting value. Each leaf of the tree specifies the

Fig 4 .

 4 Fig 4. Screenshot of live system: (a) scatter plot of performance output variables; (b) scatter plot of design input variables (showing clustering results); (c) text representation of decision rules; (d) treemap visualization of decision tree; (e) adjustable sliders for ranges of input and output variables (Yan et al., 2011, 2012)

Fig 5 .

 5 Fig 5. Screenshot of Rave tool (Daskilewicz & German, 2012)

Fig 6 .

 6 Fig 6. Screenshots of Combined Table

Fig 7 .

 7 Fig 7. Screenshots of Parallel Coordinates Plot

Fig 8 .

 8 Fig 8. Screenshots of Radar Chart

  Finally, to determine the influence of the graph on the level of confidence, we used an adapted Aloysius et al. (2006) method. For this, we have two variables: Level of certainty in the chosen solution and the Preference for one graph. As for the cognitive load, answers are given on a scale of 0 to 100. This set of measures is summarized in Fig 10.

Fig 10 .

 10 Fig 10. The experiment's measurements



  is maximized: fk(A)= the minimum value of fk and fk(I)= the maximum value. If fk is minimized: fk(A)= the maximum value of fk and fk(I)= the minimum value.

Fig 11 .

 11 Fig 11. Illustration of the solution indexes: a) specific index and b) global basic index.

  use 8 variables for 6 experimental measurements (3 variables for the cognitive load): Time to completion (TIME); Quality index (QUALITY); Number of actions (NBR_ACTION); Time spent between actions (TIME_ACTION); Cognitive load: Effort of attention (ATTENTION); Visual effort (VISUAL);Perceived stress (STRESS); and finally Level of certainty (CERTAINTY).

Fig 12 .

 12 Fig 12.Mean and Standard Error for the variable TIME during each sequence So we have to perform a post-hoc analysis thus we carry out t-test pairwise comparison:  [sequence 1]vs[sequence 2] t(29)=2.30 p< 0.03  [sequence 1]vs[sequence 3] t(29)=2.86 p< 0.01  [sequence 2]vs[sequence 3] t(29)=1.04 p< 0.31

Fig 13 .

 13 Fig 13. Mean and Standard Error for the variable TIME during sequences 2 and 3

Fig 15 .

 15 Fig 15. Mean and Standard Error for the variable NBR_ACTION (on the left) and TIME_ACTION (on the right)

Fig 16 .

 16 Fig 16. Plots of CERTAINTY vs. TIME per graphs

Fig 17 .

 17 Fig 17. Plots of ATTENTION vs. TIME (on the left), VISUAL vs. TIME (on the center) and STRESS vs. TIME (on the right)

  [START_REF] Keim | Designing pixel-oriented visualization techniques: Theory and applications[END_REF] (amongst others) about graph characteristics, we identify the values table, the table with heatmap, the combined values-heatmap table, radar chart and parallel coordinate plot (PCP).[START_REF] Gettinger | A comparison of representations for discrete multi-criteria decision problems[END_REF] have already compared the values table, the heatmap and the PCP for multicriteria decisions and they conclude that it is necessary to check the efficiency of the combined values heatmap table. Also, these authors do not consider the radar chart in their work. Thus, we propose comparing the following three graphs to select an optimal solution (design decision) in design space exploration: the Combined Table (CT), the Parallel Coordinates Plot (PCP) and the Radar Chart (RC). The interactive graphs are available at the following address: http://these.aaa.alwaysdata.net/expe1/

4.1 Combined Table (CT) The Combined Table is a combination of table values and a table with a heatmap (Fig

  Abi Akle A., Minel S., Yannou B. (2016) 'Information Visualization For Selection In Design By Shopping', Research in Engineering Design, 28(1), pp.

	99-117, doi: 10.1007/s00163-016-0235-2

Table 1 .

 1 ANOVA results for scenario and learning effects Abi Akle A., Minel S., Yannou B. (2016) 'Information Visualization For Selection In Design By Shopping', Research in Engineering Design, 28(1), pp.

		ANOVA results for scenario effect	ANOVA results for learning effect
	TIME	F(2,58)=0.42 p=0.66	F(2,58)=5.91 p<0.01
	QUALITY	F(2,58)=2.43 p<0.10	F(2,58)=0.58 p=0.57
	NBR_ACTION F(2,58)=1.59 p=0.21	F(2,58)=1.16 p=0.32

Table 3 .

 3 Results of the Wilcoxon tests for the three indicators of the cognitive load

	Wilcoxon pairwise	ATTENTION	VISUAL	STRESS
	comparison
	[PCP] vs [RC]	W=-193 p<0.03	W=-270 p<0.01	W=-254 p<0.01
	[PCP] vs [CT]	W=57 p=0.28	W=6 p=0.48	W=144 p>0.05
	[RC] vs [CT]	W=201 p<0.04	W=205 p<0.02	W=296 p<0.01
	The results of these tests show a significant difference for RC with the most effort with 63.28/100 attention
	(compared to 51.44 for PCP and 48.61 for CT), visual effort of 67.67/100 (compared to 54.39 for PCP and 53.00
	for CT) and a perceived stress of 64.12/100 (compared to 48.83 for PCP and 41.13 for CT). We cannot conclude
	any difference between PCP and CT.

Table 4 .

 4 Results of the Wilcoxon signed-rank test for the two indicators of preference

	10 12	9.67			80 100		72.96
	8				60		50.29
	6 4		3.97	3.60	40	26.19
	2				20
	0				0
		PCP	RC	CT		PCP	RC	CT

Table 5 .

 5 Significant results for correlations between TIME and CERTAINTY

	Data	N	Mean CERTAINTY	Mean TIME	R	R²	t	df	p-value
	All	90	56.63	215.21	-0.18	0.03	-1.75	88	0.04
	CT	30	63.83	211.00	-0.63	0.40	-4.32	28	<0.01

Table 6 .

 6 Significant results for correlations between TIME and ATTENTIONWe obtain significant results for the correlation between TIME and ATTENTION (see Table6). The measurements on the data linked to the RC and CT graphs give correlations. For the Radar Chart, it appears a negative correlation with 12.5% of variability in TIME linked with variability in ATTENTION (low link) and a positive correlation with 32% of variability in TIME linked with variability in ATTENTION for the Combined Table (seeFig 17).

	Data	N	Mean ATTENTION	Mean TIME	R	R²	t	df	p-value
	RC	30	63.28	241.87	-0.35	0.13	-2	28	<0.03
	CT	30	48.61	211.00	0.57	0.32	3.64	28	<0.01

Table 7 .

 7 Synthesis of results

		Path	Selected solution	Designer satisfaction
	Best graphs	PCP	none	PCP ≡ CT
	Worst graphs	CT ≡ RC		RC

99-117, doi: 10.1007/s00163-016-0235-2

Acknowledgments

We wish to thank all those took part in our experiment.

Conflict of interest:

The authors declare that they have no conflict of interest.