Tel

Prof Author Louis De Mesnard
email: louis.de-mesnard@u-bourgogne.fr

Qualitative methods of structural analysis: Layer-based methods are informationally trivial

Keywords: classification. C67, D57, R15. KEYWORDS. Boolean, Structural Analysis, Qualitative, QIOA, MFA

Some methods of qualitative structural analysis, as MFA, are based on the analysis of layers (flow matrices generated at each iteration when the equilibrium of an input-output model is computed). MFA mixes the analysis of the pure structure of production (the technical coefficients) and of the final demand. I have demonstrated that all column-coefficient matrices (or row-coefficient matrices) computed from each layer are the same in MFA: the information brought by one layer is identical to those of another layer. For a given structure of production, the only element of variability over layers is caused by the flows that final demand generates.If the new definition of layers proposed by the creators of MFA is adopted, the method becomes similar to a quantitative method of structural analysis.

Qualitative methods of structural analysis are useful to analyze the structures of production and exchange, for example in space, as the structure of exchange between countries, regions or cities but also for non-space systems as structure of production. In this last application, when structures of production are considered in an input-output framework, these methods are often called Qualitative Input-Output Analysis or QIOA '.

Among qualitative methods of structural analysis, some are based on "layers", as Schnabl's MFA or Minimal Flow Analysis [START_REF] Holub | Qualitative Input-Output Analysis with Variable Filter[END_REF], (Holub and Schnabl, 1985), [START_REF] Schnabl | Ökonomische Strukturanalyse Mit Hilfe Von Verflechtungsschichten. Ein Neues Verfahren Der Qualitativen Input-Output Analyse[END_REF](Schnabl, , 1994(Schnabl, , 1995)), (Weber and Schnabl, 1998): for that, they can be considered as intermediate between qualitative and quantitative methods. What are layers? Layers are intermediary flow matrices generated when the equilibrium of an input-output model is computed. They are analyzed by the mean topologic tools (after being transformed into boolean matrices). This idea seems to be seductive. However, the information carried by layers is in question: if each layer carries a specific information then it is interesting to study layers, but if all layers carry the same information the interest of the method is challenged.

After recalling what exactly are layer-based methods, this paper examines the problem of the information carried by layers.

I. Introduction

1 Bon (1989) has presented the basic principles of QIOA, but its paper is also a good introduction to qualitative methods of structural analysis, in general.

II. Recall

The general framework is the Leontief model [START_REF] Leontief | Input-Output Economics[END_REF]): i = A x + y , where x is the output vector, y is the final demand vector, and A is the matrix of fixed technical coefficients accounting system. The solution of the model is x = (I -A ) 1 y, with ( I -A )-1 = 2^0 A*.

Denote H any input-output matrix, that can be Z or A or B (the allocation coefficient matrix), and hy the terms of this matrix. In qualitative terms, a sector j is influencing sector i if hy > (|), where < | ) is the value of the filter. In other terms, a boolean matrix W (1)(<j>) is deduced from matrix H: W ÿ^) = 1 <=> hy > < | > .

In ordinary boolean methods, the matrix W (2)(<|)) is computed as a boolean product, W (2)(<()) = W (1)(<|)) * W (1)(<(>), * denoting the boolean product. wf\ §) = 1 if and only if there exists at least one sector i such that there is a direct path between i and /, i.e., *$ }(< | > ) = 1 and a direct path between i and j, i.e., = 1. Again, W (3)(<j>) is calculated from W (2)(< | > ) following the same rule, W*3*(<f>) = W^((|>) * W^(<|)), etc. This is generalized by:

(1) W w(< | > ) = W(1)(< | > ) * W ^1^)
Then, a dependency matrix is computed: D = 2 * W (t).

In MF A, one starts from Z = A (x). As, calculated by A = Z x 1 where Z is the matrix of transaction flows given by the national

(2)

one can write:

(3) Z = A <y) + A <A y) + A <A2 y) +... + A (A* y) +.

The layers, i.e., the matrices Zo = A (y), Zi -A (A y), Z 2 -A (A2 y ) , ..., Z* -A (A* y), can be interpreted as the successive flow matrices generated by an initial demand vector y at steps 1, 2, 3,..., k, respectively. The matrix W*, built from each layer Z*, indicates if there is a link between vertices in Z *. By analogy with formula (1), matrices W* are combined by the recursive formula (Schnabl, 1994, p. 52, eq. 2):

(4) W('> = W w * W 'w > with W(0) = I. The result of (2) is the following at the first step: W (1) = Wo * W (0) = Wo , i.e.

= 1 and only if there exists at least one sector i such that there is a direct path with a length equal to 1 between j and i, i.e. (wo),y=l. In the second step, W(2) = W 1 * W(1) = W 1 * Wo, i.e., wf1 = 1 if and only if:

• there exists at least one sector i such that there is a direct link between j and i, (wo),y = 1 in the first layer Zo (in the matrix of flows after an impulsion of final demand),

• and i is in direct relation with I i.e. (wi)/( = 1 in the second layer Z i .

At the step k, one has: W(t) = W/t_t * W*_2 * ... * W i. A dependency matrix is also computed by boolean summation of the W(i).

III. Informational problem with layer-based methods

Remark The same output x* can be generated either by Z*-i s (s is the sum vector) or by A* y because one have:

x* = A* y = A* (y) s = Zk -i s = A (x*_i) s = A x^-i
so A* corresponds to the same iteration than Z*_i, but as A k (y) *= Z*-i, analyzing the power matrix A* is not the same as analyzing the layer Z*-i. ■

When they are considered in relative terms, all layers carry the same information. To prove this, first one must define column coefficients. Column coefficients are not exactly technical coefficients, they are similar but do not count the value-added of each sector; A* is such that:

/c\ -k _ (5) ' • -* *
where (zy)k is a term of Zk and (z,j)k = X (zy)k-That is in matrix terms:

n i i = A <A* y) (A* y)-i <s/ A)-1 A . A » A t = A (s' A) , for all k, so A* is a constant. ■
MFA is not a method of pure structural analysis because it analyzes flows that depend on the production structure and of final demand. At this step, one argument appears: MFA is concerned with the structure of flows including the structure of final demand, while the above demonstration concerns the structure of production only. In other words, to save MF A, one could say: the above developements demonstrate only that the pure structure o f production is stable from one layer to the other, but, as MFA studies flows by mixing the structure of production and final demand, the stability over layers of the pure production structure is not a problem. And MFA could continue to work fine. This argument is not acceptable on a Cartesian view point.

• First, MFA cannot study any other thing than the pure production structure and the effect of final demand on flows that pass by this production structure.

• Second, in the Leontief model, technical coefficients are assumed to be stable while only final demand is variable: the Hawkins-Simon theorem shows that changes in the structure of production are independent to changes in final demand (Hawkins and Simon, 1949).

• Third, MFA studies flows in layers, that depend on the production structure and on final demand by the formula Z* = A (A* y ).

• Fourth, I have demonstrated that the pure production structure -measured by the column coefficients -is stable in MFA.

To summurize, final demand, and flows in layers change, but the pure production structure remains stable from one layer to the other. Then, logically, for a given structure of production A, the only element of variability over layers is caused by final demand and the flows that it generates at each layers. , * 2 = 567.5 N 1022.5 1182.5

' 1 0 0 ' ' 1 0 0 ' Wi = 0 1 1 => w (2) = Wi * w (1) = 0 1 1 0 1 1 0 1 1 A, = A 2 = 0.
" 1 0 0 " ' 1 0 0 " W 2 = 0 1 0 => w (3)=w 2 * w (2) = 0 1 1 0 1 1 0 1 1 a 2 = 0.
= Ai = Ao , W (3) = 0 0 0 0 0 0 0 1 1
Remark. Since the first column in Zi is greater than the first column in Zo, the influence of sector 1 seems to be increasing from step 2 to step 3, but it is only an artefact of MFA because in terms of column coefficients, nothing at all has changed from one step to the other:

A k - 2) Now, consider exactly the same example but with another vector of demand:

Os

' 1 0 0 ' ' 1 0 0 " W0 = 0 1 0 => W(1)=Wo = 0 1 0 0 1 1 0 1 1 _ but the pure production structure is the same as above, that is: As it can be seen, the second computation with another final demand vector, but the same matrix A, gives different results than the first, the layers are not the same, the boolean matrices of MFA are not the same, while the pure structure is unchanged. It seems that only the influence of final demand is caught by MFA. What happens with an unchanged final demand vector but another matrix A?

3) Consider the same final demand vector than in case 1) but another matrix A: • Etc. As it can be seen, with another matrix A the results are not the same than in the first case, but, again, from one layer to the other, the column coefficients remain unchanged.

Let's summarize. Another matrix A, another result; another final demand vector, another result; but for a given production structure and a given final demand vector, the kth layer brings no new information compared to the (&+ l)rt layer. ■

IV. Epilogue

In their recent paper, Weber and Schnabl (1998) claim to apply MFA to the analysis of the energy sector and its role in the German economy. Intermediary flows of energy are studied by premultiplying equations ( 2) by Ex and replacing y by (y):

X e = Ex ( I -A )"1 <y> » X e =E x <y> + EX A (y) + Ex A2 <y> +... + EX A* (y> +...

where "the columns of the subsystem matrix Xe indicate the composition of the cumulated energy requirements of product group j by sector i of (final) energy consumption", Ex is the energy input coefficient matrix: Ex = S* (x)-1, and Sx is the matrix of energy input (Weber and Schnabl, 1998, pp. 3398-339 and 350). Apart this, MFA is claimed to be applied in a similar way.

However, if this has the taste of MFA, this is not MFA! The terms Ex A* (y) are not layers in the sense of MFA as defined in [START_REF] Schnabl | Ökonomische Strukturanalyse Mit Hilfe Von Verflechtungsschichten. Ein Neues Verfahren Der Qualitativen Input-Output Analyse[END_REF](Schnabl, ,1994(Schnabl, ,1995)). As said, a layer Z* = A (A* y) is the flow matrix that appears at the iteration k+ 1, or if k is seen as a time index, it is the matrix of intermediary flows exchanged at the date k: A k y is the output generated at the date k, and the product A (A* y) is the quantity of intermediate commodities demanded to each sector in order to produce this output.

A term of Ak (y) is not a flow generated at date k. A term aff of matrix A k is the result of the summation of all the successive coefficients found on paths of a length equal to k. For n example, the term a y of A 2 is equal to £ au aij ' ■ n paths of length 2 between j and i are of

V. Conclusion

Layer-based methods as MFA belong to the general category of qualitative methods of structural analysis -or qualitative input-output analysis -even if they are a hybrid between qualitative and quantitative methods. MFA mixes the analysis of the pure structure of production (the technical coefficients) and of the final demand. I have demonstrated that all column-coefficient matrices (or row-coefficient matrices) computed from each layer are the same in MFA: it is illusory to believe that the information brought by one layer is different to those of another layer and MFA's approach is not informationally discerning. In other terms, for a given structure of production, the only element of variability over layers is caused by the flows that final demand generates at each layers. Finally, MFA is not a method of strcutural analysis. If the new definition of layers proposed by the creators of MFA is adopted, the method becomes similar to a quantitative method of structural analysis.

VI. References

A

  Property 1. All matrices A k of column coefficients, deduced from layers Z* at each step k, are A A identical: A* = Ao, for all k. ■ So, all layers carry the same information because all boolean matrices W* found from matrices A A* are the same. The information carried out by matrices Zk is trivial in terms of column coefficients. Proof. Â* = Zk W Z * r ' = A (A* y) <s' A (A* y))"1 A j <=> A* = A (A* y) [(s' A) (A* y)] (because if b and c are two vectors, (b '< c» = (b)(c>)

,

i=i type 7

 7 -> / -> i. The term a f yj of the matrix Ak (y) is the intermediate demand addressed to sector i in commodity i to respond to the final demand of commodity j when an indirect relation of length k is considered. It is not the flow exchanged at step k\ Weber and Schnabl have tried to transform MFA to respond to the objections developed in this paper 2. Anyway, transformed in this new way, the new layers become very close to a simple power matrix as A*: this concept is analyzed by quantitative methods when the strongest paths of all possible lengths are searched in the structure; see, for example,[START_REF] Lantner | Théorie de la dominance économique[END_REF].Remark. Note that A ** A* for all k * l generally: the series of power matrices is not informationally trivial. Consider W w defined as the boolean matrix found from A* by the relation:*[?><!> « * < ? = !where af* is a term of the power matrix A*. Generally, W w * W w .When one considers the Leontief model, only the matrix A plays a role from one iteration to the following. One can write: Z*+t = A (Z* s), because x*+i = Z* s = A*+1 y , and so the only change between z t+1 and Z* is brought by A, but it is also the case for A*+1 and A*, A*4 -1 = A A* 3. So, the lack of information between any two iterations of layers, in relative terms, does not rely on this fact. ■ 2 It's not a temporal paradox: they were exposed to them at the Twelfth International Conference on Input-Output Techniques, New-York City 3 Similar reasoning comes from matrix B.

Frontiers in input-output analysis,

  BON Ranko, 1989, "Qualitative Input-Output Analysis" in MILLER Ronald E., Karen R. POLENSKE and Adam Z.ROSE, Eds., 1989, Oxford University Press, New-York and Oxford, 222-231. HAWINS D. and H. A. SIMON, 1949, "Note: Some conditions o f macro-economic stability",

	Econometrica, 17, 245-248.

 WEBER, C. and H. SCHNABL, 1998, "Environmentally important flows: insights from main contributions identification and minimal flow analysis", Economic Systems Research, Vol 10,No 4,[337][338][339][340][341][342][343][344][345][346][347][348][349][350][351][352][353][354][355][356]


Furthermore, unlike the usual theory, the goods capable of satisfying needs are assumed to be scattered in a supply space and transportable.The agent is located at a given point of demand.

-Notation:

Only ordinary (non fuzzy) mathematical concepts are underlined.

For example A C X is a fuzzy subset of the X referential. However, when there can be no possible ambiguity, this convention will be dropped/ 04 -A supply space is characterized by a set, denoted L, of m points of supply, denoted 1 . L = �1 p � , p = 1,...,m. The set L is at most countable.

A point of demand is chosen (location of the consumer), denoted l1'

The set of n located goods is represented by a vector, denoted X : !=l!�1 ' i = 1,...,n. The quantities of the goods are denoted j(' 05 -From the set of goods X the set of possible consumptions,denotes K , is defined. An element of K is a vector whose components are the quantities of goods xP the consumer must choose. A vector of K is denoted C hk ' . .

One writes therefore: Kin r "-Fi k 1 î \ , h = 1,... ,g,... ,j,... ,1 K is assumed to be countable. A consumptions vector is written k Pl .

Careful distinction must be made between the set of possible consumptions and the supply space of goods. Goods being mobile, the set of possible consumptions is defined at point 1 and is dissociated from the supply space. It should be noted that the expression"consumptions space"

which is so often use should not lead to any confusion: it describes a set with given properties.

06 -Taking these definitions as a starting point, firstly a general study will be made of the structures of spatial preference in an imprecise context.

Next, the particular properties will be introduced. These properties must be united within this analytical framework in order to define a behaviour of attainable choice. Then the conditions which guarantee the existence of a completely preordered topological space are imposed in order to verify that the consumer's imprecise behaviour is coherent. Finally these conditions permit us to define the existence and continuity of a fuzzy utility (a numerical representation of the imprecise preference).

I -CHOICE IN IMPRECISE PREFERENCE STRUCTURES.

11 -The consumer draws a comparison between any two complexes of goods, gk and ik at a more or less high level of preference or indifference. This level expresses the imprecision of his comparison.

Hence a fuzzy binary relation, denoted 3l,between the elements of K2, is defined:

where y»a (9k,Jk) denotes the level of preference or indifference between the two complexes and M is a completely ordered membership set. A non numerical set suffices.

In other words, to each given preference-indifference system, a fuzzy subset,denoted K,is associated. K is an element of the set of applications from K2 to M, of the type: 121 -The consumer prefers more or less a complex gk to a complex gk.

One writes ,tt�(9k.jk) � .l"0t(jk,9k). /�(gk�jk) will be called the degree of strong préférence for 9k relative to �k and M?a{ k, 9k) the degree of weak preference for ik relative to gk.

The structure (K, » has the property of antisymetry as defined by KAUFMANN (7, Vol.l, page 105) since V(9k,Jk) 6 K2 ,g�j, one has:

Nevertheless, the assumption that y*^,(9k, k) = f-fci k,9k) = 0

will only be introduced in paragraph (1222) within the framework of the definition of the indifference structure, where it will be taken as a particular case of fuzzy symmetry. In the present case it is important for the structure (K,�) that V(�k�k)6j�� , g^j, f (9k,jk) such that Remark.

In the classical theory of consumer behaviour, one has:

The consumer prefers or does not prefer one complex to another.

122 -The consumer can be more or less indifferent between 9k and jk.

Thus an indifference structure can be defined (K.,u). One has:

By definition, the structure (K,,v) has the property of fuzzy symmetry.

Remarks.

(1221): Classical theory only retains the special case in which these membership functions equal 1.

(1222): The degree of indifference can equal zero. In which case, the complexes of goods are considered to be useless.

123 -The imprecise preference-indifference structure (K,1t) is complete since ��(9k,jk) = 0 if (9k',Jk) = 0 (non comparability assumption). The comparability axiom has not been introduced here and the situation of non comparability does not create any particular problems as it does in the classical theory of consumer behaviour.

13 -Now if the fuzzy relation(/lis reflexive and transitive, the set K has a fuzzy complete preordering structure.

131 -We have first: V (gk,9k) F-12 : ?�(gk,gk) = 1

The relation Jtis reflexive.

Ü 132 -Next if the preference for gk relative to lk is at least equal to the maximum for ik of the smaltest values of the preference for 9k relative to ik and for jk relative to lk, then the relation� ïs transitive.

One has:

Remarks.

(1321) We have a property of weak transitivity. It follows tnat the structure (K,tv) is not an equivalence structure.

In fuzzy preordering, all the non fuzzy subsets, denoted K r , K Q K , » r = 1,...,h, such that: form similarity sub relations, denoted Ks, Ks C K , s 6 1 , le � . We know that a similarity subrelation is maximal if it 's not a similarity subrelation of any other subrelation of the same kind in the relation being examined.

The combined properties of fuzzy symmetry, reflexivity and Max Min transitivity do not necessary imply that the maximal subrelations are disjoint.

141 -Two conditions must be imposed on the structure (�,�) so that the maximal similarity subrelations form similarity classes.

Consider two maximal similarity relations: Conversely, if any pair of complexes (9k,1k) satisfies the axioms

(1) and ( 2) its elements do not form a similarity subrelation since /�6l(gk,lk) 0 ��L(lk,gk).

Therefore they can only belong to different maximal similarity subrelations and the maximal similarity subrelations form similarity classes.

K /,�, denotes the quotient set.

144 -Conditions ( 1) and ( 2) give the structure (�'I() the properties of congruence to the right and left of indifference relative to strong preference.

We check that: V (gk,lk,mk) G K3 : A set of better elements of J is denoted S(J) and will be called the set of imprecise choices. set 152. Introducing thevS(J) enables us to formulate the consumer's behaviour which is associated with the structure (K,�).

The application S of all the fuzzy parts of K in K itself is defined such that:

The choice is still possible since application S is defined everywhere and the imprecise behaviour of choice is attainable.

-THE CONTINUITY OF PREFERENCES AND II�9PRECISE UTILITY

21 -At this point we must specify what conditions are to be imposed on the fuzzy individual preference structure (K, �) to ensure that a completely preordered topological space exists and that the consumer's imprecise behaviour is consistent. Finally, these conditions enable us to guarantee the existence and continuity of a fuzzy utility.

From now on, we choose M = [0,11 and if:

where K is a fuzzy subset of K associated with a system of imprecise preference such that: i u K (gk) and A*K (jk) will be called the relative degrees of preference for one complex of goods compared to the others.

211 -A fuzzy topology on K , denoted � , is defined.

Let h � K (gk) be the relative degree of preference for the complex gk and ( M (hk))n ' n n6!N , an infinite sequence of relative degrees of preference for the complexes of goods which tend to u K (°k), where °k is an element of K , when n tends to infinity. /.

If for any integer n, ( M K (hk))n is at most as great (resp. at least as great) as A�K (9k), then M (°k) is at most as great (resp. at least as great) as �kK (9k), which means that °k is at most as preferred (resp. at least as preferred)as gk.

In other words, for every gk and for every hk in K the fuzzy subsets:

The topological space (K,� � ) is said to be an imprecise preference space that is the preferences are continuous for � if and only if:

212 -Consider a given system of individual imprecise preference with which a fuzzy subset K is associated. We shall now indicate which convexity condition must be imposed^yin order to guarantee that the structure (K, f, g) is a completely preordered topological preference space.

2121 -A fuzzy subset�s convex (weak convexity) if and only if the membership function is quasi-concave:

If the individual is indifferent to or prefers the complex gk to complex ik , to the respective degrees tK (9k) and/ K (�k) , then the membership function of any weighted average with positive coefficients for gk and ik is greater or equal to the membership function of the complex Jk.

Remark.

-

The following preliminary result is needed to establish the existence of a completely preordered fuzzy topological space.

Let t'S, g) be a fuzzy topological space. If K , K C K is closed in (K,� ) then K contains the limit of all its convergent sequences.

Indeed, let (Xn) be a sequence of fuzzy parts of K, converging to a fuzzy subset Xo of K. By definition any neighbourhood V(Xo ) of Xo countains fuzzy subsets of(Xn)and hence of K, which means that Xi is adherent to K, and therefore Xo C K , since K is closed.

2123 -UZAWA's result obtained within the framework of the classical theory may be generalized L 5 ) -Let (�, 1; , � ) be an imprecise preference space, K convex... if: ithe fuzzy relation is complete and satisfies the convexity condition, ii -the fuzzy relation of strict preference is Max-Min transitîve, then (K.,^,% ) is a completely preordered topological préférence space.

Remark.

UZAWA's theorem only holds if the (non fuzzy) relation satisfies the strong convexity condition (in the usual meaning of the word).

-Proof. The property of totality implies that of reflexivity.The fuzzy relation can be proved to be Max-Min transitive by reductio ad absurbum. We can find O{.. é:: "\ 0,1[ such that

If not,for any ri.. f. J 0,1[ we would have:

i " " i 1 _ (see ( 211)).

' / " that since K � is closed, it contains the limit of its convergent sequences (see ( 2122)) which contradicts gk C K*" .

Since A� (gk) � � K (jk) , the convexity condition of imprecise preferences implies: Finally, relations (i) and (ii) and then Max-Min transitivity property of imprecise preference imply that: which contradicts the assumption: r The fuzzy relation of imprecise preference is therefore Max-Min transitive.

22 -Given a fuzzy structure of complete preference (�, �), by translating the qualitative description of the consumer's imprecision decision into a numerical formula we can define a fuzzy utility.

Indeed the function which relates the element of the interval C 0,1] associated with the corresponding degree of imprecise indifference to each similarity class K 6 K 1,,, has been built on to the quotient set K le-j (see( 21)).

Let f be that function

The function is strictly incresing. The function u, which associates an element of the interval 10,11 with each element of the set K is called an imprecise utility function.

Complexes of goods belonging to a same similarity class produce the same degree of utility. If one similarity class contains a complex which is more or less preferred to a complex belonging to another similarity class, then the function associated with the first is greater than that associated with the second and conversely.

In other words, the function u is such that:

The fuzzy utility function allows the complexes of goods to be classified according to the order in which they are chosen by the consumer. to ( For a structure (K, �) a utility is any homomorphism from (K, � ) 221 -It is certain that a utility defined in this way exists, because

we have here a case of the application of the theorem which states that a utility exists for a preference structure (�,�) if it is a completely preordered countable set f3, page 178).

-CONCLUSION

31 -The theory of fuzzy subsets proves relevant to the formalisation of the analysis of preferences and utility in a spatial world where consumers behaviour is imprecise.

The resulting economic theory is neither ordinal nor cardinal: we propose to call it "a valuable theory of préférence".

-

The assumption that the possible consumptions set is countable will have to be dropped. This is critical since the fuzzy subsets theory only has a weak theorem of separation at its disposal.

33 -The analysis of the structure of imprecise individual preference should be followed by a spatial theory of consumer demand in a fuzzy world, where C.I.F. prices and wealth constraints are introduced.

34 -The present model is static. In order to introduce time it will have to be developed further. For a constant system of imprecise preference structures, introducing incertitude would lead to a formulation in terms of probabilized fuzzy subsets. A dynamic analysis of the passage from one system to another would have to make use of differential topology.

  : Théorème du point fixe. Une étude topologique générale (.juin 1974) N°2 Daniel LEBLANC : L'introduction des consommations intermédiaires d?ns le r10dèle de t_EFE3ER (juin 1974) N°3 Colette BOUMON : Spatial Equi 1 i bri um of the Sector in Quasi-Pcrfoct Compctition (sept,mbcr 1974) N°4 Claude PONSARD : L'imprécision et son traitement en analyse économique (septembre 1974) ?5 Claude PONSARD : Economie urbaine et espaces métriques (ssptGmDre 1974) ?6 Michel PREVOT : Convexité (mars 1975) N°7 Claudr RONSARD : Contribution z une �h�o,:°�;: des espaces économiques imprécis (avril 1975) ^l°8 �IOGT -Analyse factoriclle en composantes principales d'un c«ï roc tir; de dimension-n (juin 1975) , N°9 ,)acques THISSE et Jacky PERREUR : Relation hctwecn the Point of "':,ximum Profit and the Point of Minimum Total Transportation Cost : A Rcstatemcnt (�ui�l�t 1975) . N"10 Bernard FUSTIER : L'attraction des points de vente dans des espaces précis et imprécis (juillet 1975) N°ll R29is DELOCHE : Théorie des sous-ensembles flous et classification 2n analyse économique spatiale (juill,:t 1975) N°12 Gérard LASSIBILLE et Catherine PARRr.�1 : Analyse multicritère dans un contexte imprécis (juillet 1975) ?13 Cleude PONSARD : On the Axiomatization of Fuzzy Subsets Theory (ju1yl975) N°14 miches PREVOT : Probability Calculatic:n and Fuzzy Suhsets Theory (august 1975) U°15 Clause RONSARD : Hi érêl rch Í c ctes s pl ac.-s centrales et graphes 5-flous (avril 1976) -N° 16 Jean-Pierre auras et Gérard JURU :Introduction a la théorie des espaces multiflous (avril 1976) ? 17 Rr);al1d LANTNER. Bernard PETITJEAN et '�ari e-C1 aude PICHERY : Jeu de simulation du circuit, économique moût 1975) HO 18 Claude PONSARD : Esquisse de simulation d'une économie r6"ionale : l'apport de la théorie des systèmes flous (s�pt�r.tr2 1976) N° 19 �1arie-Claude PICHERY : Les systèmes complets de fonctions do d^m4r,a� (avril 1 1977) n° Gérard LASSIBILLE et plain rlINGAT : L'cst-ination de modèles à v;.•wi::w. dépendants dichotomique -La sélection univ::rsi1:{'l1\ 2t la réussite en première année d'économie (avril 1977) N° Claude PONS,'1RD : La régicn en analyse spatiale (irai 1977) N° Dan RALESCU : Abstract Modcls for Sy�t2m� Identification ( juin 197 7-iV° 23 J.r1ARCI1AL et F.POULON : multiplicateur, graphes et chaines de flarkov (décembre 1977). N° 24 Pietra BALESTRA : Determinant and inverse of a sum of matrices with applications in economics and statistics (avril 1978) N° 25 Bernard FUSTIER : Etude empirique sur la notion de région homogène(avril 1978)

12-

  The structure (�,1t) is called an imprecise individual preference-indifference structure. It is broken down into two structures denoted respectively ( K, � ) and (�.N) according to whetherthe relation ais a strict fuzzy preference relation or a fuzzy indifference relation.

( 1322 )

 1322 Replacing operator Max and operator Min (which are equivalent boolean operators) by boolean addition and boolean multiplication we obtain the same formulation in the classical theory. 133 -Finally, from 123, we have the property of completness. 14 -Consider a fuzzy individual preference structure written (�,�). It does not have the properties which guarantee the consistency of the consumer's preferences. The reflexivity and Max-Min transitivity properties do not suffice to ensure that indifference is congruent to the right (resp.to the left) with strict and strong preference, contrary to the result obtained when using a theorem from the classical theory {3,page 36j (see an example,Appendix 41).

  1411 -Axiom (1): Strong preferences for the indifferent complexes 9k and ik relative to the indifferent complexes lk and mk are at least equal to the level of indifference between the complexes 9k and ik Hence: 1412 -Axiom (2): The weak preferences for the indifferent: complexes lk and mk relative to the indifferent complexes 9k and ik are strictly equal to the level of indifference between the complexes lk and can verify immediately that these conditions satisfy the property of Max-Min transitivity since by construction �61(gk,lk) is greater or equal to the maximum for Jk of the smallest value of the preference levels s between, on the one hand, the complexes gk and ik and, on the other hand, the complexes �k and lk. 143 -It is easy to prove that the structure (K,.v) is a fuzzy equivalence structure if and only if it satisfies axioms 1 and 2. Indeed, a pair of complexes 9k and jk belong to a same similarity subrelation if and only if /^(9k,Jk) = g/^(J'k,9k) = J* . It follows from this that all similarity subrelations which have symetrical pairs of elements and the same membership functions M are identical and form a maximal similarity subrelation. From axioms (1) and (2), the pairs 9k and lk are such that �81(gk,lk) 1 ��(lk,gk); therefore their elements do not belong to a same maximal similarity subrelation: the maximal similarity subrelations are disjoint and form similarity classes.

  If g is called the canon ic projection: the ocomposed function (in the usual meaning) u = f o g is written:

Table of

 of

	Contents.
	0 -INTRODUCTION.
	-Introduction
	-Choice in imprecise preference structures 01 -Faced with a set of needs of different intensities and which he perceives
	-The continuity of preferences and imprecise utility. more or less indistinctly, a consumer is not normally capable of selecting
	-Conclusion among the elements belonging to his set of possible consumptions, those he
	-Appendix prefers or is indifferent to and those from which he is likely to derive
	-Bibliography. utility.
	Moreover the goods and services are attainable to different degrees
	(available in supply space) and his knowledge is perfect only in border-line
	cases with the result that his world is generally imprecise. Even someone
	with an exceptional gift for discrimination is not capable of formulating
	for any pair of goods, his preference or indifference according to binary
	logic.
	02 -The purpose of this paper is to remove the usual assumption (which it
	must be added is essential to the classical theory of consumer behaviour)
	that the consumer is capable of discriminating perfectly between different
	goods. It is well known that his behaviour, like any human behaviour,is by
	nature imprecise.

222 -Finally, a continuous fuzzy utility is defined as any continuous homomorphism from (J�, £ , % ) to (f0,l� ,� "j�%) where ¥i the natural l topology of �0,11 [_'l�,. A fuzzy utility is continuous if and only if a utility for (JK»£) which is continuous application from (K,�;) to A similar result to that of the classical theory is found: a countable preference structure (K, � ) for which a given utility exists, this utility is a continuous application from (K,15) to ( [0,'] ,j(�')

where*^ and S are the respective intrinsic topologies.

-APPENDIX

41 -A very simple example is given to il lustra te proposition (14).

Consider the following structure (K,� ) the The Max-Min transitivity property is verified. BuÏYfuzzy indifference is not congruent on the right relative t vstrong preference. For example one has:

Similarly, fuzzy indifference is not congruent on the left relative toYstrong preference. We have for example: Remark.

It is recalled that the above numbers are not measures of utility.

A non numerical membership set suffices (see ( 11)).


Robert Myerlas

KEY PARAMETERS OF PULSED LASER DEPOSITION FOR SOLID ELECTROLYTE THIN FILM GROWTH

Keywords: solid electrolyte, thin films, memory device, pulsed laser deposition (PLD)

The pulsed laser deposition (PLD) technique has been a common method to grow thin films such as solid electrolyte (SE). The effects of substrate temperature and laser fluence on the thin film properties and the device performance are analyzed. For the first time, a quantitative analytical model dealing with the energy conversion in the process when laser hits the target is presented, which provides a solution that is crucial in correlating the formation of high quality and uniform thin films to the experimental design. The migration speed of the ablated particles, which determines the quality of the deposited films, is found to be directly related with the laser fluence. Specifically, a threshold fluence is required to generate high purity single form thin film. This model provides the opportunities to improve experimental design and quality control.

I. INTRODUCTION

Nowadays the rapidly increasing demands in electronic devices have promoted the design and development of specialty materials in a variety of forms and structures [START_REF] Meredith | Electronic and optoelectronic materials and devices inspired by nature[END_REF][START_REF] Lipomi | Stretchable, elastic materials and devices for solar energy conversion[END_REF][START_REF] Cai | 2.4GHZ Class AB Power Amplifier for Wireless Medical Sensor Network[END_REF][START_REF] Cai | Class E Power Amplifier for Wireless Medical Sensor Network[END_REF][START_REF] Cai | 2.4 GHz Class F Power Amplifier for Healthcare Application[END_REF][START_REF] Cao | Nano-modification for high performance cement composites with cellulose nanocrystals and carbon nanotubes[END_REF]. For example, the substrate and thin film materials are experiencing an evident trend into biodegradable and environmental friendly paradigms, such as chitosan and cellulose nanomaterials [START_REF] Irimia-Vladu | Green and biodegradable electronics[END_REF]. These biodegradable have found extensive practical functionalities in a variety of areas, e.g., medical care [START_REF] Rees | 3D Bioprinting of Carboxymethylated-Periodate Oxidized Nanocellulose Constructs for Wound Dressing Applications[END_REF], gas separation [START_REF] Matsumoto | Ultraselective Gas Separation by Nanoporous Metal-Organic Frameworks Embedded in Gas-Barrier Nanocellulose Films[END_REF], and construction materials [START_REF] Cao | The relationship between cellulose nanocrystal dispersion and strength[END_REF][START_REF] Cao | The influence of cellulose nanocrystal additions on the performance of cement paste[END_REF][START_REF] Cao | The influence of cellulose nanocrystals on the microstructure of cement paste[END_REF][START_REF] Cao | Performance-enhanced cementitious materials by cellulose nanocrystal additions[END_REF], while their use in electronics is still in a preliminary phase. Satisfying the requirements for sustainability, the performance of the devices is achieved via a few approaches, one of which is to fine-tune the performance of the individual components as well as the devices via employing nanomaterials with extremely small scale, inspired by the concept of controlling materials properties from the most fundamental level [START_REF] Sobolev | The influence of mechanical activation by vibro-milling on the early-age hydration and strength development of cement[END_REF][START_REF] Sun | Jet mill grinding of portland cement, limestone, and fly ash: Impact on particle size, hydration rate, and strength[END_REF]. Thin film solid electrolyte (SE) composed of nanoparticles is a class of materials which have considerably high ionic conductivity and variable electron conductivity. SE thin films have broad applications, such as solid state batteries [START_REF] Tarascon | Issues and challenges facing rechargeable lithium batteries[END_REF], sensors [START_REF] Maskell | Progress in the development of zirconia gas sensors[END_REF], fuel cells [START_REF] Haile | Fuel cell materials and components[END_REF], and memory cells [START_REF] Sakamoto | Nanometer-scale switches using copper sulfide[END_REF]. The major advantages of SE thin films are the large resistive ratio between high and low resistivity states and the high ionic conductivity. When making memory cells, the two states are assigned as "1" and "0" in a memory device. The resistivity of the two states should be significantly different (by orders of magnitude) to design a robust memory. The ionic conductivity determines the efficiency of work of the memory device. An SE memory device consists of a large amount of memory cells. Figure 1 shows a typical structure of the cell. In this example, Pt/TiO2/SiO2/Si(1 1 1) substrate is covered with a SiO2 (a few hundred nm thick) layer using plasma-enhanced chemical vapor deposition. Afterwards focused ion beam was performed to drill a hole to expose the bottom electrode. Then deposition of Cu2S films is completed with pulsed laser deposition (PLD). The final step is to deposit Ag electrode using a DC sputtering technique. Figure 2 shows a working principle of an SE memory cell. When applying a positive voltage across this structure (Pt/TiO2/SiO2/Si(111) substrate as the anode while Cu as the cathode), due to the redox reaction, Cu 2+ will be deposited onto the substrate as Cu atoms. Then the Cu atoms form a filament across this structure, from bottom to top, depicted as the red filament connecting the two electrodes in the figure. Because the conductivity of SE is quite low, before the Cu filament reaches the cathode, this memory cell is at the high-resistive state, i.e., "off" state for a memory. When the Cu filament touches the top Cu electrode, as a result of the high conductivity of Cu, the cell suddenly changes into a low-resistive state, namely the "on" state. On the other hand, when applying a negative voltage across it, the filament is resolved into the SE immediately, and hence makes the cell turn into "off" state again. As a result, one can simply control the resistivity of the memory cell by changing the direction of voltage on it. After assigning the "off" state as "1", "on" state as "0", a memory cell is established.

Fig. 2. Cu atoms grow from the bottom to top as a filament when applying a positive voltage

The property of solid electrolyte is the most critical parameter in determining the performance of the cell as well as the memory device. In this paper, PLD, the state-of-the-art technique to grow SE thin films is evaluated in depth. A major advantage of PLD is the capability to prepare stoichiometric multicomponent films [START_REF] Ohring | Materials Science of Thin Films[END_REF], which largely determines the robustness and reliability of a memory device. In addition, PLD is able to deposit many different types of materials, regardless of the physical and chemical properties. Although the most intriguing advantage of PLD is to maintain the stoichiometric composition. The experimental parameters, however, demand a careful design to generate exact targeted stoichiometry (as the source material) for the thin films. In this present paper, two significant parameters, substrate temperature and laser fluence for PLD are discussed; then a quantitative analytical model is proposed to correlate the laser fluence with particle migration speed (time) to present the importance in creating high quality stoichiometric thin films.

II. SUBSTRATE TEMPERATURE

The temperature during the deposition is an important parameter. It was reported [START_REF] Yang | Characterization of RbAg4I5 films prepared by pulsed laser deposition[END_REF] that a proper temperature is significant for a pure composition of RbAg4I5 thin films. At room temperature, the main XRD peaks represent RbAg4I5, with some impurity of AgI. The crystallization with RbAg4I5 phase happens when the substrate temperature reaches 80 °C. While the films are deposited at 120°C, some diffraction peaks from AgI with a large intensity appear. It shows that the range of the substrate temperature for depositing the pure RbAg4I5 films is not very wide (around 80 °C).

At room temperature, the sizes of crystal particles are between 1.2 μm and 0.7 μm and the surface is rough. As the temperature reaches 80 , the surface becomes obviously uniform and flat, and the average size of the particles decreases to 0.7 μm. For films deposited at 120 , the surface of the films is even flatter. The change in the surface morphology with the substrate temperature may be resulted from the large mobility of ions. For SE, ionic mobility is high at first. With increased temperature, the ions are excited with more kinetic energy and more mobile, and hence the surface is less coarse. In this report, with increased substrate temperature, comes pure RbAg4I5 films at 80 , however when temperature reaches 120 , the undesired AgI phase is formed again. As a result, the proper temperature frame targeting RbAg4I5 is a narrow window around 80

In addition, when temperature increases, the surface becomes smoother, which may be due to the ionic mobility. This is also a significant parameter for memory device, because with large ionic mobility in solid electrolyte, the metal filament can grow and resolve very fast, and hence shorten the reaction time of the device.

III. LASER FLUENCE DEPENDENCE

Light of intensity is also an important parameter of PLD. It denotes how much energy the laser hits the target in a designated area and time (J/(m 2 s)). A similar parameter referred as laser fluence, which has a different unit of J/m 2 , indicating the energy focused on the area within each pulse. It was reported [START_REF] Yang | Characterization of RbAg4I5 films prepared by pulsed laser deposition[END_REF] that larger intensity tends to form the crystallization of a desired form of RbAg4I5, and increase the resistance of the "off" state, and lower intensity results in the undesired AgI phase in the films and reduces the resistance of the "off" state. When the energy fluence is less than 1.7 J cm -2 , there are some peaks of impurity phase from AgI, such as AgI(1 0 1). When the fluence surpasses 1.7 J cm -2 pure RbAg4I5 films are obtained. This denotes a higher fluence favors the formation of the pure RbAg4I5 phase. The possible reason is that the higher fluence provides materials ejected from the target with sufficient kinetic energy, which shortens the migration of species to the substrate and minimize the possibility of dissociation and form changes. Because of the significance of the migration speed resulted determined by the laser influence, here an analytical approach is provided to evaluate the relationship between fluence and kinetic energy of particles. Given that the beam energy is converted into thermal, chemical, and mechanical energy [START_REF] Ohring | Materials Science of Thin Films[END_REF] (shown in Fig. 3), resulting electronic excitation of target atoms, ablation and some other processes, the following model is presented [START_REF] Meredith | Electronic and optoelectronic materials and devices inspired by nature[END_REF] In this model, Npulse is the total number of pulse, s the crosssection area of the laser, m the mass of a single particle, Ethermal proportional to f, while Echemical proportional to Nparticle -the total number of particles ablated and deboned from the target, which relates with the number of particles deposited onto the substrate, i.e., N'particle:

(

Where θ0 is the angle that covers the entire area of the deposited thin film (see Fig. 3), and 8<n<12 [START_REF] Ohring | Materials Science of Thin Films[END_REF]. N'particle can be readily estimated as:

(3) Where S is the deposited area, t the thickness, V the volume of a single particle, where t and V can be found out from the SEM images. As a summary, Equation (1) can be rewritten as the following formula relating fluence and migration speed:

(4) In the work by Yang et al. [START_REF] Yang | Characterization of RbAg4I5 films prepared by pulsed laser deposition[END_REF], f=1.7 J/cm 2 , the thin film deposited at 80 °C has particles of average size of about 0.7 μm. While it may be challenging to the measure the density of single particles, one can assume that the density of particle is about that of the target, given that the target is finally pressed into a dense pellet under high pressure. In the computation the density of a single particle is assumed as 10 g/cm 3 . In this work [START_REF] Yang | Characterization of RbAg4I5 films prepared by pulsed laser deposition[END_REF], the thickness of film is shown as 900 nm. With the area of total thin film area, and average size of particles, one can obtain the total number of particles in the thin film with the known size of a single particle. The last step is to calculate the number of particles each pulse deposits on the substrate with the total deposition time, repetition rate, and the time of each laser pulse. With Equation ( 4), the relationship between laser fluence and particle migration speed is given in Fig. 4. V. CONCLUSIONS

Fig. 1 .

 1 Fig. 1. A typical structure of memory device cell

Fig. 3 .

 3 Fig. 3. Energy conversion of incident laser into different forms

Fig. 4 .

 4 Fig. 4. Relationship between laser fluence and particle migration speed

Fig. 5 .

 5 Fig. 5. Relationship between laser fluence and particle migration time

Pulsed laser deposition (PLD) shows excellent capability of growing thin films such as solid electrolyte, which shows huge potential in non-volatile memory devices. Laser fluence and substrate temperature are two critical parameters during the deposition process, influencing the purity and stoichiometry of the resulting films. A reasonable fluence value is needed to excite a particle migration speed to ensure the integrity of the targeted materials. This work examined the two parameters and established a universal analytical model that relates the migration speed (time) and laser fluence. It shows that the migration speed can be quantitatively determined and the minimum speed for different material systems can be designated as such. This approach will be useful in process design and quality control, as well as theoretical analysis of thin film deposition via PLD when the thin film composition for a uniform stoichiometry is a critical requirement. In addition, the threshold migration time is characteristic of the time needed for a phase/form change, which helps enhance the understanding in the molecular dynamics in the fast-evolving process of PLD.

AUTHORS BIOGRAPHY

Robert Myerlas was born in Miami, USA, in 1993. He received the Bachelor in Science degree from the Florida Institute of Technology, Melbourne Florida, in 2015. He is currently pursuing the Master's degree with the Department of Education, University of Montana Western, Dillon Montana. His research interests include electrical engineering education, thin film characterization, and high performance flexible electronics.


If the corresponding line contains 1 at least once,let yj'" yl be the values of y such that �s(,(X, y) = 1.

We choose one of the values }l;j (x, y.) )^^(x' y,) equal to 1 and all the other values equal to orles than *( .

We cross out the correspondin9 column (or colums) and we use the reduced table to iterate the process taking the values of J.'�(x, zl) increasing by.

In the various iterations three cases may occur i ) we reach a line madeYf zeros, the problem is impossible.

ii ) all the columns are crossed out before reaching the highest value of � � , the probleme is impossible Go on with the algorithm until the bottom of the table.

The solutions are obtained as expected but it thould be noted that,if a alune of � � (y, zl) belongs to a marked column its value must at least be equal to Y,�P-1 (x, y) = iu°P (x, z1).

Justification of the algorithm

The only case to study is line

We have seen that the crosse�vfi'ümbers play no part.

As �he�� ' s i ncrease

But we must take into account

We must choose �3� (x, Y2) plays no part and can be replaced by zero.

By assumption y£ (x, YI) = ��2 (x, zl) o t The crossed�olumns play no part and for all the others values of y

To fulfill the condition it suffices that The solutions are (0,6, 0,5, �0, l], 0,4) They are belong to (0,6, 0,5, 1 , 0,4)

The solutions are 11 0 -1] , foil, 0,6, 0,6 j They are belong to � 1, 1, 0,6, 0,6 j

The solutions are 0,6 � 0 -13 , [ 0, 5, 1] , 0,4 3 They belong to

� 0,6, 1, 1, 0,4] 7
3) The values �h� (x, zl) are not all distinct

We begin thé algorithm as in (1).

We come to two equal values of /�g(x, zl) = J�*� (x', zi)

Two cases are to be distinguished .

--- r � _ .... - 1
If Y1 n Y2 = 0 we go on with thecalculation without taking the equality of the values of � � into account.

We take ^^ (y,, zl) = fqj (x, zl) and we take higher values.

Justification of the algorithm :

If YI n Y2 = 0 to détermine -J� everything happer as if the }!�'s are different and we can apply the general method.

/*3 (y,, zi) -^^ (x, zl) satisfies both lines and we can proceed with the following iteration. Examples :

The solution is ( 0,4, 0,4, 0,6)

  that the problem had no solution If the line contains value one, at least once, we can classify values of P g) and add a column giving the values of )-k ce (x, zl) transform the table by replacing conserving the value if �� (x, y) = )J(;'(x, zl) We use the algorithm of 1 until we reach a line with values of ^�^ different from 0 and 1. Two cases are to be distinguished i) In this line there are now crossedY15 ; replace f'5{(XI�)= y� (x, zl) by zero and go on with the algorithm. ii) This line contains only non crossed)1êros cross out the line and mark the corresponding column.

  the above algorithm enables us to determine the solution of fuzzy equations ; it can of course be applied to fonctional relations ; unfortunately it does not enableVto obtain necessary and sufficient conditions of solutions.

  Sanchez formulated conditions and theoretical methods to resolve fuzzy relations. The purpose of this study is to give an algorithm which would actually enable us to determine the functions of appartenance of unknown relations.

	TRAVAUX DEJA PUBLIES
	N"1 Michel PREVOT: Théorème du point fixe. Une étude topologique générale(juin	1974)
	N°2 Daniel LEBLANC; L'introduction	des consommations	intermédiaires	dans le
	�	modèle de LEFEBER (juin 1974) and � being known, we have to détermine �'	such that
	N°3 Colette BOUNON: Spatial Equilibrium of the Sector in Quasi-Perfect	Compétition
	(september 1974) Let us notice that, given the operation symmetry of o min, and since %£, e
	N°4 Claude PONSARD: L'imprécision N°21 Claude PONSARD: La région en analyse spatiale (mai 1977) et son traitemebt en analyse économique are known,�can be determined in the same way.
	(septembre 1974) ?5 Claude PONSARD: Economie urbaine et espaces métriques (septembre 1974) N°22 Dan RALESCU: Abstract Models for Systems Identification (juin 1977) N°23 Jean MARCHAL et François POULON: Multiplicateur, graphes et chaines de It suffices to formxl. the order columns of are (y, zl), J*-determined in a similar way
	N°6 Michel PREVOT: Convexité (mars 1975) ' Markov (décembre 1977)
	N°7 Claude PONSARD: Contribution N°24 Pietro BALESTRA: Déterminant and Inverse of a Sum of Matrices with Applica-à une théorie des espaces économiques imprécis
	-	(avril 1975) tions in Economics and Statistics (avril 1978) write the values of J^S^ in a table and put
	N°8 Aimé VOGT: Analyse factorielle en composantes principales d'un caractère N°25 Bernard FUSTIER: Etude empirique sur la notion de région homogène (avril 1978) the values of J*-^ (x, zl) into columns
	de dimension -n (juin 1975) N°26 Claude PONSARD: On the Imprecision of Consumer's Spatial Preferences -transform the table by replacing
	N°29 ALGORITHME POUR LA RESOLUTION DES SYSTEMES FLOUS N°9 Jacques THISSE et Jacky PERREUR: Relation between the Point of Maximum Profit and the Point of Minimum Total Transportation Cost: A Restatement (avril 1978) N°27 Roland LANTNER: L'apport de la théorie des graphes aux représentations /6L(x, y) by zero if )Xq (x, zl) (x, y) ��lze de }.\Sl(x, y) by one if^g^ (x, y) ��À�e (x, zl) (juillet 1975) N°10 Bernard FUSTIER: L'attraction des points de vente dans des espaces précis l'espace économique (avril 1978) N°28 Emmanuel JOLLES: La théorie des sous-ensembles flous au service de la -choose the smallest value of �,» ce (x, zl) let }'exo' zl) = 0{
	1 décision: deux exemples d'application.(rnai 1978) et imprécis (juillet 1975) If the corresponding line is s that o�PP.5�(x. , = 0 V y) y1the
	Michel PREVOT N°11 Régis DELOCHE: Théorie des sous-ensembles problem is impossible and the calculation stops. flous et classification	en
		Mai 1978 analyse économique spatiale (juillet 1975)
	N°12 Gérard LASSIBILLE et Catherine PARRON: Analyse multicritère	dans un contexte
		imprécis (juillet 1975)
	N°13 Claude PONSARD: On the Axiomatisation	of Fuzzy Subsets Theory (july 1975)
	N°14 Michel PREVOT: Probability Calculation and Fuzzy Subsets Theory (August 1975)
	Le but de cette Collection est de diffuser rapidement une N°15 Claude PONSARD: Hiérarchie des places centrales et graphes flous
	.	première version de travaux afin de provoquer des (avril 1976)
	discussions scientifiques. N°16 Jean-Pierre AURAY et Gérard DURU: Introduction à la théorie des espaces Les lecteurs désirant entrer en
		rapport avec un auteur sont priés d'écrire à l'adresse multiflous (avril 1976)
	suivante: INSTITUT DE MATHEMATIQUES fd°17 Roland LANTNER, Bernard PETITJEAN, Marie-Claude ECONOMIQUES PICHERY: Jeu de simulation
		4 -Boulevard Gabriel -21000 Dijon " France du circuit économique (Août 1976)
	N°18 Claude PONSARD: Esquisse de simulation d'une économie régionale: l'apport
		Cette étude fera l'objet d'une communication de la théorie des systèmes flous (septembre 1976) au Colloque
	International sur la Théorie et les Applications N°19 Marie-Claude	des Sous-
		Ensembles Flous, Marseille, 20-22 septembre 1978.

PICHERY: Les systèmes complets de fonctions de demande (avril 1977 N°20 Gérard LASSIBILLE et Alain MINGAT: L'estiamtion de modèles à variable dépendante dichotomique. La sélection universitaire et la réussite en première année d'économie

(avril 1977)

Let us study the last three cases If ft (y,, zl) is not defined, the restriction imposed on its variation implies that:

and we can remove the line.

Examples :

1)

The solutions are They belong to


To say that an economic space is precise means:

(1) that this space has, or else, has not given constituent characteristics and (2) that the agents located there prefer, or else, do not prefer one possible action to another. (1.5.4.2.5.):

(V e S, V �s E �8, � r , t G S , Af Il At = 0) 00 =�- U A . E
The pair r I I:,(',�7 de fi nos a measurable set.

A positive measure on a measurable set is a mapping, m, with domain the a -a l g eh 1'3 �. anal d \..: i the 0 -dom a i n IR +, such that:

(1 .3.4.2.4.) : Axiom 1 : m(0) = 0 or

�1.3.4.2.5.) : Axiom 2 : m( Û.As) - = E m(Ag) s=1 -s s=1 -s
The triplet (E,�,m) defines a measure set on E.

We note especially that a measure has the additive property given by ( 1 ( is the interior of its complement (pseudo-complement).

1.4.1.1.1.) : 0 E � ( E ) (1.4.1.1.2.) : E E � ( E ) II .4. 1 . 1 .3.1 : ['V r, t � S, Ar G % (E) and At e K (E.) 1 ==?�:� (Ar n At) E Z(E) (1 .4. 1 . 1 . 4.) : CAs E � (E.), s E S,
We have:

ext(A) = Â -"Â"1 . 1.4.1.6.
The fuzzy frontier of a fuzzy subset A is a fuzzy subset X E e 'D 3 (E) , such that: Generally, an economic agent can be more or less indifferent between two possible actions. Thus, the structure (H,rv) is equiped with the property of fuzzy symmetry.

a o 0 0 0 A = A U ext(A) = A U A = A U A 1.4.
We have:

The property of fuzzy symmetry implies that of fuzzy reflexivity when we have 1 = k and q = p. We have:

The degree of indifference can be zero. Vo1.46, 1974, 316-328.

  true value of the surface T, of area A, bounded by the axes, y = 0, x = a, x = b and the curve such , the smallest true value of the points on the frontier of the given surface, which is the curve y -= g(x).

A probability is a particular measure.

We thus verify the generality of the "valuation theory of preference".. Vol.1, n°1, 1978, 3-28.


Prof Author Louis De Mesnard
email: louis.de-mesnard@u-bourgogne.fr

Keywords: JEL classification. C67, D46, D57. KEYWORDS. Input-Output, Demand, Supply, Ghosh

The Ghosh model assumes that, in an input-output framework, each commodity is sold to each sector in fixed proportions. This model is strongly criticized because it seems implausible in the traditional input-output field. To answer to these critics, Dietzenbacher stresses that it can be reinterpreted as a price model: the Leontief price model is equivalent to the Ghosh model when this one is interpreted as a price model. This paper shows that the interpretation of the Ghosh model as a price model cannot be accepted because Dietzenbacher makes a strong assumption, dichotomy, while the Ghosh model does not determine prices.

I. Introduction

The Ghosh model assumes that, in an input-output framework, each commodity is sold to each sector in fixed proportions. This model is strongly criticized because in the traditional input-output field it seems implausible for many '.To answer to those critics, [START_REF] Dietzenbacher | In vindication of the Ghosh model: a reinterpretation as a price model[END_REF] stresses that it can be reinterpreted as a price model. Considering the demand-driven (Leontief) and supply-driven (Ghosh) models, Dietzenbacher demonstrates that the Leontief price model --the Leontief model solved by columns, i.e., the dual of the traditional Leontief model solved by rows -is equivalent to the Ghosh model when this one is interpreted as a price model. This is seductive because it could vindicate strongly the Ghosh model, even if this new interpretation removes its quality of quantity model. However, I will show that the interpretation of the Ghosh model as a price model cannot be accepted, by returning first to the foundations of the Leontief and Ghosh models.

II. The foundations of the linear production models

A. The hypothesis

One consider n sectors, producing n commodities, and a set of final consumers. Each sector produces one and only one commodity and each commodity is produced by one and only one sector2. The physical quantity bought by sector j to sector i when j produces the commodity j 1 See a discussion in [START_REF] Miller | Stability of Supply Coefficients and Consistency of Supply-Driven and Demand-Driven Input-Output Models: a Comment[END_REF] [START_REF] Oosterhaven | On the Plausibility of the Supply-Driven Input-Output Model[END_REF], 1989, 1996) (Rose and Allison, 1989).

2

To stay in the Dietzenbacher's framework, here I focus only of the more simple linear is denoted x,y. By hypothesis x,y > 0. The model is closed: for each commodity i, the total that is sold is equal to the total that is produced: it is denoted x,-, with x, = + //, where 7, > 0 denotes the final demand. Note that if homogeneity of rows of matrix X is granted, sums by columns cannot be done. Commodities i have a price /?, . Initially, the value of each commodity i is equal to x, pt and each sector j has in hand xj of money. The model is monetary closed: the agents have the same quantity of money before and after the exchange, i.e., the model is at equilibrium sector by sector, that is x, p, = x, for all i. So, after exchange, the value of each product i is disposed of completely:

(1) HiXij +fi = x, <=> xy + /, = x , for all i j where fi = f j pt is the final demand in value, and each sector j spends completely the money that he has in hand:

(2) X xy + Vj = Xj <=> X,xy pi + VjW = Xj pj for all j i where v, = vj w is the added-value measured in value, while v7 is the amount of labor used by sector j and w is the wage rate (the profits are assimilated to the remuneration of the manager). Then, two main hypotheses of behavior can be done: either the demand drives the model (Leontief) either the supply does it (Ghosh). models, where one product is produced by one and only one sector, and conversely, leaving aside the more complicated cases, as the "von Neumann model".

When the demand-driven model is considered, one assumes that each sector buys each commodity in fixed proportions, but there are two possibilities: coefficients can be defined in physical terms or in value terms.

X "

If coefficients are defines in physical terms, it is assumed that = -J-for all i and j are stable

x .i

3. In matrix terms, this writes: Â = X (x)_1 .The economy must be at equilibrium by row and by columns. By rows (I call this the primal), the accounting identity (1) becomes Xy ay Xj + f i = x ,, that is:

(3) Â x + f = x< = > ^I-Â jx = ?
This Cramer system has a non trivial solution only if the determinant 11 -Â | is not equal to zero; this solution is x = ^1 -f . By columns (I call this the dual), the accounting identity

(2) becomes X , â y x ,p, + vj\v = xjpj or X,-âijP>+!j w=Pj, where lj = =-are the input Xj coefficients of labor in quantity and w is the wage rate. This can be noted in matrix terms:

(4) p/ A + w I/ = p/

B. The demand-driven model

Note that nothing prevents the coefficients to be higher than 1, their magnitude depends on the chosen scale but the determinant is independent to the scale, and the result also, after the appropriate conversion of scale.

When coefficients are defined in value, are assumed to be stable the coefficients: a y -^ = ay By rows, (1) becomes: X ay */ +f i = x, for all i, that is:

J xj X j P j J P j j (5) Ax + f=x<=>(I-A)x = f
This will have a solution x = (I -A)-1 f if |I -A| * 0. By columns, (2) becomes X,ay + lj= 1 where /,■ = ^, that is:

xj (6) s' A + l/ = s/ <=>s/ (I-A ) = l/
where s; = ^ 1 ... 1 j . This is not a Cramer system of equation to be solved in s, but only an accounting identity that must be respected: if |I -A| = 0 then s' = 0, what is impossible, so ( 6

C. The supply-driven model

Each commodity is assumed to be sold in fixed proportions to each sector. Again there are -X" two possibilities. In physical terms, by = is assumed to be stable for all i,j; in matrix this

Xi

writes B = (x)-1 X. By rows, (1) becomes X by + dj = 1 for all i where di = =L, that is:

J * ' • 4
The demonstration of the condition of productivity x > A x can be found in [START_REF] Gale | The theory of linear economic models[END_REF], p. 296-...).

(7)

B s + d = s o ( l -®) s = d
Again, this is an identity, not a system of equations to be solved, what imposes | l -B | ^ 0 to be true. By columns, ( 2) is transformed into X by 3c, + v7-= xj for all i, that is:

i (8) x/ B + v/ = x/ <»x/ ( i -B ^v '
what has always a solution x' = f ' ^I - § j since | l -B | *0. In money terms, the stable

III. Discussion

Let's summarize. The primal (conventionally computed by rows) of the quantity Leontief model solves in outputs (measured in quantity) while its dual (conventionally computed by columns) solves in prices. The primal of the value Leontief model solves in outputs (measured in value) while the dual is an identity. The primal of all Ghosh models (in quantity as well as in price) is an identity while the dual solves in outputs (in quantity or in value) and is the dual of the primal of the Leontief value model (the same value added generates the same output). So, prices are undetermined in all Ghosh models and in the value Leontief model, while they are not in the quantity Leontief model compute by columns 5. To summarize, stricto sensu, you have true prices only in the dual of the Leontief quantity model. Now, let's discuss Dietzenbacher's argument. He uses the formula (Dietzenbacher, 1997, p. 633, Eq. ( 9)):

(9) ji/ = ic/ A0+ v / L o ^n ^v ' W o i l -A o ) " 1
to compute the price ratios n from the wage ratios v / = vi v©1 (assuming the dichotomy between prices and quantities, the variation of wages in money will generate only variations in prices), where Lo = vo Xq1 and where the subscript 0 refers to the data before wage raise and the subscript 1 to the data after wage raise. Dietzenbacher retrieves the fact that the table generated by the variation of the added value with the Ghosh value model is the same that the table generated by the same variation of the added value with the dual of the Leontief value model: this is true because, in value, one model is the dual of the other. However, formula ( 9) is presented as a price ratio equation 6, but this is not true because it only corresponds to the equation of the dual of the Leontief value model (formula 6).

To be convinced of this, just replace v by s, (wages are fixed) in formula ( 9): you obtain formula (6) where it is replaced by s (prices are fixed). This is why the result found by

Dietzenbacher is equal to a ratio between outputs (in value), that is %' = x{ xo: introducing a Note that the supply-driven model in quantity is not the dual of the demand-driven model in quantity, while they are in their value version.

6 "... the new price ratios (n') for the products are such that they satisfy the accounting equations." (Dietzenbacher, 1997, p.633).

change in the value added generates a change in the output, but it is only by a supplementary assumption that Dietzenbacher is able to say that it is a change in prices.

This supplementary hypothesis is dichotomy. True dichotomy comes naturally in the quantity Leontief model: solving the primal gives outputs in quantity but prices are fixed, while solving the dual provides prices but quantities are fixed. But in the other models since prices are never determined, dichotomy could be only a strong assumption. In other words, in the value Leontief model, you see that outputs (in value) vary and you say: as quantities are fixed by hypothesis, this is a variation of prices... Obviously, the reverse hypothesis could have been chosen: prices are fixed by hypothesis and variations o f quantities are found. As well as, dichotomy could have be abandoned for a mixed hypothesis: one percent of variation of value outputs is decomposed into a variation of X percent of quantities and of 1 -X percent of prices. All would have be possible... Note that the same reasoning holds in duality for formula:

(10) 0 = B 6 + D0 (p <=> 0 = ( I -B ) -1 D0 cp (Dietzenbacher, 1997, p. 639, Eq. (12)) where D0 =io' fo, <P = fo' {2 and the subscript 2 refers to a new value; it is found that 0 = X q 1 X 2 : 0 is again a ratio of outputs. Formula (10) corresponds to the primal of the Ghosh model, what could be verified by replacing <p by s (final demands are fixed): this gives formula (7) where 0 is replaced by s (outputs are fixed).

IV. Conclusion

Only the dual of the traditional Leontief quantity model (that is, solved by columns) can be considered as a price model, while the primal is the traditional Leontief model that computes the output of sectors. Prices remain always undetermined in the Leontief value model and all Ghosh models (in quantity and in value). As the Dietzenbacher's demonstration is based on the Leontief value model and Ghosh value model, prices cannot be determined, even under the form of prices indexes, except is a supplementary assumption about dichotomy is made: these price indexes introduced by Dietzenbacher depend on an hypothesis, the fixity of quantities in relation to prices in the value models. In other words, dichotomy is here a strong assumption, while dichotomy comes naturally without any supplementary hypotheses when the Leontief quantity model is solved by rows and generates outputs (in the primal) or when it is solved by columns and generates prices (in the dual). So, as is, the Ghosh model cannot be reinterpreted as a price model.

Have we to abandon the Ghosh model? Probably no, because of many reasons. First, technical coefficients are not much stable than allocation coefficients over time [START_REF] Bon | Comparative Stability Analysis of Demand-Side and Supply-side Input-Output Models[END_REF](Bon , 2000)), [START_REF] De | A biproportional filter to compare technical and allocation coefficient variations[END_REF], so the Ghosh hypothesis does not seem less reliable than the Leontief hypothesis. Second, even if the Ghosh quantity model (supply push) seems implausible in the traditional input-output field, one must consider that it is more accepted outside, for example in business economics 1.

1

Many firms compute allocation coefficients to know what is the usual share of their products that is sold to each of its main customers (these main customers are often firms themselves, but also administrations). When these shares divert, they are alerted: if the share of one customer increases, this could be satisfactory; but perhaps it is the sign that the price is too low for him; and conversely if the share decreases. So, it could be asserted that the firm "pushes" its supply toward the customer. So, in business economics, one can say that this

  ) requires | I -A | * 0 . Hence the demand-driven model in value (5) always has a non-trivial solution. And as | I -A | = | l -p A p -l | = p | l -A | p-1, if |I -A| = 0 then | l -A | =0 unless prices are null. Then the demand-driven model in quantity (3) will have a non-trivial solution always unless prices are null. Moreover, if f > 0 and as f = x -A x, then x > A x : matrix A is always productive4.

  as follows: b y = t ~-m -~ ' = ■ =2-= b y for all i,j. As they are equal to 1 Xi Pi Xi the coefficients in quantities, the model is exactly the same, either by rows, either by columns. Note that as A = x B x _l, one has I -A = I -x B x-1 =x (x-Ix -B ) x-1 = x ( I -B ) x-1 and then |I -A| = |I -B | * 0 , either in quantities or in prices.

Journal 62-3, pp. 750-759. ROSE, Adam and Tim ALLISON 1989. "On the Plausibility of the Supply-Driven Input-Output Model: Empirical Evidence on Joint Stability," Journal of Regional Science 29,[451][452][453][454][455][456][457][458]