Jean-François Ponsot

New Financial Architecture and Regional Monetary Integration in Latin America 1

Keywords: regional monetary arrangements, foreign exchange, currency union, Keynes's plan, dollar standard, payment systems JEL codes: E42, F31, F32, F33, F53

Reducing transaction costs and the need for international reserves is a primary objective to the establishment of regional payment agreements. Another objective, especially in the case of Latin America where the Ecuadorian promoters of the Bank of the South and the New Architecture are planning the implement of a regional clearing system, is to reduce member countries dependence on the US dollar as an international standard and reserve currency. To help improving the design of such agreements this paper refers to the plan Keynes designed for the Bretton Woods Conference. First, it observes that cases were made against this plan from which useful lessons may still be drawn. Second, it shows that Keynes defined a system for exchanging domestic currencies for each other that can be improved and help design currency unions in accordance with their promoters' objectives.

Introduction

Ecuadorian promoters of the Bank of the South and the New Financial Architecture in Latin America are proposing to implement a regional clearing system designed to allow a multilateral offsetting of the liabilities and assets generated in the reciprocal transactions of member countries (Government of Ecuador, 2008). In order to think up this system, they refer to previous experiments of international settlement systems and regional currencies / units of account in Latin America but also in other areasthe former Sovietic Ruble and the European Monetary System and the current CFA Franc Zone arrangements. They also mainly refer to the 'Proposals for an International Clearing Union" that Keynes presented at the Bretton Woods Conference (Keynes, 1942, pp. 33-40). In words that the British economist certainly would not have denied, they advocate the creation of a Central Clearing Union (Unidad Central de Compensacion -UCC) endowed with a currency unit they call "sucre" (Sistema Único de Compensación Regional de Pagos). This currency unit would be designed to keep member countries' accounts with the UCC. Each of them, operating through its central bank, would be credited or debited with amounts labelled in sucre in accordance with its transactions with other member countries.

The resulting assets and liabilities in sucres would be offset by the UCC, and then imbalances should be settled in domestic currencies through accounts held with the Bank of the South.

The reference to Keynes's proposals is no doubt relevant when one comes to consider schemes relating to international payments and new monetary and financial architecture (cf [START_REF] Davidson | What international payments scheme would Keynes have suggested for the twenty-first century?[END_REF][START_REF] Davidson | Financial Markets, Money and the Real World[END_REF][START_REF] Cartapanis | The Reconstruction of the International Financial Architecture: Keynes' Revenge[END_REF][START_REF] Gnos | Reforming the international monetary system; an assessment[END_REF][START_REF] Gnos | Reforming the international financial and monetary system: from Keynes to Davidson and Stiglitz[END_REF]. However, these proposals have been subject to objections that still deserve consideration. This paper aims at drawing lessons from the main cases made against Keynes's plan, in order to help improving the design of regional monetary agreements. It is also intended to show that as part of his proposals Keynes drafted a system for exchanging currencies that, if it had been completed, would have allowed him to deal more convincingly than he did with the main objections arisen. Section 2 will present, as a starting point, the regional payments and unit of accounts agreements in Latin America: the Reciprocal Payments and Credits Agreement of the Latin American Integration Association or ALADI, the 'Peso Andino" set up by the Latin American Reserve Fund (FLAR) and the Payment System on Local Currency agreement (SML) recently launched by Brazil and Argentina. Section 3 will examine the objections arisen against Keynes's plan. It will also emphasise their relevance to the sucre plan or to any similar plan. Section 4 will focus on the exchange scheme drafted by Keynes and on how to supplement it in order to get over the objections under examination. Section 5 will conclude.

Regional Payment Agreements in Latin America

Regional payment agreements are international systems set up to facilitate payments between countries. Why are such agreements needed? To answer this question, we can give an example inspired from [START_REF] Chang | Regional Monetary Arrangements for Developing Countries[END_REF]. If a resident of Bolivia wants to purchase goods from a resident of Colombia, the Bolivian has to find a way to pay for the goods with a currency that the Colombian is willing to accept. Such a currency can be the Colombian one, or some international currency such as the US dollar.

However, in both cases, the Bolivian is confronted with the cost of obtaining a currency different from his own in order to pay for the Colombian goods. While the cost may be small for an individual transaction, it may be large for the country, since in a modern economy there will typically be a need for large numbers of similar payments. Given such a situation, two countries may reduce "transaction costs" by having their central banks act as clearing houses for payments between them. Both central banks may agree to record and pay their own residents for eligible purchases from residents of the other country, thus extending credit to each other, and settle the accumulated net differences periodically, at the end of each quarter for example (cf. [START_REF] Rossi | A Common Currency for Middle Eastern and North African Countries? Lessons from the European Monetary Union[END_REF].

Currency flows, and the associated transaction costs, are actually saved on at least two ways. First, if the periodic settlement of the two accounts is done in a net basis, the amount of each settlement would only reflect the difference between accumulated sales and purchases during the settlement period. In contrast, if all transactions are paid individually, as would happen in a decentralized system, all sales and purchases would involve an international currency flow. Second, in order to pay for the bilateral trade between the two countries, the central bank of the country in deficit only needs to transfer a reserve currency at the end of the settlement period. In comparison, in a decentralized system each central bank would need to maintain enough international reserves to finance bilateral payments continuously during the period. Coupled with net settlement, this feature implies that each central bank can safely reduce its holdings of reserve assets. Reducing the need to hold international reserves was the primary motivation for the establishment of a number of regional payments systems, starting with the European Payments Union in the 1950s.

A first example of a similar system for a group of developing countries is the Reciprocal Payments and Credits Agreement of the Latin American Integration Association or ALADI. While the ALADI payments agreement was subscribed in 1982, ALADI itself dates back to the Montevideo Treaty of 1960, which set the agenda for creating a Latin American Free Trade area. This means, in particular, that the payment agreement was created to support the wider agenda for increased economic integration between Latin American economies, and consequently the payments agreement was assigned a somewhat subsidiary role. In terms of its functioning, the member central banks agreed to act as clearing houses for trade related payments of each member country vis-à-vis the others, settling the balances only every four months. Hence, the ALADI payment system worked essentially as we described above. Whether or not an eligible transaction goes through the ALADI payment system is voluntary and left to decide to the parties involved. A distinctive and noteworthy feature is that the ALADI system includes a system of guarantees for payments. In particular, there is a guarantee (the Reimbursement Guarantee) to exporters that the system will pay them the monies owed for their eligible exports, even if the importers eventually default on what they owe to the system. The system, after improving up to 1990, has decreased sharply since, before a new increase took place since 2003 (see figure 1)

Figure 1

Usage of ALADI's Payments Systems (Millions of US dollar)

Source: ALADI In a number of cases, a group of countries have set up a common pool of international reserves and, at the same time, granted each individual country the right to borrow from the pool under specified circumstances. Such circumstances have been mostly related to temporary balance of payments problems, although it is conceivable for an agreement of this kind to allow members to borrow for longer term purposes.

We can now give the example of the Latin American Reserve Fund (FLAR). FLAR evolved from the Andean Reserve Fund, which operated between 1978 and 1991. Its current members are Bolivia, Colombia, Ecuador, Peru, and Venezuela; in addition, Costa Rica's parliament is currently considering formal incorporation into FLAR. FLAR's capital is coming from members' contributions. A second source of FLAR's funds is given by demand deposits and time deposits of member central banks.

FLAR has used the resources thus obtained to grant loans to financial institutions of member countries in order to support trade and, in some circumstances, to assist member central banks experiencing a need for international liquidity. 0.00 2,000.00 4,000.00 6,000.00 8,000.00 10,000.00 12,000.00 14,000.00 16,000.00

While the preceding discussion suggests that FLAR may have played a stabilizing role, the question remains of whether FLAR can play such a role effectively given the size of its resources. And it seems that the answer, at least so far, has been negative. Indeed FLAR member central banks cannot produce convertible currencies on demand. Under such circumstances, it is not surprising that FLAR economists have argued that FLAR should aim to become a "Latin American Monetary Fund" (FLAR 2000).

The most important aspect of this new regional payment system is the creation of a common unit of account, the Andean Peso. Indeed, since 1983, Andean countries recognised the importance of a common unit of account used for regional commercial transactions. There are, nonetheless, important challenges. Indeed, for those who first proposed this initiative, the most challenging of hurdles was to determine how this new unit of account would be defined. At first, some suggested the use of SDR issues by the IMF. But it quickly became evident that the regional unit of account could not be based on a basket of local currencies, largely because of the inherent instability of these currencies. As such, the solution was to have this new regional unit of account be based on a basket of goods, that is the value of the regional staples.

In the end, the solution that was retained, on December 17, 1984, was the emission of regional assets denominated in Andean Pesos, which would distributed by those central banks that signed the Cartegena Accord (Government of Ecuador, 2008). The objective was to encourage regional trade and to accelerate regional integration. Only he FLAR had the responsibility of emitting this new regional unit of payment. At first, more than 80 million Andean Pesos were distributed to the central banks of regional member-countries.

Yet, we believe that as appealing as is the idea of the Andean Peso, it remains nevertheless a partial solution. Its major drawback is its dependence on the US dollar. First, the value of the Andean Peso is defined as on par with the US dollar.

Second, when regional central banks use the Andean Peso for the purpose of regional payments, they must reimburse the FLAR in US dollars, within 180 days. At its end, FLAR is committed to supply US dollars to any central bank wishing to convert its Andean Peso-denominated assets into US dollars. Therefore, the Andean Peso is a double-edge sword: on the one hand, we introduce a new unit of account in which to carry out regional trade, but on the other, the Andean Peso remains fundamentally tied to the US dollar.

More recently, Brazil and Argentina launched a new payment system of bilateral transaction with their local currencies, aimed at eliminating the U.S. dollar as an intermedium. The Payment System on Local Currency (SML) was agreed in October 2008 to end decades of mandated trade in dollars. Under this system, exporters and importers from both countries can settle their exchanges with Brazilian (real) and Argentine (peso) currencies. According to the Central Bank of Argentina, the trade between Brazil and Argentina is about 25 billion U.S. dollars per year. Although this new system seeks to gradually eliminate the dollar from the bilateral trade, the dollar will continue at the exchange. Indeed, the central banks of Brazil and Argentina will set the exchange rate between the reais and pesos with respect to the dollar. If the mechanism works out without incidents between Brazil and Argentina, it could be adopted by other countries of the Mercosur, like Paraguay and Uruguay. Once again, the dependence on the dollar as an international standard and reserve currency is preserved. The hegemonic status of the US dollar is still effective [START_REF] Ponsot | Dollarization and the Hegemonic Status of the US Dollar[END_REF].

Lessons from the cases made against Keynes's plan

In section 2, we have emphasised that a main asset to the regional agreements already implemented in Latin America and to the regional clearing system that Ecuadorian have in mind, is that it would help member-countries to economize on currency flows and the associated transactions costs. We have also noticed that these agreements do not break free from the dollar as an international standard and reserve currency. The case of the FLAR agreement just mentioned is quite symbolic of this situation. Could these agreements work another way? This is where the reference to Keynes's plan may prove useful. Actually, the relevance of the reference we can make to the latter plan is twofold. In the present section we propose to show that cases were made against it from which useful lessons may still be drawn. In section 4 below, we show that Keynes defined a system for exchanging domestic currencies for each other that can be improved and help get over the objections we are now going to examine.

To make our point, let us consider the stylized case of an international (regional) clearing union endowed with its own currency that will be an international common currency for member countries. This common currency would be designed to keep member countries' accounts with the clearing union. Beforehand, member countries would be endowed with drawing rights on the union. Of course, there is no chance that accounts would clear on a regular time basis. To settle their debts, member countries would then have to draw on their foreign currency reserves. And to help member countries to meet their commitments, promoters of clearing unions usually plan to build up a common pool of international reserves. The FLAR we have mentioned comprises this type of arrangements. In the Ecuadorian proposal, the Bank of the South would play a similar role. As for him, Keynes did not consider the building up of a common pool of reserves as a necessity (cf. Keynes 1943a). In his scheme, any country holding drawing rights or creditor accounts in the international currency (he named "bancor") would be simply entitled to ask any other member country to exchange bancors for the latter's currency. Anyhow, whether reserves assets are gathered together in a common pool or supplied on demand by creditor countries, the creation of drawing rights raises cases that commentators did not fail to make against Keynes's plan.

Let us consider a first case that US negotiators brought up at the Bretton Woods conference. The latter were namely concerned about the difficulty creditor countries, and specifically their country, would likely have in limiting and controlling their commitment to supply their currencies against the international currency. Allied to this, they also feared '[…] that the open-ended obligation to provide credit, involved in the Clearing Union, might have an inflationary effect in the United States, with adverse results on the dollar' (Horsefield, 1969, p. 49).

Keynes's response was that the US would not become a creditor to the Union, and therefore supply finance in dollars to deficit countries, except if she shows a trade surplus. He added that far from being a burden, becoming a creditor to the Union would be a facility: it would allow countries to keep balances in the international currency they earn when exporting goods and assets, until they spend them on foreign goods and assets (Keynes, 1943b, p. 276). He concluded that if the US were to accumulate excessive balances in bancors, it would mean that 'she has failed to solve her own problems' (p. 277).

Keynes did not convince the US negotiators, and the latter actually had good reasons for remaining sceptical. On the one hand, there is no doubt that in making drawing rights available to countries the clearing union grant credit to deficit countries and allow them to buy goods and assets from other member countries. To the deficit of some countries there necessarily corresponds a surplus in other countries. Then, contrary to Keynes's argument, it is not up to creditor countries to reduce their surplus. On the other hand, the funding of deficit countries would result in inflation in creditor countries and, moreover, in deflation in debtor countries. According to Keynes's proposal, amounts in the international money would be credited to surplus countries' central banks that would in turn, through commercial banks, credit exporters in the domestic currency. This means that domestic money would be created as a counterpart to the amounts of the international currency credited to the account of exporters' countries. Then, while the output available is reduced by exports, the quantity of money available in the economy is increased. Symetrically, deficit countries' central banks would meet a debit in the international currency, the counterpart of which would be made up of the sums of domestic currency spent by the importers and that in this way would be no more available in the economy for funding purchases on the goods available. In the latter case, the quantity of goods is increased by imports while the quantity of money is reduced.

In the 1970s, after the collapse of the Bretton Woods system, the International Monetary Fund member countries began thinking about reshaping the international monetary system. One of their objectives was to ensure short-run stability in exchange rates whereas the latter would be allowed to change in the long run. But, as McKinnon emphasised, the existence of 'hot' money made long-term flexibility in exchange rates incompatible with their short-run stability, except if foreign exchange controls were introduced, which was not considered acceptable at that time:

As long as world financial markets remained (modestly) open, speculative hot money flows would tend to anticipate any discrete change in official par values. And, certainly by 1973-74, the negotiators did not want a return to the draconian exchange controls that Keynes had in mind in 1943. (McKinnon, 1993, p. 25) It is true that in order to ensure exchange rate stability Keynes recommended implementing strict exchange controls, especially in the early versions of his plan. In subsequent versions, he was more moderate but there is no doubt that foreign exchange controls were part of his plan: "[…] the universal establishment of a control of capital movements cannot be regarded as essential to the operation of the Clearing Union; and the method and degree of such control should therefore be left to the decision of each member state" (Keynes, 1942, pp. 185-86).

These cases are all still relevant with reference to regional clearing unions. On the one hand, drawing rights would be allotted to member countries that would worsen balance of payments disequilibria in the region and induce inflation and deflation pressures on creditor and debtor countries respectively. On the other hand, the system would prove unable to stabilize member countries exchange rates and in this way it would enforce the use of the dollar or any other key-currency as a standard and an ever convertible asset among member countries. As for foreign exchange controls, if applied, they would run counter to a major goal of regional unions, to wit, to enhance the economic and financial integration of member countries.

4.

The need for a novel system for exchanging currencies: Keynes's scheme and how to supplement it.

In this section we argue that Keynes defined a system for exchanging domestic currencies for each other that can be improved and help get over the objections we have just examined.

As Keynes puts it in preamble to his plan, the bancor would have been a currency used by countries operating through their central banks or Treasuries while private individuals and companies would continue to use their domestic currencies:

We need an instrument of international currency having general acceptability between nations, so that blocked balances and bilateral clearings are unnecessary; that is to say, an instrument of currency used by each nation in its transactions with other nations, operating through whatever national organ, such as a Treasury or a central bank, is most appropriate, private individuals, businesses and banks other than central banks, each continuing to use their own national currency as heretofore. (Keynes,p. 168) He also made it clear that Within any member-country […] the provision of foreign exchange to be concentrated in the hands of its central bank which would deal with the public through the usual banks. That is to say, a member of the public here desiring to obtain dollars for a specified purpose would instruct his bank to make application to the Bank of England. […] Central banks would buy and sell their own currencies amongst themselves only against debits and credits to their accounts at the Clearing Bank (Keynes,.

Unfortunately, this scheme does not counter the objection relating to the funding of deficit countries and its inflationist and deflationist effects. Central banks in creditor countries would supply their own currencies against credits in bancors while central banks debtor countries would buy their own currencies against debits in bancors. This scheme, as well, would not be able to ensure exchange rates stability. Since foreign exchange would be concentrated in the hands of central banks, we may expect that the latter would be in a position to apply fixed exchange rates between their own currencies and the international currency. The difficulty, actually, would be to maintain exchange rates that would be less and less in accordance with inflation differentials, the system inducing inflation in creditor countries and deflation in debtor countries. Maintaining exchange rates stable would be all the more dif ficult that allocations of drawing rights would worsen payment imbalances. Then, just as McKinnon emphasised, speculative money flows would tend to anticipate any discrete change in official par values.

How are we to solve the problem? The analysis we are developing suggests that the issue lies in the way domestic currencies are managed per contra debits and credits in the international currency. Therefore, we may think along the same line as some Post Keynesian economists who have suggested that the national organs through which countries would operate should borrow or lend in the financial markets of the countries they are trading with, the very sums of foreign currency they spend or receive as a consequence of their residents' foreign transactions. In other words, an importing country A should borrow in the exporting country B the very sums of money B that is needed in country B to pay for the goods it is importing. The exporting country B, on its part, should lend in country A the sums of money A it earns. This latter proposal is actually in accordance with the logic of international payments, by which a country pays for its imports of goods and securities by means of the export of goods and securities (cf. [START_REF] Rossi | Money and Payments in Theory and Practice[END_REF]. As a consequence, the clearing of debits and credits in the international currency could take place on a very short term basis and would not result in an accumulation of imbalances. This would be so because any country importing or exporting goods would, as a counterpart, export or import securities. Deflation and inflation resulting from trade imbalances would be avoided and consequently exchange rates of domestic currencies against the common currency could be maintained fixed by the central banks without need to implement capital controls.

A further issue to be considered relates to payments between member countries and non-member countries. How should the proposed system be connected to the international monetary system? In order to maintain the benefits of it, particularly the fixity of local currencies' exchange rates against the common currency and thus against each other, the member countries should behave as a whole vis-à-vis third countries. This means that the union should hold member countries' reserves in nonmember countries' currencies (notably the US dollar). The union would then act as an interface in payments involving member and non-member countries, just as it would in member countries' mutual transactions. In this way member countries' currencies would never happen to be directly exchanged against other currencies and their fixity against the common currency could not be challenged. It is the common currency that would be exchanged for non-member countries and it would be up to the union to fix its exchange rate against, say, the US dollar, with reference to the balance of its member countries' payments, that is, with reference to a criterion that ensures the solvency of the union.

Conclusion

In the context of the present financial and economic crises, the need for a new Bretton Woods Conference has more often than not been evoked at the global level.

There is no doubt, however, that the time for implementing a renewed international monetary system along the lines designed by Keynes, has not yet come, to wit, the US dismissal of a recent proposal by Chinese authorities to consider the creation of an international currency. Then, the implementation of regional currency unions appears more than ever as a solution to rule regional payments on a sound basis despite the asymmetries in the international monetary system. In this paper, we have laid focus onto a key element of Keynes's plan, the system proposed for exchanging currencies for each other. We have seen that as it stood, this system was unsatisfactory since it would have allowed the aggravation of disequilibria and imposed foreign exchange controls within the regional union. With reference to a Keynesian framework we have endeavoured to show that this issue admits a solution that confirms the feasibility of a clearing union intended to rule regional payments in a sound way.


Paul Alain Kaldjob Kaldjob
email: paul-alain.kaldjob-kaldjob@dauphine.psl.eu

Brice Mayag
email: brice.mayag@dauphine.psl.eu

Denis Bouyssou
email: bouyssou@lamsade.dauphine.fr

On the robustness of the sign of nonadditivity index in a Choquet integral model

Keywords: Robustness, Nonadditivity index, Binary alternatives, Choquet integral model, Numerical representation

In the context of Multiple Criteria Decision Making, this paper studies the robustness of the sign of nonadditivity index for subset of criteria in a Choquet integral model. In the case where the set of alternatives is discrete, the use of the nonadditivity index proposed in the literature often leads to interpretations which are not always robust. Indeed, the sign of this nonadditivity index can depend on the arbitrary choice of a numerical representation in the set of all numerical representations compatible with the ordinal preferential information given by the Decision Maker. We characterize the ordinal preferential information for which the problem appears. We also propose a linear program allowing to test the non robustness of the sign of nonadditivity index for subset of criteria.

Introduction

The theory of value functions in Multiple Criteria Decision Making (MCDM), consists in assigning a real number to each alternative, so that the order on the alternatives induced by these real numbers is compatible with the ordinal preferential information of the Decision Maker (DM). Preferential independence is a necessary condition for these numbers to be obtained using an additive model 2 . But this property may not always be satisfied [START_REF] Grabisch | Fuzzy Measures and Integrals in MCDA[END_REF] . In this case it therefore becomes interesting to use a more general models such as the Choquet integral model. It was popularized by the work of Michel Grabisch [START_REF] Grabisch | Fuzzy integral in multicriteria decision making[END_REF][START_REF] Grabisch | The application of fuzzy integrals in multicriteria decision making[END_REF] and is now considered as a central tool in MCDM when one wants to escape the preferential independence hypothesis [START_REF] Grabisch | Set Functions, Games and Capacities in Decision Making[END_REF][START_REF] Grabisch | A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid[END_REF][START_REF] Grabisch | Fuzzy Measures and Integrals in MCDA[END_REF] . The use of Choquet integral requires that criteria are "commensurate" [START_REF] Grabisch | On the extension of pseudo-Boolean functions for the aggregation of interacting criteria[END_REF] .

When a set of ordinal preferential information is not compatible with an additive model, it is common to interpret this situation by the existence of some interactions between criteria. It is well known that the interaction among criteria caused by the nonadditivity of a capacity can be measured by a cardinal probabilistic interaction index, in particular the Shapley interaction index [START_REF] Grabisch | k-order additive discrete fuzzy measures and their representation[END_REF][START_REF] Shapley | A Value for n-Person Games[END_REF] . More details in the literature on axiomatic properties of all kinds of cardinal probabilistic interaction indices can be found in [START_REF] Fujimoto | Cardinal-Probabilistic Interaction Indices and their Applications: A Survey[END_REF][START_REF] Fujimoto | Representations of Importance and Interaction of Fuzzy Measures, Capacities, Games and Its Extensions: A Survey[END_REF][START_REF] Fujimoto | Axiomatic characterizations of probabilistic and cardinal-probabilistic interaction indices[END_REF][START_REF] Grabisch | Set Functions, Games and Capacities in Decision Making[END_REF] . Besides the Shapley interaction index [START_REF] Wu | Nonadditivity index and capacity identification method in the context of multicriteria decision making[END_REF] suggests to interpret this lack of compatibility with an additive model using the notion of nonadditivity, which is the subject of this paper.

After having modeled the preferences by a Choquet integral model, we are interested to the interactions between criteria. For a subset of criteria, when its Shapley interaction index is negative (resp. null, positive), it is usual to conclude that the interaction for this set of criteria is negative (resp. null, positive). However, it is possible to build another representation for which the conclusions are totally contradictory. Thus in [START_REF] Kaldjob Kaldjob | A characterization of necessary and possible interaction among more than two criteria in a Choquet integral model[END_REF] , we showed that, in practice, when we elicit a capacity on the basis of preferential information, the interaction is not easy to interpret and requires caution. The objective of this present paper, is to extend this analysis to the nonadditivity index, which is used to take into account some interaction phenomenon.

In the case where all the alternatives are binary, we characterize the ordinal preferential information for which the use of the nonadditivity index proposed in the literature leads to interpretations that are not robust. Since, in practice, wag only binary alternatives is restrictive, so we propose a linear program, inspired by [START_REF] Mayag | Necessary and possible interaction between criteria in a 2-additive Choquet integral model[END_REF] , allowing to test whether the interpretation of the nonadditivity index is ambiguous or not.

This paper is organized as follows. We recall in Section 2 some basic elements of Choquet integral model in MCDM. In Section 3, we give an example to motivate the introduction of the notion of necessary and possible nonadditivity. This notion is the subject of Section 4. In Section 5, we give our main results. Section 6 proposes a linear programming model allowing to test the existence of necessary and possible nonadditivity. A final section concludes.

Notation and preliminaries

The framework

Let X be a set of alternatives evaluated on a set of n criteria N = {1, 2, . . . , n} (n ≥ 2). For each subset A ⊆ N , throughout this paper we use the notation

A ⊆ ≥t N (resp. A ⊆ =t N ) if |A| ≥ t (resp. |A| = t)
, where t is an integer in {1, 2, . . . , n}. The set of all alternatives X is assumed to be a subset of a Cartesian product X 1 × . . . × X n , where X i is the set of possible levels on criterion i ∈ N . The criteria are recoded numerically using, for all i ∈ N, a utility function u i from X i to [0, +∞[. Using these functions, we assume that the various recoded criteria are commensurate and, hence, the application of the Choquet integral model is meaningful [START_REF] Grabisch | On the extension of pseudo-Boolean functions for the aggregation of interacting criteria[END_REF] .

As in [START_REF] Mayag | Elaboration d'une démarche constructive prenant en compte les interactions entre critères en aide multicritère à la décision[END_REF][START_REF] Mayag | A representation of preferences by the Choquet integral with respect to a 2-additive capacity[END_REF] , we assume that the DM is able to identify on each criterion i ∈ N two reference levels 0 i and 1 i :

• the level 0 i in X i is considered as a neutral level and we set u i (0 i ) = 0,

• the level 1 i in X i is considered as a good level and we set u i (1 i ) = 1.

For all x = (x 1 , . . . , x n ) ∈ X and S ⊆ N, we will sometimes write u(x) as a shorthand for (u 1 (x 1 ), . . . , u n (x n )) and we define the alternatives a S = (1 S , 0 -S ) ∈ X such that a i = 1 i if i ∈ S and a i = 0 i otherwise. We shall often work on the set B g which we define as follows.

Definition 1. The set of generalized binary alternatives is defined by:

B g = {a S = (1 S , 0 -S ) : S ⊆ N }.

Choquet integral

The Choquet integral [START_REF] Choquet | Theory of capacities[END_REF][START_REF] Grabisch | k-order additive discrete fuzzy measures and their representation[END_REF][START_REF] Grabisch | Set Functions, Games and Capacities in Decision Making[END_REF][START_REF] Grabisch | Fuzzy Measures and Integrals in MCDA[END_REF][START_REF] Keikha | An Approach Based on Combining Choquet Integral and TOPSIS Methods to Uncertain MAGDM Problems[END_REF][START_REF] Olgun | A Cosine Similarity Measure Based on the Choquet Integral for Intuitionistic Fuzzy Sets and Its Applications to Pattern Recognition[END_REF][START_REF] Ünver | Cosine and Cotangent Similarity Measures Based on Choquet Integral for Spherical Fuzzy Sets and Applications to Pattern Recognition[END_REF] is an aggregation function known in MCDM as a tool generalizing the weighted arithmetic mean. The Choquet integral uses the notion of capacity [START_REF] Choquet | Theory of capacities[END_REF][START_REF] Pignon | A methodological approach for operational and technical experimentation based evaluation of systems of systems architectures[END_REF] defined as a function µ from the power set 2 N into [0, 1] such that:

• µ(∅) = 0, • µ(N ) = 1, • ∀S, T ∈ 2 N , S ⊆ T =⇒ µ(S) ≤ µ(T ) (monotonicity).
For an alternative x = (x 1 , . . . , x n ) ∈ X, the expression of the Choquet integral w.r.t. the capacity µ is given by:

C µ u(x) = C µ u 1 (x 1 ), . . . , u n (x n ) = n i=1 u σ(i) (x σ(i) ) -u σ(i-1) (x σ(i-1) ) µ N σ(i) ,
where σ is a permutation on N such that:

N σ(i) = {σ(i), . . . , σ(n)}, u σ(0) (x σ(0) ) = 0 and u σ(1) (x σ(1) ) ≤ u σ(2) (x σ(2) ) ≤ . . . ≤ u σ(n) (x σ(n) ).
We often suppose that the DM gives his preferences by comparing some elements of B g . We then obtain the binary relations P and I defined as follows.

Definition 2. An ordinal preferential information {P, I} on B g is given by: P = {(x, y) ∈ B g × B g : DM strictly prefers x to y}, I = {(x, y), (y, x) ∈ B g × B g : DM is indifferent between x and y}.

The next definition makes explicit the compatibility of {P, I} with a Choquet integral model. Definition 3. An ordinal preferential information {P, I} on X is representable by a Choquet integral model if we can find a capacity µ such that: for all x, y ∈ X, we have,

x P y =⇒ C µ u(x) > C µ u(y) , x I y =⇒ C µ u(x) = C µ u(y) .
The set of all capacities that can be used to represent the ordinal preferential information {P, I} at hand will be denoted by C Pref (P, I). When there is no ambiguity on the underlying ordinal preferential information, we will simply write C Pref . As in [? ? ], we add to this ordinal preferential information a binary relation M modeling the monotonicity relations between generalized binary alternatives, and allowing us to ensure the satisfaction of the monotonicity condition: S ⊆ T =⇒ µ(S) ≤ µ(T ) . Definition 4. For all a S , a T ∈ B g , a S M a T if not(a S (P ∪ I)a T ) and S ⊇ T . Remark 1. For all S ⊆ N , we have C µ (u(a S )) = µ(S).

Remark 2. For all S, T ⊆ N , we have a S M a T =⇒ C µ (u(a S )) ≥ C µ (u(a T )).

In the sequel, we need the following two basic definitions in graph theory [START_REF] Lidl | Applied Abstract Algebra[END_REF] . Definition 5. There exists a strict path in (P ∪ M ∪ I), from x to y if there exists the elements x 0 , x 1 , . . . , x r of B g such that x = x 0 (P ∪M ∪I)x 1 (P ∪M ∪I) . . . (P ∪M ∪I)x r = y and for a least one i ∈ {0, . . . , r -1}, x i P x i+1 . In this case, we note x T C P y. We speak of a strict cycle when x = y. Definition 6. x T C M ∪I y if there exists elements x 0 , x 1 , . . . , x r of B g such that

x = x 0 (M ∪ I)x 1 (M ∪ I) . . . (M ∪ I)x r = y. Hence, T C M ∪I is the transitive closure of the binary relation M ∪ I.
In the next subsection, we recall the definition of the nonadditivity index 30 .

Nonadditivity index

Our work is based on the nonadditivity index, for which the definition and axiomatic properties can be found in [START_REF] Wu | Nonadditivity index and capacity identification method in the context of multicriteria decision making[END_REF] .

Definition 7. For all A ⊆ ≥2 N, the nonadditivity index [START_REF] Huang | Nonadditivity Index Based Quasi-Random Generation of Capacities and Its Application in Comprehensive Decision Aiding[END_REF][START_REF] Wu | Nonadditivity index and capacity identification method in the context of multicriteria decision making[END_REF][START_REF] Wu | Nonadditivity Index Oriented Decision Preference Information Representation and Capacity Identification[END_REF] w.r.t. a capacity µ is defined as follows:

η µ A = 1 2 |A|-1 -1 (B, A\B) ∅⊊B⊊A µ(A) -µ(B) -µ(A \ B) (1)
For all A ⊆ ≥2 N, for each partition (B, A \ B) of A with ∅ ⊊ B ⊊ A, we compute the difference µ(A) -µ(B) + µ(A \ B) . Thus η µ A corresponds to the arithmetic mean of these differences over all such partitions. Remark 3. We have η µ ij = µ ij -µ i -µ j , therefore the nonadditivity index coincides with the Shapley interaction index I µ ij , for pairs {i, j}. We recall that, the Shapley interaction index [START_REF] Shapley | A Value for n-Person Games[END_REF] of A ⊆ ≥2 N is given by

I µ A = K⊆N \A (n -|K| -|A|)!|K|! (n -|A| + 1)! L⊆A (-1) |A|-|L| µ(K ∪ L).
The following remark gives two equivalent expressions of η µ A that can be found on pages 3 and 4 in 30 .

Remark 4. Given a capacity µ on N and A ⊆ ≥2 N , Equation (1) is equivalent to each of Equations (2) and (3).

η µ A = 1 2 |A| -2 ∅⊊B⊊A µ(A) -µ(B) -µ(A \ B) (2)
η µ A = µ(A) - 1 2 |A|-1 -1 ∅⊊B⊊A µ(B) (3)
In the next section, we give an example, motivating the introduction of the concept of necessary and possible nonadditivity.

A motivating example

This example is inspired by [START_REF] Grabisch | Fuzzy Measures and Integrals in MCDA[END_REF] . Four students are evaluated on three subjects Mathematics (M), Statistics (S) and Language skills (L). All marks are taken from the same scale, from 0 to 1. The evaluations of these students are given in the Table 1 To select the best students, the Dean of the faculty expresses his/her preferences where the notation x P y means x is strictly preferred to y. For a student bad in Mathematics, Statistics is more important that Language, so that a P b.

(4)

For a student good in Mathematics, Language is more important that Statistics, so that

d P c. (5)
It is not possible to model the two preferences a P b and d P c by an arithmetic mean model. Indeed let us denote by q M , q S and q L the weights associated to Mathematics, Statistics and Language. We have:

a P b =⇒ u M (0.3)q M + u L (0.25)q L + u S (0.6)q S > u M (0.3)q M + u L (0.6)q L + u S (0.25)q S . d P c =⇒ u M (0.7)q M + u L (0.6)q L + u S (0.25)q S > u M (0.7)q M + u L (0.25)q L + u S (0.6)q S .
Adding up the previous two inequalities leads to the contradiction 0 > 0.

Let us assume that the scale of evaluation [0, 1] corresponds to the utility function associated to each subject, i.e., u M (0.3) = 0.3, u M (0.7) = 0.7, u L (0.25) = 0.25, u L (0.6) = 0.6, u S (0.25) = 0.25 and u S (0.6) = 0.6. In this case, the strict preferences a P b and d P c, are now representable by a Choquet integral model w.r.t. any capacity given in Table 2. We choose six capacities compatible with these preferences (Cap. for short in Table 2) in order to illustrate the fact that the sign of nonadditivity index is strongly dependent upon the chosen capacity.

In this example, the interpretation of the nonadditivity index between criteria is not easy. For instance, the nonadditivity index between Language and Statistics, η µ LS , could be strictly positive (Cap. 1, Cap. 3) or null (Cap. 5), or strictly negative (Cap. Depending on the choice of a capacity µ, the nonadditivity index can be null, or strictly positive, or strictly negative. This leads us to the definition of the notion of necessary and possible nonadditivity. This notion is similar at the notion of, necessary and possible interaction introduced in the case of a 2-additive Choquet integral model [START_REF] Mayag | Necessary and possible interaction between criteria in a 2-additive Choquet integral model[END_REF] , and is somewhat similar to the notion of necessary and possible preference relations introduced in robust ordinal regression [START_REF] Giarlotta | Necessary and possible preference structures[END_REF][START_REF] Greco | Ordinal regression revisited: Multiple criteria ranking using a set of additive value functions[END_REF] , replacing preferences with interaction indices.

η µ M L 1 -0.2 0.8 0 0.1 0.2 η µ M S 0 -0.5 0.2 -0.3 -0.4 -0.4 η µ LS 0.1 -0.4 0.2 -0.3 0 -0.1 η µ M LS 0.23 -0.13 0.1 -0.23 -0.1 0 Table 2: A set

Necessary and possible nonadditivity

The following definition of necessary and possible nonadditivity will be central in the rest of this text.

Definition 8. Let A ⊆ ≥2 N and {P, I} an ordinal preferential information. We say that:

1. There exists a possible positive (resp. null, negative) nonadditivity for A if there exists µ ∈ C Pref such that η µ A > 0 (resp.

η µ A = 0, η µ A < 0), 2.
There exists a necessary positive (resp. null, negative) nonadditivity for

A if η µ A > 0 (resp. η µ A = 0, η µ A < 0) for all µ ∈ C Pref . Remark 5. Let A ⊆ ≥2 N .
• If there exists a necessary positive (resp. null, negative) nonadditivity for A, then there exists a possible positive (resp. null, negative) nonadditivity for A.

• If there is no necessary positive (resp. null, negative) nonadditivity for A, then there exists a possible negative or null (resp. positive or negative, positive or null) nonadditivity for A.

If we have a possible but not necessary nonadditivity, then the interpretation of the nonadditivity is difficult because it depends on the capacity chosen in C Pref . Indeed, the interpretation of the nonadditivity only makes sense in the case of the necessary.

In [START_REF] Kaldjob Kaldjob | Study of the Instability of the Sign of the Nonadditivity Index in a Choquet Integral Model[END_REF] , we treated the case where preferential information does not contains indifference. The next section treats the second case. Under some conditions, positive and negative nonadditivity are always possible.

Results when I is not empty

In the framework of generalized binary alternatives, we proved in [START_REF] Kaldjob Kaldjob | A characterization of necessary and possible interaction among more than two criteria in a Choquet integral model[END_REF] , that, an ordinal preferential information {P, I} on B g is representable by a Choquet integral model if and only if the binary relation (P ∪ M ∪ I) contains no strict cycle. In this section, we assume that this condition holds and the set of ordinal preferential information {P, I} can contain an indifference. Given a subset A ⊆ ≥2 N , Proposition 1 gives a sufficient condition on {P, I} such that negative nonadditivity is always possible for A. Indeed, she shows that, if the DM is not indifferent between the worst alternative a 0 and another alternative, then negative nonadditivity is always possible for A.

Proposition 1. Let {P, I} be an ordinal preferential information on B g representable by a Choquet integral model and A ⊆ ≥2 N . If for all i ∈ A, not(a 0 T C M ∪I a i ), then there exists a capacity µ ∈ C Pref such that η µ A < 0.

Proof. Let A ⊆ ≥2 N , we assume that for all i ∈ A, not(a 0 T C M ∪I a i ). Since {P, I} is representable by a Choquet integral model, then (P ∪ M ∪ I) contains no strict cycle, hence we can build a partition {B 0 , B 1 , . . . , B m } of B g using a suitable topological sorting on (P ∪ M ∪ I) (see proof of Proposition 4 on [START_REF] Kaldjob Kaldjob | A characterization of necessary and possible interaction among more than two criteria in a Choquet integral model[END_REF] ). Let us define the capacity µ: 2 N -→ [0, 1] as follows:

for all S ⊆ N , µ(S) =        0, if a S ∈ B 0 ℓ + 1 ℓ + 2 , if a S ∈ B ℓ , ℓ ∈ {1, 2, . . . , m -1} 1, if a S ∈ B m
Let a S , a T ∈ B g .

• If a S I a T , then a S , a T ∈ B ℓ , thus µ(S) = µ(T ).

• If a S P a T , then there exists r, q ∈ {0, 1, . . . , m} such that a S ∈ B r , a T ∈ B q since {B 0 , B 1 , . . . , B m } is a partition of B g . As a S P a T , then r > q. We have

C µ (u(a S )) = µ(S) = r + 1 r + 2 (if 1 ≤ r ≤ m -1) or µ(S) = 1 (if r = m), we then have C µ (u(a S )) ≥ r + 1 r + 2 , since 1 ≥ r + 1 r + 2 . • If q = 0, then C µ (u(a T )) = C µ (u(a 0 )) = µ(∅) = 0 < r + 1 r + 2 ≤ C µ (u(a S )). • If q ≥ 1, C µ (u(a T )) = µ(T ) = q + 1 q + 2 , since 1 ≤ q ≤ m -1. But r > q then r + 1 r + 2 > q + 1 q + 2
, since the sequence (f n ) n∈N is strictly increasing, where

f n = n + 1 n + 2 for all n ∈ N. Then C µ (u(a S )) > C µ (u(a T )).
Hence, in both cases we have

C µ (u(a S )) > C µ (u(a T )). We deduce that µ ∈ C Pref . Let ∅ ⊊ B ⊊ A, as ∀i ∈ A, not(a 0 T C M ∪I a i ), then ∀i ∈ A, we have a i / ∈ B 0 , so a B / ∈ B 0 . Hence µ(B) ≥ ℓ + 1 ℓ + 2 with 1 ≤ ℓ ≤ m -1. Thus µ(B) ≥ 2 3 > 1 2 , then ∅⊊B⊊A µ(B) > 1 2 (2 |A| -2) = 2 |A|-1 -1, i.e., 1 2 |A|-1 -1 ∅⊊B⊊A µ(B) > 1 ≥ µ(A). Thus η µ A < 0.
Remark 6. The sufficient condition of Proposition 1 is a necessary condition for A ⊆ =2 N (see Proposition 2 below) but not necessary for A ⊆ ≥3 N . Indeed, let us consider N = {1, 2, 3}, P = {(a 13 , a 2 )}, I = {(a 0 , a 1 )} and A = N . {P, I} is representable by the capacity given by the Table 3.

S {1} {2} {3} {1, 2} {1, 3} {2, 3} {1, 2, 3} µ(S) 0 0.5 1 1 1 0.5 1 Table 3: A capacity µ ∈ C Pref We have 1 ∈ A and a 0 T C M ∪I a 1 , but η µ 123 = 1- 1 3 (µ 1 +µ 2 +µ 3 +µ 12 +µ 13 +µ 23 ) = - 1 3 < 0.
Given {i, j} ⊆ N , the following Proposition 2 gives a necessary and sufficient condition on {P, I} such that negative nonadditivity that is always possible for {i, j}.

Proposition 2. Let {P, I} be an ordinal preferential information on B g representable by a Choquet integral model and i, j ∈ N . There exists a capacity µ ∈ C Pref such that η µ ij < 0 if and only if not(a 0 T C M ∪I a i ) and not(a 0 T C M ∪I a j ).

Proof. Let i, j ∈ N , we assume that {P, I} is representable by a Choquet integral model.

Necessity. Assume that there exists a capacity µ ∈ C Pref such that η µ ij < 0. If a 0 T C M ∪I a i or a 0 T C M ∪I a j , then η µ ij = µ ij -µ j ≥ 0 or η µ ij = µ ij -µ i ≥ 0 respectively. Therefore η µ ij ≥ 0 in the both cases. Contradiction since we assume that η µ ij < 0. Sufficiency. According to the Proposition 1 it sufficient to consider A = {i, j}.

Let A ⊆ ≥2 N and ∅ ̸ = B ⊊ A, suppose that DM is indifferent between alternatives a A and a A\B . This would suggest that subset B is quite unimportant for the DM, so that a B is indifferent with a 0 . We translate this idea by the following Definition 9. Definition 9. Let A ⊆ ≥2 N . We call Monotonicity of Ordinal Preferential Information for A, the following property

(denoted A -MOPI): for all ∅ ̸ = B ⊊ A, a A ∼ a A\B =⇒ not(a B T C P a 0 ).
Given a subset A ⊆ ≥2 N , the Proposition 3 gives a sufficient condition on {P, I} such that null or positive nonadditivity that is always possible for A. for all S ⊆ N, µ(S) =

   0, if a S ∈ B 0 (2n) ℓ (2n) m , if a S ∈ B ℓ , ℓ ∈ {1, 2, . . . , m} Let a S , a T ∈ B g .
• If a S I a T , then a S , a T ∈ B q , therefore µ(S) = µ(T ).

• If a S P a T , then a S ∈ B q and a T ∈ B r with q > r. Therefore µ(S) = (2n) q (2n) m and

µ(T ) = 0 (if r = 0) or µ(T ) = (2n) r (2n) m (if r ≥ 1). But (2n) q (2n) m > max 0, (2n)
r (2n) m since q > r ≥ 0, so µ(S) > µ(T ).

Hence, we have µ ∈ C Pref . Let A ⊆ ≥2 N , we consider the set Ψ = {∅ ̸ = B ⊊ A : not(a A ∼ a B ) and not(a A ∼ a A\B )}.

We can write:

(2 |A| -2)η µ A = ∅̸ =B⊊A µ(A) -µ(B) -µ(A \ B) = B∈Ψ µ(A) -µ(B) -µ(A \ B) + B / ∈Ψ µ(A) -µ(B) -µ(A \ B) . Let ∅ ̸ = B ⊊ A. If B /
∈ Ψ, then a A ∼ a B or a A ∼ a A\B , therefore we have a A ∼ a B and not(a A\B T C P a 0 ) or a A ∼ a A\B and not(a B T C P a 0 ) since by hypothesis, the property A-MOPI is satisfied. Thus, µ(A) = µ(B) and µ(A \ B) = 0 or µ(A) = µ(A \ B) and µ(B) = 0 respectively, i.e., µ(A) -µ(B) -µ(A \ B) = 0 in the both cases, and we have

B / ∈Ψ µ(A) -µ(B) -µ(A \ B) = 0. Hence (2 |A| -2)η µ A = B∈Ψ µ(A) -µ(B) -µ(A \ B) . • If Ψ = ∅, then (2 |A| -2)η µ A = 0, i.e., η µ A = 0.
• If Ψ ̸ = ∅, then for all B ∈ Ψ, we have a B ∈ B r , a A\B ∈ B s and a A ∈ B q with q > r and q > s. Hence µ(B) ≤ (2n) r , µ(A \ B) ≤ (2n) s and µ(A) = (2n) q = (2n)(2n) q-1 > 2(2n) q-1 = (2n) q-1 +(2n) q-1 ≥ (2n) r +(2n) s since q -1 ≥ r and q -1 ≥ s. Therefore

µ(A) > (2n) r + (2n) s ≥ µ(B) + µ(A \ B), i.e., µ(A) -µ(B) -µ(A \ B) > 0 for all B ∈ Ψ. Hence B∈Ψ µ(A) -µ(B) -µ(A \ B) > 0, so η µ A > 0.
In the both case, we deduce that η µ A ≥ 0. 4.

S {1} {2} {3} {1, 2} {1, 3} {2, 3} {1, 2, 3} µ(S) 0 0 0.5 1 0.5 0.5 1 Table 4: A capacity µ ∈ C Pref
We have a 12 ∼ a 123 and a 3 T C P a 0 so the {1, 2, 3} -MOPI property is not satisfied, while

η µ 123 = 1 - 1 3 (µ 1 + µ 2 + µ 3 + µ 12 + µ 13 + µ 23 ) = 1 6 ≥ 0.
Given a subset A ⊆ ≥2 N , we are interested in the class of alternatives for which the DM is neutral for all the criteria not belonging to A. The alternative a A is the best of them. Proposition 4 shows that, if the DM is not indifferent between this best alternative a A and another alternative a B (with B ⊊ A), then positive nonadditivity is always possible for A. Proof. Let A ⊆ ≥2 N . We assume that for all i ∈ A, not(a A\{i} T C M ∪I a A ). We define the set Ψ as in the proof of Proposition 3. Since, for all i ∈ A, not(a A\{i} T C M ∪I a A ), then for all ∅ ⊊ B ⊊ A, we have not(a B ∼ a A ) and not(a A\B ∼ a A ). Therefore each subset ∅ ̸ = B ⊊ A is an element of Ψ, hence Ψ ̸ = ∅. According to the proof of Proposition 3, we can build µ ∈ C Pref such that η µ A > 0.

Remark 8. The sufficient condition of Proposition 4 is a necessary condition for A ⊆ =2 N (see Proposition 5 below) but not necessary for A ⊆ ≥3 N . Indeed, let us consider N = {1, 2, 3}, P = {(a 12 , a 3 )}, I = {(a 13 , a 123 )} and A = N . {P, I} is representable by the capacity given by the Table 5.

S {1} {2} {3} {1, 2} {1, 3} {2, 3} {1, 2, 3} µ(S) 0 0 0 1 1 0 1 Table 5: A capacity µ ∈ C Pref We have 2 ∈ A and a 13 T C M ∪I a 123 , but η µ 123 = 1- 1 3 (µ 1 +µ 2 +µ 3 +µ 12 +µ 13 +µ 23 ) = 1 3 > 0.
Given a pair of criteria {i, j}, Proposition 5 gives a necessary and sufficient condition on {P, I} such that positive nonadditivity that is always possible for {i, j}.

Proposition 5. Let {P, I} be an ordinal preferential information on B g representable by a Choquet integral model and i, j ∈ N . There exists a capacity µ

∈ C Pref such that η µ ij > 0 if and only if [not(a i T C M ∪I a ij ) and not(a j T C M ∪I a ij )]. Proof. Let i, j ∈ N .
Necessity. Assume that there exists a capacity µ

∈ C Pref such that η µ ij > 0. If a i T C M ∪I a ij or a j T C M ∪I a ij , then η µ ij = -µ j ≤ 0 or η µ ij = -µ i ≤ 0 respectively. Therefore η µ
ij ≤ 0 in the both cases. Contradiction since we assume that η µ ij > 0. Sufficiency. According to the Proposition 4, it is sufficient to consider A = {i, j}.

The results of this section show that, under some conditions, it is possible to represent the preferences of DM, in such a way as to have indices of nonadditivity with non-constant sign. Therefore, the interpretation of nonadditivity between criteria requires caution.

All previous results are based on the set of generalized binary alternatives B g . As this set is restrictive in practice, we propose in the next section, a process based on linear programming, allowing us to test the existence of a necessary positive (resp. negative) nonadditivity for a subset of criteria A ⊆ ≥2 N .

A LP model testing for necessary nonadditivity

In [START_REF] Kaldjob Kaldjob | A characterization of necessary and possible interaction among more than two criteria in a Choquet integral model[END_REF] we proposed outside the framework of generalized binary alternatives, a linear program allowing to test the existence of some necessary interactions when I = ∅. We show how to test the existence necessary positive or negative nonadditivity on the basis of information given on a subset of X that is not necessarily B g , when I = ∅. We assume that the DM provides at least one strict preference, (i.e., P ̸ = ∅) and an indifference I relations on a subset of X. Our approach consists in testing first, the compatibility of this ordinal preferential information with a Choquet integral model, and then, in the second step, the existence of a necessary positive or negative nonadditivity for a subset A ⊆ ≥2 N .

The process

Step 1. At this step, we test the compatibility of {P, I} with a Choquet integral. This is similar to what is done in [START_REF] Angilella | Stochastic multiobjective acceptability analysis for the Choquet integral preference model and the scale construction problem[END_REF][START_REF] Mayag | Necessary and possible interaction between criteria in a 2-additive Choquet integral model[END_REF] . The following linear program LP 1 models each preference of P by introducing a non negative slack variable ε in the corresponding constraint (Equation (1a)). The Equation (1c) (resp. (1d)) ensures the normalization (resp. monotonicity) of capacity µ. The objective function Z 1 maximizes the non negative variable ε introduced in (1a).

Maximize Z 1 = ε LP 1 Subject to C µ (u(x)) -C µ (u(y)) ≥ ε ∀x, y ∈ X such that x P y (1a) C µ (u(x)) -C µ (u(y)) = 0 ∀x, y ∈ X such that x I y (1b) µ(N ) = 1 (1c) µ(S ∪ {i}) ≥ µ(S) ∀S ⊊ N, ∀i ∈ N \ S. (1d) ε ≥ 0
We have one of the following two cases:

1. If the linear program LP 1 is not feasible or feasible with an optimal solution Z * 1 = 0, then there is no Choquet integral model compatible with {P, I}.

2. If the linear program LP 1 is feasible with an optimal solution Z * 1 > 0, then ordinal information {P, I} is representable by a Choquet integral model.

Step 2. At this step, we suppose that the preference information {P, I} is representable by a Choquet integral model, i.e., Z * 1 > 0. In order to know if the nonadditivity index for subset of criteria A is necessarily negative, at LP 1 , we add the constraint (1e) and we obtain the following linear program denoted by LP A N N .

Maximize

Z 2 = ε LP A N N Subject to C µ (u(x)) -C µ (u(y)) ≥ ε ∀x, y ∈ X such that x P y (1a) C µ (u(x)) -C µ (u(y)) = 0 ∀x, y ∈ X such that x I y (1b) µ(N ) = 1 (1c) µ(S ∪ {i}) ≥ µ(S) ∀S ⊊ N, ∀i ∈ N \ S (1d) ε ≥ 0 η µ A ≥ 0. ( 1e
)
To know if the nonadditivity index for subset of criteria A is necessarily positive, we change the constraint (1e) by η µ A ≤ 0 and we obtain the linear program denoted LP A N P . After a resolution of the linear programs, we have one of the following three possible conclusions:

1. If LP A N N (resp. LP A N P ) is not feasible, then there is a necessary negative (resp. positive) nonadditivity index for the subset A. Indeed, as the program LP 1 is feasible with an optimal solution Z * 1 > 0, the contradiction about the representation of {P, I} only comes from the introduction of the constraint η µ A ≥ 0 (resp. η µ A ≤ 0).

If LP A

N N (resp. LP A N P ) is feasible and the optimal solution Z * 2 = 0, then the constraint C µ (u(x)) -C µ (u(y)) ≥ ε ∀x, y ∈ X such that x P y is satisfied with ε = 0, i.e., it is not possible to model strict preference by adding the constraint η µ A ≥ 0 (resp. η µ A ≤ 0) in LP A N N (resp. LP A N P ). Therefore, we can conclude that there is a necessary negative (resp. positive) nonadditivity index for A.

If LP A

N N (resp. LP A N P ) is feasible and the optimal solution Z * 2 > 0, then there is no necessary negative (resp. positive) nonadditivity index for A.

Note that this process can be done in three steps [START_REF] Mayag | Necessary and possible interaction between criteria in a 2-additive Choquet integral model[END_REF] . This way of doing it saves from having to specify an arbitrary parameter ε. But here we chose to do it in two steps.

For each of the previous linear programs, we have n(2 n-1 -1) monotonicity constraints. Furthermore, the Table 6 gives an idea of the decision variables and Table 7 gives an idea of number of variables and number of monotonicity constraints. In practice, the number of criteria generally does not exceed 12. Thus, with a standard LP solver, we are able to deal with these linear programs.

Decision variables

LP 1 ε, µ(S) (∅ ⊊ S ⊊ N ) LP A N N ε, µ(S) (∅ ⊊ S ⊊ N ) LP A N P ε, µ(S) (∅ ⊊ S ⊊ N )

Example

In this section, we illustrate our decision procedure with an example given by Brice Mayag in [START_REF] Mayag | Elaboration d'une démarche constructive prenant en compte les interactions entre critères en aide multicritère à la décision[END_REF] . Six young artists without a producer take part in a high-audience singing competition program, where a winner will see his/her work produced by a famous record company. Each candidate performs, in front of a jury his own song. The jury is subdivided into three groups: a group of choreography professionals, another of professional singers and vocals, and the last group is formed by professional musicians. The following three criteria are used to classify candidates.

1. Choreography: the choreography chosen by the candidate during his performance.

Evaluations are given as a number of vertical bars |. There are four members of the sub jury and each marks from 0 to 5 bars. The best candidate in choreography will be the one who will collect the greatest number of bars.

2. Singing: the quality of the song performed, taking into account the voice of the performer. The Singing sub jury evaluates the candidates in a classic way, assigning them marks between 0 and 20. The best candidate in singing will be the one who will collect the greatest number of marks.

3. Music: the quality of the music used to accompany the chosen song. The ability to play musical instruments is also taken into account at this level. The evaluations are given between 0 and 100. The best candidate in music will be the one who will collect the greatest number of marks.

The evaluations obtained by the candidates are given in the In this example we have N = {1, 2, 3} and X = {a, b, c, d, e, f }. To choose the winner, the jury establishes the following two rules:

• When two candidates have good marks in singing and in music, the jury will prefer the one who has a better evaluation in choreography, even if it means being less good in singing or in music. Therefore he strictly prefers b to a.

• When two candidates have a poor performance in singing, the jury will prefer the one who has the best rating in music. Therefore he strictly prefers c to d.

Besides, the jury finding the evaluations of candidates e and f very similar, considers them indifferent. The ordinal preferential information on X provided by the jury will therefore consist of the four following binaries relations P = {(b, a), (c, d)}, I = {(e, f ), (f, e)} To apply the Choquet integral to our example, we need to define commensurable scales. We assume that the construction of the utility functions u 1 , u 2 and u 3 is done simply by normalization the evaluations according to each criterion by reducing them to marks between 0 and 20. Thus, the evaluations of the choreography criterion will be done by counting just the number of bars obtained by each candidate. On the music criterion, the evaluations will be normalized by a division by 5. We then obtain the Table 9 Step 1. The linear program corresponding to the test of the existence of a capacity µ compatible with {P, I} is the following.

Maximize

Z 1 = ε LP 1 Subject to C µ (u(b)) -C µ (u(a)) ≥ ε C µ (u(c)) -C µ (u(d)) ≥ ε C µ (u(e)) -C µ (u(f )) + α + ef -α - ef = 0 C µ (u(a)) = 7 + 7µ 23 + 3µ 2 C µ (u(b)) = 9 + 3µ 23 + 5µ 2 C µ (u(c)) = 7 + µ 23 + 6µ 3 C µ (u(d)) = 8 + µ 13 + 3µ 3 C µ (u(e)) = 9 + µ 12 + µ 1 C µ (u(f )) = 9 + µ 12 + 2µ 1 µ 1 ≥ 0; µ 2 ≥ 0; µ 3 ≥ 0 µ 12 ≥ µ 1 ; µ 12 ≥ µ 2 ; µ 13 ≥ µ 1 ; µ 13 ≥ µ 3 ; µ 23 ≥ µ 2 ; µ 23 ≥ µ 3 µ 123 ≥ µ 12 ; µ 123 ≥ µ 13 ; µ 123 ≥ µ 23 µ 123 = 1 ε ≥ 0.
The linear program LP 1 is feasible with an optimal solution Z * 1 = 0.8 > 0, then we can conclude that, {P, I} is representable by a Choquet integral model.

Step 2. In order to know if the nonadditivity for {1, 2, 3} is necessarily negative (resp. positive). We obtain the LP 123 N N (resp. LP 123 N P ) by adding at the previous linear program LP 1 the constraints η µ 123 ≥ 0 (resp. η µ 123 ≤ 0) with η µ 123 = 1-1 3 (µ 12 +µ 13 +µ 23 +µ 1 +µ 2 +µ 3 ).

• The linear program LP 123 N N is feasible with an optimal solution Z * 2 = 0.8 > 0. Then the nonadditivity for {Choreography, Singing, Music} is not necessarily negative. Moreover, the results obtained by solving LP 123 N N are given by the Tables 10 and11

Conclusion

In the Choquet integral model, the capacity elicited to represent the preferences of the decision maker is not likely to be unique. This contrasts with the "continuous case" studied in [START_REF] Timonin | Axiomatization of the Choquet integral for 2-dimensional heterogeneous product sets[END_REF][START_REF] Timonin | Conjoint axiomatization of the Choquet integral for heterogeneous product sets[END_REF] . This non-uniqueness complicates the interpretation of the nonadditivity index. Indeed, we give some examples in which the sign of the nonadditivity index depends upon the arbitrary choice of a capacity within the set of all capacities compatible with the preferences that were obtained. We define the concept of necessary and possible nonadditivity. This concept is similar at necessary and possible interaction, introduced in 22 in the case of 2-additive capacities. Necessary nonadditivity is the only nonadditivity that can safely be interpreted since its sign does not vary within the set of all compatible capacities. We have given conditions under which preferences on binary alternatives can be represented using a capacity in a Choquet integral model. We do the same by adding the extra conditions so that one of the representative capacities induces strictly positive nonadditivity indices for all groups of criteria, and another representative capacities induces strictly negative nonadditivity indices for all groups of criteria. These results show that, in practice, when we elicit a capacity on the basis of preferential information, it is not easy to interpret what we find using the nonadditivity index. Therefore, the interpretation of nonadditivity between criteria requires caution.

Our results leave some important questions open. The first one would be to develop tools allowing to analyze "necessary nonadditivity" for a large class of aggregation models, including the Choquet integral model. The second would be to study of aggregation models using bipolar scales [START_REF] Grabisch | Set Functions, Games and Capacities in Decision Making[END_REF] .

  .

  of six capacities compatible with the preferences a P b and d P c.

Proposition 3 .

 3 Let {P, I} be an ordinal preferential information on B g representable by a Choquet integral model. Let A ⊆ ≥2 N . If the A-MOPI property is satisfied, then there exists a capacity µ ∈ C Pref such that η µ A ≥ 0. Proof. Since {P, I} is representable by a Choquet integral model, then (P ∪ M ∪ I) contains no strict cycle, hence we can build a partition {B 0 , B 1 , . . . , B m } of B g using a suitable topological sorting on (P ∪ M ∪ I) (see proof of Proposition 4 on 17 ). Let us define the capacity µ: 2 N -→ [0, 1] as follows:

Remark 7 .

 7 The sufficient condition of Proposition 3 is a necessary condition for A ⊆ =2 N (see Theorem 4 in 22 ) but not necessary for A ⊆ ≥3 N . Indeed, if we consider N = {1, 2, 3}, P = {(a 23 , a 1 ), (a 3 , a 0 )}, I = {(a 12 , a 123 )} and A = N . The ordinal preferential information {P, I} is representable by the Choquet integral model w.r.t the capacity given in the Table

Proposition 4 .

 4 Let {P, I} be an ordinal preferential information on B g representable by a Choquet integral model and A ⊆ ≥2 N . If for all i ∈ A, not(a A\{i} T C M ∪I a A ), then there exists a capacity µ ∈ C Pref such that η µ A > 0.

  (with η µ 123 = 0.3 > 0).S{1} {2} {3} {1, 2} {1, 3} {2, 3} {1, µ (u(x)) 11 12.2 9.8 9.6 9.5 9.5

Table 1 :

 1

Evaluations of the four students on the three criteria

  2, Cap. 4, Cap. 6). This conclusion is still valid concerning the nonadditivity index η µ M S between Mathematics and Statistics, the nonadditivity index η µ M L between Mathematics and Language, the nonadditivity index η µ M LS between Mathematics, Language and Statistics. Moreover, all nonadditivity indices are strictly positive w.r.t. Cap. 3 but strictly negative w.r.t. Cap. 2. Cap. 1 Cap. 2 Cap. 3 Cap. 4 Cap. 5 Cap. 6

	µ M	0	0.5	0	0.7	0.6	0.6
	µ L	0	0.4	0	0.3	0.2	0.1
	µ S	0.4	0.6	0.5	0.5	0.4	0.4
	µ M L	1	0.7	0.8	1	0.9	0.9
	µ M S	0.4	0.6	0.7	0.7	0.6	0.6
	µ LS	0.5	0.6	0.7	0.5	0.6	0.4
	C µ (a)	0.39	0.46	0.43	0.43	0.4	0.4
	C µ (b)	0.3	0.40	0.29	0.39	0.35	0.32
	C µ (c)	0.39	0.51	0.49	0.56	0.52	0.5
	C µ (d)	0.6	0.54	0.53	0.67	0.62	0.62

Table 6 :

 6 Decision variables Number of variables µ(S) Number of constraints of monotonicity

	n = 3	6	9
	n = 4	14	28
	n = 5	30	75
	n = 6	62	186
	n = 7	126	441
	n = 8	254	1 016
	n = 9	510	2 295
	n = 10	1 022	5 110
	n = 11	2 046	11 253
	n = 12	4 094	24 564

Table 7 :

 7 Number of variables µ(S) and number of monotonicity constraints with 3 ≤ n ≤ 12

Table 8 .

 8

	Candidates 1: Choreography 2: Singing 3: Music
	a: Anita	||||| ||	17	70
	b: Bertrand	||||| ||||	17	60
	c: Carine	||||| ||	8	70
	d: Didier	||||| ||||	8	60
	e: Elise	||||| ||||| |	10	45
	f : Fabien	||||| ||||| ||	10	45

Table 8 :

 8 Evaluation matrix

  .

		1	2	3
	u i (a)	7	17	14
	u i (b)	9	17	12
	u i (c)	7	8	14
	u i (d)	9	8	12
	u i (e)	11	10	9
	u i (f )	12	10	9

Table 9 :

 9 Utility function u i scaled between 0 and 20

Table 11 :

 11 Choquet integral corresponding at the capacity µ of Table10

•

  The linear program LP 123 N P is feasible with an optimal solution Z * 2 = 0.8 > 0. Then the nonadditivity for {Choreography, Singing, Music} is not necessarily positive. Moreover, the results obtained by solving LP 123 N P are given by the Tables12 and 13(with η µ 123 = -0.2 < 0).

	S	{1} {2} {3} {1, 2} {1, 3} {2, 3} {1, 2, 3}
	µ(S)	0	0.7 0.7	0.8	0.7	0.7	1

Table 12 :

 12 A capacity µ ∈ C Pref

	x	a	b	c	d	e	f
	C µ (u(x)) 14 14.6 11.9 10.8 9.8 9.8

Table 13 :

 13 Choquet integral corresponding at the capacity µ of Table12


Sara Dorregaray-Oyaregui

César Martín-Gómez

Ignacio Hernando Gil

Mónica Aguado

Technical, constructive and economical feasibility to turn off-grid an existing building

Keywords: off-grid, plug&play, existing, building, efficiency, saving, refurbishment, integration, renewable, replicable

Existing educational buildings built with old normatives suffer a lack of technology, even though they are the frame of reference for our future society's architects(/builders). These buildings, usually promoted by the public sector, don't have significant economic investment, even if they are going to affect our children's world perception. The object construction of this study is a building from 1978. It has an educational use located at the Pamplona campus of the Universidad de Navarra. The building is part of the Living Lab of the Campus, where technologies, solutions and strategies can be proved It is a protected building by the "Documentation and Conservation of buildings, sites, and neighborhoods of the Modern Movement" (Do. Co,Mo.Mo.), so all actions that can affect its aesthetic aspects need to be justified because of the value of the building. The methodology presented concerns the development of a replicable technical, constructive, and economical feasibility model to reach an off-grid disconnection of an existing building. The steps followed for this study are classified in seven main steps. This proposal aims to define a replicable solution that is going to be applied to other buildings of campus Universidad de Navarra in Pamplona. Even that the first solution is going to be limited by the aesthetic aspect, the final objective is to develop a Plug & Play solution following the methodology, answering the energetical deficiency and complexity of existing constructions.

INTRODUCTION

Buildings are responsible for 40% of the total energy consumption, according to the United Nations Environmental Program (UN Environment 2019). Thus, the concern around energy usage in buildings has been under focus; fossil fuels still being the world´s most used energy source, alternatives to their consumption are among the primary policies in consideration. Buildings need to be refurbished, and not for aesthetical or economic reasons. It is just part of their normal life cycle. This necessity is explained in the document of the "renovation wave," where they argue that 85% of the building stock has been built before 2001. Furthermore, 85-95% of the existing building will still be in use in 2050. A proposal of the Climate Target Plan 2030 has been the reduction of greenhouse gas emissions in the European Union by 55% at least by 2030 (European Commission 2020).

Energy efficiency is essential for action, with the building sector as one of the areas where efforts must be ramped up.

This demonstrates and measures the contributions of refurbishing our buildings. Energy technical saving shows that there can be around 30-60% in various Member States. The saving on energy consumption and the reductions of CO₂ (?) [START_REF] Mata | Contributions of Building Retrofitting in Five Member States to EU Targets for Energy Savings[END_REF].

The public sector, in particular educational institutions, needs to step up and be a referent. In this context, it is important to remember that this type of building influences students directly through its interior and exterior layout, and could foster the creation of an atmosphere conductive to learning, as well as an awareness of what we need to do . [START_REF] Le | Sustainable Refurbishment for School Buildings: A Literature Review[END_REF]. This is already possible at new buildings, since they have the technology of control, actual building services and elevated enclosure requirements. However, historical buildings used by many education institutions would require an energy efficiency evaluation, as well as effectiveness and sustainability refurbishment solutions [START_REF] Balocco | Modelling of Reversible Plant System Operations in a Cultural Heritage School Building for Indoor Thermal Comfort[END_REF]. 17% of the nonresidential buildings in the EU, are educational constructions under public control. They have high impact, public visibility, and can show a best-practice example (Österreicher 2018).

The main problem of a successful building retrofit is the complexity of the whole process. As Ma et al. [START_REF] Ma | Existing Building Retrofits: Methodology and State-ofthe-Art[END_REF]) explains, it is not just about money. There are some key elements that have significant impact on building retrofits, including policies and regulations, client resources and expectations, retrofit technologies, building specific information, human factors and other uncertainty factors.

OBJECTIVES

The Universidad de Navarra's 2025 Strategy aims to contribute to the attainment of the challenges posed by society through its work researcher, professor and assistant, and in collaboration with other actors and institutions. Sustainable development and care for people and the environment are the point of reference for the orientation of its projects. Thus, sustainability in its triple dimension -environmental, economic and social-becomes the transversal purpose of the entire Strategy 2025 of the Universidad de Navarra and the projects to which it gives rise will be aligned with the Sustainable Development Goals. (Navarra n.d.) This study is focused on the School of Architecture at Universidad de Navarra from Pamplona, a city in the north of Spain where the climate regarding the Döppen Geigger classification is considered Cfb (warm temperature, fully humid and warm summer), has an average of 27 ºC (80,6 ºF) of maximum and 2 ºC (35,6 ºF) of minimum. Built atop a tufa -a variety of limestone-plateau, with an average solar radiation of 1.000-1.300 kW/m2 and main northern wind of 5-10 m/s. The building of 1978 is located near a small river, in the south part, down the hill, of Pamplona (Fig. 1). The building orientation is NNE-SSW and is surrounded by trees of high elevation with a car park at the West side. (Fig. 2) Following the Living Lab project on the Campus, this building will become the seed for diverse actions, showroom of solutions and technologies that are going to be replicated in other buildings of the Universidad de Navarra. Even though the building is 44 years old, it maintains its contemporary character (Fig. 3, Fig. 4).. The construction follows a perimetral brick enclosure with external concrete pillars that support the metallic roof. The section shows the complexity of the building (Fig. 5). The internal surface distribution is homogeneous with a big and open central space. In total it has 9.486 m 2 (102106,5 ft 2 ) distributed in four floors. All the complete information of the building characteristics can be found at XL book of Tárrago and Cidoncha [START_REF] Tárrago | Thermal Energy Performance in Working-Spaces from Biomorphic Models: The Tuna Case in an Office Building[END_REF]. Other relevant and conditioning aspect of the building is that it is catalogued at the Do.Co.Mo.Mo. It means that actions related to the aesthetic aspect that affect the image have to be limited and will need to be justified. The final objective is to create and design a replicable Plug & Play solution for existing campus buildings at the Universidad de Navarra. The School of Architecture is going to be the pilot project to study and incorporate different solutions replicable in other buildings and thus becoming off-grid constructions.

METHODOLOGY DEVELOPMENT

To create a replicable solution, it is necessary to develop a methodology that defines the best technical and economical solution for building grid disconnection.

1. Energetical analysis. It can be gathered from the data of the electrical and thermal consumption, from an evaluation of heat losses and from a digital simulation. Thus, construction and building services are analyzed and evaluated from different levels of precision. 2. Reduction of energy requirements. It's the reduction of the effort that needs to be done to maintain certain conditions in the interior space. Adjusting the operating mode of the building and users into a more energy-efficient one, as well as following technical recommendations of referential institutions, affects positively into energy needs of the construction. The reductions can be seen in electrical and thermal consumption and in interior space requirements. 3. Space distribution for current and future needs. A correct space distribution adapted to the use of the building, can be perceived when minimum energy is required. This minimum is going to be quantified by comparing the effort required to maintain a comfortable range in other building spaces. 4. Renewable energy resources. Study compatible and more efficient energy sources depending on the characteristics of the building and its surrounding space. Production and accumulation, commercial solutions, and disruptive technologies. 5. Match requirement and production. Calculations of demand and production to determine storage needs of thermal and/or electricity energy.

6. Control development. Optimization of energy control system of the building, with a correct control of energy use and needs. 7. Integration of plug and play solution. Creation of a replicable element, with its relative parametrizations that can "off-grid"-existing buildings. Technology implementation with capacities to develop more.

CASE STUDY

The replication of the developed methodology in a timeline can be seen in the following image (Fig. 6).

Starting from an energy audit, followed by energy reduction actions, a study of renewable energies, planning of spatial distributions and the development of control and a plug & play solution. As a result, if the methodology is correct, by 2025 the School of Architecture will be off-grid. Thus, the followed steps are going to be implemented at the Campus Universidad de Navarra by 2030.

Energy consumption

The first step is the energy audit of the building. The process concerns the construction and building services of the School of Architecture. The electrical consumption (lights, computers, devices and HVAC system) during 2021 has an average of 60 kW, the maximum being 195,76 kW and the minimum 8,52 kW. Figure 7 shows the energy needs during 2021. Adjacent to it, the gas consumption is detailed at figure 8 There is a particular highlight found during the energy audit, the schedule of HVAC machines and temperatures; during the heating period, the air handling unit injected air into the ducts at 60ºC (140ºF), which is a unnecessary (if the system works correctly) and risky action due to burns and flames. Some findings, such as faulty machinery, uninsulated and broken ducts were found as well. More details of HVAC system can be found at the publication "El aire acondicionado como factor de diseño en la arquitectura española: Energía materializada" pag.316-321 (Martín-Gómez 2014), , In conclusion, we could observe and demonstrate the high energy loss due to the inefficient properties of the building's enclosure (lack of insulation, thermal bridges and ancient windows). This result can be extrapolated to other buildings that also have those deficiencies. Furthermore a lack of adequate use of building services, such as poor condition of ducts and machines (Fig. 9) are contributing to this consumption data, forcing the need to overheat the spaces to attain a correct temperature at the building. The system management is the same in the whole campus, so the findings can be applied to other buildings. With demand response actions, for example, adjusting working hours on the schedule, the temperature of heated water, heating temperature, lighting, and conscious actions, it is possible to reduce energy consumption. In this case, the building reached a 23% of electric energy consumption decrease compared to the same year period , 2022-2021, achieved with an investment of zero €; Constructive elements haven't been modified. These actions are based in ASHRAE and the Código Técnico de la Edificación, CTE, (Spanish Construction Technical Code). The conscious actions are: The revision of the use of lighting, implantation of free-cooling, detection of refuge zones, and natural and cross-ventilation to avoid stratification.

The following constructive elements have been evaluated: Lights, walls, windows, and roof. Lights will be changed by LEDs control with presence detectors in representative zones of the building. The budget is 25.000€ . The walls could be improved with blown insulation, not modifying the aesthetic part, and with an approximate cost of 50.000€. Windows can be replaced with ones recommended by the CTE and with a new type of glass but maintaining the original aspect, and an estimated cost of 600.000€. The roof can be prepared to have Photovoltaic systems and absorb NOx, after reaching a better thermal property, which would be the most expensive renovation at an approximate price point of 1.500.000€. These solutions are commercial ones, viable, and their results could be quantified and compared with the ones of the energy audit. The possibility of comparing the results can allow the investigation and development of possible combinations. This control is essential and relevant to knowing if a solution is more or less viable before implementing it to other educational buildings of the Campus of Universidad de Navarra. It is detected that the space distribution is not the optimal one too. Studies explain how a proper space occupation can make users more efficient and consume less energy (Zuazua-Ros et al. 2016). At the School of Architecture, the areas more exposed to sub-optimal interior conditions are those that have the longest period of occupation. It is recommended that offices be distributed on the ground floor and punctual activities on the 3 rd floor.

Renewable energies

Regarding Pamplona's natural resources, (water, sun, and wind), the School of Architecture can generate between 500-800 MWh with photovoltaic panels and 11,2 W/m 2 with wind turbines. Information have been obtained from different budgets required, thus can be extrapolated to the rest of the Campus. The objective of this first approach is to have an array of scalable solutions in one building, the Living Lab. The current market and technology offer a wide range of solutions. Table 1 shows a comparative analysis of each of them from a technical, constructive, and economic point of view.

-Technical feasibility: technological level of development of the solution.

-Constructive feasibility: installation of the system and its problems or barriers.

-Economic feasibility: Comparison of the final cost of the installations. Usually related to the return period.

Technical feasibility

Constructive feasibility

Economic feasibility

Price As it can be seen, the best solutions in our case are photovoltaic, aerothermal, and biomass. Of course, not being viable solution for this building doesn't mean they aren't for others. The following lines show some of the incompatible solutions for this case study: -Production of gas through organic waste: in this case, there is a low quantity of dust (?) production to be affordable.

1 2 3 1 2 3 Wind √ √ √ 160.000€
-Generation with wind energy is discarded due to the airport's proximity, which limits the allowed height of elements.

-Solar thermal production: elevated weight for the existing roof. Regarding storage, the market offers less product diversity, as can be seen in Table 2. The evaluation criteria of it is the same as in Table 1.

Technical feasibility

Constructive feasibility

Economic feasibility

Price 2 is the low economic feasibility of energy storage. The most feasible options are the thermal accumulation and generic storage. The School of Architecture has a big surrounding area and building services allocation that makes it possible to insert a wide range of accumulators, from a buried tank to a series of batteries.

1 2 3 1 2 3 1 2 3 Generical accumulation √ √ √

Conclusion of Table

Control and energy require-production is not developed because the study is still in a theoretical phase.

In any case, the charge cycle will be deeply analyzed to improve energy production maxims, and for that a connection with a weather database is going to be needed. Thus intelligently (?) the energy requires aren't going to be affected thanks to a good control system.

PLUG & PLAY

Last phase of the methodology is the integration and development of a Plug & Play solution. This technologycal result can be extrapolate to the rest of the campus thanks to the spaces that surrounds the buildings. This solution isn't going to affect the buildings aesthetic.

Knowing that compatible solutions of energy productions for the School of Architecture are photovoltaic, biomass and aerothermal the next hypotheses is proposed: Installations of 400kWp of photovoltaic panels to feed building requirements, with the surplus being stored. Photovoltaic panels are going to fulfill the electrical needs of the building during its academic period, and during summer are capable of generating a surplus.

Main energy storage will be done with interseasonal thermal accumulation. This is going to be materialized with a tank of 10.000 m2 that is going to pre-heat the water with the electrical surplus. Of course, it will maintain the energy until the heating period. As a last point, even the feasibility study shows that hydrogen is not the first feasible solution: it is going to be part of the solution. This technology can give a constant and adjustable energy source thanks to its capability of be stored for long periods of time.

The following image show the concept of installing the three technologies around the School of Architecture at Universidad de Navarra. (Fig. 10) Photovoltaic panels at the parking and on the roof of the building, interseasonal thermal accumulation on the back side next to the chiller machines room, and finally the hydrogen in the left back side as an annex to the School of Architecture. These three technologies can complement each other without conflict in this theoretical hypothesis. Even if energy require-production and control are not developed, the idea allows the opportunity to adjust if needed.

CONCLUSIONS

In this paper, we present a methodology developed to analyse the viability to turn a building off-grid. Implemented actions respect the aesthetic aspect of this building but it is not a condition in the case of other buildings, where other implementations can be possible,. In general, existing buildings are complex and there are hundreds of variables that have a significant influence in the behavior of them. A real evaluation of the actual state gives an idea of quickly affordable solutions, especially knowing that the decarbonization process is going to be slow.

A correct space and interior distribution of buildings allows a reduction of demand. So, it is recommended that the distribution of the building of the School of Architecture be changed due to the existing one not being the optimal one, and even being detrimental for its users' health. Thus, a distribution involved from the design point of the building will have better results of comfort and energy consumption (Zuazua-Ros et al. 2016) One key point of the methodology is the development of a control and energy management system, that is going to allow the modification and adjustment of the developed solution.

We conclude that the best solution is the combination of photovoltaic panels with thermal interseasonal accumulation and hydrogen. The idea of the industrialized solution of green energy generator and accumulation can be interesting for society. If this production is managed with an efficient control, the processes can be optimized without affecting the user's comfort.

The working group of researchers and campus managers considers the implementation of plug&play solution next to buildings. Universidad de Navarra's campus offers free spaces that is going to helps into use new technology implantation. If the results of the School of Architecture are favorable, into 2030 the work is going to be developed.

Fig. 1

 1 Fig.1,2. Left image of Pamplona and the Campus Universidad de Navarra. Right, image of School of Architecture.

Fig

  Fig 3, 4. Image of the exterior and interior of the School of Architecture at Universidad de Navarra.

Fig. 6 .

 6 Fig.6. Process of the working scheme of this proposal at the School of Architecture at Universidad de Navarra.

  , needed to boilers. November, 9-11,2022 | Valencia 2022, Universitat Politècnica de València

Fig. 7 .

 7 Fig.7. Electrical consumption during 2021 at School of Architecture. Hourly energy consumption during 2021.

Fig. 8 .

 8 Fig. 8. Gas consumption during 2021 at School of Architecture. Monthly gas consumption during 2021.

Fig. 9 .

 9 Fig. 9. HVAC duct distribution at the School of Architecture. Orange offices, blue workshop, pink classroom and green laboratory.

Fig. 10 :

 10 Fig.10: Draft of technology implantation at the School of Architecture of Universidad de Navarra.

Table 1 .

 1 Table of evaluated solutions of production of energy. 1 not feasible, 2 low feasible, and 3 high feasible.

Table 2 .

 2 Table of evaluated solutions of storage of energy. 1 not feasible, 2 low feasible and 3 high feasible.

	Not available

ACKNOWLEDGEMENTS

Cátedra Universidad de Navarra Fundación Saltoki for the economical afford of this work.