Fuzzy systems theory for decision making in strategic resources management

The Strategic Water Resources Management (SWRM) is inherent in the Systemic approach in Water Resources utilization and protection plans. The multi-criteria approach as well as other approaches such as groups consisting of stakeholders that participate in the ordering and choosing solutions for the future, as well as the complexity of the natural and human systems impose the use of Systemic approach methods and especially multi-criteria methods. The main object of this doctoral thesis is the incorporation of the uncertainty which inheres in the decision-making procedure according to the principles and the methodologies of Fuzzy Logic. The doctoral thesis could be structured in five parts: In the first part, the introduction is included, where some basic concepts and speculations for the SWRM are given, as well as the Systemic approach emphasizing to the multi-criteria analysis, a first approach of the uncertainty and also the orientations and objects of the doctoral thesis. In the second part, an extended introduction to the fuzzy sets, the fuzzy numbers and the arithmetic operations on fuzzy numbers, the axiomatic basis of fundamental operations of fuzzy sets and fundamental principles of fuzzy logic compared to respective principles of classic logic is made. Furthermore, comparison methods of the fuzzy numbers are deployed. The development of the fuzzy logic principles took place having as main criterion the applications which were built up next. The third part of the thesis is referring to fuzzy methods of multi-criteria decision aid. The third part could be compartmentalized into methods development using fuzzy logic and sets, which conclude in finding the optimum solutions, and into fuzzy methods, which conclude in the comparison of discernible alternatives. In this part, main emphasis was given to the development of fuzzy methods in multi-criteria decision making orientating towards the SWRM cases. In the framework of the doctoral thesis, according to the existing methods of multi-criteria analysis fuzzy methods were developed, which were compared with the existing ones and original conclusions were proposed. Subsequently, based on the conclusions, new innovative methods were developed. The doctoral thesis focused on the multi-criteria analysis trends, such as the Multi-criteria mathematical programming - especially the flexible fuzzy programming and the possibilistic programming - and the multi-criteria comparsion of discernible alternatives - especially the Multi-criteria with the value function approach, emphasizing on the proposed innovative multi-level composition model, the multi criteria decision aid fuzzy principles, the distance based methods and the outranking relation theory (French school), in which an assiduous development was carried out. In the same time, a new method with fuzzy opinion aggregation in group decision making problems was examined. In the fourth part, various characteristic cases of SWRM are cited, where for the first time in the terms of the doctoral thesis (not supported by referenced examples), the mentioned methods from the References or proposed innovative methods were tested. In fact, the problems themselves drove mainly the choice of methods, the conclusions and in many cases the new innovative methods. In many cases Matlab software were developed in order to apply the proposed methods. In this way methodologies in characteristic cases of SWRM are produced. In the last part the conclusions of the thesis and the future research orientations are presented. The methodologies developed encounter the following characteristic cases of SWRM: - Design of pressured irrigation network by the innovative use of flexible integer programming, adopting fuzzy constraints for the pressure head and integer switch length constraints. - Multi water resources management in the agricultural sector by the use of fuzzy goal programming, for the conjunctive use of surface waters and groundwater, taking into consideration the socioeconomic criterion. The proposed by the doctoral thesis innovative hybrid approach while a compensatory approach of goal programming is applied. - Water Resources Management scenario development in over pumped groundwater table and classification, with the case study of the groundwater table of Velestino. Scenarios with optimization techniques are formulated, where the use of flexible programming, which meets the goals conflict and the fuzzy constraints in order to produce a compromise solution, is examined. The scenarios formulated are ordered according to the fuzzy multi-criteria classification techniques based upon the binary comparisons and the scoring functions. - Possibilistic model of strategic management in water shortage conditions, adopting the possibilistic programming with fuzzy numbers, giving the possibility to adaptive decisions for extreme hydrological conditions. A timely matter that is highlighted is the interpretation method of the fuzzy inequalities, where various models are formulated in the framework of the doctoral thesis. - Examination of the karstic connection between two water basins by applying the fuzzy linear regression (FLR) with modifications for the FLR model, which were proposed in the framework of the doctoral thesis. - Rational Programming of Deficit Irrigation using Fuzzy Logic, adopting the proposed model of multi-level composition and the choice of appropriate fuzzy operators. For the last level of the composition the use of binary comparisons based on the scoring functions is also examined. - Estimation of the Water Potential of a Karstic Saline Source, case study: the karstic saline source of Almyros (Heraklion - Crete). Critical parameter is proved to be the water source quality, which is confronted with multi-criteria analysis. For the determination of the source potential the min - cross section methodology for potable water is used as well as the fuzzy classification in predefined categories methodology. The selected methodologies refer to a wide range of characteristic cases which arise in SWRM, such as water distribution systems, conjunctive use of surface water and groundwater in the agricultural sector, deficit irrigations, case of depleted water table, water distribution in extreme conditions, saline sources, karstic water basins interconnection and aggregate water management for an integrated section. The main conclusions are presented next, which constitute the contribution of the doctoral thesis in the progress of the scientific research. The flexible linear integer programming application on the pressured irrigation network planning, taking into consideration the innovative approach of fuzzy constraints for the pressure head, instead of crisp constraints, ensures a flexible and less expensive planning. The fuzzy efficient solution concept may be used for the formulation of balanced scenarios in SWRM, given the multiplicity of the goals that appear in SWRM as well as the fuzzy constraints such as the case of scenario development for an overpumped aquifer. Fuzzy goal programming methodologies for the management of multiple water resources in the agricultural sector are proposed. The innovation consists in the adaptation of fuzzy method goal programming for the conjunctive utilization of surface water and groundwater in the agricultural sector, where apart from the criterion of water consumption compared to the monthly water availability, socioeconomic criteria are also taken into consideration. Complex models of multi-level composition are proposed, while apart from the other methods, a mixed technique of goal programming and fuzzy approach which accomplishes balanced solutions was invented, while, in the same time, weights in the last level of composition are avoided. An innovative methodology for the flexible strategic water management was developed in water shortage conditions, in the aid of possibilistic programming. The fact that the decision variables are Fuzzy Symmetrical Triangular Numbers (FSTN), giving the possibility to have an adaptable decision in the changing conditions of water availability, constituted an innovative choice. It was also noted that during the application of the possibilistic programming, the feasible solution range depends on the interpretation mode of the fuzzy inequalities. Moreover, innovative techniques were proposed and their advantages and conditions of application in regard to the water availability nodes were investigated. The fuzzy linear regression model may be applied for the interpretation and interconnection between two water basins effort by applying an elementary water balance. The simple water balance model was proposed to determinate the fuzzy linear regression model by modifying the objective function and adopting non-negativity constraints for the fuzzy coefficients. The proposed innovative model of fuzzy multi-level composition may be applied for the rational deficit irrigation programming by the conceptual choice of appropriate fuzzy operators. In this way, a holistic dealing for the deficit irrigation programming is achieved, which also incorporates the fuzziness in the deficit irrigation production. Subsequently, for the last level of the composition the use of binary comparisons is used according to the score functions. Useful observations derived for the fuzzy multilevel composition model and the fuzzy operators use in the several levels of the model, as well as for the critical condition of the choice of a fuzzy operator at the last composition level. Alternatively the construction of complex non linear value functions was proposed in the aid of the formulation of fuzzy multi-criteria rules. The fuzzy rules may also be applied in others not precisely determined concepts and procedures. However, the fuzzy operators use in complex multi-criteria cases in more complex systems is rather simplifying and not reliable. On the other hand, the existence of multi-criteria presupposes an important number of fuzzy rules, for which many doubts may exist for their mathematical simulation, but especially in Greece we are short of data for the documentation of such a learning procedure. Therefore, from the theoretic analysis as well as the multi-criteria methods application in the SWRM cases, which are in their nature multi-criteria, it is evident that the fuzzy methods based on the French school achieve a more holistic dealing, ensuring more the balanced satisfaction of the criteria. The proposed from this doctoral thesis multi-level composition model cannot give reliable results in complex cases with a big criteria number. The proposed decision aid system model, consisting of fuzzy rules is not easily applicable in Greece due to the SWRM systems complexity and the experience shortage for integrated water resources management. The distance based methods cannot incorporate the balanced criteria objective, whereas the definition of ideal and anti-ideal point has a very important role in the final result, a choice which cannot be but absolutely subjective. The veto right as well as the implication rule for the whole composition, and also the binary comparison nature stand in diminishing the subjectivity in decision making and convey more rational decisions. It should be mentioned, following the text's coherence, that from the recent bibliography review it derives that the French school methods are based on binary comparisons, while they incorporate many fuzzy logic concepts. In the aid of the fuzzy logic and sets, the strict preference and indifference is founded as a function of outranking relation. Moreover the overall composition of concordance and discordance is rationalized using fuzzy operators. In the doctoral thesis framework, the final veto index was proposed, derived from the use of wider approaches for the fuzzy negation or alternatively from the use of fuzzy implications which emphasize on the veto right. The interpretation and generalization of the French school in multi-criteria with the use of fuzzy logic is timely for the exploitation of outranking relations, i.e. for the extraction of final conclusions. It was ascertained that the exploitation of outranking relations was simpler using fuzzy logic in regard to the classic methods, since the strict preference and indifference relation is fuzzily defined and not in the classic interpretation way used by Electre for the strict preference and indifference. An innovative model was proposed based on the rule of "negation of the implication that the rule of the criteria majority implies the overall discordance" for the overall composition. In short, an alternative with high compensative performance with respective imbalanced criteria satisfaction is cancelled. Actually, it was noted that in regard to other approaches, the implication rule approach for the outranking relation and by applying the proposed scoring function approach of net flows gave preference in the most compromising scenario that derived from the fuzzy linear multi-criteria programming. Oftentimes, the economic or another criterion is drawn to high solutions which do not ensure though the balanced development as well as the environmental goals. The implication composition in the SWRM cases ensures in a strict way the achievement of balanced solutions. A way to exploit the fuzzy outranking relations is the classification in predetermined categories, which is distinguished between ordered categories and non- ordered categories. In the framework of the doctoral thesis, it was noted during the application that it is closer to reality and more interpretative choice that the preference and equivalence thresholds to be defined as a function of the thresholds for each virtual alternative (filter). The use of strict preference for the sorting in ordered categories was also justified. According to the already mentioned points, innovative multi-criteria methodologies for the karstic saline source water potential estimation of a variable quality were developed. The source's water quality is proved to be a critical parameter, the estimation of which is confronted by multi-criteria approach. For the determination of the source potential the appropriate participation and min-cross section function is used, especially for the potable water and the fuzzy classification methodology into predetermined categories for the water destined for irrigation use incorporating the fuzzy veto rules of the French school. The fuzziness arises from the uncertainty in the standards comparison with the samples or in the quality range between the Maximum Acceptable Concentration (MAC) and the Guide Value (GV). The min cross section approach is more appropriate for the estimation of quantities for potable water, while the multi-criteria method is more flexible for the determination of water which is meant for irrigation. Two alternatives integrated innovative methodologies were formulated for the exploitation of the fuzzy outranking relations, that is the classification according to the concept of nondomination degree and integrated value functions. The use of scoring functions incorporating the veto right even in the case of fuzzy data was proposed, while the overall composition derived from the fuzzy logic and is the one that determines the flows. The preference to net flow of outranking relation for the scoring functions was documented by the characteristic theorems demonstration. Alternatives ranking was invented according to the flow of nondomination degree, which, as it was proved theoretically and through the applications, is appropriate only for a first selection of the alternatives. The scoring function method may drive in the development of an innovative programming model with variables 0/1 for the objective function, and then by the appropriate constraints adoption a model that determines a set of alternatives in water resources system is formulated. For the issue's optics examination the implication of many groups in multi-criteria classification is proposed. An innovative method of fuzzy "Delphi" was developed, i.e. a repeating procedure of weights or other parameters in the form of fuzzy numbers determination (for the multi-criteria approach of SWRM) and the use of similitude measures.Η Στρατηγική Διαχείριση Υδατικών Πόρων είναι σύμφυτη με τη Συστημική θεώρηση σε έργα αξιοποίησης και προστασίας Υδατικών Πόρων. Η προσέγγιση με πολλαπλά κριτήρια αλλά και ευρύτερες θεωρήσεις όπως, οι πολλαπλές ομάδες ενδιαφερομένων που συμμετέχουν στην ιεράρχηση και επιλογή των λύσεων για το μέλλον αλλά και η περιπλοκότητα των φυσικών και ανθρώπινων συστημάτων επιβάλλουν τη χρήση μεθόδων Συστημικής θεωρίας και ειδικότερα πολυκριτηριακών μεθόδων. Ο κύριος σκοπός της Διδακτορικής Διατριβής είναι η ενσωμάτωση στη θεωρία της Πολυκριτηριακής Ανάλυσης αλλά και των άλλων μεθόδων της Συστημικής θεωρίας της αβεβαιότητας που ενυπάρχει σε όλη τη διαδικασία λήψης αποφάσεων με βάση τις αρχές και τις μεθοδολογίες της Ασαφούς Λογικής. Η διδακτορική διατριβή θα μπορούσε να χωρισθεί σε πέντε μέρη: Στο πρώτο μέρος περιλαμβάνεται η εισαγωγή όπου δίνονται μερικές βασικές έννοιες και προβληματισμοί για τη ΣΔΥΠ, τη Συστημική θεώρηση με έμφαση στην πολυκριτηριακή ανάλυση, μια πρώτη προσέγγιση της αβεβαιότητας καθώς και οι προσανατολισμοί και στόχοι της διδακτορικής διατριβής Στο δεύτερο μέρος γίνεται μία εκτεταμένη εισαγωγή στα ασαφή σύνολα, τους ασαφείς αριθμούς και την ασαφή αριθμητική, την αξιωματική θεμελίωση βασικών πράξεων των ασαφών συνόλων και βασικών αρχών της ασαφούς λογικής σε σύγκριση με αντίστοιχες αρχές της κλασικής λογικής. Επίσης, αναπτύσσονται τρόποι σύγκρισης των ασαφών αριθμών. Η ανάπτυξη των αρχών της ασαφούς λογικής έγινε με γνώμονα τις εφαρμογές που αναπτύχθηκαν στη συνέχεια. Το τρίτο μέρος της διατριβής αναφέρεται σε ασαφείς μεθόδους πολυκριτηριακής στήριξης αποφάσεων. Το τρίτο μέρος θα μπορούσε να διαχωριστεί σε ανάπτυξη μεθόδων με τη βοήθεια της ασαφούς λογικής και συνόλων, που καταλήγουν στην εύρεση της βέλτιστης λύσης και σε ασαφείς μεθόδους που καταλήγουν στην ταξινόμηση διακριτών εναλλακτικών. Σε αυτό το μέρος κύρια έμφαση δόθηκε στην ανάπτυξη ασαφών μεθόδων στη λήψη αποφάσεων με πολυκριτήρια με προσανατολισμό τα προβλήματα της ΣΔΥΠ. Στα πλαίσια της διδακτορικής διατριβής με βάση τις υπάρχουσες μεθόδους της πολυκριτηριακής ανάλυσης έγινε ανάπτυξη ασαφών μεθόδων, οι οποίες συγκρίθηκαν με τις υπάρχουσες μεθόδους και προτάθηκαν πρωτότυπα συμπεράσματα και κατόπιν, με βάση τα συμπεράσματα, έγινε ανάπτυξη νέων καινοτόμων μεθόδων. Τα ρεύματα της πολυκριτηριακής ανάλυσης στα οποία επικεντρώθηκε η διδακτορική διατριβή είναι ο Πολυκριτηριακός μαθηματικός προγραμματισμός - με υποκατηγοριοποίηση τον εύκαμπτο ασαφή προγραμματισμό και τον προγραμματισμό δυνατοτήτων - και η πολυκριτηριακή ταξινόμηση διακριτών εναλλακτικών - με υποκατηγοροποποίηση την Πολυκριτηριακή θεωρία ολικού κριτηρίου με ιδιαίτερη έμφαση στο προτεινόμενο καινοτόμο μοντέλο της πολυεπίπεδης σύνθεσης, τους ασαφείς κανόνες στήριξης αποφάσεων με πολλαπλά κριτήρια, τις μεθόδους αποστάσεων και τη θεωρία των σχέσεων υπεροχής (Γαλλική σχολή) στις οποίες έγινε ενδελεχή μελέτη. Παράλληλα, διερευνήθηκε η ανάλυση αποφάσεων με πολλαπλούς παίκτες και πολλαπλά κριτήρια. Στο τέταρτο μέρος παρατίθενται διάφορα χαρακτηριστικά προβλήματα της ΣΔΥΠ, όπου για πρώτη φορά στα πλαίσια της διδακτορικής διατριβής (δε στηρίχθηκαν σε παραδείγματα από τη βιβλιογραφία), δοκιμάσθηκαν οι αναφερόμενες από τη βιβλιογραφία ή προτεινόμενες καινοτόμες μέθοδοι. Στην πραγματικότητα τα προβλήματα τα ίδια καθοδήγησαν σε μεγάλο βαθμό την επιλογή των μεθόδων, τα συμπεράσματα και πολλές φορές τις καινοτόμες νέες μεθόδους. Σε πολλές περιπτώσεις αναπτύχθηκε λογισμικό σε γλώσσα Matlab προκειμένου να εφαρμοσθούν οι προτεινόμενες μέθοδοι. Με αυτόν τον τρόπο παράγονται μεθοδολογίες σε χαρακτηριστικά προβλήματα της Σ.Δ.Υ.Π. Στο τελευταίο μέρος ανήκουν τα συμπεράσματα της διατριβής και οι μελλοντικές κατευθύνσεις για έρευνα. Οι μεθοδολογίες που ανεπτύχθησαν αντιμετωπίζουν τα παρακάτω χαρακτηριστικά προβλήματα της Σ.Δ.Υ.Π: - Σχεδιασμός αρδευτικού δικτύου υπό πίεση με την καινοτόμο χρήση του ασαφούς (εύκαμπτου) ακέραιου προγραμματισμού, υιοθετώντας ασαφείς περιορισμούς για το ύψος πίεσης και περιορισμούς ακέραιους διακόπτες. - Διαχείριση πολλαπλών υδατικών πόρτου στον αγροτικό τομέα, με τη χρήση του ασαφούς προγραμματισμού των στόχων, για τη συνδυασμένη χρήση επιφανειακών και υπογείων νερών, λαμβάνοντας υπ' όψιν και το κοινωνικό - οικονομικό κριτήριο. Εφαρμόζεται η προτεινόμενη από τη διδακτορική διατριβή καινοτόμος υβριδική προσέγγιση και η αντισταθμιστική προσέγγιση του προγραμματισμού των στόχων. - Ανάπτυξη σεναρίων Δ. Υ.Π. σε υπεραντλούμενο υδροφόρο ορίζοντα και ταξινόμησή τους, με περίπτωση εφαρμογής τον υδροφόρο ορίζοντα του Βελεστίνου. Διαμορφώνονται σενάρια με τεχνικές βελτιστοποίησης όπου εκτός των άλλων, εξετάζεται η χρήση του εύκαμπτου προγραμματισμού που αντιμετωπίζει τη σύγκρουση των στόχων και την ασάφεια στους περιορισμούς προκειμένου να παραχθεί συμβιβαστική λύση. Τα σενάρια που διαμορφώθηκαν ταξινομούνται με τεχνικές ασαφούς πολυκριτηριακής ταξινόμησης με βάση τις διμερείς συγκρίσεις και τις συναρτήσεις βαθμολόγησης. - Ευέλικτο μοντέλο στρατηγικής διαχείρισης σε συνθήκες έλλειψης νερού, υιοθετώντας τον προγραμματισμό δυνατοτήτων με τη δυνατότητα οι μεταβλητές απόφασης να είναι ασαφείς αριθμοί, δίνοντας τη δυνατότητα σε ευέλικτες αποφάσεις για ακραίες υδρολογικές συνθήκες. Καίριο θέμα που αναδεικνύεται είναι ο τρόπος ερμηνείας των ασαφών ανισοτήτων, οπότε διαμορφώνονται στα πλαίσια της διδακτορικής διατριβής διάφορα μοντέλα. - Διερεύνηση της καρστικής διασύνδεσης μεταξύ δύο υδρολογικών λεκανών εφαρμόζοντας την ασαφή γραμμική παλινδρόμηση με τροποποιήσεις για το μοντέλο της α.γ.π. που