Approximation and optimization of discrete and differential inclusions

[No abstract available]Publisher's Versio

ISTANBUL’S THIRD AIRPORT IN TERMS OF TRANSPORTATION GEOGRAPHY: GEOPOLITICS, REGIONAL AND ECONOMIC EFFECTS

More passengers have come to use airplanes in parallel with the developing aircraft technology. The remarkable and rapid development of commercial airlines after the World War II positively affected the costs to turn airborne transportation into a global activity taken up by masses. Doubtlessly, immense is the effect of developing airborne transportation on the emergence and dissemination of “globalization” herein.As one of the major components of airborne transportation, airports serve as spaces allowing for direct and connecting flights in a short time and make considerable contributions to the economy of the country they are located in. A region’s airborne accessibility enables the transportation of any goods and services to that particular region in a short while. Thus, accessibility is indispensable to the growth of a region’s economy. Taking the abovementioned as a premise, some airports have acquired new functions not only to regulate ordinary flight services and airborne transportation but also to turn into hubs to orchestrate interregional transportation. Such airports also function as tourism embassies.  Among the hubs are the airports in Dubai, Bangkok, Frankfurt, London, Paris, and New York. In recent years, Istanbul has set out to be listed among the above hubs thanks to its geographical location and developing economy. Istanbul’s Third Airport, which has reached almost 70% of its construction and will come into service soon, has been designed to be the largest airport of the world.  Istanbul is distinguished from the metropolitans listed above for the historical role it played and the diversity of its historical and cultural landmarks. Throughout its history of around 8000 years, Istanbul has served as a capital to the Roman, Byzantine, and Ottoman Empires, which exerted substantial influences on the world history, and is teemed with cultural heritage of these civilizations. Every year, the city hosts some 12 million tourists. The present study discusses the probable global effects of Istanbul’s Third Airport under construction in terms of transportation geography and its geopolitical, regional, and economic effects in consideration of its projected mission

Multicriteria analysis of technological innovation investments using fuzzy sets

Innovation is the way of transforming the resources of a company through the creativity of people into new resources and wealth. Innovation investments are essential instruments in a company’s competitive productivity and profitability strategy. Evaluation of innovation investments is a multicriteria decision making problem with many conflicting tangible and intangible criteria. Vague nature of this evaluation requires a fuzzy multicriteria methodology. In this paper we propose a fuzzy multicriteria method to evaluate technological innovation investments using eight different criteria. Fuzzy TOPSIS method is used in this evaluation and a sensitivity analysis is given. First published online: 07 Sep 201

A comparison of fuzzy multicriteria decision making methods for intelligent building assessment

The methodology, Multicriteria Decision Making (MCDM), refers to finding the best alternative from all of the feasible alternatives in the presence of multiple, usually conflicting, decision criteria. Nowadays, intelligent buildings’ performance that is increasingly evidenced in building design and construction has been analyzed by using MCDM techniques. Intelligent buildings (IBs) are also under assessment according to their IB related characteristics and actual circumstances as a MCDM problem. In this paper, two most known MCDM methodologies, Analytic Hierarchy Process (AHP) and TOPSIS (Technique for Order Preference by Similarity to Ideal Solution), are used for intelligent building assessment under fuzzy environment for dealing with the evaluations’ uncertainty and imprecision in which the expert's comparisons that are represented as fuzzy numbers. For this aim, three intelligent building alternatives for a business centre in Istanbul are evaluated by using these two fuzzy MCDM methods and the obtained ranking results are compared

Display design of passenger cars based on fuzzy axiomatic design

Aksiyomlarla Tasarım (AT) yöntemi, tasarım süreçleri için bilimsel ve sistematik bir temel oluşturmak amacıyla geliştirilmiş bir yöntemdir. Yöntem “bağımsızlık aksiyomu” ve “bilgi aksiyomu” olmak üzere iki aksiyom içerir. Yöntemin ana amacı; tasarımcıları daha yaratıcı yapmak, rastsal tarama süreçlerini azaltmak, deneme yanılma süreçlerini minimize etmek ve en iyi tasarımın seçimine yardımcı olmaktır. Yöntem, literatürde ürün tasarımından sistem tasarımına kadar geniş bir uygulama alanına sahiptir. Bu çalışmada, tasarım aksiyomlarının bulanıklaştırılarak uygulama alanlarının genişletilmesi hedeflenmiştir. Bağımsızlık aksiyomunun bulanıklaştırılması tasarım matrisindeki zayıf ve güçlü ilişkilerin dikkate alınmasına, farklı tasarım parametreleriyle oluşturulan tasarımlardan iyi olanının seçimine ve tasarım parametrelerinin önem derecelerinin belirlenmesine yardımcı olur. Çalışmada ayrıca, sadece seçim amaçlı kullanılan bilgi aksiyomu, çeşitli karar verme problemlerinin çözümünü içerecek şekilde geliştirilmiştir. Karar verme problemleri, beklenen değer, kesin değer ve sıralama problemleri olarak çeşitli sınıflara ayrılmış ve bilgi aksiyomunun kullanımı için problem türlerine göre fonksiyonel gereksinim tanımları yapılmıştır. Böylece, geliştirilen bilgi aksiyomu, karar vericinin beklentisini dikkate alan ve çeşitli karar problemlerine uygulanabilen etkin bir karar verme aracı haline dönüştürülmüştür. Ayrıca yöntem, aynı problemde yer alan rasyonel değerlendirmeleri de dikkate alarak çözüm sunabilecek şekilde geliştirilmiştir. Bir diğer iyileştirme ise; sadece kıyaslama aracı olarak kullanılan tasarıma ait bilgi içeriği değeri çalışma kapsamında geliştirilerek tasarıma ait başarım oranı tanımı yapılmasıdır. Geliştirilen yöntem binek otomobiller için gösterge tasarımına uygulanmıştır. Anahtar Kelimeler: Aksiyomlarla tasarım, bulanık mantık, gösterge tasarımı.Axiomatic Design (AD) has been proposed as a scientific and systematic basis providing structures to design processes for engineers. The primarily goal of axiomatic design is to make designers more creative, to reduce the random search process, to minimize the iterative trial-and-error process, and to determine the best designs among the proposed designs. The axiomatic approach uses two axioms named as "independence axiom" and "information axiom" to improve a design. The first axiom, independence axiom, states that the independence of functional requirements (FRs) must always be maintained, where FRs are defined as the minimum set of independent requirements that characterizes the design goals. Then, the second axiom, information axiom, states that the design having the smallest information content is the best design among those designs that satisfy the independence axiom (Suh, 2001). Axiomatic design methodology has been applied to various application areas since it was proposed to literature. Some successful applications of AD methodology are as follows: Software Design, System Design, Manufacturing System Design, and Product Design. Besides of these application studies, there are some other studies including theoretical developments in the literature.In the scope of this study, following contributions are presented; The first contribution of the study is that the relation between functional requirements and design parameters in the independence axiom has been defined by the degree of the relation. However, in the conventional axiomatic design, the relation between is symbolized by 0 or 1 such that 1 represents a relation and 0 represents no relation. Therefore, the conventional axiomatic design methodology does not take into consideration the weak relations or the design is accepted as coupled design because of weak relations. The grading of the relations helps designers to recognize design problems caused by weak relations since the developed methodology can take into consideration even weak relations. Hence, it is easily determined whether a design works properly. Moreover, the importances of the design parameters are put forward by using the developed independence axiom. Thus, the design parameters which mostly satisfy functional requirements are determined. Furthermore, the quality of a design is defined by the functional independences and dependences belonging to design matrix. Thus, the comparison of the design matrices which consist of different design parameters can be done. The second contribution of the study is that the fuzzy information axiom has been improved to be used for the solution of all types of multiciriteria decision making problems. In this perspective, multicriteria decision making problems have been classified as exact value problems, expect value problems, and ranking problems. The definition of functional requirements has been also defined based on the problem types. Hence, the developed information axiom has been used an as effective decision making tool which takes into consideration decision makers' expectations and can be used for the solution of all multicriteria decision making problems. Moreover, the information axiom has been developed to present a solution for the crisp evaluation symbolized by a real number. Furthermore, the definition of design satisfaction ratio has been proposed based on information content to explain the satisfaction level of a design while information content values obtained from information axiom have been used only as a comparison and selection tool. The third contribution of the study is that the proposed algorithm is applied to display design of passenger cars in terms of ergonomics issues. In the application, the design parameters for display design are determined and importances of the design parameters are defined. Moreover, the characteristics for display design have been presented for Turkish people by a test study. As a further aspect, the scope of the developed algorithm can be extended by using importances of the design parameters to optimize design dimensions. For instance, control buttons and air passages on the center of the dashboard can be evaluated based on their locations. Keywords: Axiomatic design, fuzzy logic, display design.

Bulanık karar ortamında karınca kolonisi optimizasyonu yöntemiyle araç rotalama

Many companies use fleets of vehicles within their activity in a range of sectors. More often than not, meeting customers' requirements, taking into account their geographical spread and delivery time windows, as well as managing the company's operating and financial constraints, turns into a nightmare. The vehicle routing problem with time windows (VRPTW) is a well-known and complex combinatorial optimization problem concerned with finding efficient routes, beginning and ending at a central depot, for a fleet of identical vehicles to serve a number of customers with capacity and time window constraints where each customer is visited exactly once by a vehicle. The capacity constraint signifies that the total load on a route cannot exceed the capacity of the assigned vehicle. The time window constraint signifies that each vehicle must start the service each customer in the period specified by the customer. The objective is to find the feasible solution (hierarchically or not) with minimal number of vehicles or with the minimal total distance. VRPTW has been a subject of intensive research focused mainly on the heuristic and the metaheuristic approaches. The VRPTW is still one of the most difficult problems in combinatorial optimization and this problem contributes directly to a real opportunity to reduce costs in the important area of logistics. Transportation management, and more specifically the vehicle routing, has a considerable economical impact on all logistic systems. However, the classical definitions of the vehicle routing problem often lack handling of uncertain parameters and flexible constraints such as the traveling times between customers and the latest delivery times for the custoners. In addition, a best/optimal solution generated by a heuristic/exact method, for the classical VRPTW do not mean any knowledge to the user about its realization when applicated. Whereas, the solutions generated with the classical models usually became infeasible when implemented and the planners are often involved to make corrections by hand. The natural approach to modeling the uncertainty is a stochastic one. Unfortunately, the stochastic models are often hard to solve. Moreover, it may be hard or expensive to assume any speci?c probability distributions for the unknown parameters. For these reasons stochastic models are behind the needs of users. Up to past decades, operational research methods seem to be inadequate for the large sized combinatorial optimization problems due to their large computational effort and long solution time. But, recent developments in data processing and communication technologies and recently proposed metaheuristic methods that can generate solutions to large sized combinatorial problems made these classical disadvantages less important for the researchers. Subsequently, validity of used models became a more important issue for the researchers. In this study, the fuzzy set and the possibility theories are utilized in order to propose a fuzzy robust programming model for the VRPTW that can be used in the uncertain decision environments. The fuzzy programming model proposed in this study exploit fuzzy sets and fuzzy intervals in order to model flexibilities on latest delivery times and uncertainties for traveling times between customers. A necessity measure is used to generate knowledge about the satisfaction of delivery time constraints when a route is realized by the vehicle. In addition, a possibility measure is used to evaluate the customer satisfaction levels. Using the necessity and the possibility measures, solutions that have the maximum risk level to be unfeasible and the minimum customer satisfaction, which are specified by the user, can be generated. Validities of the models are increased while decreasing the computational effort with fuzzy programming models. In order to generate solutions for the proposed model an ant colony optimization based algorithm is developed. The algorithm is capable of solving both the classical and the fuzzy programming models for the VRPTW. Results of the experimental studies with the benchmark problems indicate that the proposed fuzzy programming model and the ant colony optimization algorithm can be usable for solving practical problems in means of solution time and quality. The proposed approach can be integrated with a decision support system in order to generate alternative solutions achieving the planners' and the customers' preferences along with acquiring more information about the realization of the solutions. Keywords: Fuzzy sets, possibility theory, vehicle routing problem, ant colony optimization. Bu çalışmada, bulanık kümeler ve olabilirlik teorilerinden faydalanılarak zaman aralıklı araç rotalama problemi (ZAARP) için bulanık karar ortamında kullanılabilecek bir gürbüz bulanık programlama modeli önerilmiştir. Geçmiş çalışmalar incelendiğinde ZAARP'nin genellikle parametrelerindeki belirsizliklerin ve kısıtlarındaki esnekliklerin göz ardı edilerek modellendiği görülmüştür. Bu tip modellere üretilen çözümler uygulama aşamasında çoğunlukla geçerliliklerini yitirmekte ve kullanıcılar tarafından elle düzeltilmeyi gerektirmektedirler. Stokastik modellerin kullanıldığı çalışmalarda ise önerilen modellerin çok fazla hesaplama yükü gerektirdiği ve parametrelerinin belirlenmesi için problemle ilgili geçmiş verilere ihtiyaç duyulduğu görülmektedir. Bu nedenlerle stokastik modeller de gerçek hayatta karşılaşılan problemlerin çözümünde rahatlıkla kullanılamamaktadır. Bu çalışmada belirsizliklerin ve esnekliklerin modellenmesi için bulanık aralıklar ve bulanık kümeler kullanılmıştır. Gereklilik ve olabilirlik ölçütleri ile planlayıcının belirlediği eşik güvenilirlik ve müşteri tatmini düzeylerine sahip çözümler oluşturulmuştur. Bulanık programlama modeli ile yüksek veri işleme maliyeti düşürülürken modellerin geçerlilikleri de arttırılmıştır. Önerilen modellere çözüm oluşturmak amacıyla karınca kolonisi optimizasyonu tabanlı bir algoritma geliştirilmiştir. Bulanık değerler kesin değerlerin genelleştirilmiş hali olarak ele alındığından geliştirilen algoritma kesin veri ve/veya bulanık veri durumlarında çözüm oluşturabilmektedir. Örnek problemler üzerinde gerçekleştirilen deneylerde planlayıcıların tercih ve önceliklerine göre alternatif çözümlerin üretilebileceği ve oluşturulan çözümler hakkında planlayıcılara ve müşterilere daha fazla bilgi sağlanabileceği gösterilmiştir. Anahtar Kelimeler: Bulanık kümeler, olabilirlik teorisi, araç rotalama problemi, karınca kolonisi optimizasyonu

Development of fuzzy process control charts: Direct fuzzy approach

Klasik kontrol diyagramları, W.A. Shewhart tarafından 1920’lerde geliştirilmiş olmasına rağmen yeni uygulama alanları ile günümüzde hala gelişimini sürdürmektedir. Verilerin tam ve kesin olduğu durumlarda klasik kontrol diyagramlarının kullanılması uygundur; ancak subjektifliğin önemli bir rol oynadığı bazı durumlarda bu kadar kesin verilere sahip olmak neredeyse imkânsızdır. Belirsizlik altındaki durumlarda karar analizleri genellikle olasılık teorisi ve/veya bulanık kümeler teorisi kullanılarak yapılmaktadır. Bunlardan birincisi karar vermenin stokastik yapısını diğeri ise insanın düşüncesinin subjektifliğini temsil eder. Bulanık kümeler teorisi, ne rassal ne de stokastik olan insanın zihinsel yapısından kaynaklanan belirsizliğin modellenmesinde mükemmeldir. Belirsiz, kesin olmayan veya dilsel anlatımlar içeren durumlarda bulanık kümeler teorisinin kullanılması kaçınılmazdır. Bu çalışmada, bulanık kümeler teorisi kullanılarak belirsizlik içeren dilsel verilerle kontrol diyagramlarına yeni yaklaşımlar geliştirilmiştir. Belirsizlik içeren dilsel veriler, bulanık sayılarla ifade edilmiştir. Dilsel veriler için bulanık kontrol diyagramları α-kesim yaklaşımı kullanılarak geliştirilmiş ve bu suretle muayene sıklığı tanımlanmıştır. Veri ve kontrol limitlerinin temsili değerler ile klasik biçime (nümerik değerlere) dönüştürülmesi sonucu taşıdığı bilgiler yitirilmektedir. Bulanık kontrol diyagramlarının oluşturulmasında, bulanık verilerin taşıdığı bilgilerin kaybolmasını önlemek amacıyla “Direkt Bulanık Yaklaşım” geliştirilmiştir. Bu yaklaşımda veriler bulanık sayılarla ifade edilmiş ve temsili değerler kullanılmadan kontrol limitleri de bulanık sayılar olarak hesaplanmıştır. Kontrol altında, kontrol dışında kararlarına ek olarak kısmen kontrol altında, kısmen kontrol dışında gibi ara kararlar geliştirilmiştir.  Anahtar Kelimeler: Bulanık proses kontrol diyagramları, bulanık kümeler, dilsel veriler, normal olmayan davranış analizi, belirsizlik. Control charts have been widely used for monitoring process stability and capability. Control charts are based on data representing one or several quality-related characteristics of the product or service. If these characteristics are measurable on numerical scales, then variable control charts are used. If the quality-related characteristics cannot be easily represented in numerical form, then attribute control charts are useful. Even though the first classical control chart was proposed during the 1920's by W.A. Shewhart, today they are still subject to new application areas that deserve further attention. Classical process control charts are suitable when the data are exactly known and precise; but in some cases, it is nearly impossible to have such strict data if human subjectivity plays an important role. It is not surprising that uncertainty exists in the human world. To survive in our world, we are engaged in making decisions, managing and analyzing information, as well as predicting future events. All of these activities utilize information that is available and help us try to cope with information that is not. A rational approach toward decision-making should take human subjectivity into account, rather than employing only objective probability measures. A research work incorporating uncertainty into decision analysis is basically done through the probability theory and/or the fuzzy set theory. The former represents the stochastic nature of decision analysis while the latter captures the subjectivity of human behavior. The fuzzy set theory is a perfect means for modeling uncertainty (or imprecision) arising from mental phenomena which is neither random nor stochastic. Many problems in scientific investigation generate nonprecise data incorporating nonstatistical uncertainty. A nonprecise observation of a quantitative variable can be described by a special type of membership function defined on the set of all real numbers called a fuzzy number or a fuzzy interval. A methodology for constructing control charts is proposed when the quality characteristics are vague, uncertain, incomplete or linguistically defined. The binary classification into conforming and nonconforming used in the p-chart might not be appropriate in many situations where product quality does not change abruptly from satisfactory to worthless, and there might be a number of intermediate levels. Without fully utilizing such intermediate information, the use of the p-chart usually results in poorer performance than that of the x-chart. This is evidenced by weaker detectability of process shifts and other abnormal conditions such as unnatural patterns. To supplement the binary classification, several intermediate levels may be expressed by using linguistic terms. For example, the quality of a product can be classified into the following terms: 'perfect', 'good', 'medium', 'poor', or 'bad' depending on its deviation from specifications. Then, the continuous functions selected appropriately can be used to describe the quality characteristic associated with each linguistic term. In this study, the control charts for number of nonconformities are handled. The type of available data is the imprecise number of nonconformities such as "between 5 and 8" or "approximately 6". The statistical model is based on the classical Shewhart control charts. In the literature, there exist few papers on fuzzy control charts, which use defuzziffication methods such as fuzzy mod, fuzzy midrange, fuzzy median, and fuzzy average in the early steps of their algorithms. The use of defuzziffication methods in the early steps of the algorithm makes it too similar to the classical analysis. Linguistic data in those works are transformed into numeric values before control limits are calculated. Thus both control limits as well as sample values become numeric. This transformation may cause biased results due to the loss of information included by the samples. For example, two fuzzy samples with the equal fuzzy mod may explain very different characteristics. A new approach called direct fuzzy approach to fuzzy control charts is modeled in order to prevent the loss of information of the fuzzy data during the construction of control charts. In this approach, linguistic or uncertain data are represented by means of triangular and/or trapezoidal fuzzy numbers. Using fuzzy arithmetics, control limits based on the fuzzy data are also determined as fuzzy numbers. The decision about the process control is based on the area measurement method. The proposed approach directly compares the linguistic data in fuzzy space without making any transformation. The percentage area of the fuzzy sample behind the fuzzy control limits is used in the decision and intermediate decision levels are defined. Keywords: Fuzzy control charts, fuzzy sets, linguistic data, unnatural pattern analysis, uncertainty.

Fuzzy real options valuation for oil investments

Traditional valuation methods are less viable under uncertainty. Hence, other methods such as real options valuation models, which can minimize uncertainty, have become more important. In this study, the hybrid approach suggested by Carlsson and Fuller is examined for the case of discrete compounding as this approach better models risky cash flows. A new real options valuation model that will evaluate the investment in a more realistic way is suggested by postponing the defuzzification of parameters in early stages. The suggested model has been applied to the data of an oil field investment and in conclusion the loss of information caused by early‐defuzzification has been determined. Santrauka Tradiciniai vertinimo metodai yra mažiau patikimi esant neapibrėžtumams. Vadinasi, kiti metodai, tokie kaip realių pasirinkčių vertinimo modeliai, kurie gali minimizuoti neapibrėžtumus, tampa svarbesni. Šiame straipsnyje nagrinėjamas hibridinis Carlsson ir Fuller metodas, kuris buvo panaudotas diskrečiajam rizikingų pinigų srautų modeliavimui. Pasiūlytas naujas realių pasirinkčių vertinimo modelis, kuris realistiškiau įvertins investicijas, rodiklius apibūdinančią neapibrėžtą informaciją apdorojant ankstyvojoje stadijoje. Pasiūlytas modelis buvo pritaikytas investicijoms į naftos verslą modeliuoti, nustatytas informacijos nuostolis, kuris atsiranda dėl ankstyvo neapibrėžtų duomenų apdorojimo. First published online: 21 Oct 2010 Reikšminiai žodžiai: neapibrėžtos aibės, realių pasirinkčių vertinimas, neapibrėžtos pasirinktys, investavimas

Internal rate of return decision rules in case of uncertainty

İç verim oranı; Şimdiki Değer (ŞD), Kar/Maliyet oranı, Geri Dönüşüm süresi gibi yatırımları değerlendirmede kullanılan yöntemlerden biridir. İç verim oranı yöntemi, elde edilen değerin Minimum Çekici Verim Oranı (MÇVO) değeri ile karşılaştırılmasını içerir. İç verim oranının hesaplanmasında literatürde,  projenin türüne göre birden fazla iç verim oranı değerinin elde edilmesi ve bu değerlerin karşılaştırma için kullanılması üzerine çalışmalar yapılmaktadır. Ancak iç verim oranı hesaplanmasında dikkate alınması gereken bir diğer konu da belirsizlik durumunun dikkate alınması gerektiği, aksi takdirde beklenen net nakit akışı değerleri ile yapılan hesaplamaların yanlış kararlara neden olabileceği gerçeğidir. Bu düşünceden yola çıkarak çeşitli yatırım değerlendirme yöntemleri için aralarında bulanık kümeler yaklaşımının da kullanıldığı formülasyonlar gerçekleştirilmiştir. Ancak bulanık kümeler yaklaşımı dikkate alındığında iç verim oranının hesaplanmasında diğer yatırım değerlendirme yöntemlerine nazaran belirli bir formülasyon gerçekleştirilememiştir. Bu çalışmada bulanık sıralama yöntemlerinden olan t-norm ve t-conorm bulanık bağıntıları kullanılarak Şimdiki Değer eşitlikleri oluşturulmakta, herbir bulanık bağıntı için çeşitli α-kesme düzeylerine karşılık gelen iç verim oranı değerleri hesaplanarak iç verim oranı ile ilgili karar kuralları oluşturulmaktadır. Bir basit proje örneği üzerinde, önerilen karar kuralları uygulanmakta ve bu örnekle ilgili yorumlar yapılmaktadır. Ayrıca, iç verim oranı için belirlenen karar kurallarına alternatif bir yöntem olarak, t-norm ve t-conorm bulanık bağıntıları kullanılarak elde edilen üçgen bulanık sayıların durulaştırılması da önerilmekte ve bu yöntemle elde edilen sonuçların karar kuralları kullanılarak elde edilen yöntemlerle karşılaştırması yapılmaktadır. Anahtar Kelimeler: İç verim oranı, t-norm/t-conorm bulanık bağıntı, karar kuralları.Internal rate of return (IRR) method is one of the investment appraisal techniques to evaluate the acceptability of a single project which is characterized by the comparison of IRR with Minimum Attractive Rate of Return (MARR) of the company. Some other methods are Present Value (PV), Future Value (FV), Equivalent Uniform Annual Value (EUAV), Benefit/Cost Ratio (B/C Ratio), and Payback Period. To the best of our knowledge, the literature pertaining to IRR mostly revolves around the problem of multiple returns (for example, see Hartman and Schafrick, 2004; Zhang, 2005; Hazen, 2003) and the propositions how to integrate multiple rates of return into one internal rate of return or which one of the multiple rates of return to consider as a unique internal rate of return to be able to compare with the MARR of the company. On the other hand, to our mind, another crucial problem regarding IRR should be the consideration of uncertainty, i.e., the calculation of IRR by considering the net cash flows as uncertain. Since, the traditional approaches involve merely the expected values of the net cash flows for the calculation of IRR, any variation from the expected values may invalidate the decision of the project. With this concern, fuzzy sets approach is one of the approaches in the literature that considers uncertainty in investment appraisal techniques for single projects, especially for PV, FV, Capitalized Value, EUAV, B/C Ratio, and Payback Period methods (for example, see Chiu and Park, 1994; Kahraman et al.,2000; Kahraman et al., 2002). However, the use of fuzzy sets approach for the IRR method was also discussed and some authors concluded the impossibility of the applicability of the fuzzy approach to the calculation of IRR (for example, see Kahraman et al., 2002 for the review), since 0 is not a fuzzy number, and the left-hand side and 0 cannot be compared. In this paper, we assume net cash flows as fuzzy numbers, specifically as triangular fuzzy numbers, and propose to reformulate two PV equations by considering t-norm and t-conorm fuzzy relations to compare the left-hand side and the right-hand side of each PV equation. We also consider MARR value as a fuzzy number, specifically a triangular fuzzy number, to be able to compare the IRR values obtained from t-norm and t-conorm fuzzy relations with the respective MARR values for each α-cut level. Afterwards, we propose the decision rules for IRR. In the decision rules, we note that the predetermined thresholds for α-cut levels determine the final decision of the decision maker. We also note that we expect to obtain the best values and worst values of IRR for a given α-cut level, and that the decision maker may have the opportunity to see the possible range of the IRR values between these best values and worst-values. A taxonomy of fuzzy ranking methods and the details of the methods have been provided in Chen and Hwang (2002). t-norm and t-conorm fuzzy relations are the extensions of possibility and necessity measures (Ramik, 2006), while possibility and necessity measures are classified under the group "Comparison function" in the taxonomy of Chen and Hwang (2002). t-norm and t-conorm fuzzy relations and their properties are examined in detail in most reference books (for example, see Klir and Yuan, 1995; Buckley et al., 2002). Inuiguchi et al. (2003) also studies t-norm and t-conorm fuzzy relations in detail and applies t-norm and t-conorm fuzzy relations to a general Fuzzy Linear Programming (FLP) model to compare the left-hand side with the right-hand side of the inequality constraints, and also to compare the objective function with the aspiration level as a definition of the maximization of the objective function. The following paper follows the approach of Inuiguchi et al. (2003), but applies the approach to the PV equations to compare the left-hand side and right-hand side, so that t-norm and t-conorm based IRR values are obtained for each α-cut level. We illustrate the proposed decision rules with a three-period simple project example, and interpret the results. As an alternative to the proposed decision rules, we also propose to defuzzify the triangular fuzzy numbers representing IRR and MARR by Center of Gravity method, so that we can compare the singleton IRR and MARR results. We also compare the results obtained from the decision rules and from the defuzzification of the triangular fuzzy numbers. Finally, in the conclusions, we summarize the results and give the potentials for future research to apply the proposed decision rules to other types of projects. Keywords: Internal rate of return, t-norm/t-conorm fuzzy relations, decision rules

A fuzzy multiple attribute utility model for intelligent building assessment

Multi-Attribute Utility Theory (MAUT) is an evaluation scheme which is very popular by decision makers for evaluating their judgments. According to MAUT, the overall evaluation U(x) of an object x is defined as a weighted addition of its evaluation with respect to its relevant value dimensions. The recent years have witnessed a huge concentration and interest in intelligent buildings’ performance that is increasingly evidenced in building design and construction. Intelligent buildings (IBs) are also under assessment according to their IB related characteristics and actual circumstances. For this aim, in this paper a fuzzy multiple attribute utility model for intelligent building assessment is proposed and three alternative intelligent buildings for a business center in Istanbul are evaluated