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Ulaştırma problemlerinde bulanık optimizasyon

Başlık çevirisi mevcut değil.

  1. Tez No: 75360
  2. Yazar: İSMAİL ÖZDABAN
  3. Danışmanlar: DOÇ. DR. COŞKUN ÖZKAN
  4. Tez Türü: Yüksek Lisans
  5. Konular: Endüstri ve Endüstri Mühendisliği, Industrial and Industrial Engineering
  6. Anahtar Kelimeler: Belirtilmemiş.
  7. Yıl: 1998
  8. Dil: Türkçe
  9. Üniversite: İstanbul Teknik Üniversitesi
  10. Enstitü: Fen Bilimleri Enstitüsü
  11. Ana Bilim Dalı: Endüstri Mühendisliği Ana Bilim Dalı
  12. Bilim Dalı: Belirtilmemiş.
  13. Sayfa Sayısı: 119

Özet

ÖZET Yirminci yüzyılın son yarısında klasik mantığın oldukça genelleştirilmiş bir hali olarak ortaya çıkan bulanık mantık, bulanık küme teorisine dayanan bir matematiksel disiplin olup, doğruluğun ve yanlışlığın derecesini konu almaktadır. İlk olarak konuyu bulanık küme disiplini içerisinde ele alan ve bunu 1965 yılında yazdığı“Fuzzy Sets”adlı makalesiyle açıklayan L. A. Zadeh olmuştur. Tüm dünyada genel kabul gören bu anlayış, beraberinde, konunun önem kazanarak günümüze gelmesini sağlamıştır. Günümüzde tüm bilim dallarında geniş bir uygulama alanı bulan bulanık mantık, problemlerin çözümünde, gerek bilim adamı gerekse uygulamacılara büyük kolaylık sağlamaktadır. Bulanık kümeler, birçok araştırma alanında olduğu gibi ulaştırma problemlerinin çözümünde de en başta yer alan bir konu haline gelmiştir. Bu noktadan hareketle yapılan çalışmada bulanık ulaştırma modeli üzerinde yoğunlaşılmış ve konunun teorik esaslarının yanında bir de uygulamaya yer verilmiştir. Bu çerçevede hazırlanan tez beş bölümden oluşmaktadır. Birinci bölüm olan giriş bölümünde, konunun genel esasları belirtilerek, konu tanıtılmaya çalışılmıştır. ikinci bölümde, bulanık mantık kavramının tanımı, tarihçesi, uygulama alanları ve bulanık mantığın avantaj ve dezavantajları belirtilmiştir. Üçüncü bölümde, bulanık küme kavramına gelinceye kadar geçen sürece uygun olarak, küme kavramı ve kesin kümeler teorisinden bahsedilmiş ve daha sonra ise, bulanık küme, bulanık sayılar ve bulanık bağıntı konulan açıklanmıştır. Dördüncü bölümde, ulaştırma modeli tanıtılarak, bulanık verilerle ulaştırma optimizasyonu kapsamlı bir şekilde ele alınmıştır. Beşinci bölümde ise, bulanık ulaştırma problemlerinden, bulanık maliyet verileri ile ulaştırma optimizasyonu konusunda bir uygulamaya yer verilmiştir. Sonuç ve öneriler kısmında ise, konu genel hatları ile ortaya konarak, uygulama sonuçlan değerlendirilmiş ve bu sonuçlara bağlı olarak öneriler sunulmuştur.

Özet (Çeviri)

SUMMARY FUZZY OPTIMIZATION IN TRANSPORTATION PROBLEMS At the turn of the century, reducing complex real-world systems into precise mathematical models was the main trend in science and engineering. In the middle of this century, Operations Research began to be applied to real-world decision-making problems and thus became one of the most important fields in science and engineering. Unfortunately, real-world situations are often not so deterministic. Thus precise mathematical models are not enough to tackle all practical problems. To deal with imprecision/uncertainty, the concepts and techniques of probability theory are usually employed. In the 1960s, meanings of the probability theory has been reconsidered and criticized when modelling practical problems. Around the same time as the development of chaos theory to handle non-linear dynamic systems in physics and mathematics, fuzzy set theory was developed in 1965 by L. A Zadeh. Since then, fuzzy set theory has been applied to the fields of operations research, management science, artificial intelligence/expert system, control theory, statistics etc.. Fuzzy set theory is a theory of graded concepts (a matter of degree), but not a theory of chance. Therfore, figures and numerical tables are considered paramount in the study of fuzzy set theory. They, unlike confusing mathematical slang, difficult functions, etc., provide the best ways to communicate with outsiders. This is an important concept in the new generation of operations research. In 1965, Zadeh formally published the famous paper“Fuzzy Sets”. The fuzzy set theory is developed to improve the oversimplified model, thereby developing a more robust and flexible model in order to solve real-world complex systems involving human aspects. Furthermore, it helps the decision maker not only to consider the existing alternatives under given constraints (optimize a given system), but also to develop new alternatives (design a system). Advances in science and technology have made our modern society very complex, and with this, decision processes have become increasingly vague and hard to analyze. The human brain possesses some special characteristics that enable it to learn and reason in a vague and fuzzy enviroment. It has the ability to arrive at decisions based on imprecise, qualitative data in contrast to formal mathematics and formal logic which demand precise and quantitative data. Modern binary computers possess capacity but lack human-like ability. Undoubtedly, in many areas of cognition, human intelligence far excels the computer“intelligence”of today, and the development of fuzzy concepts is a step forward towards the development of tools capable of handling humanistic types of problems.We do have sufficient mathematical tools and computer-based technology for analyzing and solving problems embedded in deterministic and uncertain (probabilistic) enviroment. Here uncertainty may arise from the probabilistic behaviour of certain physical phenomena in mechanistic systems. We know the important role mat vagueness and inexactitude play in human decision making, but we did not know until 1965 how the vagueness arising from subjectivity which is inherent in human thought processes can be modelled and analyzed. In effect, frizzy set theory is a body of concepts and techniques that gave a form of mathematical precision to human cognitive processes that in many ways are imprecise and ambiguous by the standards of classical mathematics. Today, these concepts are gaining a growing acceptability among engineers, scientists and mathematicians. Since its inception, research in the fuzzy set field has faced an increasing exponential growth. This fuzzy field has blossomed into a many-faceted field of inquiry, drawing on and contributing to a wide spectrum of areas ranging from pure mathematics to human cognition, perception and judgement. Its influence in science, engineering and social sciences has been felt already, and is certain to grow in the decade to come. A set is a collection of objects that are well specified and possess some common properties. These objects may represent some abstract concept, or may be a collection of some physical properties. It can be finite or infinite, enumerable or non- enumerable.. This is an intuitive definition of a set, although mathematicians prefer to use more sophisticated definitions. The numbers 0 and 1 define the membership of each element of the subset, where 1 means the element belongs to the subset and 0 means the element does not belong to the subset. If x is an element of A we denote the corresponding function as The function u,A(x) is called the“characteristic function'' or ”membership function“. In the theory of sets the main operators are defined as follows where a means ”minimum“ and v ”maximum“. Consider a referential set E with x as its element. The ”characteristic function“ or ”membership function“ of x is where [0,1] is the segment or closed interval from 0 to 1. A subset is a fuzzy subset (or fuzzy set). Thus, a fuzzy subset has a membership function with not only values of 0 (does not belong to) or 1 (belongs to), but any number over the interval 0 and 1.Along with the expression of this equation, Zadeh also proposed the following notations. When E is a finite set {xh x%..., Xn }, a fuzzy set A is then expressed as: When E is not a finite set, A then can be written as: The intersection (minimum), union (maximum) and complementation operators defined earlier can be used also for fuzzy sets. Linguistic variables represent another useful concept in fuzzy set theory. A linguistic variable has values which are not numbers but words (or phrases) of a natural language. The variable values represent restrictions on the values of a base variable. As an example, consider the linguistic variable : temperature, defined on the base variable : centigrade degrees. Possible values of this variable include : hot, very hot, cold, mild, and not cold. Typical values of the linguistic variables contain not only primary terms (e.g., hot, cold) but also hedges such as ”somewhat“ or ”very“, fuzzy connectives : ”and“, ”or“, and negation : ”not“. The support of a fuzzy set A is the crisp subset of E and is presented as : The a-level set (a-cut) of a fuzzy set A is a crisp subset of E and is denoted by : A fuzzy set E is normal if and only if Sup* |AA(x) = 1, that is, the supreme of Ha(x) over E is unity. A fuzzy set is subnormal if it is not normal. A non-empry subnormal fuzzy set can be normalized by dividing each u,A(x) by the factor Sup, u,A(x). A fuzzy set A in E is convex if and only if for every pair of point Xi and x2 in E, the membership function of A satisfies the inequality : where X,e[0,l]. Alternatively, a fuzzy set is convex if all a-level sets are convex. We can define a number AeR over an interval of confidence [ai, as] asAn interval of confidence in R is an ordinary subset of R which represents a type of uncertainty. We know that A cannot be smaller than ai and it cannot be greater thana3. A triangular fuzzy number can be defined by a triplet (ai, a2, a3 ). For the triangular fuzzy number, the membership function is defined as Ha(*) = Alternatively, defining the interval of confidence at level a, we characterize the triangular fuzzy number as The properties of trapezoidal fuzzy numbers are smiliar to those of the triangular fuzzy numbers. A trapezoidal fuzzy number can be represented competely by a quadruplet A = (ai, a2, a3, m ) We can characterize also a trapezoidal fuzzy number by the interval of confidence at level a. Thus, The membership function of a trapezoidal fuzzy number is charecterized as Fuzzy relations are fuzzy subsets of XxY, that is, mapping from X-»Y. Applications of fuzzy relations are widespread and important.Let X, Y c R be universal sets, then R - my), Mw)) \(*,y) * XxY} is called a fuzzy relation on XxY. Let R and Z be two fuzzy relations in the same produet space; the union/intersection of R with Z is then defined by liRf*(x,y) = mm{yK(x,y),Mx>y)}, (x,y)eXxY Fuzzy relations in different product spaces can be combined with each other by the operation ”composition“. Different versions of ”composition“ have been suggested which differ in their results and also with respect to their mathematical properties. The max-min composition has become the best known and the most frequently used one. Let Ri(x,y), (x,y) e XxY and K2(y, 2), (y, z)? YxZ be two fuzzy relations. The max- min composition Ri max-min R2 is then the fuzzy set Ri° R2 ={[(*, z), max {min{nRi(**y), um (y,z)}}] I *eX, yeY, z&Z } y The transportation problem is one of the earliest applications of linear programming problems. Efficient methods of solution derived from the simplex algorithm were developed in 1941, primarily by Dantzig and then by Koopmans et al. The transportation problem can be modeled as a standard linear programming problem, which can then be solved by the simplex method. A great part of practical applications of linear programming falls into the field of network flow problems. The transportation problem is of special interest. As is well known, transportation problem is formulated as follows: A product is to be transported from each of m sources to any of n destinations. The sources are production facilities (supply points) characterized by available capacities ai,,a”. The destinations are consumption facilities characterized by required levels of demand bi,,b“. There is a cost c^ associated with transporting a unit of the given product from the i-th source to the j-th destination. One must determine the amounts x% of the product to be transported from all sources i to all destinations j so that the total operation cost will be minimized. m n It is usual to impose and with this assumption transportation problem can be formulated as the following linear programming problem:Zrtin^ £2CiJ*iİ O»”Zmin=:Cn*ll+Ci2Jfl2+ + Cmn*Wi s.t : a J^x^m i=l,2,3,....,m 2>s = bj j = l,2,3,....,n This formulation has a clear economic interpretation from which the great interest in this type of problems arises. On the other hand, methods exist (the stepping stone or the Hungarian one) which permit the problem to be solved more easily than by the simplex algorithm. These reasons justify, extraordinary importance transportation problem has for operations research. Moreover, there are many problems, not being exactly transportation problem, which can be solved in the same way because of their mathematical formulation. All these areas of interest in transportation problem can be enlarged when it is assumed that some parameters taking part in the formulation of the problem are fuzzy. Thus, the attempt is to solve problems, such as those mentioned above, assuming that some fuzziness is present in them. Fuzzy transportation problem was first studied by Prade (1980); Oheigeartaigh (1982), Verdegay (1983) and Delgado and Verdegay (1984) should also be mentioned. A parametric approach to solve fuzzy transportation problem has been proposed by Chanas, Kolodziejczyk and Machaj (1984). In thesis fuzzy transportation problems were examined in two parts. The first part is fuzzy transportation problem with a fuzzy cost data and the second part is fuzzy transportation problem with fuzzy supply and demand data. The stepping-stone method, a very well known classical approach, is used extensively for the solution of transportation and scheduling problems. It was introduced almost four decades ago by the American mathematician George Dantzig. This method was used with non-fuzzy data and then extend the method to a situation where the cost data are fuzzy. I explained the method by means of two examples in thesis. A parametric approach and resolution method were used with a fuzzy supply and demand and non-fuzzy objective function in thesis. Using the parametric approach to solve fuzzy mathematical programming and if Si, iel, bj?J are fuzzy numbers or more particularly, trapezoidal fuzzy numbers, as in Chanas, Kolodziejczyk and Machaj, transportation problem becomes : XVIwhere [%(a), Aj(a)], [bj(a), Bj(a)], a ?(0,1] is the a-cut of level a of ai, (bj), iel, (jeJ). I explained this formulation by means of an example in thesis. Application was done in three tile fabrics. Fuzzy transportation problem was considered with fuzzy cost data and then problem was solved with stepping-stone method.

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