A Survey on Interactive Approaches for Multi-Objective Decision ...

Savunma Bilimleri Dergisi The Journal of Defense Sciences Mayıs/May 2016, Cilt/Volume 15, Sayı/Issue 1, 231-255. ISSN (Basılı) : 1303-6831 ISSN (Online): 2148-1776

A Survey on Interactive Approaches for Multi-Objective Decision Making Problems Fatih KASIMOĞLU1 Abstract Having a vast area of implementation, Multi-Criteria Decision Making gets interest of many scientists and consequently there are a lot of methods developed for the solution of the problems in this field. The diversity and the multitude of the methods make it necessary to develop a systematic approach to analyze them in a holistic manner. This study gives an overview of the interactive Multi-Objective Decision Making methods developed for continuous problems. The most common way, in which these methods are classified, the fundamental aspects of the associated algorithms and their main assumptions are discussed. Among many methods the ones that are thought to be laying the theoretical philosophy of the field and substantially differing from the others have been given priority. It has been observed that when classifying the methods, two different approaches are most commonly adopted in the literature: classifying the methods either according to the interaction style or according to the technical aspects of the algorithm. More recently, evolutionary algorithms emerged as a different type especially in non-linear and NP-hard domain. Each method is built upon the specific features of a problem. Thus, linearity/nonlinearity, convexity/ concavity, differentiability and NP-hardness are among the defining factors to choose a method. Nowadays human factor is also getting a defining role. Keywords: Multi-Criteria Decision Making, Multi-Objective Decision Making, Interactive Multi-Objective Decision Making Methods.

1

Address: Dr., Turkish Military Academy Defense Sciences Institute, 06654, Bakanlıklar, Ankara, Turkey, [email protected]. Makalenin geliş tarihi: 13.03.2015 Kabul tarihi: 28.12.2015

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Çok Amaçlı Karar Verme Problemleri İçin İnteraktif Yaklaşımlar Üzerine Bir Araştırma Öz Geniş bir uygulama alanına sahip olan Çok Kriterli Karar Verme, birçok bilim adamının ilgisini çekmekte ve sonuç olarak da bu alanda problemlerin çözümü için geliştirilmiş bol sayıda metot bulunmaktadır. Geliştirilen metotların çeşitliliği ve çokluğu bunların bütüncül olarak, sistematik bir şekilde analiz edilmesini gerekli kılmaktadır. Bu çalışma sürekli yapıdaki problemler için geliştirilen interaktif Çok Amaçlı Karar Verme metotları hakkında bir çerçeve sunmaktadır. Çalışmada metotların en yaygın tasnif şekilleri, varsayım ve algoritmalarındaki temel hususlar ele alınmaktadır. Birçok yöntem arasından, alanın teorik temellerini oluşturanlar ve birbirlerinden önemli oranda farklılık arz edenler seçilmiştir. Bu tip problemlerin tasnifinde, literatürde yaygın olarak iki yaklaşımın kullanıldığı görülmektedir: metotların interaktiflik şekline göre ya da kullanılan algoritmaların teknik özelliklerine göre sınıflandırılması. Son zamanlarda evrimsel algoritmalar da ayrı bir tür olarak özellikle doğrusal olmayan ve NP-zor sahasında öne çıkmıştır. Her metot, bir problemin belli özellikleri üzerine inşa edilmiştir. Bu nedenle doğrusal/doğrusal olmama, konvekslik/ konkavlık, türevlenebilme, NP-zor yapı gibi özellikler hangi metodun seçilmesi gerektiği konusunda belirleyici role sahiptir. Bugünlerde insan faktörünün de belirleyici hâle geldiğini söyleyebiliriz. Anahtar Kelimeler: Çok Kriterli Karar Verme, Çok Amaçlı Karar Verme, İnteraktif Çok Amaçlı Karar Verme Metotları. Introduction Multi-criteria Decision Making (MCDM) has received great interest for the last three decades due to the huge are of applicability in industry, government , military, economy etc. A lot of papers have been published in the refereed journals, many conferences have been held on the topic since 1970s (See Steuer et. al 1996). There are two basic theoretical trends regarding this area. The first is the one that deals with the problems having continuous decision space; this is also referred as Multi-Objective Decision Making (MODM) and requires very sophisticated mathematical theory. The second one deals with the problems that have discrete decision space in nature, which is also referred as Multi-Attribute Decision Making (MADM) (Triantaphyllou, 2000). A lot of different methods have been developed for both leaning.

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When it comes to classifying the methods developed in MCDM, a lot different ways have been observed in the literature. A summary of general classifications of methods is given in Figure 1.

-Deterministic -Stochastic -Fuzzy

-Linear -Nonlinear

-Single Decision Making -Group Decision Making

-No preference -Apriori -Aposteriori -Interactive

Figure 1. A Summary of General Classification Approaches in MCDM

In Figure 1 one way to classify these methods is according to the type of the data used. In this regard Multi-Criteria Decision Making methods are classified as deterministic, stochastic and fuzzy methods. Another way is to classify the methods depending on the number of decision makers (decision makers), that is methods requiring either a single decision makers or more than one decision makers (group) (Triantaphyllou, 2000). Multi-Criteria Decision Making methods can also be classified as those adopting nonlinear approaches (Multi-Objective Nonlinear Programming (MONLP)) and those requiring linear approach (Multi-Objective Linear Programming (MOLP)) (Sayin, 2003). One other way is according to the role of the decision makers in the decision process; Multi-Criteria Decision Making methods can be classified into four groups in this respect: nopreference methods, a priori methods, a posteriori methods and interactive methods. Next we will discuss this last classification in a little bit more detail, since our main focus throughout the study is on interactive methods. When there is no decision maker to articulate his or her opinion nopreference methods are used. These methods try to find a neutral compromise solution without using preference information from the

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decision makers. In priori methods the decision maker articulates his or her preference or aspiration beforehand. The solution process then uses this information in order to find a solution that satisfies the preferences of the decision maker best. Even though this is a straightforward method, the difficulty is that the decision makers may not know the limitations and possibilities regarding the problem and therefore he/ she may be either too optimistic or too pessimistic in his/her expectations. In a posteriori method first a set of efficient solutions is generated and then the decision maker is referred to in selecting the better alternative. In these methods it may be difficult for the decision makers to get through a large amount of alternatives and even the computation of these alternatives may be tiresome. Typically evolutionary algorithms are developed to solve the problems falling into this class (Miettinen, 2008). The class of interactive methods, which is also the main concern of this paper, is an extensive one. In this method the decision makers is supposed to be involved in the solution process. At each iteration of the solution, the decision maker is asked about his or her preference information. The methods falling into this class try to realistically determine the preference structure of the decision makers in an interactive way. The solution process of Multi-Criteria Decision Making problems usually requires active involvement of decision makers. This is because of the difficulty of combining multiple objectives into a single objective that enables using regular optimization techniques. Multi-Criteria Decision Making problems usually have objectives that are defined in different units, and these units are most of the time incommensurable. For example, when deciding to buy a car the attributes like cost, security and comfort are all defined in terms of different units and cannot be combined into a single objective easily. Structuring the preference tendency of decision makers is usually cumbersome. Interactive methods solve this problem, simply because there is not any need for the decision makers to explicitly define his or her global preference structure. The value function is neither assumed to be known nor tried to be estimated explicitly. Decision maker’s responses to specific questions are used to guide the solution process towards the most preferred solution (Miettinen et al., 2008). In this sense interactive methods overcome some weaknesses of apriori and aposteriori methods. Thus, it is necessary to interact with the decision makers in order to figure out the actual relative values of the objectives and the global preference structure of the decision makers. This is especially important when human behavior has to be taken into account in the decision making process.

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In this study our specific focus is on interactive Multi-Objective Optimization methods for continuous problems. Throughout this survey the related publications are scrutinized in a manner to figure out those that are prevailing in the area and contain the philosophic and theoretical essences of the topic. Instead of giving the details of each method, we intend to put forward the fundamental aspects of the methods and depict the general framework. General Overview of Interactive Approaches Generally speaking Multi-Criteria Decision Making problems can be stated as follows: Maximize z = f(x) s.t. x Є X Where, f(x) = (f1(x),……. ,fp(x)) is a set of objective functions (a column vector), fi(x) represents the ith objective function, x = (x1,…….,xn) is the decision column vector, X represents the feasible decision space in Rn, z = f(x) represents the objective (criterion) function, Z=f(X) represents the feasible objective (criterion) space in Rn. Notice that objective function in the model is consisted more than one different objection functions. It should be noted that for a problem to be a Multi-Criteria Decision Making problem at least two of the objective functions should be in conflict (Tabucanon, 1988). When two objective functions are in conflict an improvement in one of them is not possible without a dis-improvement (deterioration) in the other. It should be noted that in Multi-Criteria Decision Making an optimal solution in terms of a single objective function is not usually the solution to look for. That is why the term efficient solution is used more often. An efficient solution can be

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simply stated as a solution in which an improvement in a specific objective function cannot be achieved without causing some deterioration in one or more other objective functions. So, essentially what is searched for in MultiCriteria Decision Making is the most preferred efficient solution. Another important point worth to mention about the mathematical model is that there are two different spaces, one being decision space (X) and the other being objective (criterion) space (Z). Decision space contains feasible set of our courses of action, whereas the objective space contains set of all achievable values for different objectives. As an example the two different spaces in R2 is sown in Figure-2. Every point in decision space has its counterpart in objective space as depicted in Figure-2. Most of the time the considerations in Multi-Criteria Decision Making problems, are made in objective (criterion) space. Set Z may be convex or nonconvex, it may be bounded or unbounded, it may be precisely known or unknown, and it may consist of a finite or infinite number of alternatives. All these considerations affect the algorithms developed for Multi-Criteria Decision Making methods. When Z consists of a finite number of elements explicitly known at the beginning of the solution process, we have the class of problems which are sometimes called Multiple Criteria Evaluation Problems. Those problems are also referred to as Discrete Multiple Criteria Problems or as mentioned earlier Multi-Attribute Decision Making problems (Karhonen, 2005). When the number of alternatives in Z is infinite, the alternatives are usually defined using a mathematical formulation, and this time the problem is referred to as Continuous Multi-Objective Problem. In this case we say that the alternatives are only implicitly known. In the literature this kind of problems are referred as Multiple Criteria Design Problems or Continuous Multiple Criteria Problems (Karhonen, 2005) or Continuous MultiObjective Optimization (CMOO) problems (Miettinen, 2008). The methods developed for this group of problems differs hugely from those developed for discrete type problems described above.

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x1

f1

X

.(x

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Z

.(f

1, f2)

1, x2)

x2

f2

Figure 2. Decision and Objective Spaces

When the number of alternatives in Z is infinite, the alternatives are usually defined using a mathematical formulation, and this time the problem is referred to as Continuous Multi-Objective Problem. In this case we say that the alternatives are only implicitly known. In the literature this kind of problems are referred as Multiple Criteria Design Problems or Continuous Multiple Criteria Problems (Karhonen, 2005) or Continuous MultiObjective Optimization (CMOO) problems (Miettinen, 2008). The methods developed for this group of problems differs hugely from those developed for discrete type problems described above. Interactive methods in general has two main phases; one is when the decision maker learns about the possible feasible efficient solutions and the other is when the decision makers makes up his or her decision about the preferred solution. Solution algorithms are formed in an iterative manner. Its steps are repeated and the decision makers specify preference information progressively. After every iteration, some solutions are proposed to the decision makers, and the decision makers having this solution set is asked to specify his or her preferences. This information obtained from decision makers is then used to structure the preference function of the decision makers and used to generate new solutions. In this way, the decision maker take active participation in the solution process thereby explicitly form his or her preference structure. Sometimes it is even possible that decision

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makers correct their own selections, better realizing what they really want. The main steps of a general interactive method can be described as below (Miettinen et al 2008): (1) Start with initial best and worst solutions at hand (calculate ideal and nadir solutions and inform the decision makers), (2) Generate an efficient start point solution (this can be found as a neutral compromise solution or can be given by the decision makers), (3) Ask for preference information from the decision makers, (4) Generate new efficient solutions according to the preferences of the decision makers, (5) Ask the decision makers to select the best solution found so far, (6) If the decision makers are satisfied stop the algorithm. Otherwise go to step 3. One of the difficulties encountered in Multi-Criteria Decision Making is that usually it is not possible to figure out the preference function. Interactive methods make the job of analysts easier since the preference is directly obtained from the decision makers. Of course by doing so the preference is more accurately reflected in decision making process, since the procedure is repeated until the decision makers are satisfied. This is actually one of the important aspects of interactive methods and the reason why these methods get so much interest. There are lots of studies related to the Continuous Multi-Objective Optimization methods in the literature. The methods developed in these studies can be classified depending on the style of interaction with the decision makers in the solution process and the technical aspects of the method. Style of interaction includes the form of the information related to the possible solutions passed to the decision makers and the reaction the decision makers give to that. In other words, style of interaction specifies the way we inform the decision makers and as a result of this, the type of preference information obtained from the decision makers. Technical elements on the other hand are much more related with the mathematical assumptions made for the problem, convergence of the method, how optimal the found solution is (weakly efficient or strongly efficient etc.) and the kind of scalarizing function used (Miettinen et al 2008).

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Based on the style of interaction, most of the Continuous MultiObjective Optimization methods in the literature (though not necessarily all of them) can be classified into three main types as follows (Miettinen et al 2008): Methods based on trade off information, methods using reference point approach, Classification-based methods. The Continuous Multi-Objective Optimization methods can also be classified depending on the technical aspects of the method. Steuer (1986) classifies these methods as follows: Feasible region reduction methods, Weighting vector space reduction methods, Criterion cone contraction or line search methods. Shin and Ravindran (1991) have also similar approach to that of Steur when classifying these methods. Even though they give some more types in addition to the types given by Steuer (1986), their classes can be categorized into these three types. One possible way of classification of Continuous Multi-Objective Optimization methods that we propose in this study is based on two main streams emerged on this field: classical methods and evolutionary methods. Classical methods try to find a single solution in an exact manner, whereas evolutionary methods find a population of solutions in each iteration (Deb, 2011). Classification methods and the types in each classification for MultiObjective Optimization problems are summarized in Table-1. Table 1. Classification Methods and Types in Multi-Objective Optimization Classification Method Style of Interaction

Technical Aspect

Main Stream

Types 

Methods based on trade off information



Methods using reference point approach



Classification-based methods



Feasible region reduction methods



Weighting methods



Criterion cone contraction or line search methods



Classical methods



Evolutionary methods

vector

space

reduction

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Interactive Continuous Multi-Objective Optimization Methods In this section we present several interactive methods developed to solve Continuous Multi-Objective Optimization problems. There are a lot of them but we try to cover those that are prevailing in the area and contain the philosophic and theoretical essences of the topic. STEM STEM is one of the first interactive methods developed originally for Multi-Objective Linear Programming problems by Benayoun et al. (1971), the main assumptions of STEM in this respect relate to the linearity of the objective functions as well as the constraints. It is a classificationbased method in which the decision makers is supposed to classify the objective functions at the current iteration as those having satisfactory values and those not having satisfactory values. It uses the minimum weighted Tchebycheff distance from the ideal point when finding an efficient solution in the feasible region at each iteration. Being a feasible region reduction method from technical aspect it tries to converge to the most preferred solution by reducing the feasible region according to the information given by the decision makers. The procedure continues until the decision makers is satisfied with the current solution and does not want to change the objective function values any more. Geoffrion-Dyer-Feinberg (GDF) Method GDF method proposed by Geoffrion-Dyer-Feinberg (1972) is one of the most well-known interactive methods and it is based on the maximization of the implicitly known value function. The main assumptions of GDF method can be summarized as below (Miettinen, 2001): 

The decision space is compact and convex,



Objective functions are continuously differentiable and convex,



An implicit value function exists, and it is assumed to be continuously differentiable, monotonically decreasing and concave.

In terms of the interaction style this method is a tradeoff-based method. Given the current solution, the decision makers is asked to provide marginal rates of substitution, which are used to determine an ascent direction for the value function. The Frank-Wolf algorithm is gotten used of

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in order to solve the intermediate problems that are formed when applying the algorithmic steps of this method. Thus from the technical aspect we can categorize this method as a line search method. The optimal step-length is approximated using an evaluation scheme, and then the next iteration is generated. Interactive Weighted Tchebycheff Procedure Interactive weighted Tchebycheff method discussed by Steur (1986) is a weighting vector space reduction method from the technical aspect. In this method the objective functions do not have to be linear and the decision space does not have to be a convex set. However, the main assumptions are as follows (Steuer 1986): 

Each objective is bounded over the decision space S,



There does not exist a point in S at which all the objectives are simultaneously maximized.

This method brings about some advantages like: It can converge to final solutions that is not extreme in Multi-Objective Linear Programming problems and it can be generalized to integer and Multi-Objective Nonlinear Programming problems. However, the flexibility of the method is reduced by the fact that the thrown away parts of the weighting vector space cannot be restored if the decision maker changes her or his mind. Thus, consistency is important in this case. The weakness of the Tchebycheff method is that a great deal of calculation is needed at each iteration and many of the results are discarded. For large and complex problems, the Tchebycheff method may not be the appropriate choice (Miettinen, 2001). In terms of the interaction style this method can be classified as a reference point approach method. Visual Interactive Approach of Korhonen and Laakso This method is proposed by Korhonen and Laakso (1986) and has been developed in an effort to relax the assumptions concerning the decision maker’s behavior. The method does not require any specific knowledge about the properties of the utility function. Theoretically this method can be

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considered as an extension of GDF method. The line search is analogous to that of GDF method, but this method uses reference directions suggested by Wierzbicki (1980), which reflect the preference of the decision makers. The method is designed for both linear and nonlinear multiple objective programs. Given that the decision makers’ utility function is pseudo concave, and the decision space is formed by linear constraints (convex set) and bounded, a necessary and sufficient condition exists for determining whether a given point is optimal (Steuer, 1986). With respect to the interaction style this method can be classified as a reference point approach method. From the technical aspect it can be categorized as a line search method. One drawback related to this method is that its performance depends on the degree at which the decision makers specify the reference directions correctly, which leads to improved solutions. Zionts-Wallenius Method This method introduced by Zionts and Wallenius (1976) is based on piecewise linearization of the problem. It is a typical weighting vector space reduction method for solving Multi-Objective Linear Programming problems. The assumptions of the method are as follows (Miettinen, 2008):  An implicit value function exist and it is assumed to be concave,  The objective functions and the feasible region are convex The method was then updated (Zionts and Wallenius, 1983) for a class of pseudo concave value functions. It converges to the efficient extreme point of greatest utility when the decision maker’s utility function is pseudo concave. The method operates by iteratively asking the decision makers questions about extreme points or trade off vectors (Steuer, 1986). The decision maker’s responses may be a preference/no preference for a tradeoff or it may be indifference. In terms of interaction style this method is a trade-off based method. The Sequential Proxy Optimization Technique (SPOT) The basic idea of this interactive method developed by Sakawa (1982) assumes the existence of an implicit value function of the decision makers. This function is to be maximized over the feasible region. Maximization is done using a feasible direction scheme. Trade-off

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information at each current solution is obtained from the decision makers to improve the solutions at hand. The main assumptions of this method are as follows (Miettinen, 2008):  The implicit value function exists, and it is continuously differentiable,  All objective functions are convex and differentiable,  The feasible region convex,  Optimal Karush- Kuhn-Tucker information about tradeoffs.

(KKT)

multipliers

give

At each iteration the decision makers is asked to give his/her preferences, which are used to find a search direction. The method can be classified as a trade-off method in terms of interaction style and a line search one with respect to its technical structure. Guess Method The GUESS method is a simple interactive method related to the reference point method. The method is also sometimes called a naive method and it is presented in Buchanan (1997). It is assumed that the ideal and the nadir points are available. The decision maker specifies a reference point (or a guess) below the nadir point and then the minimum weighted deviation from the nadir point is maximized. Later the decision maker specifies a new reference point and the iteration continues until the decision makers is satisfied with the solution produced. The decision maker can also reduce the feasible region by specifying upper and lower bounds at each iteration. Heavy reliance on the availability of the nadir point is the weakness of the GUESS method since the nadir point is not easy to determine and it is usually only an approximation (Miettinen, 2001). Satisfcing Trade-Off Method (STOM) This classification-based method developed by Nakayama (1995) is based on ideas similar to the GUSESS method. The difference is the way the information obtained from the decision makers is utilized. The

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functioning of STOM is shortly as the following: After an efficient solution has been obtained by optimizing a scalarizing function, it is presented to the decision makers. On the basis of this information the decision makers is asked to classify the objective functions into three classes. The classes are the unacceptable objective functions whose values are to be improved, the acceptable objective functions whose values may be relaxed and the acceptable objective functions whose values are to be kept as they are. From the technical aspect this method can be classified as a feasible region reduction method. The objective and the constraint functions are assumed to be differentiable. Trade-off information can be obtained from the KarushKuhn-Tucker (KKT) multipliers related to the scalarizing function (Miettinen, 2001). Nondiferentiable Interactive Optimization System (Nimbus)

Multi-Objective

Bundle-Based

NIMBUS presented by Miettinen and Makela (1995), is an interactive multi-objective optimization method designed especially to be able to handle non-differentiable functions efficiently. For this reason, it is considered to be capable of solving complicated real-world problems. NIMBUS is a classification based method in terms of interaction style where the decision makers can easily indicate what kind of improvements are desirable and what kind of impairments are tolerable. The decision maker examines at each iteration the values of the objective functions found at the current solution and classifies the objective functions into five. These are objective functions whose values should be definitely decreased / should be decreased to a certain level / are satisfactory / are allowed to increase to a certain upper bound / are allowed to change at will (Miettinen, 2001). We can say that the reduction of the feasible region is utilized in this method. The strength of this method is that it can handle indifferantiable functions and it is not dependent much on how well the decision maker is in specifying the classification. If the decision makers do not like the current solution for some reason, he or she can explore other solutions between the old and the current one. Then a search direction is calculated and more solutions are provided by taking steps in this direction. Thus, this method uses also line search concept.

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Interactive Multi-Objective Evolutionary Algorithms So far the methods described are the classical methods in which the multiple objectives are transformed into a single objective and an exact solver is used for a solution. However, especially in non-linear and NP-Hard problems an exact solver might not be available. In such cases evolutionary algorithms can be used as a very practical metaheuristic solution strategy (Jaszkiewicz and Branke, 2008). In fact after 2000 there has been an increasing trend in the studies that has been done in this field. Deb (2011) gives an overview of evolutionary multi-objective optimization (EMO) algorithms and argues some fundamental principles of the topic putting emphasis on the current prevalence of these methods due to simplicity, flexibility and applicability. Additional introductory information about EMO can be found in Deb (2001), Coello et al. (2002), Osyczka (2002) and Obayashi et al. (2007). More recently EMO algorithms using interactive approaches have obtained attention of many researchers. These algorithms are considered to be more efficient for problems with large number of objectives (Sinha et al., 2014). Typically, the decision maker is integrated into the process by expressing his/her preference when generating solutions through an evolutionary algorithm. As one of the initial predominant studies in this area, Phelps and Koksalan (2003) use the information taken repeatedly from the decision maker to create a weighted objective function. Koksalan and Karahan (2010), attempt to find the most preferred point focusing on designated territories of preferences. Deb et al. (2010) proposes a method that helps to depict the preference function of the decision maker. Sinha et al. (2010) uses the information from decision maker to define better domination criteria. Said et al. (2010) and Kaliszewski et al. (2011) use the preference information to define upper and lower bounds for the best solution. Sinha et al. (2014) bring into focus one of the shortcomings of interactive methods, the availability of decision makers to be involved in the solution process. Thus in the study the decision maker is given a maximum number of availability times to be provide preference information. The procedure has been successfully applied in both constrained and unconstrained cases for multi-objective problems having up to five objectives.

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Conclusion There are a lot of methods developed for Multi-Criteria Decision Making problems and a lot of ways of classifying these methods. In this study interactive approaches for Continuous Multi-Objective problems, have been focused on, based on the fact that a systematic approach needs to be developed to get a better understanding of the fundamental aspects of the methods on the field. It has been observed that the main difference between methods lies in the interaction style and technical aspect of the algorithms. Recently evolutionary algorithms in a relatively different stream have also emerged as an efficient metaheuristic technique especially to solve nonlinear and NP-Hard problems. A summary of the methods covered in this respect is given in Table 2. It is worth to note that none of the methods mentioned can be considered as superior to the other. Each method is built upon the specific features of a problem in concern. In other words, whether a problem has a linear or non-linear structure, whether a set (or a function as well) is convex, concave, nonconvex or non-concave, whether a function is differentiable or non-differentiable, whether the problem is NP-Hard are the defining factors to choose which method to use in a certain case. So, problem specifications are defining factors at the usage of the methods in subject. Another remarkable point to note is that when finding a solution to a Multi-Objective problem the interaction methods are more likely to reflect the human behavior and usually nonlinearity is in the nature of most reallife problems. On the other hand, interactive methods require that decision makers spend time in the solution process during the interaction phase to get information from them. In practice this is not easy, simply because of the fact that decision makers usually have not enough time for that. It seems that the trend of future researches will be towards the models in which human factor is integrated in a better way. Thus it would be valuable to design user-friendly methods that will still get information from decision makers, but at the same time will make life easier for them. These models should involve decision makers into the solution process in such a way that minimum amount of their time is used.

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Table 2. Summary of the Methods STEM

A classification based feasible region reduction method. Applicable to problems with linear objective functions.

Geoffrion-Dyer-Feinberg (GDF) A trade-off based line search method. Applicable to Method problems with differentiable objective functions in case of a implicitly known value function. Interactive Weighted A vector space reduction method using reference point Tchebycheff Procedure approach. Applicable to problems with both linear and non-linear objective functions. Visual Interactive Approach of A reference point method using line search. Applicable to Korhonen and Laakso problems with both linear and non-linear objective functions. Zionts-Wallenius Method

A vector space reduction, trade-off based method. Applicable to problems with linear objective functions.

The Sequential Proxy A trade-off based, line search method. Applicable to Optimization Technique (SPOT) problems with both linear and non-linear objective functions in case of a implicitly known value function. GUESS Method

A feasible region reduction method using reference point. Applicable to problems with both linear and non-linear objective functions.

Satisfcing Trade-Off Method A trade-Off based feasible region reduction method. (STOM) Applicable to problems with both linear and non-linear objective functions. Nondiferentiable Interactive A feasible region reduction method using line search. Multi-Objective Bundle-Based Applicable to problems with non-differentiable objective Optimization System (NIMBUS) functions. Evolutionary Multiobjective Metaheuristic search based on trade-off information. Optimization (EMO) Applicable to problems with no specific exact solver (i.e. NP-Hard).

References Benayoun, R., de Montgolfier, J., Tergny, J., Laritchev, O. (1971). Programming with multiple objective functions: Step method (STEM). Mathematical Programming, Vol 1, pp. 366–375.

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Buchanan, J.T. (1997). A naive approach for solving multi-criteria decision making problems: The guess method. Journal of the Operational Research Society 48, pp. 202-206. Coello, C.A.C., VanVeldhuizen, D.A., Lamont, G. (2002). Evolutionary algorithms for solving multi-objective problems. Boston, MA: Kluwer. Deb, K. (2001). Multi-objective optimization using evolutionary algorithms. Chichester, UK: Wiley. Deb, K., Sinha, A., Korhonen, P., Wallenius, J. (2010). An interactive evolutionary multi-objective optimization method based on progressively approximated value functions. IEEE Transactions on Evolutionary Computation, 14 (5), 723–739. Deb,

K. (2011). Multiobjective Optimization Using Evolutionary Algorithms: An Introduction. KanGAL Report Number 2011003.

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Genişletilmiş Özet Çok Amaçlı Karar Verme Problemleri İçin İnteraktif Yaklaşımlar Üzerine Bir Araştırma Çok Amaçlı Karar Verme problemlerinin çözümü için geliştirilmiş birçok yöntem bulunmaktadır ve bu yöntemlerin farklı şekillerde sınıflandırılmaları mümkündür. Bu çalışmada geliştirilen yöntemler arasındaki temel farklılıkları ortaya koymak maksadıyla, sürekli yapıdaki çok amaçlı problemler için geliştirilmiş interaktif yöntemler incelenmiştir. Bu yöntemlerin iyi anlaşılması için sistematik ve bütüncül bir yaklaşıma ihtiyaç olduğu düşüncesinden yola çıkılmıştır. Çalışma neticesinde bu yöntemlerin literatürde iki temel yaklaşımla sınıflandırıldığı görülmektedir. Birinci sınıflandırma şekli interaktifliğin nasıl yapıldığı (gerçekleştirildiği) ile ilgilidir. İkinci sınıflandırma şekli ise geliştirilen algoritmanın teknik yapısı ile ilgilidir. Son zamanlar da doğrusal olmayan ve NP-Zor türündeki problemlemlerin çözümü için ayrı bir sınıf olarak evrimsel algoritmalar da öne çıkmaktadır. Çalışma kapsamında konunun teorik olarak gelişimine önemli katkı sağlayan metotlar öncelikle ele alınmıştır. Bu metotlar arasındaki temel farklılıklar incelenerek ortaya konmuştur. Genel olarak bir Çok Kriterli Karar Verme problemine ait matematiksel modeli aşağıda sunulduğu gibi ifade edebiliriz. x Є X olmak üzere, Maksimum z = f(x) Burada, f(x) = (f1(x),……. ,fp(x)) amaç fonksiyonlarının oluşturduğu bir fonksiyon kümesini (bir kolon vektörü), fi(x) i’ninci amaç fonksiyonunu, x = (x1,…….,xn) karar kolon vektörünü, X, Rn `de uygun çözüm kümesini (karar uzayını), z = f(x) modelin amaç (kriter) fonksiyonunu,

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Z=f(X) ise Rn `de uygun amaç kümesini (kriter uzayını) ifade etmektedir. Modelin en önemli özelliklerinden biri iki farklı uzaya sahip olmasıdır. X karar uzayını ifade ederken, Z kriter veya amaç uzayını ifade etmektedir. Karar uzayındaki herbir noktanın amaç uzayında bir karşılığı vardır. Kritik olan konu, amaç uzayında çözüm için sunulan noktanın karar vericinin tercihine uygun olup olmadığıdır. Matematiksel model birden çok amaç fonksiyonundan ibarettir. Problemin Çok Kriterli Karar Verme modeli olarak değerlendirilebilmesi için amaç fonksiyonlarından en az ikisinin birbiriyle çatışması gerekmektedir. Yani amaçlardan en az birinin iyileştirilmesi başka bir amaçta kötüleşmeye sebep olmalıdır. İnteraktif metotlar genel olarak iki safhadan oluşur. Birinci safha karar vericilerin uygun etkin çözümleri öğrendikleri safhadır. İkinci safha ise karar vericilerin tercih edilen çözümle ilgili karar verme safhasıdır. Çözüm algoritmaları döngülü bir yapı içerisinde oluşturulmaktadır. Algoritmanın basamakları, karar vericilerin adım adım tercihlerini belirtmeleri için tekrar edilir. Her bir çözümde karar vericilere bulunan çözümler sunulur ve karar vericilerin bu çözümlerden hangisini tercih ettiği sorulur. Karar vericilerden temin edilen bu bilgi tercih fonksiyonunu şekillendirmek ve yeni bir çözüm üretmek için kullanılır. Bu şekilde karar vericilerin çözüm bulma sürecine aktif olarak katılımları sağlanmış olur ve tercih fonksiyonunun yapısı ortaya konur. Bazen karar vericilerin kendi kararlarını (tercihlerini) değiştirdikleri de görülür. Genel olarak Çok Kriterli Karar Verme problemlerinin interaktif bir metotla çözümü için aşağıda sunulan adımlar takip edilebilir (Miettinen et al 2008). (1) En ideal ve en kötü çözümleri bul ve karar vericileri bu çözümler hakkında bilgilendir. (2) Bir başlangıç etkin çözüm noktası üret. (Bu çözüm doğal bir uzlaşma noktası olabileceği gibi karar vericiden de istenebilir.) (3) Karar vericiden tercih bilgisi al. (4) Karar vericinin tercihlerine göre yeni etkin çözümler üret. (5) Karar vericiden bulunan çözümler içinden en iyisini seçmesini iste.

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(6) Karar verici bulunan çözümden memnun olursa algoritmayı durdur. Bulunan çözüm karar vericiyi tatmin etmediyse üçüncü adıma git. Çok Kriterli Karar Verme sürecinde karşılaşılan önemli bir güçlük tercih fonksiyonunun doğru bir şekilde tespit edilmesidir. İnteraktif metotlarda bu bilgi karar vericilerden direk olarak temin edildiği için analizcilerin işi kolaylaşmış olur. İnteraktif süreç, karar vericinin çözümden memnun olması garanti edilene kadar takrarlandığından, tercih fonksiyonun doğru bir şekilde karar verme sürecine yansıması mümkün olmaktadır. Bu Çok Kriterli Karar Verme problemlerinde interaktif yöntemlerin tercih edilmesinin önemli sebeplerinden biridir. Literatürde sürekli yapıdaki Çok Amaçlı Optimizasyon problemlerinin çözümüne yönelik geliştirilen bir çok metot bulunmaktadır. Geliştirilen bu metotlar, çözüm aşamasında karar verici ile interaktiflik şekline göre ve metodun teknik olarak yapılandırma şekline göre sınıflandırılabilir. İnteraktiflik şekli, elde edilen muhtemel çözümlerin karar vericiye iletilmesi ve karar vericinin buna verdiği tepkiyi içermektedir. Başka bir ifade ile interaktiflik, çözüme yönelik bilginin karar vericiye ulaştırılma şeklini ve bunun sonucunda da karar vericiden tercihe ait bilginin temin edilmesini belirtmektedir. Öte yandan çözüm ile ilgili teknik hususlar ise kullanılan matematiksel varsayımlar, metodun optimal sonuca yakınsaması, bulunan çözümün ne kadar optimal olduğu konularını kapsamaktadır (Miettinen vd. 2008). İnteraktifliğin şekline göre, literatürde sürekli yapıdaki Çok Amaçlı Optimizasyon metotları aşağıda belirtildiği gibi üç ana gurup altında toplanabilir (Miettinen vd. 2008).  Takas / ödün verme bilgisine dayalı metotlar  Referas noktası yaklaşımı kullanan metotlar  Gruplandırmaya dayalı metotlar. Sürekli yapıdaki Çok Amaçlı Optimizasyon metotlarının bir diğer sınıflandırma şekli, bu metotların kullandıkları çözüm tekniklerine göredir. Kullanılan çözüm tekniğine göre metotların sınıflandırması aşağıda sunulduğu gibi yapılabilir (Steuer, 1986).

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 Uygun çözüm bölgesini azaltma metodu,  Ağırlık vektör uzayı azaltma metodu,  Kriter konunu daraltma veya doğru araştırma metodu. Son yıllarda özellikle doğrusal olmayan ve NP-Zor türünde problemlerin çözümü için klasik yöntemlerin dışında evrimsel algoritmalara da ağırlık verildiği görülmektedir. Bu kapsamda Çok Amaçlı Optimizasyon metotlarının bir başka sınıflandırma yönteminin de klasik ve evrimsel algoritmalar olarak ifade edebiliriz. Çalışmada kapsamında irdelenen sürekli yapıdaki Çok Amaçlı Optimizasyon metotları özet olarak aşağıda sunulmuştur. STEM Metodu: Uygun çözüm bölgesi azaltma metodu Geoffrion-Dyer-Feinberg (GDF) Metodu: Takasa dayalı doğrusal araştırma metodu İnteraktif Ağırlıklandırılmış Tchebycheff Prosedürü: Referans noktası yaklaşımı kullanan bir vektör uzayı azaltma metodu Korhonen ve Laakso Görsel İnteraktif Yaklaşım Metodu: Referans noktası kullanan bir doğrusal araştırma metodu Zionts-Wallenius Metodu: Takasa dayalı vektör uzayı azaltma metodu Ardışık Proxy Optimization Tekniği (SPOT): Takasa dayalı doğrusal araştırma metodu GUESS Method: Referans noktası yaklaşımı kullanan bir uygun çözüm bölgesi azaltma metodu Tatminkar Takas Metodu (STOM): Takasa dayalı uygun çözüm bölgesi azaltma metodu Türevlenemeyen İnteraktif Çok Amaçlı Optimizasyon Sistemi (NIMBUS): Doğrusal araştırma kullanan uygun çözüm bölgesi azaltma metodu Evrimsel Çok Amaçlı İnteraktif Algoritmalar: Takasa dayalı metahüristik araştırma metodu Üzerinde durulması gereken önemli bir nokta söz konusu yöntemlerin hiçbiri bir diğerine üstün değildir. Her bir metot ilgilenilen bir problemin özelliklerine istinaden geliştirilmiştir. Başka bir ifade ile

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problemin yapısını belirleyen doğrusal veya doğrusal olmama durumu, fonksiyon veya uygun çözüm kümelerinin konveks veya konkav olması, fonksiyonların türevlenebilir olup olmaması, problem yapısının NP-Hard türünde olması gibi hususlar, belli bir durumda uygulanacak yöntemi belirleyen temel faktörlerdir. Dolayısıyla bir problemi çözerken hangi yöntemin ihtiyacımıza en uygun olduğunu, ancak problemin yapısını ve özelliklerini doğru bir şekilde analiz ederek anlayabiliriz. Konu kapsamında ifade edilmesi gereken başka bir konu da interaktif çözümlerin çok amaçlı problemlerin çözümünde insan davranışını dikkate alarak çözüm sürecine dâhil etmesidir. Bu yöntemler aynı zamanda birçok gerçek problemin doğasında var olan doğrusal olmama durumu için de uygun çözüm yöntemleri olarak karşımıza çıkmaktadır. Gelecek te doğrusal olmayan tekniklerin bu kapsamda yaygın olarak kullanılması kaçınılmaz olarak görülmektedir. Öte yandan interaktif yöntemler karar vericilerin kısıtlı olan zamanlarının kullanılmasını gerektirmektedir. Araştırmacılar, karar vericilerden bilgi temin ederken onların zamanlarını da minimum seviyede kullanmaya gayret etmelidirler. Bu kapsamda karar vericilerin işlerini kolaylaştırmak ve zamanlarını fazla almamak için daha kullanıcı dostu teknikler geliştirilmesi gerekir. Gelecekte, bu tür insan faktörlerini dikkate alan ve modele uygun bir şekilde yansıtan modeler geliştirilmesinin faydalı olacağı değerlendirilmektedir.