Çok amaçlı karar vermede yeni bir yöntem ve uygulaması
Application working on trade banking of a new multiple objective decision making method
- Tez No: 46304
- Danışmanlar: PROF.DR. RAMAZAN EVREN
- Tez Türü: Doktora
- Konular: Endüstri ve Endüstri Mühendisliği, Industrial and Industrial Engineering
- Anahtar Kelimeler: Çok amaçlı karar verme, Multiobjective decision making
- Yıl: 1995
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
ÖZET ÇAKV Yöntemlerinin İncelenmesi ve Yeni bir ÇAKV Yönteminin Ticari Bankalara Uygulama Çalışması Bu çalışma 5 bölümden oluşmaktadır. Birinci bölümde, Karar Verme Teorisinden kısaca bahsedilmiş ve karar vermeye sistem yaklaşımı yapılmaya çalışılmıştır. Daha sonra karar verme sürecindeki temel unsurlar sıralanmış, amaç sayısı ve bilgi düzeyine göre karar verme türleri tablo halinde ifade edilmiştir. ikinci bölümde ÇAKVnin kısa tarihçesi ve terminolojisine değinilmiştir. Aynı bölümün gereği ÇAKV yöntemleri sınıflandırılmış ve özetlenmiştir. Üçüncü bölümde, yeni yöntemin geliştirilmesinde yararlanılan temel bazı ÇAKV yöntemleri örnek çözümleri ile birlikte ele alınmıştır. Dördüncü Bölümde yeni yöntem; formülasyonu, algoritması, temel prensipleri, Lineer ve Non-Lineer sayısal örnekleri, fayda ve sakıncaları ile birlikte genişçe ele alınmıştır. Beşinci bölümde ise; yeni yöntem sermaye yeterliliği, risk faktörü ve karlılıktan oluşan üç amaçlı, 12 kısıtlı ve 16 karar değişkenli gerçek bir finansal probleme uygulanmış ve bu yeni çalışmanın genel bir değerlendirilmesi yapılmıştır. xııı
Özet (Çeviri)
SUMMARY APLICATION WORKING ON TRADE BANKING OF A NEW MULTIPLE OBJECTIVE DECISION MAKING METHOD Decision making is the process of selecting a possible course of action from all the available alternatives. In almost all such problems the multiplic ity of criteria forjudging the alternatives is pervasive. That is, for many such problemls, the decision maker wants to attain more than one objective or goal in selecting the course of action while satisfying the constrains dictated by environment, processes, and resources. Another characteristic of these pro blems is that the objectives are apparently noncommensurable mathemati cally, these problems can be represented as: Max Lfi (x), f2 (x),..., fk(x)j Subject to: gi(x)<0, i=l,..., m Where x is an n dimensional decision variable vector. The problem con sists of decision variables, m constraints and k objectives. In Literature, this problem is often referred to as a vector Maximum problem. Traditionally, there are two approaches for solving the vector maxi mum problem. One of them is to optimize one of the objectives while appending the oth er objectives to a canstraint set so that the optimal solution would satisfy these objectives at least up to a predetermined level. The problem is given as: Max fj (x) Subject to: gj(x)<0,j=i,...,m fjCx^aL, L=i,..., k, and k*iWhere aLis any acceptable predetermined level objective L. The other approach is to optimize a super-objective function created by multiplying each objective function with a suitable weight and then by adding them to gether. This aproach leads to the solution of the following problem: k Max5>ifi(x) i=l Subject to: gi(x)<0, i=i,....,m k The weights are usually normalized so that zA j = 1 Often both of the i=l above approaches lead to a solution which may not be the best or most satis factory. Because of the incommensurability and the conflincting nature of the multiple criteria, the problem becomes complex and it becomes difficult to choose the acceptable levels, aL 's, that will result in a nonempty con straint set in the first attempt for the solution. In the first approach, the im plied value trade-off between fL and fj is as follows: Value Trade-off 0, k^L oo This may not be the actual value structure of the decision maker and this value structure is sensitive to the level of a^. In the second approach, the major problem is to determine the proper weights, X{. The X,j s are sensitive to the level of the particular objective as well as the level of all other objectives. Multiple objective decision making methods are the result of the desire to eliminate the above difficulties as well as to treat the objectives indepent- ly. Most of the progress in this area has taken place within the last decade even though one of the earliest considerations of multiple objectives can be found in 1951 by kuhn-Tucker. The rapid progress in such a short time neces sitates a thorough review of the existing literature and a systematic classifi cation of methods for the guidance of future user. This thesis consists of for chapters; In the first section, decision making thery has been shortly stated out and system aproach has been tried to be ap plied to decision making. Later on basic factors have been listed m the deci sion making process. Finally the vericties of decision making have been sche matized. xvIn chapter 2, basic concepts of mutiple criteria decision making are pre sented. There are some main terms Which have not a Universal definition used in multiple criteria decision making literature. These words are: At tributes, Objectives, Goals, Criteria. Attributes are the characteristics, qualities or performance parame- teres of alternatives. Objectives are the directions“to do better”as perceived by the decision maker. Goals are things desired by the decision maker expressed in terms of a specific state in space and time. Criteria are standards of judgement or rules to test acceptability. This is the dictionary definition of the term. However, as used in the Multiple Cri teria Decision Making Literature, it indicates attributes and/or objectives. Some other main concepts such as feasible region, optimal solution, the best compromise solution, nondominated solution, dominated solution, the preferred solution, etc. Have also been exlained with examples in this chap ter. In addition to this, multiple objective decision making theories have been applied to a general classification. One of the deterministic methods“The method of Electre and Zionts-Wallenius”, which includes alternative in limited number and where out put are deterministic and constraints are im plicit, has been told. In this chapter, finally the methods of multiple objective mathematic programmes where outputs are deterministic, constraints are implicit and includes endless number of alternatives have been given in the form of table- ue. After that all the methods in this group have been shortly mentioned. All the methods in this group can be classified according to as follows: 1- No articulation of preference information. 2- A prior articulation of preference information. 3- Progressive articulation of preference information (intiractive met-hods). 4- A pasteriori articulation of preference information (Non-dominated solution generating method). In the first group, the methods for solving multiple criteria decision making problems in which no articulation of preference information is asked from decision maker is presented. This means that this approach do not need any inter objective or other subjective preference informatiton from the decision maker once the problem constraints and objectives have been de fined. Thus this approach requires that the decision maker is able to accept the solution obtained from the method. The advantage of this route is that in the process of obtaining the solution, the decision maker will not be disturbed by the analyst which is preferable from the point-of-view of the decision mak er. But a major disadvantage then is the necessity for the analyst to make many assumptions about the decision maker's preferences. xviThis is difficult to be done by the best several knowledgeable analysts. The major method of this category is the method of global criterion which is explained. In second group, the methods in which a priori articulation of prefer ence information is asked from decision maker is presented. In this method,“a priori”means the preference information is given by the decision maker to the analyst before he/she actually solves the problem. The decision maker provides the information during or after the actual mathema-tical formulation of the problem. The information may be either: (1) cardinal information, or (2) mixed (ordinal and cardinal) information. In the case of cardinal information, the decision maker must give some judge ment about specific objective preference levels or specific trade-offs. But if the information is mixed, the decision maker must rank the objec tives in order of their importance. Utility function methods and bounded objective methods, under cardi nal information category are prensented. The basic logic of the utility func tion method is to attain utilities apporiate to the objectives. In the con strained objective method decision maker must define an acceptable lower bounds for each objective. The main techniques which can be applied when the mixed information given are Lexicographic method, Goal Programming method and Goal At tainment method. In lexicographic method, all the objectives are sequenced according to their degree of importance by decision maker. The“preferred so lution”is the solution which maximizes the objectives by starting from the most important one and progressing according to degree of importance of them. In goal attainment method which is another type of the linear goal pro gramming, a goal vectur of the weighting factors related to negative or pozi- tive devations of the goals is determined by the decision maker. This method also finds the nondominated set. The third category includes the methods used when progressive articu lation of preference information is given. In such kind of methods, the pre ferred solution is obtained after some iterations. These methods are the in teractive searching methods. In every step of this searching, decision maker finds the preferred solu tion by seeking answers to some questions. There are an optimization and a decison phase in the searching process. The method of surrogate worth trade-off is other interactive technic. This method is proposed by Haimes, Hall, and Freedman. It recognizes that given any current set of objective levels attained, it is much easier for the dec ision maker to assess the relative value of the trade-off of marginal increases and decreases between any two objectives than their absolute values. The last method which is called as interactive compromise program ming. The method attempts to reduce the complexity of information required from the decision maker. No prior information required from the decision maker at each iteration is also simple to provide. xvnThe solutions in terms of the degree of closeness to the ideal solution are pre sented to the decision maker, and he is only asked his most preferred and/or least preferred solution. The method does not require significantly more da ta than the pure linear programming does. One of the important method in the third category is STEM (STEP- method). The STEM, the progressive orientation procedure and the method of constraints are for solution of multiple objective linear programming pro blems. The Interval criterion method is MOLP method. The method, proposed by Steuer, is an extension of the multiple objective linear programming method. Multiple objective linear programming problems, even moderately sized ones, often have an unworkably large number of nondominated ex treme points. Steuer's interactive Multiple objective linear programming method presents to the decision maker 2k+l nondominated extreme points at each iteration (k is the number of objectives); the decision maker has only to indicate the most preferable solution from this set. Once this solution is identified, the nondominated extreme point in the neighborhood are ex plored and a new set of nondominated solutions are identified and presented to the decision maker. The fourth category is a category which consists of some MOLP tech nics. When a pasteriori articulation of preference information is given, it is not wanted a preference information related to the objectives from the deci sion maker at the begining. First, the subset of the inferior solutions is ob tained and then, decision maker selects the best compromise solution from this subset. Then, decision maker is wanted to give the trade-off information. The methods which belong to this group are the weighting method, e con straint searching method. vt> In the Weighting method, all the objectives are weighted to generate a non dominated set and so that, it's possible to express the opposite objective values in the same unit. £ constraint method supposes that a profit which is upon the maximum level defined associated to the objectives by decision maker, is harmful. The aim of another method which is called Multiple objective linear programming is to optimise the objectives under definite accepts by associat ing the best combinations of the qualities. Finally, when a decision maker is only related to inferrior values in the solution of the problem, adjusting searching method can be used. In third chapter, some basic multiple objective decision making meth ods have been mentioned with their examples, and Interactive compromise Goal Programming which is the subject matter of this thesis and which has been told in fourth chapter made use of this theory. One of the widely used methods for multiple Criteria Decision making method, Goal programming is presented. The method requires the decision maker to set goals for each objective that he/she wishes to attain. A preferred solution is then defined as the one which minimizes the deviations from the set goals. XVlllThe most common form of Goal programming formulation requires that the decision maker, in addition to setting the goals for the objectives, is also able to give on ordinal ranking of the objectives. The Goal programming for mulation of the Vector maximum problem for such case is: Min [pihiCd-, d+), p2h2(d-, d+),...., pLhL(dr, d+)] Subject to: gj(x)<0,j=l,2,...,m fi(x) + di--di+=bi+,i=l,2,....,k d-, dj+ > 0, Vi di-.dj^O.Vi Where bj, j= l, 2,..., k are the goals set by the decision maker for the ob jectives; dj“ and d+ are respectively the under- achievement and over- achievement of the jth goal, h^d”, d+), are linear functions of the devational variables and are called achievement functions. The pj's are preemptive weights; that is, Pj » pi+1. The solution is that h1(d“, d+) is minimized first; let min h1=h1*. Next h2(d”, d+) is minimized, but in no circumstances can hj be grater then. l^*. Thus a lower ranking achievement function. This process continues until hi(d“, d+) is minimized. Goal programming method is quite similar to lexico graphic method; the difference is that Goal programming requires goals for the objectives which are set by the decision maker and achievement func tions to be minimized in the order they are formed. Advante ges of Goal Progranrming are that the decision maker does not need to give the numerical weights for the objectives. He/She is obliged to give only an ordinal ranking of them. There are mainly three methods for the solution of linear goal program ming: * Graphical solution method. * Iterative solution method * The modified simplex method. xixSome special computer programs for linear models are available. The modified simplex algorithm approach for a moderate size problem is time consuming, and it needs a large capacity computer. The same problem can be solved iteratively by the basic simplex algorithm. If any of f j(x) and gj(x) functions are nonlinear, the problem becomes a nonlinear goal programming problem. The following methods could be used for solving nonlinear goal programming problems: * Iterative solution method * Method of Griffith and stewart * Pattern search method. Other important and interactive method is method of Geoffrion-Dyer- Feinberg; Interactive methods rely on the progressive definition of the deci sion maker's preferences along with the exploration of the criteria space. Much work has been done recently on these methods. The progressive defini tion takes place through a decision maker-analyst or decision maker-ma chine dialogue at each iteration. At each such dialogue, the decision maker is asked about some trade-off or preference information based upon the current solution (or the set of cur rent solutions) in order to determine a new solution. The method proposed by Geoffrion, Dyer and Feinberg demonstrates that a large-step gradient algorithm can be used for solving the vector maxi mum problem if the decision maker is able to specify an overall utility func tion defined on the values of the objlectives. However, the method never actu ally requires this funciton to be identified explicitly. Instead, it asks only for such local information as is needed to perform the computations. The proce dure is described in the context of the Frank- Wolfe algorithm Which is a spe cific nonlinear programming method. The problem is formulated as follows: MaxUtfiCx),^)...,^!)) Subject to: xe X The objective functions f^x) and the set X, X= (x I g(x)<0) are assumed to be explicitly known, but the utility function U(f) is assumed to be only impli citly known. Finally the third, interactive Goal programming method is presented. This method is another mathematical exression of the concept of the Geoffri on method, and the computational procedures are the same. xxIn section four, it is mentioned about ICGP method which is a new MODM method. The basic principals of this method are as follows:. - If the relationship between DM objectives is explained in the way U=f1(x)+f2(x)+ fmfe) > f°r these objectives ideal values are avaliable in order to make the function of xe X. DM for each objective function can do sensitivity analysis in order to be in the way f(x), xe X by putting added constraint. Added constraint is set by reducing Af value from ideal value. This Af value is changed constantly by DM. These changings are in equal parts. When the number of these parts in crease, they get away from ideal value. The value which is reduced as monotonous value from ideal value of ob jective function can not be smaller than anti-ideal value (minimum) Thus, the more sensitivity analysis is done, further it gets away from ideal value. This operation can be until the last f(x) value becomes equal to anti-ideal value. - DM and DM's can be easily directed by computer if they are prepared by persons who know objective and constraint equations. Thus, even only the DM which can turn on and off the computer, the method will operate the com puter program and can get result. - If DM can weight the objective functions as verbal and numerical, ob jective equation number will be reduced from. Multiple objectives to single objective. So with the constraint set which is given to equated objective func tion, solution can be found with the condition of xe X. These solutions can be renewed by changing their order or weighting ratio. - If DM prefers the solution set in any process, or determines as compro mise solution, the operation ends. If this condition is not provided, termina tion criterion can determine how much iteration will continue. - Criterion depends on the condition of operation ending when it reach es the value of oc determined by DM beforehand. Absolute value of ideal value of i. objective is calculated, i. iteration val ue is substracted from the gotten value. Then the gotten value is diveded to absolute value of ideal value. The gotten result is substracted from one. - By converting objective funtions to single objective, all the solutions produced by optimizations with the condition of xe X are non dominated. But while objective function values between two points are determined with the increases of At, and if the begining point is not a non dominated point, the prefered compromise solution can be non dominated solution ei ther. In this point the method can provide a dominance in favour of objective reference that DM will choose. But this is possible by adding the above men tioned reference objective to constraint equation. When in minimization problems, it is > and in maximization problems, it is <. But here the right side of constraint is mentioned by DM considering ideal value. xxiIf the utility function of DM is summed up as Usf1(x)+f2(x)+ fmfe) ' ^ can De solved as VMP, xs X by using the chain rule. DM only mentions with real values that how much it will decrease from which objective, and howmuch increase it will demand from which ob jective. So instead of rational weighting, by doing weighting by real values, multip le objectives are reduced to single objectives. The provided solutions can be renewed by changing real-value - changing by using chain rule of VMP. If DM can produce solutions by prioriting its objectives and weighting, or by doing value - changing as xe X, and operation can be renewed by chang ing the data, in this situation the renewing system is possible between weighting, prioriting and value-changing processes. That means triple transfer can be obtained among solution technique. - Work done so far, computer programs are developed which can solve NLGP and LGP problems and applied to real business administiration. In the same way, at a university DM's, that don't have any idea about problems have done some value changes by the direction of computer and reached the correct result. [30J That is why it is obvious that ICGP method can be applied to real linear and NLP problems. DM may not want any value of objective to be smaller than a certain minimum limit. In this situation, DM determines the least satisfying goals for its desired objectives. In this way it can reach (Compromise Solution) and (Preferred Solution) and also solution process can be shortened. DM determines goal values for its objectives. The goal values gain im portance in verbal weighting in the mathematical way. They are not important in numerical weighting and in value-changing processes. This can only be useful for DM and its user in order to give mission, vision and motivation. Another possible advantage can be in the field of in vestment, production, enlargment and planning. In the fifth section, a numerical example of a multiobjective matemacti- cal programming model of commercial bank balance sheet management is developed. The purpose of the model is to demonstrate the formulation of a multiobjective matematical programming model and to present an analysis of the results obtained from the procedure. As a result, the model is for expos itory purposes and thus considerably simplified in that the asset and liability categories used are somewhat aggregated and the model is single-period in nature. Although more detail and multiple time periods will frequently be more useful in practical applications, the prototype model presented here fa cilitates the exposition considerably. Even though the model is simplified, it does contain the essential elements of a working model and could be expand ed to include greater disaggregation, multiple time periods and additional objectives. xxnBank managers are assumed to have a utility function that contains as arguments profit and solvency. Utility is positively related to both argu ments. The actual specification of the model requires that operational meas ures of profitability and solvency be developed. First, the specification of the profit function is straightforward. Net after- tax profit is assumed to be the objective and the net after-tax rates of return on the decision variables are shown in Figure 5-2. Based on these returns, the profit function to be maximized is shown in line (1) of Fig ure 5-4. Second, the specification of operational measures of solvency is concep tually more difficult than that of profitability. Since the primary goals of bank managers, other than profitability, are generally stated in terms of li quidity and risk, measures of liquidity and risk would seem to reflect the bank's solvency objective accurately. The problem thus becomes one of find ing measures of liquidity and risk. Obviously, there are many measures of li quidity and risk that could be employed, just as there are many measures used by different banks and regulatory authorities. For purposes of the mod el presented in this thesis, liquidity and risk are jointly measured by two re lated ratios: The capital adequacy ratio employed by bank examiners of the Federal Reserve System in USA, and the risk-asset to capital ratio. The capital-adequacy ratio is a comprehensive measure of the bank's li quidity and risk because both asset and liability composition are considered when determining the value of the ratio. The explicit measure of capital adequacy employed here is the ratio of required to actual bank capital. Banks have considerable discretion in deter mining the value of the ratio of required to actual capital (hereafter referred to as the CA ratio) and, by choosing higher or lower values for the CA ratio, choose to be less liquid and bear more risk or vice versa. Since liquidity di minishes and risk increases as the CA ratio increases, banks can maximize liquidity and minimize risk by minimizng the CA ratio. The capital-adequa cy objective function is shown in line (2) of Figure 5-4. A discussion of the ac tual derivation of the CA objective is presented in a part 5-4. A second objective reflecting the bank's solvency is the risk-asset to cap ital (RA) ratio. The RA ratio is a type of capital-adequacy ratio, and, as such, is related to and included in the more comprehensive measure of capital adequacy pre sented above. It is reasonable to believe, however, that the CA ratio does not totally reflect all the relevant solvency considerations and the inclusion of the RA ratio may provide additional information on the bank's liquidity and risk po sition. Risk-assets are the most illiquid assets held by banks and have the highest default risk. In addition, the rate of loan default tends to follow the business cycle, and this part of the portfolio is thus affected by a molp consid erable amount of nondiversifiable risk. Therefore, bank managers may wish to maintain some control over the ratio of risk-assets to capital, and the CA ratio does not necessarily reflect this information. XXlllAs will be seen part 5-5 tradeoffs are possible between the CA and RA ratios, and this is a tradeoff that may concern bank management. Using the RA ratio as a risk measure, the bank is assumed to incur greater risk as the RA ratio increases. Therefore, in order to minimize risk, the RA ratio is minimized. The RA objective is shown in Figure 5-4 as line (3). It should be kept in mind that the objectives presented here are by ne cessity only examples of possible objectives that bank managers may choose in a practical setting. Any number and specification of profit, liquidity and risk objectives can be used depending on the preferences of bank manage ment. The objectives used here, however, do seem to be a realistic set of man agerial objectives and provide a good framework for a multiobjective model of bank management. If the model presented in this thesis were specified as a conventional linear programming model, constraint values would be required for the CA and RA ratios. These constraints would generally be considered policy or managerial constraints. Although the MOLP model does not require that ex plicit policy constraints be specified for the liquidity and risk objectives, it will generally be convenient to place minimum and/or maximum constraint values on certain decision variables in addition to the environmental con straints. By specifying limits on certain decision variables, implicit limits are, at the same time, placed on the objective functions. A procedure similar to this will be desirable in most cases since the MOLP procedure will otherwise arrive at a number of non-dominated solu tions that greatly exaggerate some objectives at the expense of others. As a result, a MOLP model will also require certain policy constraints; however, these constraints do not play the crucial role they do in convention al linear programming models. As Figure 5-33 indicates, the ICGP procedure yields a wide range of possible solutions to the problem. The objective values range from a high of 4.345 million for profit with accompanying CA and RA ratios of 1.468 and 9.23 respectively, to a low of 4.203 million in profits and CA and RA ratios of 1.321 and 8.43 respectively. These results of the ICGP model present the de cision maker with an efficient set of solutions from which to choose. The ”best" overall solution, i.e., the utility maximizing solution, depends on the utility function of bank managers and their evaluation of the tradeoffs that can be derived from the solutions presented in Figure 5-31, 5-33 xxix
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