Dominance and potential optimality in multiple criteria decision analysis with imprecise information

This paper discusses multiple criteria models of decision analysis with finite sets of alternatives. A weighted sum of criteria is used to evaluate the performance of alternatives. Information about the weights is assumed to be in the form of arbitrary linear constraints. Conditions for checking dominance and potential optimality of decision alternatives are presented. In the case of testing potential optimality, the proposed appoach leads to the consideration of a couple of mutually dual linear programming problems. The analysis of these problems gives valuable information for the decision maker. In particular, if a decision alternative is not potentially optimal, then a mixed alternative dominating it is defined by a solution to one of the LP problems. This statement generalizes similar results known for some special cases. The interpretation of the mixed alternative is discussed and compared to its analogue in a data envelopment analysis context.     

A new method for ranking intuitionistic fuzzy values and its application in multi-attribute decision making

In this paper, we propose a new method for ranking intuitionistic fuzzy values (IFVs) by using the similarity measure and the accuracy degree of IFVs. Then we apply the proposed ranking method to multi-attribute decision making.     

Multi-person multi-attribute decision making models under intuitionistic fuzzy environment

Intuitionistic fuzzy numbers, each of which is characterized by the degree of membership and the degree of non-membership of an element, are a very useful means to depict the decision information in the process of decision making. In this article, we investigate the group decision making problems in which all the information provided by the decision makers is expressed as intuitionistic fuzzy decision matrices where each of the elements is characterized by intuitionistic fuzzy number, and the information about attribute weights is partially known, which may be constructed by various forms. We first use the intuitionistic fuzzy hybrid geometric (IFHG) operator to aggregate all individual intuitionistic fuzzy decision matrices provided by the decision makers into the collective intuitionistic fuzzy decision matrix, then we utilize the score function to calculate the score of each attribute value and construct the score matrix of the collective intuitionistic fuzzy decision matrix. Based on the score matrix and the given attribute weight information, we establish some optimization models to determine the weights of attributes. Furthermore, we utilize the obtained attribute weights and the intuitionistic fuzzy weighted geometric (IFWG) operator to fuse the intuitionistic fuzzy information in the collective intuitionistic fuzzy decision matrix to get the overall intuitionistic fuzzy values of alternatives by which the ranking of all the given alternatives can be found. Finally, we give an illustrative example.     

A fuzzy closeness approach to fuzzy multi-attribute decision making

The aim of this paper is to develop a new fuzzy closeness (FC) methodology for multi-attribute decision making (MADM) in fuzzy environments, which is an important research field in decision science and operations research. The TOPSIS method based on an aggregating function representing “closeness to the ideal solution” is one of the well-known MADM methods. However, while the highest ranked alternative by the TOPSIS method is the best in terms of its ranking index, this does not mean that it is always the closest to the ideal solution. Furthermore, the TOPSIS method presumes crisp data while fuzziness is inherent in decision data and decision making processes, so that fuzzy ratings using linguistic variables are better suited for assessing decision alternatives. In this paper, a new FC method for MADM under fuzzy environments is developed by introducing a multi-attribute ranking index based on the particular measure of closeness to the ideal solution, which is developed from the fuzzy weighted Minkowski distance used as an aggregating function in a compromise programming method. The FC method of compromise ranking determines a compromise solution, providing a maximum “group utility” for the “majority” and a minimum individual regret for the “opponent”. A real example of a personnel selection problem is examined to demonstrate the implementation process of the method proposed in this paper.     

Decision-aid for discrete multiple criteria decision making problems with imprecise data

We describe ways of aiding decision making with a discrete set of alternatives. In many decision situations, it is not possible to obtain explicit preference information from the decision makers. Instead, useful decision-aid can be provided to the decision makers by describing what kind of weighting of the criteria result in certain choices of the alternatives. The suggested treatment is based on the basic ideas of the ELECTRE III method. The modelling of the preferences by pseudo-criteria is especially helpful in case the data, that is, the criterion values are imprecise. Unlike ELECTRE III, no ranking of the alternatives is produced. Based on a minimum-procedure in the exploitation of the outranking relations, we provide information about the weights of the criteria that make a certain alternative the best. We also present an interactive searching procedure in the weighting space. The auxiliary optimization problems to be solved are nondifferentiable. Cases with both single and multiple decision makers are considered.     

Exponential operations for intuitionistic fuzzy numbers and interval numbers in multi-attribute decision making

Intuitionistic fuzzy numbers (IFNs) have already been applied to many fields, especially in multi-attribute decision making (MADM). Based on the basic operational laws and information aggregation methods of IFNs, MADM with intuitionistic fuzzy information has become more and more popular. In this paper, we investigate the MADM problems where the attribute values take the form of interval numbers and the weight information on the attributes are expressed as IFNs. We first propose a novel exponential operational law based on IFNs and interval numbers, and then study some of its desirable properties. Based on the exponential operational law, we put forward an intuitionistic fuzzy weighted exponential aggregation operator, and utilize it to develop a MADM method. Finally, we apply our method to solve the decision making problem under uncertainty.     

Fuzzy optimality based decision making under imperfect information without utility

In the realm of decision making under uncertainty, the general approach is the use of the utility theories. The main disadvantage of this approach is that it is based on an evaluation of a vector-valued alternative by means of a scalar-valued quantity. This transformation is counterintuitive and leads to loss of information. The latter is related to restrictive assumptions on preferences underlying utility models like independence, completeness, transitivity etc. Relaxation of these assumptions results into more adequate but less tractable models. In contrast, humans conduct direct comparison of alternatives as vectors of attributes’ values and don’t use artificial scalar values. Although vector-valued utility function-based methods exist, a fundamental axiomatic theory is absent and the problem of a direct comparison of vectors remains a challenge with a wide scope of research and applications. In the realm of multicriteria decision making there exist approaches like TOPSIS and AHP to various extent utilizing components-wise comparison of vectors. Basic principle of such comparison is the Pareto optimality which is based on a counterintuitive assumption that all alternatives within a Pareto optimal set are considered equally optimal. The above mentioned mandates necessity to develop new decision approaches based on direct comparison of vector-valued alternatives. In this paper we suggest a fuzzy Pareto optimality (FPO) based approach to decision making with fuzzy probabilities representing linguistic decision-relevant information. We use FPO concept to differentiate “more optimal” solutions from “less optimal” solutions. This is intuitive, especially when dealing with imperfect information. An example is solved to show the validity of the suggested ideas.     

Z probabilistic linguistic term sets and its application in multi-attribute group decision making

Fuzzy Optimization and Decision Making - Probabilistic linguistic term set solves the problem of probabilistic distribution of linguistic terms. Due to the objective and subjective factors such as...     

Robust optimization analysis for multiple attribute decision making problems with imprecise information

This paper considers ranking decision alternatives under multiple attributes with imprecise information on both attribute weights and alternative ratings. It is demonstrated that regret results from the decision maker??s inadequate knowledge about the true scenario to occur. Potential optimality analysis is a traditional method to evaluate alternatives with imprecise information. The essence of this approach is to identify any alternative that outperforms the others in its best-case scenario. Our analysis shows that potential optimality analysis is optimistic in nature and may lead to a significant loss if an unfavorable scenario occurs. We suggest a robust optimization analysis approach that ranks alternatives in terms of their worst-case absolute or relative regret. A robust optimal alternative performs reasonably well in all scenarios and is shown to be desirable for a risk-concerned decision maker. Linear programming models are developed to check robust optimality.     

A new methodology for fuzzy multi-attribute group decision making with multi-granularity and non-homogeneous information

The aim of this paper is to develop a new methodology for solving fuzzy multi-attribute group decision making problems with non-homogeneous information, including multi-granular linguistic term sets, fuzzy numbers, interval values and real numbers. In this methodology, different distances are defined to measure differences between alternatives and the ideal solution as well as the negative ideal solution. A relative closeness method is developed by introducing the multi-attribute ranking index based on the particular measure of closeness to the IS. The proposed method determines a compromise solution for the group, providing a maximum “group utility” for the “majority” and a minimum of an individual regret for the “opponent”. The implementation process, effectiveness and feasibility of the method proposed in this paper are illustrated with a real example of the missile weapon system design project selection.     

On the maximum number of feasible ranking sequences in multi-criteria decision making problems

In many decision problems the focus is on ranking a set of m alternatives in terms of a number, say n, of decision criteria. Given are the performance values of the alternatives for each one of the criteria and the weights of importance of the criteria. This paper demonstrates that if one assumes that the criteria weights are changeable, then the number of all possible rankings may be significantly less than the upper limit of m!. As a matter of fact, this paper demonstrates that the number of possible rankings is a function of the number of alternatives and the number of criteria. These findings are important from a sensitivity analysis point of view or when a group decision making environment is considered.     

Some interval-valued intuitionistic uncertain linguistic Choquet operators and their application to multi-attribute group decision making

In this study a generated admissible order between interval-valued intuitionistic uncertain linguistic numbers using two continuous functions is introduced. Then, two interval-valued intuitionistic uncertain linguistic operators called the interval-valued intuitionistic uncertain linguistic Choquet averaging (IVIULCA) operator and the interval-valued intuitionistic uncertain linguistic Choquet geometric mean (IVIULCGM) operator are defined, which consider the interactive characteristics among elements in a set. In order to overall reflect the correlations between them, we further define the generalized Shapley interval-valued intuitionistic uncertain linguistic Choquet averaging (GS-IVIULCA) operator and the generalized Shapley interval-valued intuitionistic uncertain linguistic Choquet geometric mean (GS-IVIULCGM) operator. Moreover, if the information about the weights of experts and attributes is incompletely known, the models for the optimal fuzzy measures on expert set and attribute set are established, respectively. Finally, a method to multi-attribute group decision making under interval-valued intuitionistic uncertain linguistic environment is developed, and an example is provided to show the specific application of the developed procedure.     

Reserving relief supplies for earthquake: a multi-attribute decision making of China Red Cross

Earthquake usually causes grievous damages especially in the earthquake prone areas. Building a fast and effective relief system is very important. China Red Cross is the largest humanitarian organizations in China. To improve its disaster response system, reserving relief supplies gets more and more attentions. In this article, a decision making programming based on the data envelopment analysis method is proposed in the construction of reserve network for China Red Cross with three factors, including cost, risk and utility. Four warehouses are selected and the optimal allocation strategies are obtained under the principles of equity and effectiveness which should be obeyed in humanitarian assistance.     

Power average operators of trapezoidal intuitionistic fuzzy numbers and application to multi-attribute group decision making

Trapezoidal intuitionistic fuzzy numbers (TrIFNs) is a special intuitionistic fuzzy set on a real number set. TrIFNs are useful to deal with ill-known quantities in decision data and decision making problems themselves. The focus of this paper is on multi-attribute group decision making (MAGDM) problems in which the attribute values are expressed with TrIFNs, which are solved by developing a new decision method based on power average operators of TrIFNs. The new operation laws for TrIFNs are given. From a viewpoint of Hausdorff metric, the Hamming and Euclidean distances between TrIFNs are defined. Hereby the power average operator of real numbers is extended to four kinds of power average operators of TrIFNs, involving the power average operator of TrIFNs, the weighted power average operator of TrIFNs, the power ordered weighted average operator of TrIFNs, and the power hybrid average operator of TrIFNs. In the proposed group decision method, the individual overall evaluation values of alternatives are generated by using the power average operator of TrIFNs. Applying the hybrid average operator of TrIFNs, the individual overall evaluation values of alternatives are then integrated into the collective ones, which are used to rank the alternatives. The example analysis shows the practicality and effectiveness of the proposed method.     

Intuitionistic uncertain linguistic powered einstein aggregation operators and their application to multi-attribute group decision making

The intuitionistic uncertain fuzzy linguistic variable can easily expressthe fuzzy information, and the power average (PA) operator is a usefultool which provides more versatility in the information aggregation procedure.At the same time, Einstein operations are a kind of various t-normsand t-conorms families which can be used to perform the corresponding intersectionsand unions of intuitionistic fuzzy sets (IFSs). In this paper, wewill combine the PA operator and Einstein operations to intuitionistic uncertainlinguistic environment, and propose some new PA operators. Firstly,the definition and some basic operations of intuitionistic uncertain linguisticnumber (IULN), power aggregation (PA) operator and Einstein operationsare introduced. Then, we propose intuitionistic uncertain linguistic fuzzypowered Einstein averaging (IULFPEA) operator, intuitionistic uncertain linguisticfuzzy powered Einstein weighted (IULFPEWA) operator, intuitionisticuncertain linguistic fuzzy Einstein geometric (IULFPEG) operator and intuitionisticuncertain linguistic fuzzy Einstein weighted geometric (IULFPEWG)operator, and discuss some properties of them in detail. Furthermore, we developthe decision making methods for multi-attribute group decision making(MAGDM) problems with intuitionistic uncertain linguistic information andgive the detail decision steps. At last, an illustrate example is given to showthe process of decision making and the effectiveness of the proposed method.     

Decision making,new information,ranking and structure

The rank of a set of alternatives can change if a new criterion is introduced into the set of criteria, but it can also change if the importances of the criteria depend on the number of alternatives and on the strength of their ranking. As a result a new alternative may change the relative order of the previous set. This is allowed to happen by making paired comparisons of alternatives with respect to criteria. It is particularly useful in a complex structure where the importances of the criteria are not well established. If on the other hand they are, then the alternatives can be scored with respect to intensities of the criteria and a new alternative would not change the relative rank of the old ones. Copies and near copies of alternatives can be treated by applying the same weight to them or by comparing them with the others respectively. Near copies are identified by a % variation measure.     

Efficiency ranking of decision making units in data envelopment analysis by using TOPSIS-DEA method

Data envelopment analysis methods classify the decision making units into two groups: efficient and inefficient ones. Therefore, the fully ranking all DMUs is demanded by most of the decision makers. However, data envelopment analysis and multiple criteria decision making units are developed independently and designed for different purposes. However, there are some applications in problem solving such as ranking, where these two methods are combined. Combination of multiple criteria decision making methods with data envelopment analysis is a new idea for elimination of disadvantages when applied independently. In this paper, first the new combined method is proposed named TOPSIS-DEA for ranking efficient units which not only includes the benefits of both data envelopment analysis and multiple criteria decision making methods, but also solves the issues that appear in former methods. Then properties and advantages of the suggested method are discussed and compared with super efficiency method, MAJ method, statistical-based model (CCA), statistical-based model (DR/DEA), cross-efficiency—aggressive, cross-efficiency—benevolent, Liang et al.’s model, through several illustrative examples. Finally, the proposed methods are validated.     

A new method for ranking fuzzy numbers and its application to group decision making

In this paper, a new method for comparing fuzzy numbers based on a fuzzy probabilistic preference relation is introduced. The ranking order of fuzzy numbers with the weighted confidence level is derived from the pairwise comparison matrix based on 0.5-transitivity of the fuzzy probabilistic preference relation. The main difference between the proposed method and existing ones is that the comparison result between two fuzzy numbers is expressed as a fuzzy set instead of a crisp one. As such, the ranking order of n fuzzy numbers provides more information on the uncertainty level of the comparison. Illustrated by comparative examples, the proposed method overcomes certain unreasonable (due to the violation of the inequality properties) and indiscriminative problems exhibited by some existing methods. More importantly, the proposed method is able to provide decision makers with the probability of making errors when a crisp ranking order is obtained. The proposed method is also able to provide a probability-based explanation for conflicts among the comparison results provided by some existing methods using a proper ranking order, which ensures that ties of alternatives can be broken.     

A grey relational projection method for multi-attribute decision making based on intuitionistic trapezoidal fuzzy number

With respect to multiple attribute decision making (MADM) problems in which the attribute value takes the form of intuitionistic trapezoidal fuzzy number, and the attribute weight is unknown, a new decision making analysis methods are developed. Firstly, some operational laws and expected values of intuitionistic trapezoidal fuzzy numbers, and distance between two intuitionistic trapezoidal fuzzy numbers, are introduced. Then information entropy method is used to determine the attribute weight, and the grey relational projection method combined grey relational analysis method and projection method is proposed, and to rank the alternatives are done by the relative closeness to PIS which combines grey relational projection values from the positive ideal solution and negative ideal solution to each alternative. Finally, an illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness.