The technique for order performance by similarity to ideal solution （TOPSIS） is one of the major techniques in dealing with multiple criteria decision making （MCDM） problems, and the belief structure （BS） model ...The technique for order performance by similarity to ideal solution （TOPSIS） is one of the major techniques in dealing with multiple criteria decision making （MCDM） problems, and the belief structure （BS） model has been used successfully for uncertain MCDM with incompleteness, impreciseness or ignorance. In this paper, the TOPSIS method with BS model is proposed to solve group belief MCDM problems. Firstly, the group belief MCDM problem is structured as a belief decision matrix in which the judgments of each decision maker are described as BS models, and then the evidential reasoning approach is used for aggregating the multiple decision makers＇ judgments. Subsequently, the positive and negative ideal belief solutions are defined with the principle of TOPSIS. To measure the separation from ideal solutions, the concept and algorithm of belief distance measure are defined, which can be used for comparing the difference between BS models. Finally, the relative closeness and ranking index are calculated for ranking the alternatives. A numerical example is given to illustrate the proposed method.展开更多
There is an increased interest in the extraction of nucleic acids from various environmental samples since culture-independent molecular techniques contribute to deepen and broaden the understanding of a greater porti...There is an increased interest in the extraction of nucleic acids from various environmental samples since culture-independent molecular techniques contribute to deepen and broaden the understanding of a greater portion of uncultivable microorganisms. Due to difficulties to select the optimum DNA extraction method in view of downstream molecular analyses, this article presents a straightforward mathematical framework for comparing some of the most commonly used methods. Four commercial DNA extraction kits and two physical-chemical methods (bead-beating and freeze-thaw) were compared for the extraction of DNA under several quantitative DNA analysis criteria: yield of extraction, purity of extracted DNA (A260/280 and A260/230 ratios), degradation degree of DNA, easiness of PCR amplification, duration of extraction, and cost per extraction. From a practical point of view, it is unlikely that a single DNA extraction strategy can be optimum for all selected criteria. Hence, a systematic Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) was employed to compare the methods. The PowerSoil? DNA Isolation Kit was systematically defined as the best performing method for extracting DNA from soil samples. More specifically, for soil:manure and soil:manure:biochar mixtures, the PowerSoil?DNA Isolation Kit method performed best, while for neat soil samples its alternative version gained the first rank.展开更多
To overcome the limitations of the traditional surrogate worth trade-off （SWT） method and solve the multiple criteria decision making problem more efficiently and interactively, a new method labeled dual worth trade...To overcome the limitations of the traditional surrogate worth trade-off （SWT） method and solve the multiple criteria decision making problem more efficiently and interactively, a new method labeled dual worth tradeoff （DWT） method is proposed. The DWT method dynamically uses the duality theory related to the multiple criteria decision making problem and analytic hierarchy process technique to obtain the decision maker＇s solution preference information and finally find the satisfactory compromise solution of the decision maker. Through the interactive process between the analyst and the decision maker, trade-off information is solicited and treated properly, the representative subset of efficient solutions and the satisfactory solution to the problem are found. The implementation procedure for the DWT method is presented. The effectiveness and applicability of the DWT method are shown by a practical case study in the field of production scheduling.展开更多
This paper presents an effective decision making framework for software selection in manufacturing industries using a multiple criteria decision making method, Preference Ranking Organization Method for Enrichment Eva...This paper presents an effective decision making framework for software selection in manufacturing industries using a multiple criteria decision making method, Preference Ranking Organization Method for Enrichment Evaluations (PROMETHEE). The method is improved in the present work by integrating with analytic hierarchy process (AHP) and the fuzzy logic. Fuzzy logic is introduced to handle the imprecision of the human decision making process. The proposed decision making framework is practical for ranking competing software products in terms of their overall performance with respect to multiple criteria. An example is included to illustrate the approach.展开更多
This paper explores a decision making model for a multidisciplinary problem in nature. This problem considers the role of energy use in sustainable development and the potential sources to increase energy efficiency d...This paper explores a decision making model for a multidisciplinary problem in nature. This problem considers the role of energy use in sustainable development and the potential sources to increase energy efficiency during its whole life cycle;it also deals with multicriteria decision making of plastic materials used in a day to day basis. Exergy analysis of plastic materials used to the manufacture of disposable polyethylene bags comparing them with other materials that can be used for substitution will be important to take decisions. We are also interested in plastic poly (ethylene Terephthalate or PET) bottles. The calculation of the incoming and outgoing Exergy flows during the production processes are carried out. The Exergy loss considering the sustainability concept, Green House Gases emissions, real energy flows needed to the chain of processes, material balances in the productions chains and value added, are presented as a set of criteria to make decisions of alternative materials including the actual ones. A case study for Mexico’s market will be developed in order to prove the methodology. It offers some interesting data about consumption and production of bags and bottles.展开更多
Researchers have been active in the field of software engineering measurement over more than 30 years. The software quality product is becoming increasingly important in the computerized society. Target setting in sof...Researchers have been active in the field of software engineering measurement over more than 30 years. The software quality product is becoming increasingly important in the computerized society. Target setting in software quality function and usability deployment are essential since they are directly related to development of high quality products with high customer satisfaction. Software quality can be measured as the degree to which a particular software program complies with consumer demand regarding function and characteristics. Target setting is usually subjective in practice, which is unscientific. Therefore, this study proposes a quantity model for controlling and measuring software quality via the expert decision-making algorithm-based method for constructing an evaluation method can provide software in relation to users and purchasers, thus enabling administrators or decision makers to identify the most appropriate software quality. Importantly, the proposed model can provide s users and purchasers a reference material, making it highly applicable for academic and government purposes.展开更多
The Paraconsistent Many-Valued Similarity (PMVS) method for multi-attribute decision making will be incomplete as a decision model if it is not extended to the realm of group decision-making. Therefore, in this articl...The Paraconsistent Many-Valued Similarity (PMVS) method for multi-attribute decision making will be incomplete as a decision model if it is not extended to the realm of group decision-making. Therefore, in this article, our primary objective is to show how the paraconsistent many-valued similarity method can be used to solve group decision-making problems involving choice making or ranking of a finite set of decision alternatives. Moreover, since weights are very important parameters in multi-attribute decision-making, we have introduced the Borda rule to calculate the weights of experts and that of every criterion under consideration. To demonstrate how the proposed method works, a numerical example on energy sources of an economy from the points of view of a group of experts is investigated. Further, we compare the results of this new approach with that of fuzzy TOPSIS group decision-making method to illustrate the robustness and effectiveness of the former.展开更多
Group decision making plays an important role in various fields of management decision and economics. In this paper, we develop two methods for hesitant fuzzy multiple criteria group decision making with group consens...Group decision making plays an important role in various fields of management decision and economics. In this paper, we develop two methods for hesitant fuzzy multiple criteria group decision making with group consensus in which all the experts use hesitant fuzzy decision matrices （HFDMs） to express their preferences. The aim of this paper is to present two novel consensus models applied in different group decision making situations, which are composed of consensus checking processes, consensus-reaching processes, and selection processes. All the experts make their own judgments on each alternative over multiple criteria by hesitant fuzzy sets, and then the aggregation of each hesitant fuzzy set under each criterion is calculated by the aggregation operators. Furthermore, we can calculate the distance between any two aggregations of hesitant fuzzy sets, based on which the deviation between any two experts is yielded. After introducing the consensus measure, we develop two kinds of consensus-reaching procedures and then propose two step-by-step algorithms for hesitant fuzzy multiple criteria group decision making. A numerical example concerning the selection of selling ways about ＇Trade-Ins＇ for Apple Inc. is provided to illustrate and verify the developed approaches. In this example, the methods which aim to reach a high consensus of all the experts before the selection process can avoid some experts＇ preference values being too high or too low. After modifying the previous preference information by using our consensus measures, the result of the selection process is much more reasonable.展开更多
The limitations of traditional approaches to selection problems are examined. A problemsolving strategy is presented in which decision-support and knowledge-based techniques play complementary roles. An approach to th...The limitations of traditional approaches to selection problems are examined. A problemsolving strategy is presented in which decision-support and knowledge-based techniques play complementary roles. An approach to the representation of knowledge to support the problem-solving strategy is presented which avoids commitment to a specific programming language or implementation environment. The problem of choosing a home is used to illustrate the representation of knowledge in a specific problem domain. Techniques for implementation of the problem-solving strategy are described. Knowledge elicitation techniques and their implementation in a development shell for application of the problem-solving strategy to any selection problem are also described.展开更多
There are many practical decision problems where decision makers＇ preferences may be inconsistent and contradictory. In this paper, new methods for ordering and classifying multi-attribute objects by discordant colle...There are many practical decision problems where decision makers＇ preferences may be inconsistent and contradictory. In this paper, new methods for ordering and classifying multi-attribute objects by discordant collective preferences are suggested. These methods are based on the theory of multiset metric spaces. The proposed techniques are applied to ranking companies and a competitive selection of projects, which are estimated by several experts upon multiple qualitative criteria.展开更多
In the decision-making process,the decision information provided by decision makers over alter-natives may take the form of intuitionistic fuzzy numbers and come from different periods.The weight of information on dec...In the decision-making process,the decision information provided by decision makers over alter-natives may take the form of intuitionistic fuzzy numbers and come from different periods.The weight of information on decision makers,criteria,periods is usually completely unknown.To this issue,we first utilise hesitation degree information and introduce the concept of confi-dence degree function to determine the decision maker’s weights.Then we aggregate individual evaluation information into group evaluation information through intuitionistic fuzzy number weighted arithmetic averaging operator.We construct a nonlinear optimisation model to gain the criterion weights and apply the aggregate operator to gain the integrated rating value of alternatives in different periods,calculating the deviations of the integrated rating values with respect to their average.Then the period weights are been obtained by using the entropy method.According to the closeness coefficient between alternatives and ideal solution to sort the alternatives and select the optimal one.展开更多
The application of data envelopment analysis （DEA） as a multiple criteria decision making （MCDM） technique has been gaining more and more attention in recent research. In the practice of applying DEA approach, the...The application of data envelopment analysis （DEA） as a multiple criteria decision making （MCDM） technique has been gaining more and more attention in recent research. In the practice of applying DEA approach, the appearance of uncertainties on input and output data of decision making unit （DMU） might make the nominal solution infeasible and lead to the efficiency scores meaningless from practical view. This paper analyzes the impact of data uncertainty on the evaluation results of DEA, and proposes several robust DEA models based on the adaptation of recently developed robust optimization approaches, which would be immune against input and output data uncertainties. The robust DEA models developed are based on input-oriented and outputoriented CCR model, respectively, when the uncertainties appear in output data and input data separately. Furthermore, the robust DEA models could deal with random symmetric uncertainty and unknown-but-bounded uncertainty, in both of which the distributions of the random data entries are permitted to be unknown. The robust DEA models are implemented in a numerical example and the efficiency scores and rankings of these models are compared. The results indicate that the robust DEA approach could be a more reliable method for efficiency evaluation and ranking in MCDM problems.展开更多
Facility layout selection is a multi-criteria decision making (MCDM) problem,since it has a strategic impact on the efficiency of manufacturing system.In view of the interdependency among selection criteria,analytic n...Facility layout selection is a multi-criteria decision making (MCDM) problem,since it has a strategic impact on the efficiency of manufacturing system.In view of the interdependency among selection criteria,analytic network process (ANP) is proposed to analyze the structure of the facility layout selection problem and determine the weights for each criterion.A network structure is constructed that shows all elements and clusters and their interactions.Limit priorities are also calculated which help decision maker evaluate the relative importance among criterion in the alternative selection process.Moreover,a hybrid MCDM approach that employs ANP and technique for order preference by similarity to an ideal solution (TOPSIS)method to rank the optimal facility layout alternatives.Finally,an application of a new aeronautic component assembly workshop facility layout selection is conducted.To further illustrate the advantage of the proposed approach,the difference between ANP-TOPSIS and AHP-TOPSIS methods are compared and discussed.Results have demonstrated the effectiveness and feasibility of the proposed method.展开更多
More attention has been drawn to the selection of contractors in recent years,but little research has been conducted on distinguishing representation of performance values for qualitative and quantitative criteria in ...More attention has been drawn to the selection of contractors in recent years,but little research has been conducted on distinguishing representation of performance values for qualitative and quantitative criteria in one decision-making(DM) problem yet.Thus a novel method based on fuzzy number is proposed to solve the representation issue,in which linguistic terms and fuzzy memberships are used to separately describe qualitative and quantitative criteria,and then the linguistic terms are translated into fuzzy numbers which will be defuzzified and unified with fuzzy memberships.The novel method is used to develop a synthetic decision model for selection of contractors.The model is developed to verify the feasibility of the novel method,which(1) builds a twolevel criteria system based on survey and literature summary;(2)utilizes the analytic network process(ANP) to determine criteria weights;(3) uses the novel method to make different representations of performance values for qualitative and quantitative criteria in one DM problem and then unifies different representations for decision matrices;(4) utilizes three multi-criteria decision-making(MCDM) methods containing simple additive weighting(SAW),technique for order preference by similarity to ideal solution(TOPSIS) and complex proportional assessment(COPRAS) to deal with decision matrices.Besides,a case of contractor selection is given to check the novel method by applying developed model,and the case results show that the novel method can suitably solve the representation issue of the performance values for two types of criteria as well as simplify the processes of TOPSIS and COPRAS.展开更多
基金supported by National Natural Science Foundation of China (No.70971131, 70901074)
文摘The technique for order performance by similarity to ideal solution （TOPSIS） is one of the major techniques in dealing with multiple criteria decision making （MCDM） problems, and the belief structure （BS） model has been used successfully for uncertain MCDM with incompleteness, impreciseness or ignorance. In this paper, the TOPSIS method with BS model is proposed to solve group belief MCDM problems. Firstly, the group belief MCDM problem is structured as a belief decision matrix in which the judgments of each decision maker are described as BS models, and then the evidential reasoning approach is used for aggregating the multiple decision makers＇ judgments. Subsequently, the positive and negative ideal belief solutions are defined with the principle of TOPSIS. To measure the separation from ideal solutions, the concept and algorithm of belief distance measure are defined, which can be used for comparing the difference between BS models. Finally, the relative closeness and ranking index are calculated for ranking the alternatives. A numerical example is given to illustrate the proposed method.
文摘There is an increased interest in the extraction of nucleic acids from various environmental samples since culture-independent molecular techniques contribute to deepen and broaden the understanding of a greater portion of uncultivable microorganisms. Due to difficulties to select the optimum DNA extraction method in view of downstream molecular analyses, this article presents a straightforward mathematical framework for comparing some of the most commonly used methods. Four commercial DNA extraction kits and two physical-chemical methods (bead-beating and freeze-thaw) were compared for the extraction of DNA under several quantitative DNA analysis criteria: yield of extraction, purity of extracted DNA (A260/280 and A260/230 ratios), degradation degree of DNA, easiness of PCR amplification, duration of extraction, and cost per extraction. From a practical point of view, it is unlikely that a single DNA extraction strategy can be optimum for all selected criteria. Hence, a systematic Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) was employed to compare the methods. The PowerSoil? DNA Isolation Kit was systematically defined as the best performing method for extracting DNA from soil samples. More specifically, for soil:manure and soil:manure:biochar mixtures, the PowerSoil?DNA Isolation Kit method performed best, while for neat soil samples its alternative version gained the first rank.
基金This project was supported by the National Natural Science Foundation of China(79870030)
文摘To overcome the limitations of the traditional surrogate worth trade-off （SWT） method and solve the multiple criteria decision making problem more efficiently and interactively, a new method labeled dual worth tradeoff （DWT） method is proposed. The DWT method dynamically uses the duality theory related to the multiple criteria decision making problem and analytic hierarchy process technique to obtain the decision maker＇s solution preference information and finally find the satisfactory compromise solution of the decision maker. Through the interactive process between the analyst and the decision maker, trade-off information is solicited and treated properly, the representative subset of efficient solutions and the satisfactory solution to the problem are found. The implementation procedure for the DWT method is presented. The effectiveness and applicability of the DWT method are shown by a practical case study in the field of production scheduling.
文摘This paper presents an effective decision making framework for software selection in manufacturing industries using a multiple criteria decision making method, Preference Ranking Organization Method for Enrichment Evaluations (PROMETHEE). The method is improved in the present work by integrating with analytic hierarchy process (AHP) and the fuzzy logic. Fuzzy logic is introduced to handle the imprecision of the human decision making process. The proposed decision making framework is practical for ranking competing software products in terms of their overall performance with respect to multiple criteria. An example is included to illustrate the approach.
文摘This paper explores a decision making model for a multidisciplinary problem in nature. This problem considers the role of energy use in sustainable development and the potential sources to increase energy efficiency during its whole life cycle;it also deals with multicriteria decision making of plastic materials used in a day to day basis. Exergy analysis of plastic materials used to the manufacture of disposable polyethylene bags comparing them with other materials that can be used for substitution will be important to take decisions. We are also interested in plastic poly (ethylene Terephthalate or PET) bottles. The calculation of the incoming and outgoing Exergy flows during the production processes are carried out. The Exergy loss considering the sustainability concept, Green House Gases emissions, real energy flows needed to the chain of processes, material balances in the productions chains and value added, are presented as a set of criteria to make decisions of alternative materials including the actual ones. A case study for Mexico’s market will be developed in order to prove the methodology. It offers some interesting data about consumption and production of bags and bottles.
文摘Researchers have been active in the field of software engineering measurement over more than 30 years. The software quality product is becoming increasingly important in the computerized society. Target setting in software quality function and usability deployment are essential since they are directly related to development of high quality products with high customer satisfaction. Software quality can be measured as the degree to which a particular software program complies with consumer demand regarding function and characteristics. Target setting is usually subjective in practice, which is unscientific. Therefore, this study proposes a quantity model for controlling and measuring software quality via the expert decision-making algorithm-based method for constructing an evaluation method can provide software in relation to users and purchasers, thus enabling administrators or decision makers to identify the most appropriate software quality. Importantly, the proposed model can provide s users and purchasers a reference material, making it highly applicable for academic and government purposes.
文摘The Paraconsistent Many-Valued Similarity (PMVS) method for multi-attribute decision making will be incomplete as a decision model if it is not extended to the realm of group decision-making. Therefore, in this article, our primary objective is to show how the paraconsistent many-valued similarity method can be used to solve group decision-making problems involving choice making or ranking of a finite set of decision alternatives. Moreover, since weights are very important parameters in multi-attribute decision-making, we have introduced the Borda rule to calculate the weights of experts and that of every criterion under consideration. To demonstrate how the proposed method works, a numerical example on energy sources of an economy from the points of view of a group of experts is investigated. Further, we compare the results of this new approach with that of fuzzy TOPSIS group decision-making method to illustrate the robustness and effectiveness of the former.
基金Project supported by the National Natural Science Foundation of China （Nos. 61273209, 71501135, 71571123, and 71532007）
文摘Group decision making plays an important role in various fields of management decision and economics. In this paper, we develop two methods for hesitant fuzzy multiple criteria group decision making with group consensus in which all the experts use hesitant fuzzy decision matrices （HFDMs） to express their preferences. The aim of this paper is to present two novel consensus models applied in different group decision making situations, which are composed of consensus checking processes, consensus-reaching processes, and selection processes. All the experts make their own judgments on each alternative over multiple criteria by hesitant fuzzy sets, and then the aggregation of each hesitant fuzzy set under each criterion is calculated by the aggregation operators. Furthermore, we can calculate the distance between any two aggregations of hesitant fuzzy sets, based on which the deviation between any two experts is yielded. After introducing the consensus measure, we develop two kinds of consensus-reaching procedures and then propose two step-by-step algorithms for hesitant fuzzy multiple criteria group decision making. A numerical example concerning the selection of selling ways about ＇Trade-Ins＇ for Apple Inc. is provided to illustrate and verify the developed approaches. In this example, the methods which aim to reach a high consensus of all the experts before the selection process can avoid some experts＇ preference values being too high or too low. After modifying the previous preference information by using our consensus measures, the result of the selection process is much more reasonable.
文摘The limitations of traditional approaches to selection problems are examined. A problemsolving strategy is presented in which decision-support and knowledge-based techniques play complementary roles. An approach to the representation of knowledge to support the problem-solving strategy is presented which avoids commitment to a specific programming language or implementation environment. The problem of choosing a home is used to illustrate the representation of knowledge in a specific problem domain. Techniques for implementation of the problem-solving strategy are described. Knowledge elicitation techniques and their implementation in a development shell for application of the problem-solving strategy to any selection problem are also described.
文摘There are many practical decision problems where decision makers＇ preferences may be inconsistent and contradictory. In this paper, new methods for ordering and classifying multi-attribute objects by discordant collective preferences are suggested. These methods are based on the theory of multiset metric spaces. The proposed techniques are applied to ranking companies and a competitive selection of projects, which are estimated by several experts upon multiple qualitative criteria.
基金supported by the Natural Science Foundation of China[grant number 71601059],[grant number 71673069].
文摘In the decision-making process,the decision information provided by decision makers over alter-natives may take the form of intuitionistic fuzzy numbers and come from different periods.The weight of information on decision makers,criteria,periods is usually completely unknown.To this issue,we first utilise hesitation degree information and introduce the concept of confi-dence degree function to determine the decision maker’s weights.Then we aggregate individual evaluation information into group evaluation information through intuitionistic fuzzy number weighted arithmetic averaging operator.We construct a nonlinear optimisation model to gain the criterion weights and apply the aggregate operator to gain the integrated rating value of alternatives in different periods,calculating the deviations of the integrated rating values with respect to their average.Then the period weights are been obtained by using the entropy method.According to the closeness coefficient between alternatives and ideal solution to sort the alternatives and select the optimal one.
文摘The application of data envelopment analysis （DEA） as a multiple criteria decision making （MCDM） technique has been gaining more and more attention in recent research. In the practice of applying DEA approach, the appearance of uncertainties on input and output data of decision making unit （DMU） might make the nominal solution infeasible and lead to the efficiency scores meaningless from practical view. This paper analyzes the impact of data uncertainty on the evaluation results of DEA, and proposes several robust DEA models based on the adaptation of recently developed robust optimization approaches, which would be immune against input and output data uncertainties. The robust DEA models developed are based on input-oriented and outputoriented CCR model, respectively, when the uncertainties appear in output data and input data separately. Furthermore, the robust DEA models could deal with random symmetric uncertainty and unknown-but-bounded uncertainty, in both of which the distributions of the random data entries are permitted to be unknown. The robust DEA models are implemented in a numerical example and the efficiency scores and rankings of these models are compared. The results indicate that the robust DEA approach could be a more reliable method for efficiency evaluation and ranking in MCDM problems.
基金supported by the National Natural Science Foundation of China (No.51575274)the National Defense Basic Scientific Research Program of China (No.JCKY2016605B006)the Key Research and Development Project of Jiangxi Province in China (No.20161ACE50004)
文摘Facility layout selection is a multi-criteria decision making (MCDM) problem,since it has a strategic impact on the efficiency of manufacturing system.In view of the interdependency among selection criteria,analytic network process (ANP) is proposed to analyze the structure of the facility layout selection problem and determine the weights for each criterion.A network structure is constructed that shows all elements and clusters and their interactions.Limit priorities are also calculated which help decision maker evaluate the relative importance among criterion in the alternative selection process.Moreover,a hybrid MCDM approach that employs ANP and technique for order preference by similarity to an ideal solution (TOPSIS)method to rank the optimal facility layout alternatives.Finally,an application of a new aeronautic component assembly workshop facility layout selection is conducted.To further illustrate the advantage of the proposed approach,the difference between ANP-TOPSIS and AHP-TOPSIS methods are compared and discussed.Results have demonstrated the effectiveness and feasibility of the proposed method.
基金National Natural Science Foundation of China(No.51178358)
文摘More attention has been drawn to the selection of contractors in recent years,but little research has been conducted on distinguishing representation of performance values for qualitative and quantitative criteria in one decision-making(DM) problem yet.Thus a novel method based on fuzzy number is proposed to solve the representation issue,in which linguistic terms and fuzzy memberships are used to separately describe qualitative and quantitative criteria,and then the linguistic terms are translated into fuzzy numbers which will be defuzzified and unified with fuzzy memberships.The novel method is used to develop a synthetic decision model for selection of contractors.The model is developed to verify the feasibility of the novel method,which(1) builds a twolevel criteria system based on survey and literature summary;(2)utilizes the analytic network process(ANP) to determine criteria weights;(3) uses the novel method to make different representations of performance values for qualitative and quantitative criteria in one DM problem and then unifies different representations for decision matrices;(4) utilizes three multi-criteria decision-making(MCDM) methods containing simple additive weighting(SAW),technique for order preference by similarity to ideal solution(TOPSIS) and complex proportional assessment(COPRAS) to deal with decision matrices.Besides,a case of contractor selection is given to check the novel method by applying developed model,and the case results show that the novel method can suitably solve the representation issue of the performance values for two types of criteria as well as simplify the processes of TOPSIS and COPRAS.
