Development of Fuzzy TOPSIS-Based Software Tool (FTST) Fadlelmoula M. Baloul Ahmed A. Al-Amayrah D.R Prince Williams E-mail: [email protected] E-mail: [email protected] E-mail: [email protected], [email protected] Department of Information Technology, College of Applied Sciences - Sohar, Sohar, Sultanate of Oman. Abstract: The aim of this paper is to describe FTST software tool for selecting the optimum alternative among many alternatives according to multi specified criteria. FTST tool is designed based on the fuzzy TOPSIS method as one of the most popular methods in multi-criteria decision analysis (MCDA). FTST, as an automated system, could be used as decision support system software for any case where the goal is to select the right and the optimum alternative from many alternatives with intended criteria. At the end of the paper, the FTST tool will be demonstrated on a real fuzzy TOPSIS-based system. Keywords: Decision Support software, TOPSIS, Academic Advising, Multi-criteria Decision Making. I. INTRODUCTION A Decision Support System (DSS) is a computer-based information system that could be used to eases and supports the decision makers in any organization to take unbiased decisions [11]. A well-designed DSS is an interactive software system designed mainly to help the decisions makers to compile multi-criteria about anything to take the right decision. Any DSS has four components which are: Input - criteria, variables, weight and rates User Knowledge - the user has be a decision maker or an expert person who has the knowledge of the decision makers. Output - generated tables from which the decision will be taken. Decisions – usually produced by DSS based on the set of input that used as criteria. Decision-making problems are an important process for any organization and it helps the management to identify the most suitable option from the available feasible alternatives. All the feasible or possible alternatives are derived from the multiple criteria. This type of problems is known as multiple criteria decision making (MCDM). The processes in MCDM problems are very algorithmic processes that based on well-defined complicated formulas that could be used in a similar way in many application areas. Many authors have studied and applied in many situations like location selection, service quality of domestic airlines selection, bus company selection, etc.[1-3,59]. In most of the literature, technique for order performance by similarity to ideal solution (TOPSIS) method is used to solve the MCMD problem. Due to the uncertainty in MCDM, fuzzy set [10] plays a vital role in solving MCDA problem. Chen [4], have combined both the TOPSIS and fuzzy sets to solve the MCDM problem. In this paper, we show and describe our software tool which developed based on the extended concept of the fuzzy TOPSIS method which was developed by Chen [4], for solving multi-criteria alternative selection problems in many application areas and environments. According to the fuzzy nature of the raw data that used as a core for taking any decision and the process of the decision itself, some linguistic variables have been used to rate the weights of all criteria and rates the different intended alternatives with respect to each criterion. The fuzzy positive-ideal solution (FPIS) and the fuzzy negative-ideal solution (FNIS) have been defined in [4]. The fuzzy TOPSIS method involves many complicated calculations in each of its steps. For example, in step 7 of the method, a specific vertex method is used to find the distance between any two triangular fuzzy ratings points and then the distance of each alternative from FPIS and FNIS, respectively. Following these calculation formulas manually could be a very difficult process and some automation is needed. Our FTST tool which we describe in this paper affords a great help in applying the fuzzy TOPSIS method in specific and MCDM in general. FTST tool performs all the complicated calculations automatically behind the scene and shows the result of the application within a few clicks. FTST tool could be implemented and used in many MCDM application areas. II. PRELIMINARIES In the following paragraphs, some basic definitions of the fuzzy sets will be reviewed. These basic definitions will be used in the framework of this paper. ~ Definition 1. As defined in [10], a fuzzy set A in a universe of discourse X is characterized by the membership function ̃ ()which associates with each element x in X a real number in the interval [0,1]. The function value ̃ () is termed the ~ grade of the membership of x in A [10]. Definition 2. As defined in [4], a triangular fuzzy number n~ can be defined by a triplet n1 , n2 , n3 shown in Fig.1 n~ x 1 0 n1 X n3 n2 Fig. 1. A triangular fuzzy number if if wij wi1 , wi 2 , wi 3 Based on the above fuzzy environment we can define a following fuzzy decision matrix and weight matrix as x11 x21 D . . xn1 ~ ~ ~ W w w 1 x13 ...x1n x23 ...x21n . . xn 3 ...xmn x12 x22 . 2 xn 2 ~ w 3 (1) xij and wi for all where and j 1, 2,..., n are linguistic variables. i 1, 2,..., m The linguistic variables can be defined as triangular fuzzy n~ . The membership function is n~ x defined as [11]: x n1 n n , 2 1 x n2 , n x n3 n 2 0, xij aij , bij , cij numbers, n1 x n 2 , n 2 x n3 , ~ xij aij , bij , cij and wi wi1 , wi 2 , wi 3 . The weight of each criterion is defined as ̃ = ( , [ ] [ ] , [ ] ) (2) i 1, 2,3,...m , j 1, 2,3,...n . Where otherwise. ~ Definition 3. As defined in [8,2] D is called a fuzzy matrix, ~ if at least an entry in D is a fuzzy number. Definition 4. As defined in [4], a linguistic variable is a variable whose values are linguistic terms. Rij max a11 , b11 , c11 , a21 , b21 , c21 ,..., amj , bmj , cmj max j 1, 2,3,...n . As defined in [4], for i=1 to m, j=1 to m, the normalized decision matrix denoted by ̂ can be calculated as ̂ = [ ̃ ]× , where ̃ = ( , , ) (3) max max max The concept of linguistic variable is very useful in dealing with situations which are too complex to be reasonably described in conventional quantitative expressions [15]. For example, "weight" is a linguistic variable; its values could be: very low, low, medium, high, very high, etc. These linguistic values can be represented by fuzzy numbers. Definition 5. As defined in [4], let ~ m , m , m and m 1 2 3 Considering the different risk level of each criterion, we can construct the weighted normalized fuzzy decision matrix as ~ rij . w j V v~ij mn , where v~ij ~ Then the fuzzy positive ideal solution (FPIS, A ) and fuzzy negative ideal solution (FNIS, A ) will be defined as n~ n1 , n2 , n3 be two triangular fuzzy numbers, then the A v~1 , v~2 ,...v~n and A v~1 , v~2 ,...v~n vertex method is defined to calculate the distance between them as ~, n~ 1 n m 2 n m 2 n m 2 d m 1 1 2 2 3 31 3 The decision maker has to determine the importance level each criterion and the rates the different intended alternatives as Where (4) v j 1,1,1 and v j 0,0,0 The distance of each alternative from A and A can be calculated using the following formula: n d i d v~ij , v~i for all i 1,2,..., m j and (5) n d i d v~ij , v~i for all i 1,2,..., m . (6) j Where d .,. is the distance between two triangular fuzzy numbers. As the last step of the method, a closeness coefficient is defined as a key to identify the ranking order of each alternative has been calculated as d i CCi * d i d i Just for comparison, we have used an example that have been used as numerical example in [4], to show how our tool could be used to choose the best alternative among a set of possible alternatives automatically as shown in next section. FTST DEMONSTARTION A company desires to hire a person for a job. After the initial evaluation, three candidates are remaining for further evaluation (A1, A2 and A3). The decision maker runs a final interview to select the most appropriate candidate for the job. The following five criteria are used for evaluation: This decision problem structure is described in Fig. 2. LINGUISTIC VARIABLES CHOSEN FOR THE IMPORTANCE WEIGHT OF EACH CRITERION (7) All these steps have been designed and implemented as integrated software tool using Visual Basic Programming Language. (1) Emotional steadiness (C1), (2) Oral communication skill (C2), (3) Personality (C3), (4) Past experience (C4), (5) Self-confidence (C5). 1- Based on the screen on Fig. 3, the user has to specify the number of criteria, the number of alternatives and number of rates/weight scales. 2- The linguistic variables of the importance weight of each criterion in table I will be entered in FTST tool as shown in Fig. 4. TABLE I. As specified in [4], in total, the algorithm for the fuzzy TOPSIS-based system will follow the following steps: 1- Set the evaluation for the given criteria. 2- Choose suitable linguistic variables for the importance weight of the given criteria and the given linguistic ratings for alternatives. 3- Estimate the weight of criteria to get the estimated fuzzy weight of each criterion, and puddle the decision makers' suggestions to get the grouped fuzzy rating of all alternative under all criterions. 4- Normalize the fuzzy decision matrix. 5- Build the weighted normalized fuzzy decision matrix. 6- Calculate the fuzzy positive ideal solution (FPIS) and the fuzzy negative ideal solution (FNIS). 7- Find the distance of FPIS and FNIS for each alternative. 8- Find the closeness coefficient (CC) of each alternative. 9- Determine the ordering rank of each alternative according to its closeness coefficient (CC). III. Our software tool (FTST) is used to solve the complicated computational steps as given in the following algorithm: VL (Very low) L (Low) ML (Medium low) M (Medium) MH (Medium high) H (High) VH (Very high) 3- (0.0,0.0,0.1) (0.0,0.1,0.3) (0.1,0.3,0.5) (0.3,0.5,0.7) (0.5,0.7,0.9) (0.7,0.9,1.0) (0.9,1.0;1.0) Based on the data in table II, the linguistic variables for alternatives ratings will be assessed and entered into the system as shown in Fig. 5. TABLE II. LINGUISTIC VARIABLES CHOSEN FOR TABLE III. ALTERNATIVE RATINGS VP (Very poor) P (Poor) MP (Medium poor) F (Fair) MG (Medium good) G (Good) VG (Very good) (0,0,1) (0,1,3) (1,3,5) (3,5,7) (5,7,9) (7,9,10) (9,10,10) 4- Using the linguistic rating variables, the decision maker evaluates and rates each criterion as shown in Fig. 6. 5- The decision maker determines the fuzzy weight of each criterion for each alternative as shown in Fig 7. 6- FTST performs all the complex computations based on the fuzzy TOPSIS method. 7- FTST generates all the intermediate tables for the user as shown in Fig. 8. 8- FTST determines and displays the best choice from the given alternatives as its final result as shown in Fig. 9. IV. CONCLUSION AND FUTURE WORK The paper has discussed developing and demonstrating a tool for fuzzy TOPSIS-based model to evaluates three alternatives and determines the best alternative. Five criteria and three alternatives were used to assess the developed tool. The developed software tool could be used by the decision maker to automate choosing the best alternative in any application area such as medicine, finance, education, human resource, software development, etc. In the real life situation there may be two or more decision makers are possible and each of them may have different views and ideas for alternatives evaluation. This idea motivates us to upgrade our tool for fuzzy TOPSIS method for multi decision makers in the future. Fig. 2. The structure of the problem Fig. 3. The Basic Input Screen Fig. 4. The linguistic variables for the importance weight for each criterion Fig. 5. The linguistic variables and values chosen for alternatives rating Fig. 6. The selection of the criterion’s importance weight Fig. 7. The decision maker rating for each alternative/criterion Fig. 8. The intermediate tables generated by FTST Fig. 9. 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