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Advances in Fuzzy Systems
Volume 2016, Article ID 6475403, 12 pages
http://dx.doi.org/10.1155/2016/6475403
Research Article

An Efficient Ranking Technique for Intuitionistic Fuzzy Numbers with Its Application in Chance Constrained Bilevel Programming

1Department of Mathematics, University of Kalyani, Kalyani 741235, India
2Department of Mathematics, Government College of Engineering and Textile Technology, Serampore 712201, India

Received 22 November 2015; Revised 24 March 2016; Accepted 3 April 2016

Academic Editor: Kemal Kilic

Copyright © 2016 Animesh Biswas and Arnab Kumar De. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The aim of this paper is to develop a new ranking technique for intuitionistic fuzzy numbers using the method of defuzzification based on probability density function of the corresponding membership function, as well as the complement of nonmembership function. Using the proposed ranking technique a methodology for solving linear bilevel fuzzy stochastic programming problem involving normal intuitionistic fuzzy numbers is developed. In the solution process each objective is solved independently to set the individual goal value of the objectives of the decision makers and thereby constructing fuzzy membership goal of the objectives of each decision maker. Finally, a fuzzy goal programming approach is considered to achieve the highest membership degree to the extent possible of each of the membership goals of the decision makers in the decision making context. Illustrative numerical examples are provided to demonstrate the applicability of the proposed methodology and the achieved results are compared with existing techniques.

1. Introduction

The concept of bilevel programming problem (BLPP) was first introduced by Candler and Townsley [1]. The BLPP is considered as a class of optimization problems where two decision makers (DMs) locating at two different hierarchical levels independently control a set of decision variables paying serious attention to the benefit of the others in a highly conflicting decision making situation. The upper level DM is termed as leader and the lower level DM as follower. There are several applications of BLPP in many real life problems such as agriculture, biofuel production, economic systems, finance, engineering, banking, management sciences, and transportation problem. Several methods were proposed to solve BLPPs by different researchers [2, 3] in the past. But these traditional approaches are unable to provide a satisfactory solution if the parameter values involved with a BLPP inevitably contain some uncertain data or linguistic information. Stochastic programming (SP) and fuzzy programming (FP) are two powerful techniques to handle such type of problems. Using probability theory, Dantzig [4] introduced SP. The SP was developed in various directions like chance constrained programming (CCP), recourse programming, multiobjective SP, and so forth. Charnes and Cooper [5] developed the concept of CCP.

Again, from the viewpoint of uncertainty or fuzziness involved in human’s judgments, Zimmermann [6] first applied fuzzy set theory [7] in decision making problems with several conflicting objectives. The concept of membership functions in BLPPs was introduced by Lai and Hwang [8]. Lai’s solution concept was then extended by Shih et al. [9] and a supervised search procedure with the use of max-min operator of Bellman and Zadeh [10] was proposed. The basic concept of this procedure is that the follower optimizes his/her objective function, taking into consideration leader’s goal. Recently, Lodwick and Kacprizyk [11] developed a methodology for solving decision making problems under fuzziness. The main difficulty of FP approach is that the objectives of the DMs are conflicting. So there is possibility of rejecting the solution again and again by the DMs and the solution process is continued by redefining the membership functions repeatedly until a satisfactory solution is obtained. This makes the solution process a very lengthy and tedious one. To remove these difficulties fuzzy goal programming (FGP) [1214] is used as an efficient tool for making decision in an imprecisely defined multiobjective decision making (MODM) arena. Baky [15] developed a FGP technique for solving multiobjective multilevel programming problem.

Also it is observed that the fuzzy sets (FSs) are not always capable of dealing with lack of knowledge with respect to degrees of membership. Realizing the fact Atanassov [1618] introduced the concept of intuitionistic FSs (IFSs) by implementing a nonmembership degree which can handle the drawback of FSs and express more abundant and flexible information than the FSs. In recent years, there is a growing interest in the study of decision making problems with intuitionistic fuzzy numbers (IFNs) [1921] and with interval-valued intuitionistic fuzzy information [2225].

The ranking of IFNs [26, 27] plays an important role in dealing with IFNs as ranking of fuzzy numbers (FNs). Grzegoraewski [28] suggested some methods for measuring distances between IFNs and interval-valued FNs, based on Hausdorff metric. The methodology for solving intuitionistic fuzzy linear programming problems with triangular IFNs (TIFNs) was developed by Dubey and Mehra [29] by converting the model into crisp linear programming problem. Li [30] developed a ratio ranking method for the TIFNs. Nehi [31] put forward a new ordering method of IFNs in which two characteristic values for IFNs are defined by the integral of the inverse fuzzy membership and nonmembership functions multiplied by the grade with powered parameters. Recently, numerous ranking methods for IFNs have been proposed in literature to rank IFNs [3234]. Although many defuzzification methods have already been proposed so far, no method gives a right effective defuzzification output. Most of the existing defuzzification methods tried to make the estimation of IFNs in an objective way. This paper proposes a method using the concept probability density function of IFNs and Mellin’s transform [35, 36] to find the ranking of normal IFNs. This ranking method removes the ambiguous outcomes and distinguishes the alternatives clearly.

In fuzzy BLPP [37] it is sometimes realized that the concept of membership function does not provide satisfactory solutions in a highly conflicting decision making situation. In this context IFNs can be used to capture both the membership and nonmembership degrees of uncertainties of both the DMs. Also there are some real world situations, where randomness and fuzziness occur simultaneously. The decision making problem having such type of ambiguous information is known as fuzzy stochastic programming problem. Many researchers [3841] derived different methods to solve such type of decision making problems. But FGP approach for solving fuzzy stochastic linear BLPP with IFNs is yet to appear in the literature.

In the present study FGP process is adopted for solving bilevel intuitionistic FP problems where the parameters are expressed in terms of normal IFNs. A ranking technique for normal IFNs is first proposed and then using the proposed technique the model is converted into a deterministic problem. The individual optimal value of each objective is found in isolation to construct the fuzzy membership goals of each of the objectives. Finally, FGP model is developed for the achievement of highest degree of each of the defined membership goals to the extent possible by minimizing group regrets in the decision making context. To explore the potentiality of the proposed approach, two illustrative examples are considered and solved and the achieved solutions are compared with the predefined technique developed by Dubey and Mehra [29].

2. Preliminaries

In this section some basic concepts on FNs, triangular FNs (TFNs), IFNs, and triangular IFNs (TIFNs) are discussed.

2.1. FN [42]

A fuzzy set defined on the set of real numbers, , is said to be an FN if its membership function satisfies the following characteristics:(i) is continuous.(ii) for all .(iii) is strictly increasing on and strictly decreasing on .(iv) for all , where .

2.2. TFN [43]

An FN is said to be TFN if its membership function is given by

The TFN can be expressed in the form of Figure 1.

Figure 1: TFN.
2.3. IFN [44]

An IFN is(i)an IFS defined on ;(ii) normal; that is, there exists such that and ;(iii) convex for the membership function ; that is, for all ;(iv) concave for the nonmembership function ; that is, for all .

2.4. TIFN [30]

Let be a TIFN. Then the membership and the nonmembership function of are expressed as That TIFN is expressed by Figure 2.

Figure 2: TIFN.

A TIFN is called normal if and for at least one .

3. Proposed Ranking Technique for Normal TIFN

Let be a normal TIFN. Then the membership and the nonmembership functions of are expressed as which is presented in Figure 3.

Figure 3: Triangular normal IFN.

It is to be noted here that if the nonmembership value decreases, acceptance of the IFN increases. As a consequence, if the value of complement of the nonmembership function increases, the acceptance possibility of the IFN increases. Considering those concepts, membership functions as well as the complement of the nonmembership functions are taken into account to rank the IFNs. The complement of the nonmembership function of a normal TIFN is given byLet us define two probability density functions and corresponding to the membership function and the complement of the nonmembership function , respectively.

Let ; then implies ;   that is,that is,ThusSimilarly, defining , the constant and the function are obtained as Let be the probability density function corresponding to the normal TIFN which is defined as A technique for defuzzification of fuzzy number using Mellin’s transform was developed by R. Saneifard and R. Saneifard [36]. There are lots of successful applications [4549] of this technique. From that viewpoint, Mellin’s transformation has been applied to find the defuzzified value of IFNs. Mellin’s transform is given by Here denotes the random variable corresponding to the normal TIFN . ThusFor , Mellin’s transform converted to the definition of expectation of a random variable. Since the target is to find the expected or defuzzified value of TIFNs, has been considered.

Thus the crisp equivalent value of the normal TIFN is found asFor any two normal TIFNs and if and represent their equivalent crisp values, then,(1) if and only if ;(2) if and only if ;(3) if and only if .It is worthy to mention here that Wang and Kerre [50] proposed seven axioms as the reasonable properties of ordering fuzzy quantities for an ordering approach. The properties of IFNs depend on two FSs, namely, membership function and nonmembership function; and it can easily be shown that all the seven axioms have been satisfied by the proposed ranking technique as described above.

The derived process of ranking of normal TIFNs is to be summarized through the following algorithm.

3.1. Solution Algorithm

Step 1. Write the membership function and nonmembership function of the normal TIFN (refer to (3)).

Step 2. Calculate the complement of the nonmembership function of the normal TIFN (refer to (4)).

Step 3. Construct the probability density function for the membership function and for the complement of the nonmembership function of the normal TIFN (refer to (7) and (8)).

Step 4. Take the convex combination of and to form the probability density function of the normal TIFN (refer to (9)).

Step 5. Use Mellin’s transform to calculate the crisp value of the normal TIFN (refer to (12)).

Step 6. Stop.

3.2. Illustrative Example

The following three TIFNs are considered to find their equivalent crisp value by the proposed technique.

Let , , and be three TIFNs.

The membership function of the TIFN is given by The nonmembership function of is presented as The complement of the non-membership function is calculated asLet and be the respective density functions of the membership function and the complement of nonmembership function which are calculated as defined in the proposed methodology. Then the density function of the TIFN is considered as , .

Using Mellin’s transform the crisp value of TIFN is calculated as . Similarly the crisp value of TIFNs and is evaluated as , , .

Thus the ordering in the proposed ranking technique is .

3.3. Comparison of the Proposed Ranking Method with the Predefined Methods

The proposed ranking technique is compared with other predefined ranking methods [2934] to explore the consistency of the proposed ranking methodology through Table 1.

Table 1: Comparison between ranking techniques.

From Table 1 it is evident that the ordering obtained using the proposed methodology is identical in comparison with other techniques. This indicates that the proposed methodology is consistent with the other predefined techniques which can be considered as an alternative technique for ranking IFNs. However, the superiority of the proposed approach would be reflected in the context of solving BLPPs with TIFNs.

Using the proposed ranking method of normal TIFN a fuzzy stochastic linear bilevel programming (FSLBLP) model is developed and solved in the following section.

4. FSLBLP Model Formulation

An LFSBLP problem with normal TIFNs as coefficients is presented as follows:where and (; ; ) are normal TIFNs and is any real number that lies within . Here denotes fuzzy random variable following normal distribution whose mean and standard deviation are normal TIFNs and , , and denote fuzzily equal, less than or equal, and greater than or equal, respectively. The decision vector is controlled by the upper level DM and is controlled by the lower level DM; and .

4.1. FP Model Formulation [5]

Applying CCP technique, the probabilistic constraints () are modified as follows:where the function represents the cumulative distribution function of the standard normal fuzzy random variate.

Hence the linear fuzzy BLPP is written in the following form as

4.2. Construction of Linear Bilevel Programming Model Using Proposed Ranking Method

In this subsection the proposed ranking function is applied to convert the FP model into a deterministic model. Using the linearity of ranking function, the above model is written asUsing the parameters of TIFNs, model (19) is expressed as Now, the DMs are trying to optimize their objective independently under the system constraints defined above. Let and be the best value of the objective of the upper and lower level DMs, respectively, obtained by solving each objective independently. Since the DMs are trying to maximize their objective and they also want that the maximum value of the objective of the other level DM cannot exceed his minimum value of the objective, therefore, the worst values and are calculated by considering the best solution point of the other level DM. That is,Hence the fuzzy goal for each objective is expressed as Thus the membership function of the upper and lower level objective is constructed asWith the help of the above membership functions, the FGP model is defined in the next section.

4.3. Weighted FGP Model Formulation

In FGP model formulation, the membership functions are first converted into flexible membership goals by introducing under- and overdeviational variables to each of them and thereby assigning the highest membership value (unity) as the aspiration level to each of them. Also it is evident that full achievement of all the membership goals is not possible in MODM context. So the underdeviational variables are minimized to achieve the best goal values of objectives in the decision making environment.

Thus a FGP model is formulated aswhere represents the numerical weights of the goals which are determined as The developed model (24) is solved to find the most satisfactory solution in the decision making environment.

4.4. Solution Algorithm

The methodology for developing the intuitionistic FSLBLP model is presented in the form of an algorithm as follows.

Step 1. Apply CCP technique to all the fuzzy probabilistic constraints, to form a fuzzy linear bilevel programming model (refer to (18)).

Step 2. Using the proposed ranking technique of IFNs, the deterministic linear bilevel model is developed (refer to (19)).

Step 3. The individual optimal value of the objective of each DM is found in isolation.

Step 4. The worst value of the objective of the DMs is evaluated at the best solution point of the other level DMs (refer to (21)).

Step 5. Construct the fuzzy membership goals of the objective of each DM (refer to (22)).

Step 6. FGP approach is used to achieve maximum degree of each of the membership goals (refer to (24)).

Step 7. Stop.

5. Numerical Example

To illustrate the proposed methodology the following FSLBLP problem is considered with coefficients of objectives and system constraints are taken as normal TIFNs: where and denote fuzzy random variables following normal distribution. Here , , , , , , , and are all normal TIFNs. Then the TIFNs can be expressed as The mean and standard deviation of the fuzzy random variables following normal distribution are given asApplying CCP methodology the above linear fuzzy stochastic BLPP is converted into linear fuzzy BLPP asBy applying the proposed ranking technique of normal TIFNs, the above IFNs are converted into the equivalent crisp values as

Hence the deterministic model of (29) is represented as Now, each objective is considered independently and is solved with respect to the system of constraints in (31) to find the individual optimal values of the objectives. The results are obtained asThe worst values of and are obtained by putting the best solution point of the 2nd and 1st objective, respectively. Thus and .

Then the fuzzy goals of the objectives are found as .

The membership function of the objectives is written asHence the FGP model can be presented by eliciting the membership goals aswhere , , , and with .

Now solving the above model using software LINGO (ver. 11) the solutions are found as , with objective values and .

The solutions achieved through FGP technique are summarized in Table 2.

Table 2: Compromise solution obtained by proposed methodology.

If the numerical example presented in this section is solved by applying the ranking technique of IFN developed by Dubey and Mehra [29], then the objective values are obtained as and . The solutions achieved by two ranking techniques are summarized in Table 3.

Table 3: Comparison of solutions between existing and proposed techniques.

This comparison can also be shown graphically as in Figure 4.

Figure 4: Comparison of solutions.

From the comparison table and from Figure 4 it is realized that better objective values are obtained if the example is solved by the proposed ranking technique. So the proposed ranking technique is more acceptable to the DMs in real life decision making context.

6. Conclusions

This paper presents a new ranking technique for normal TIFNs. The proposed methodology can be used to find the defuzzified value of intuitionistic trapezoidal FNs. Based on the proposed ranking technique an intuitionistic LFSBLP model is developed. This methodology covers a wider range of applications with enriched solutions than earlier techniques. The proposed model can also be applied to solve many real life decision making problems in which parameters are expressed in the form of IFNs. Further the proposed methodology can be used efficiently to solve multiobjective fractional programming problems, multilevel optimization problems, and other associated problems in an intuitionistic fuzzy stochastic decision making arena. However, it is hoped that the proposed methodology may open up new vistas into the way of making decision in an imprecisely defined decision making arena.

Competing Interests

The authors declare that they have no competing interests.

References

  1. W. Candler and R. Townsley, “A linear two-level programming problem,” Computers and Operations Research, vol. 9, no. 1, pp. 59–76, 1982. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. W. F. Bialas and M. H. Karwan, “On two-level optimization,” IEEE Transactions on Automatic Control, vol. 27, no. 1, pp. 211–214, 1982. View at Google Scholar · View at Scopus
  3. W. F. Bialas and M. H. Karwan, “Two-level linear programming,” Management Science, vol. 30, no. 8, pp. 1004–1020, 1984. View at Publisher · View at Google Scholar · View at MathSciNet
  4. G. B. Dantzig, “Linear programming under uncertainty,” Management Science, vol. 1, pp. 197–206, 1955. View at Google Scholar · View at MathSciNet
  5. A. Charnes and W. W. Cooper, “Chance-constrained programming,” Management Science. Journal of the Institute of Management Science. Application and Theory Series, vol. 6, pp. 73–79, 1959/1960. View at Google Scholar · View at MathSciNet
  6. H.-J. Zimmermann, “Fuzzy programming and linear programming with several objective functions,” Fuzzy Sets and Systems, vol. 1, no. 1, pp. 45–55, 1978. View at Google Scholar · View at MathSciNet
  7. L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338–356, 1965. View at Publisher · View at Google Scholar
  8. Y. J. Lai and C. L. Hwang, Fuzzy Mathematical Programming Methods and Applications, Springer, Berlin, Germany, 1996.
  9. H.-S. Shih, Y.-J. Lai, and E. S. Lee, “Fuzzy approach for multi-level programming problems,” Computers and Operations Research, vol. 23, no. 1, pp. 73–91, 1996. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. R. E. Bellman and L. A. Zadeh, “Decision-making in a fuzzy environment,” Management Science, vol. 17, pp. B141–B164, 1970. View at Google Scholar · View at MathSciNet
  11. W. A. Lodwick and J. Kacprizyk, Fuzzy Optimization, Springer, Berlin, Germany, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  12. E. L. Hannan, “Linear programming with multiple fuzzy goals,” Fuzzy Sets and Systems, vol. 6, no. 3, pp. 235–248, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  13. R. Narasimhan, “On fuzzy goal programming—some comments,” Decision Sciences, vol. 12, no. 3, pp. 532–538, 1981. View at Publisher · View at Google Scholar
  14. B. B. Pal and B. N. Moitra, “A fuzzy goal programming procedure for solving quadratic bilevel programming problems,” International Journal of Intelligent Systems, vol. 18, no. 5, pp. 529–540, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  15. I. A. Baky, “Solving multi-level multi-objective linear programming problems through fuzzy goal programming approach,” Applied Mathematical Modelling. Simulation and Computation for Engineering and Environmental Systems, vol. 34, no. 9, pp. 2377–2387, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  16. K. T. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 20, no. 1, pp. 87–96, 1986. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. K. T. Atanassov, Intuitionistic Fuzzy Sets, Springer, Heidelberg, Germany, 1999.
  18. K. T. Atanassov, “Two theorems for intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 110, no. 2, pp. 267–269, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  19. D.-F. Li, “Multiattribute decision making models and methods using intuitionistic fuzzy sets,” Journal of Computer and System Sciences, vol. 70, no. 1, pp. 73–85, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  20. D.-F. Li, “Extension of the LINMAP for multiattribute decision making under Atanassov’s intuitionistic fuzzy environment,” Fuzzy Optimization and Decision Making, vol. 7, no. 1, pp. 17–34, 2008. View at Publisher · View at Google Scholar
  21. D.-F. Li, G.-H. Chen, and Z.-G. Huang, “Linear programming method for multiattribute group decision making using IF sets,” Information Sciences, vol. 180, no. 9, pp. 1591–1609, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. S.-P. Wan and D.-F. Li, “Fuzzy mathematical programming approach to heterogeneous multiattribute decision-making with interval-valued intuitionistic fuzzy truth degrees,” Information Sciences, vol. 325, pp. 484–503, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. Y. X. Xue, J. X. You, X. D. Lai, and H. C. Liu, “An interval-valued intuitionistic fuzzy MABAC approach for material selection with incomplete weight information,” Applied Soft Computing, vol. 38, pp. 703–713, 2016. View at Publisher · View at Google Scholar
  24. S.-P. Wan and J.-Y. Dong, “Interval-valued intuitionistic fuzzy mathematical programming method for hybrid multi-criteria group decision making with interval-valued intuitionistic fuzzy truth degrees,” Information Fusion, vol. 26, pp. 49–65, 2015. View at Publisher · View at Google Scholar · View at Scopus
  25. S. H. R. Hajiagha, H. A. Mahdiraji, S. S. Hashemi, and E. K. Zavadskas, “Evolving a linear programming technique for MAGDM problems with interval valued intuitionistic fuzzy information,” Expert Systems with Applications, vol. 42, no. 23, pp. 9318–9325, 2015. View at Publisher · View at Google Scholar · View at Scopus
  26. H. B. Mitchell, “Ranking intuitionistic fuzzy numbers,” International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, vol. 12, no. 3, pp. 377–386, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  27. V. L. G. Nayagam, G. Venkateshwari, and G. Sivaraman, “Ranking of intuitionistic fuzzy numbers,” in Proceedings of the IEEE International Conference on Fuzzy Systems (FUZZ '08), pp. 1971–1974, Hong Kong, June 2008. View at Publisher · View at Google Scholar · View at Scopus
  28. P. Grzegoraewski, “The hamming distance between intuitionistic fuzzy sets,” in Proceedings of the IFSA World Congress, pp. 223–227, 2003.
  29. D. Dubey and A. Mehra, “Linear programming with triangular intuitionistic fuzzy number,” Advances in Intelligent Systems Research, vol. 1, pp. 563–569, 2011. View at Google Scholar
  30. D.-F. Li, “A ratio ranking method of triangular intuitionistic fuzzy numbers and its application to MADM problems,” Computers & Mathematics with Applications, vol. 60, no. 6, pp. 1557–1570, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  31. H. M. Nehi, “A new ranking method for intuitionistic fuzzy numbers,” International Journal of Fuzzy Systems, vol. 12, no. 1, pp. 80–86, 2010. View at Google Scholar · View at MathSciNet · View at Scopus
  32. S.-P. Wan, “Multi-attribute decision making method based on possibility variance coefficient of triangular intuitionistic fuzzy numbers,” International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, vol. 21, no. 2, pp. 223–243, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  33. S.-P. Wan, Q.-Y. Wang, and J.-Y. Dong, “The extended VIKOR method for multi-attribute group decision making with triangular intuitionistic fuzzy numbers,” Knowledge-Based Systems, vol. 52, pp. 65–77, 2013. View at Publisher · View at Google Scholar · View at Scopus
  34. S. P. Wan and Z. H. Yi, “Power average of trapezoidal intuitionistic fuzzy numbers using strict t-norms and t-conorms,” IEEE Transactions on Fuzzy Systems, In press. View at Publisher · View at Google Scholar
  35. K. P. Yoon, “A probabilistic approach to rank complex fuzzy numbers,” Fuzzy Sets and Systems, vol. 80, no. 2, pp. 167–176, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  36. R. Saneifard and R. Saneifard, “A modified method for defuzzification by probability density function,” Journal of Applied Sciences Research, vol. 7, no. 2, pp. 102–110, 2011. View at Google Scholar · View at Scopus
  37. A. Biswas and K. Bose, “A fuzzy programming approach for solving quadratic bilevel programming problems with fuzzy resource constraints,” International Journal of Operational Research, vol. 12, no. 2, pp. 142–156, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  38. B. Liu and K. Iwamura, “Chance constrained programming with fuzzy parameters,” Fuzzy Sets and Systems, vol. 94, no. 2, pp. 227–237, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  39. M. K. Luhandjula, “Linear programming under randomness and fuzziness,” Fuzzy Sets and Systems, vol. 10, no. 1, pp. 45–55, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  40. M. K. Luhandjula, “Mathematical programming in the presence of fuzzy quantities and random variables,” Journal of Fuzzy Mathematics, vol. 11, no. 1, pp. 27–39, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  41. M. K. Luhandjula and M. M. Gupta, “On fuzzy stochastic optimization,” Fuzzy Sets and Systems, vol. 81, no. 1, pp. 47–55, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  42. A. Kaufmann and M. M. Gupta, Fuzzy Mathematical Models in Engineering and Management Science, North-Holland Publishing, 1988.
  43. A. Kaufmann and M. M. Gupta, Introduction to Fuzzy Arithmetic: Theory and Applications, Van Nostrand Reinhold, 1985.
  44. G. S. Mahapatra and T. K. Roy, “Intuitionistic fuzzy number and its arithmetic operation with application on system failure,” Journal of Uncertain Systems, vol. 7, no. 2, pp. 92–107, 2013. View at Google Scholar · View at Scopus
  45. B. Epstein, “Some applications of the Mellin transform in statistics,” Annals of Mathematical Statistics, vol. 19, pp. 370–379, 1948. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  46. P. E. Zwicke and I. Kiss Jr., “A new implementation of the Mellin transform and its application to radar classification of ships,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 5, no. 2, pp. 191–199, 1983. View at Google Scholar · View at Scopus
  47. F. R. Gerardi, “Application of mellin and hankel transforms to networks with time-varying parameters,” IRE Transactions on Circuit Theory, vol. 6, no. 2, pp. 197–208, 1959. View at Publisher · View at Google Scholar · View at Scopus
  48. J. Peschon, “A modified form of the Mellin transform and its application to the optimum final value control problem,” Tech. Rep. 2102-1, Stanford Electronics Labs, Stanford University, Stanford, Calif, USA, 1965. View at Google Scholar
  49. B. A. Dolan, “The Mellin transform for moment-generation and for the probability density of products and quotients of random variables,” Proceedings of the IEEE, vol. 52, pp. 1745–1746, 1964. View at Google Scholar · View at MathSciNet
  50. X. Wang and E. E. Kerre, “Reasonable properties for the ordering of fuzzy quantities (I),” Fuzzy Sets and Systems, vol. 118, no. 3, pp. 375–385, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus