Note on “Optimal path selection approach for fuzzy reliable shortest path problem”

Abstract

There are different conditions where SPP play a vital role. However, there are various conditions, where we have to face with uncertain parameters such as variation of cost, time and so on. So to remove this uncertainty, Yang et al. [1] “[Journal of Intelligent & Fuzzy Systems, 32(1), 197-205”] have proposed the fuzzy reliable shortest path problem under mixed fuzzy environment and claimed that it is better to use their proposed method as compared to the existing method i.e., “[Hassanzadeh et al.; A genetic algorithm for solving fuzzy shortest path problems with mixed fuzzy arc lengths, Mathematical and Computer Modeling, 57(2013) 84-99” [2]]. The aim of this note is, to highlight the shortcoming that is carried out in Yang et al. [1] article. They have used some mathematical incorrect assumptions under the mixed fuzzy domain, which is not true in a fuzzy environment.

Keywords

normal fuzzy number Shortest path problem (SPP)fuzzy shortest path problem (FSPP)mixed fuzzy environment

1 Introduction

Yang et al., [1] “[Journal of Intelligent & Fuzzy Systems, 32(1), 197-205]” proposed that, if, ’A’ is normal fuzzy number and ’B’ is not a normal fuzzy number (i.e. either triangular fuzzy or trapezoidal fuzzy number), then to add these two different types of membership fuzzy numbers, they have used alpha cut principle. This stage has been clearly explained in algorithm 2 [1]. Further, in Algorithm 3, the authors have predicted the fuzzy reliable shortest path (FRSP) distance (i.e., crisp distance after adding mixed fuzzy numbers). If these predicted values are used in the further scientific process, then it may lead to error-prone results under a mixed fuzzy environment. However, after close investigation with the proposed method [1] within the given parameters,it fails to predict the crisp value under a mixed fuzzy atmosphere. Hence, The motivation for this paper is to highlight the mathematical incorrect assumptions through this short note. In support of our claim, we have investigated all the three different examples which were considered by the authors [1] (i.e., example 2,3 and 4).

This research note progresses as follows; Section 2 represents the definitions of fuzzy set theory, Section 3 includes shortcomings of the Yang et al., [1] method, Section 4 covers the critical analysis followed by the conclusion at end of the researchnote.

2 Preliminaries

Definiton 1. [3] Let $\tilde{a} = (a_{1}, a_{2}, a_{3}, a_{4})$ be a trapezoidal fuzzy number, then the grade mean integration representation by $P (\tilde{a})$ i.e., $P (\tilde{a}) = \frac{a_{1} + 2 a_{2} + 2 a_{3} + a_{4}}{6}$

Definiton 2. [4] Let $\tilde{a} = (a_{1}, a_{2}, a_{3}, a_{4})$ be a trapezoidal fuzzy number, then the expected value represented as $E (\tilde{a})$ i.e., $E (\tilde{a}) = \frac{a_{1} + a_{2} + a_{3} + a_{4}}{4}$

3 Shortcoming of the Yang et al. proposed method [1]

In this section, we have considered all the examples which are used in Yang et al. [1][Section 4, Example 2-4, pp. 201] article to justify the shortcoming of their model.

Example 3.1. After consideration of the algorithm 3 as mentioned in the author‘s article [1] [Section 4, Example 2 (Fig. 1), Table 1, pp. 201-202 with n=100] , the fuzzy reliable shortest path is 1-3-4 and approximate FRSP length is 48.4916.

Fig. 1

Network considered for Example 2 [1]

Observation: As per our observation, we found that the authors [1] have considered the arc length of (1,3) is (4,8,12,16) and the arc length of (3,4) is (5,1). After the addition of these mixed fuzzy numbers, the authors [1] concluded that the FRSP length is 48.4916 which is not true in the fuzzy environment as the fuzzy range lies between 7 to 22.

Example 3.2. In the similar way, Yang et al. [1] explored in the [section 4, Example 3 (Fig. 2) Table 2, pp. 202 with n=100] , the fuzzy reliable shortest path is 1-10-11, and approximate FRSP length is 1598.

Fig. 2

Network considered for Example 3. [1]

Observation: Here, we observe that the arc length of (1,10) is (420,450,570) and the arc length of (10,11) is (125,41). After adding these two mixed fuzzy numbers, we found that the fuzzy range lies between 480 to 659, whereas, the authors have mentioned FSRP value 1598 which drastically differs from the valid fuzzy range. Hence, the method proposed by the authors [1] is not valid.

Example 3.3. In the similar manner, Yang et al. [1] [section 4, Example 4 (Fig. 3), Table 3 pp. 202] evaluated the fuzzy reliable shortest path as 1-5-11-14-21-23 and FSRP value 142.8005.

Fig. 3

Network considered for Example 3. [1]

Observation: During re-evaluation of the above mentioned problem [Example 3], the reliable shortest path length is: = 1 to 5 + path length of 5 to 11 + path length of 11 to 14 + path length of 14 to 21 + path length of 21 to 23 = (7,8,9,10) + (7,10,13,14) + (8,9,11,13) + (11,12,13,14) + (12,15,17,18) = 142.8005. But as per our calculations, the FRSP length should be (45,54,63,69) and the range of FSRP length must be within 45 to 69. In contrary, the FRSP length evaluated by the authors is 142.8005 i.e., way beyond the fuzzy range. Hence, the method proposed by the authors is not valid for any of the cases.

4 Comparison with the Yang et al. [1] method

In this section, we have presented a comparative analysis with respect to Yang et al. (2017) [1] method. Yang et al. have considered three examples under mixed fuzzy environment that are represented in Tables 1, 2 and 3. Whereas, we have considered one (Example 3) out of three examples for the critical analysis. During our analysis, we have found that the suggested FRSP length is going beyond the fuzzy range. The same has been represented both in the tabular and graphical manner in Table 4 and Fig. 4 respectively. Fig. 4, illustrates the lowest fuzzy range by green color (Y = 45) and highest fuzzy range by orange color (Y = 69). Based on the Definition 1 and 2, the performance of Yang et al. [1] method is way beyond the actual fuzzy range.

Table 1

The Arc Length for Example 2. [1]

Arc	Membership Function
${\tilde{r}}_{12}$	(2,3,4,5)
${\tilde{r}}_{24}$	(15,4)
${\tilde{r}}_{23}$	(4,1)
${\tilde{r}}_{13}$	(4,8,12,16)
${\tilde{r}}_{34}$	(5,1)

Table 2

The Arc Length for Example 3. [1]

${\tilde{s}}_{12}$	(800,820,840)	${\tilde{s}}_{13}$	(35,11)	${\tilde{s}}_{16}$	(650,677,783)
${\tilde{s}}_{19}$	(290,300,350)	${\tilde{s}}_{110}$	(420,450,570)	${\tilde{s}}_{23}$	(180,186,293)
${\tilde{s}}_{25}$	(495,510,625)	${\tilde{s}}_{29}$	(90,30)	${\tilde{s}}_{34}$	(650,667,983)
${\tilde{s}}_{84}$	(710,730,835)	${\tilde{s}}_{87}$	(230,242,355)	${\tilde{s}}_{97}$	(120,130,250)
${\tilde{s}}_{98}$	(13,4)	${\tilde{s}}_{35}$	(730,748,870)	${\tilde{s}}_{38}$	(42,14)
${\tilde{s}}_{45}$	(190,199,310)	${\tilde{s}}_{46}$	(310,340,460)	${\tilde{s}}_{411}$	(71,23)
${\tilde{s}}_{56}$	(610,660,790)	${\tilde{s}}_{611}$	(23,7)	${\tilde{s}}_{76}$	(390,410,540)
${\tilde{s}}_{711}$	(45,15)	${\tilde{s}}_{910}$	(23,7)	${\tilde{s}}_{107}$	(330,342,450)
${\tilde{s}}_{1011}$	(125,41)

Table 3

The Arc Length for Example 4. [1]

Arc	Mem.function	Arc	Mem.function	Arc	Mem.function
${\tilde{s}}_{12}$	(12,13,15,17)	${\tilde{s}}_{13}$	(40,11)	${\tilde{s}}_{14}$	(8,10,12,13)
${\tilde{s}}_{15}$	(7,8,9,10)	${\tilde{s}}_{26}$	(35,10)	${\tilde{s}}_{27}$	(6,11,11,13)
${\tilde{s}}_{38}$	(40,11)	${\tilde{s}}_{47}$	(17,20,22,24)	${\tilde{s}}_{411}$	(6,10,13,14)
${\tilde{s}}_{58}$	(29,9)	${\tilde{s}}_{511}$	(7,10,13,14)	${\tilde{s}}_{512}$	(10,13,15,17)
${\tilde{s}}_{69}$	(6,8,10,11)	${\tilde{s}}_{610}$	(35,11)	${\tilde{s}}_{1518}$	(8,9,11,13)
${\tilde{s}}_{1519}$	(25,7)	${\tilde{s}}_{1620}$	(38,12)	${\tilde{s}}_{1720}$	(7,10,11,12)
${\tilde{s}}_{1922}$	(15,16,17,19)	${\tilde{s}}_{2023}$	(13,14,16,17)	${\tilde{s}}_{710}$	(9,10,12,13)
${\tilde{s}}_{711}$	(6,7,8,9)	${\tilde{s}}_{812}$	(5,8,9,10)	${\tilde{s}}_{813}$	(20,5)
${\tilde{s}}_{916}$	(6,7,9,10)	${\tilde{s}}_{1016}$	(40,13)	${\tilde{s}}_{1017}$	(15,19,20,21)
${\tilde{s}}_{1114}$	(8,9,11,13)	${\tilde{s}}_{1117}$	(28,9)	${\tilde{s}}_{1214}$	(13,14,16,18)
${\tilde{s}}_{1215}$	(12,14,15,16)	${\tilde{s}}_{1315}$	(37,12)	${\tilde{s}}_{1319}$	(17,18,19,20)
${\tilde{s}}_{1421}$	(11,12,13,14)	${\tilde{s}}_{1721}$	(6,7,8,10)	${\tilde{s}}_{1821}$	(15,17,18,19)
${\tilde{s}}_{1822}$	(16,5)	${\tilde{s}}_{1823}$	(15,5)	${\tilde{s}}_{2123}$	(12,15,17,18)
${\tilde{s}}_{2223}$	(20,5)

Table 4

Comparison on Example 3. [1]

Fuzzy arc length	Yang et al(2017) proposed value [1]	Definition 1[3]	Definition 2[4]
(45, 54, 63, 69)	142.8005	58	57.5

Fig. 4

Graphical representation for the shortcoming of Yang et al .[1] method.

Conclusion

The detailed analysis of the proposed method in 1], concludes that the outcome is error-prone which fails to predict the crisp objective value for all trapezoidal, triangular, and mixed fuzzy environment. Hence, it is scientifically incorrect to use this method for solving fuzzy reliable shortest path. In this paper, we have proposed a claim with the support of existing literature to falsify the method proposed by [1]. To motivate the readers, few relevant research articles [5 –12] have been suggested to provide more insight on this topic.

References

Yang

, Yan

and Zhao

, Optimal path selection approach for fuzzy reliable shortest path problem, Journal of Intelligent and Fuzzy Systems 32 (2017), 197–205.

Hassanzadeh

, Mahdavi

, Mahdavi-Amiri

and Tajdin

, A genetic algorithm for solving fuzzy shortest path problems with mixed fuzzy arc lengths, Mathematical and Computer Modelling 57 (2013), 84–99.

Deng

, Chen

, Zhang

and Mahadevan

, Fuzzy Dijkstra algorithm for shortest path problem under uncertain environment, Applied Soft Computing 12 (2012), 1231–1237.

, Iwamura

and Shao

, New models for shortest path problem with fuzzy arc lengths, Applied Mathematical Modelling 31 (2007), 259–269.

Kung

J.Y.

, Chuang

T.N.

and Lin

C.T.

, A new dynamic programming approach for finding the shortest path length and the corresponding shortest path in a discrete fuzzy network, Journal of Intelligent & Fuzzy Systems 18 (2007), 117–122.

Jiang

Z.Z.

, Jiao

Y.R.

, Sheng

and Chen

, A novel model and its algorithms for the shortest path problem of dynamic weight-varying networks in Intelligent Transportation Systems, Journal of Intelligent & Fuzzy Systems 33 (2017), 3095–3102.

Gayathri

R.G.

and Nair

J.J.

, ex-FTCD: A novel mapreduce model for distributed multi source shortest path problem, Journal of Intelligent & Fuzzy Systems 34 (2018), 1643–1652.

Enayattabr

, Ebrahimnejad

, Motameni

and Garg

, A novel approach for solving all-pairs shortest path problem in an interval-valued fuzzy network, Journal of Intelligent & Fuzzy Systems 37 (2019), 6865–6877.

Abbaszadeh Sori

, Ebrahimnejad

and Motameni

, The fuzzy inference approach to solve multi-objective constrained shortest path problem, Journal of Intelligent & Fuzzy Systems 38 (2020), 4711–4720.

10.

Zhan

, Masood Malik

and Akram

, Novel decision-making algorithms based on intuitionistic fuzzy rough environment, International Journal of Machine Learning and Cybernetics 10 (2019), 1459–1485.

11.

Zhan

, Akram

and Sitara

, Novel decision-making method based on bipolar neutrosophic information, Soft Computing 23 (2019), 9955–9977.

12.

Akram

, Habib

and Alcantud

J.C.R.

, An optimization study based on Dijkstra algorithm for a network with trapezoidal picture fuzzy numbers, Neural Computing and Applications (2020), https://doi.org/10.1007/s00521-020-05034-y.