GRA Method for Probabilistic Linguistic Multiple Attribute Group Decision Making with Incomplete Weight Information and Its Application to Waste Incineration Plants Location Problem
- DOI
- 10.2991/ijcis.d.191203.002How to use a DOI?
- Keywords
- Multiple attribute group decision making (MAGDM); Probabilistic linguistic term sets (PLTSs); GRA method; Incomplete weight information; Waste incineration plants
- Abstract
In this essay, we investigate the probabilistic linguistic multiple attribute group decision making (PL-MAGDM) with incomplete weight information. In this method, the linguistic representation developed recently is converted into probabilistic linguistic information. For deriving the weight information of the attribute, an optimization model is built on the basis of the fundamental idea of grey relational analysis (GRA), by which the attribute weights can be decided. Then, the optimal alternative is chosen through calculating largest relative relational degree from the probabilistic linguistic positive ideal solution (PLPIS) which considers both the largest grey relational coefficient (GRC) from the PLPIS and the smallest GRC form probabilistic linguistic negative ideal solution (PLNIS). In the end, a case study concerning waste incineration plants location problem is given to demonstrate the merits of the developed methods.
- Copyright
- © 2019 The Authors. Published by Atlantis Press SARL.
- Open Access
- This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).
1. INTRODUCTION
The grey relational analysis (GRA) method was initially designed by Deng [1] for tackling the MADM issue. In addition, GRA method is one of the very popular and useful tools to analyze diverse relationships among the discrete information and make decisions in different situations [2–4]. The major merits of the GRA method are that the analyzed results are depended upon the original data, the calculating process are simple and straightforward and, finally, it is one of the optimal methods to make decisions under diverse business environment [5–7]. Kung and Wen [5] employed GRA to study the grey MADM issue for venture capital enterprises. Tan, Chen and Wu [8] studied the green design alternatives and GRA connected with AHP. Chiang [9] extended GRA for dependent criteria MADM issue. Malek, Ebrahimnejad and Tavakkoli-Moghaddam [10] proposed an improved hybrid GRA method for green resilient supply. Alptekin, Alptekin and Sarac [11] assessed the low carbon development with GRA model in some countries. Zhu, Yuan and Ye [12] aimed at discerning the multi-timescales of carbon market through GRA method and empirical mode decomposition (EMD). Yazdani, Kahraman, Zarate and Onar [13] provided a good platform to ease decision process through the integration of quality function deployment (QFD) and GRA in showing main supply chain drivers under fuzzy setting. Chen [14] connected intuitionistic fuzzy GRA techniques with entropy-based TOPSIS for selecting building materials supplier. Wei [15] tackled dynamic hybrid MADM issue based on the GRA.
In real life, the decision makers (DMs) always employ linguistic terms rather than using the exact numbers because of the complex socioeconomic environment and fuzzy human beings' thinking [16–18]. Under this situations, Zadeh [19] proposed fuzzy linguistic method to depict the qualitative assessment information. In certain environments, DMs may be possible to be uncertain about some diverse linguistic terms when they depict the objects. Thus, Rodriguez, Martinez and Herrera [20] devised the tool of hesitant fuzzy linguistic term set (HFLTS) to snatch the hesitancy degree in the context of the linguistic. Gou, Xu and Liao [21] expanded the entropy and cross-entropy measures to HFLTS for MADM. Wei [22] gave the generalized dice similarity measures for MADM with HFLTS. Liao, Xu and Zeng [23] provided the VIKOR method for qualitative MADM with HFLTS. However, at certain times, the DMs may intend to express the linguistic term “medium” to the linguistic term “bad,” that's to say, these two linguistic terms' probabilities are not same [24]. It can be easily found that the HFLTS can't describe this sort of complicated qualitative information. To overcome this drawback and exactly describe this kind of complicated qualitative information, Pang, Wang and Xu [25] developed the probabilistic linguistic term set (PLTS), which permits DMs to express the preference information of themselves as a collection of several possible linguistic terms combined with probabilities. Lin, Chen, Liao and Xu [26] proposed the ELECTRE II method to deal with PLTSs for edge computing. Feng, Liu and Wei [27] constructed possibility degree comparison of probabilistic linguistic QUALIFLEX method. Bai, Zhang, Qian and Wu [28] gave comparative method and proposed a more efficient mean to tackle PLTSs. Chen, Wang and Wang [29] employed the probabilistic linguistic MULTIMOORA for cloud-based ERP system selection. Lin, Chen, Liao and Xu [26] developed the ELECTRE II method to deal with PLTSs for edge computing. Liu and Teng [30] proposed some Muirhead mean operators for PLTSs. Liu and Teng [31] defined the probabilistic linguistic TODIM method. Liu and Li [32] designed the generalized maclaurin symmetric mean aggregation operators for probabilistic linguistic information. Liu and Li [32] gave the bidirectional projection method for probabilistic linguistic multi-criteria group decision making based on power average operator. Liang, Kobina and Quan [33] designed the probabilistic linguistic grey relational analysis (PL-GRA) for MAGDM based on geometric Bonferroni mean [34–37]. Lu, Wei, Wu and Wei [38] designed the TOPSIS method for probabilistic linguistic MAGDM with entropy weight foro supplier selection of new agricultural machinery products.
But the PL-GRA [33] method have two shortcomings: (1) this method think that the optimal alternative is chosen through calculating largest relative relational degree from the probabilistic linguistic positive ideal solution (PLPIS) which considers both the largest grey relational coefficient (GRC) from the PLPIS which neglects the smallest GRC form probabilistic linguistic negative ideal solution (PLNIS); (2) this method can't tackled the probabilistic linguistic MADM (PL-MADM) or PL-MAGDM issues with incomplete weight information. As a supplement to it, we shall develop a novel GRA-based method for PL-MAGDM in this study and then apply it to select the waste incineration plants sites. The innovativeness of the paper can be summarized as follows: (1) an optimization model is built to derive the weight information of the attribute on the basis of the fundamental idea of conventional GRA method under PLTSs; (2) the PL-GRA method is proposed to solve the probabilistic linguistic MAGDM problems with incomplete weight information; (3) a case study for selecting the waste incineration plants sites is supplied to show the developed approach and (4) some comparative studies are provided with the existing methods.
The remainder of this paper is arranged in the following way: Section 2 gives some fundamental ideas connected to PLTSs. In Section 3, the GRA approach is proposed for PL-MAGDM with incomplete weight information. In Section 4, given a numerical example a case study concerning waste incineration plants location problem to illustrate the developed PL-GRA method and some comparative analysis is provided. In the end, this study makes some conclusions in Section 5.
2. PRELIMINARIES
As an essential and useful tool, The HFLTS [20] is utilized to tackle the hesitancy in the context of the linguistic. Zhao, Xu and Ren [39] designed some distance measures for HFLTS. Wei, Zhao and Tang [40] devised some operations on HFLTSs and possibility degree formulas for comparing HFLTSs. In order to strengthen the modeling capability of HFLTSs, Pang, Wang and Xu [25] proposed the definition of PLTSs to link each linguistic term with a probability value.
Definition 1.
[25] Let
At the same time,
Definition 2.
Given an LTS
In order to facile computation, Pang, Wang and Xu [25] normalized the PLTS
Definition 3.
Let
Definition 4.
For a PLTS
By using the Eqs. (4) and (5), the order of two PLTSs is defined in the following: (1) if
Definition 5.
[41] Let
3. GRA METHOD FOR PROBABILISTIC LINGUISTIC MAGDM WITH INCOMPLETE WEIGHT INFORMATION
In such section, we propose a novel PL-GRA method to tackle the MAGDM issues with incomplete weight information. Assume that
Subsequently, we shall use PL-GRA to tackle MAGDM issues with incomplete weight information. The detailed calculating steps of the proposed method are presented in the following:
Step 1. Shift cost attributes into beneficial attributes. If
Step 2. Switch the linguistic variables
Step 3. Derive the normalized assessing matrix
Step 4. Giving the definition of PLPIS and PLNIS as follows:
Step 5. Computing the corresponding GRC of each alternative from PLPIS and PLNIS by utilizing the following equation, respectively:
Step 6. Calculating the degree of GRC of all possible alternatives from PLPIS and PLNIS, respectively:
The fundamental idea of GRA method is that the optimal alternative is supposed to possess the “largest degree of GRC” from PLPIS and “smallest degree of GRC” from PLNIS. Evidently, the larger
Due to each alternative is non-inferior, for all the alternatives, there is no preference relation. Besides, the above multiple objective optimization models
By solving the model
Step 7. Derive the probabilistic linguistic relative relational degree (PLRRD) of all possible alternatives from PLPIS.
Step 8. According to
4. NUMERICAL CASE AND COMPARATIVE ANALYSIS
4.1. Numerical Case
Along with the acceleration of urbanization and the rapid growth of urban population, the output of municipal solid waste also increases rapidly. More and more city or region will create more and more waste incineration plants in accordance with “The Twelfth Five-Year Plan.” However, the waste incinerator is a NIMBY facility, and if the site selection is not scientific and reasonable, it is likely to cause NIMBY conflicts. Therefore, how to reasonably carry on the waste incineration plant scientific location is particularly important. Waste incineration plants location problem should be regarded as the corresponding MAGDM [45–51]. Thus, in this chapter we developed a case study concerning waste incineration plants location problem to demonstrate the approach presented in this essay. There are five potential waste incineration plants sites
Alternatives | G_{1} | G_{2} | G_{3} | G_{4} |
---|---|---|---|---|
A_{1} | G | VG | M | VG |
A_{2} | G | P | VG | P |
A_{3} | VG | VG | P | VG |
A_{4} | VG | VP | VP | P |
A_{5} | EG | VP | G | EG |
DM, decision maker.
Linguistic decision matrix by the first DM.
Alternatives | G_{1} | G_{2} | G_{3} | G_{4} |
---|---|---|---|---|
A_{1} | EG | VP | VG | VG |
A_{2} | G | EP | EG | M |
A_{3} | VG | M | G | EG |
A_{4} | VG | VP | P | P |
A_{5} | VG | P | G | VG |
DM, decision maker.
Linguistic decision matrix by the second DM.
Alternatives | G_{1} | G_{2} | G_{3} | G_{4} |
---|---|---|---|---|
A_{1} | EG | G | VG | VG |
A_{2} | G | EP | EG | G |
A_{3} | G | M | G | VG |
A_{4} | G | EP | P | P |
A_{5} | VG | VP | M | G |
DM, decision maker.
Linguistic decision matrix by the third DM.
Alternatives | G_{1} | G_{2} | G_{3} | G_{4} |
---|---|---|---|---|
A_{1} | VG | G | VG | VG |
A_{2} | G | P | VG | M |
A_{3} | G | EG | G | VG |
A_{4} | G | EP | VP | EP |
A_{5} | EG | P | VG | VG |
DM, decision maker.
Linguistic decision matrix by the fourth DM.
Alternatives | G_{1} | G_{2} | G_{3} | G_{4} |
---|---|---|---|---|
A_{1} | EG | VP | M | G |
A_{2} | P | EP | VG | M |
A_{3} | VG | VG | G | VG |
A_{4} | VP | VP | P | P |
A_{5} | EG | VP | VG | VG |
DM, decision maker.
Linguistic decision matrix by the fifth DM.
Following that, the PL-GRA method is utilized to select the optimal waste incineration plants sites.
Step 1. Shift cost attribute G_{2} into beneficial attribute. If the cost attribute value is
Alternatives | G_{1} | G_{2} | G_{3} | G_{4} |
---|---|---|---|---|
A_{1} | G | VP | M | VG |
A_{2} | G | G | VG | P |
A_{3} | VG | VP | P | VG |
A_{4} | VG | VG | VP | P |
A_{5} | EG | VG | G | EG |
DM, decision maker.
Linguistic decision matrix by the first DM.
Alternatives | G_{1} | G_{2} | G_{3} | G_{4} |
---|---|---|---|---|
A_{1} | EG | VG | VG | VG |
A_{2} | G | EG | EG | M |
A_{3} | VG | M | G | EG |
A_{4} | VG | VG | P | P |
A_{5} | VG | G | G | VG |
DM, decision maker.
Linguistic decision matrix by the second DM.
Alternatives | G_{1} | G_{2} | G_{3} | G_{4} |
---|---|---|---|---|
A_{1} | EG | P | VG | VG |
A_{2} | G | EG | EG | G |
A_{3} | G | M | G | VG |
A_{4} | G | EG | P | P |
A_{5} | VG | VG | M | G |
DM, decision maker.
Linguistic decision matrix by the third DM.
Alternatives | G_{1} | G_{2} | G_{3} | G_{4} |
---|---|---|---|---|
A_{1} | VG | P | VG | VG |
A_{2} | G | G | VG | M |
A_{3} | G | EP | G | VG |
A_{4} | G | EG | VP | EP |
A_{5} | EG | G | VG | VG |
DM, decision maker.
Linguistic decision matrix by the fourth DM.
Alternatives | G_{1} | G_{2} | G_{3} | G_{4} |
---|---|---|---|---|
A_{1} | EG | VG | M | G |
A_{2} | P | EG | VG | M |
A_{3} | VG | VP | G | VG |
A_{4} | VP | VG | P | P |
A_{5} | EG | VG | VG | VG |
DM, decision maker.
Linguistic decision matrix by the fifth DM.
Step 2. Transform the linguistic variables into PLTSs (Table 11).
Alternatives | G_{1} | G_{2} |
---|---|---|
A_{1} | ||
A_{2} | ||
A_{3} | ||
A_{4} | ||
A_{5} | ||
Alternatives | G_{3} | G_{4} |
A_{1} | ||
A_{2} | ||
A_{3} | ||
A_{4} | ||
A_{5} |
PLTS, probabilistic linguistic term set.
Decision matrix with PLTSs.
Step 3. Calculate the decision matrix with normalized PLTSs (Table 12).
Alternatives | G_{1} | G_{2} |
---|---|---|
A_{1} | ||
A_{2} | ||
A_{3} | ||
A_{4} | ||
A_{5} | ||
Alternatives | G_{3} | G_{4} |
A_{1} | ||
A_{2} | ||
A_{3} | ||
A_{4} | ||
A_{5} |
PLTS, probabilistic linguistic term set.
Decision matrix with Nnormalized PLTSs.
Step 4. Defining the PLPIS and PLNIS by Eqs. (7–9) (Table 13):
G_{1} | G_{2} | |
---|---|---|
PLPIS | ||
PLNIS | ||
G_{3} | G_{4} | |
PLPIS | ||
PLNIS |
PLPIS, probabilistic linguistic positive ideal solution; PLNIS, probabilistic linguistic negative ideal solution.
PLPIS and PLNIS.
Step 5. Computing the corresponding GRC of each alternative from PLPIS and PLNIS (Tables 14 and 15):
Alternatives | G_{1} | G_{2} | G_{3} | G_{4} |
---|---|---|---|---|
A_{1} | 0.6241 | 0.4864 | 0.5284 | 0.3333 |
A_{2} | 0.5631 | 0.5355 | 1.0000 | 0.4835 |
A_{3} | 0.7467 | 0.4261 | 0.4387 | 1.0000 |
A_{4} | 0.5613 | 1.0000 | 0.4261 | 0.3459 |
A_{5} | 1.0000 | 0.5826 | 0.5745 | 0.6241 |
GRC, grey relational coefficient; PLPIS, probabilistic linguistic positive ideal solution.
GRC of each alternative from PLPIS.
Alternatives | G_{1} | G_{2} | G_{3} | G_{4} |
---|---|---|---|---|
A_{1} | 0.6511 | 0.6868 | 0.5051 | 0.4427 |
A_{2} | 1.0000 | 0.4283 | 0.4124 | 0.5000 |
A_{3} | 0.6230 | 1.0000 | 0.5012 | 0.3333 |
A_{4} | 0.5405 | 0.4124 | 1.0000 | 1.0000 |
A_{5} | 0.5493 | 0.4816 | 0.5627 | 0.3910 |
GRC, grey relational coefficient; PLNIS, probabilistic linguistic negative ideal solution.
GRC of each alternative from PLNIS.
Step 6. The model
Solve this model, the weight vector of attributes can be got:
Step 7. Calculating the degree of GRC of all possible alternatives from PLPIS and PLNIS, respectively (Table 16):
Alternatives | ||
---|---|---|
A_{1} | 0.4933 | 0.5571 |
A_{2} | 0.6914 | 0.5600 |
A_{3} | 0.6356 | 0.5863 |
A_{4} | 0.5500 | 0.7860 |
A_{5} | 0.6774 | 0.5025 |
PLPIS, probabilistic linguistic positive ideal solution; PLNIS, probabilistic linguistic negative ideal solution.
Step 8. Calculating the
Alternatives | A_{1} | A_{2} | A_{3} | A_{4} | A_{5} |
---|---|---|---|---|---|
0.4696 | 0.5525 | 0.5202 | 0.4117 | 0.5741 |
PLRRD, probabilistic linguistic relative relational degree; PLPIS, probabilistic linguistic positive ideal solution.
PLRRD of each alternative from PLPIS.
Step 9. According to the
4.2. Comparative Analysis
Then, our proposed method is compared with probabilistic linguistic weighted average (PLWA) operator [25] and PL-TOPSIS method [25] as in Table 18.
Methods | Computing Results | Ordering |
---|---|---|
PL-TOPSIS [25] | ||
PLWA operator [25] | ||
PL-GRA method |
PLWA, probabilistic linguistic weighted average; PL_GRA, probabilistic linguistic grey relational analysis.
Ordering of the waste incineration plants sites by using diverse methods.
In terms of the above analysis, it can be found that these abovementioned methods have the same best waste incineration plants site
5. CONCLUSION
In this essay, the GRA method is expanded to the PL-MAGDM with incomplete weight information. First and foremost, the definition, comparative method and distance of PLTs are simply reviewed. Additionally, the extended GRA method is employed to tackle PL-MAGDM issues with incomplete weight information. We construct the multiple objective optimization models on the basis of the conventional GRA method. Besides, the multiple objective optimization models can be converted into a single-objective programming model by making use of the linear equal weighted method. By calculating the single-objective programming model, the weight information can be acquired. In the light of the conventional GRA, the optimal choice is derived by obtaining “largest degree of GRC” from PLPIS and “smallest degree of GRC” from PLNIS. Finally, a practical case study concerning waste incineration plants location problem is designed to validate the proposed algorithms and some comparative studies are also designed to verify the applicability. In our future research, the proposed methods and algorithm will be needful and meaningful for other real decision making problems [52–60] and the developed approaches can also be extended to other fuzzy [48,61,62] and uncertain information [63–73].
CONFLICT OF INTEREST
The authors declare that they have no conflict of interest.
AUTHORS' CONTRIBUTIONS
Fan Lei, Guiwu Wei, Jianping Lu, Jiang Wu and Cun Wei conceived and worked together to achieve this work, Fan Lei compiled the computing program by Excel and analyzed the data, Fan Lei and Guiwu Wei wrote the paper. Finally, all the authors have read and approved the final manuscript.
Funding Statement
The work was supported by the National Natural Science Foundation of China under Grant No. 71571128 and the Humanities and Social Sciences Foundation of Ministry of Education of the People's Republic of China (14YJCZH082).
REFERENCES
Cite this article
TY - JOUR AU - Fan Lei AU - Guiwu Wei AU - Jianping Lu AU - Jiang Wu AU - Cun Wei PY - 2019 DA - 2019/12/09 TI - GRA Method for Probabilistic Linguistic Multiple Attribute Group Decision Making with Incomplete Weight Information and Its Application to Waste Incineration Plants Location Problem JO - International Journal of Computational Intelligence Systems SP - 1547 EP - 1556 VL - 12 IS - 2 SN - 1875-6883 UR - https://doi.org/10.2991/ijcis.d.191203.002 DO - 10.2991/ijcis.d.191203.002 ID - Lei2019 ER -