doi: 10.3934/dcdss.2020175

An improved fuzzy linear weighting method of multi-objective programming problems and its application

1. 

School of Management, Hebei University, Baoding 071000, China

2. 

School of Economics and Management, Hebei University of Technology, Tianjin 300401, China

3. 

School of Information, Beijing Wuzi University, Beijing 101149, China

* Corresponding author: Zixue Guo

Received  March 2019 Revised  April 2019 Published  December 2019

Fund Project: The first author is supported by School of Management of Hebei University

Multi-objective programming problem is a branch of mathematical programming, and the general method is to transform it into a single objective programming problem. In this paper, in order to consider the different importance of each objective function, $ G1 $ method for determining the weight of each objective function is proposed. Then, the membership function of each objective function is linearly weighted after being solved, and the multi-objective programming problem is transformed into a single objective programming problem. We obtain and prove the equivalent model of the single objective programming problem, and also obtain the non-inferior solution of the original multi-objective programming problem by solving the optimal solution of the equivalent model. Finally, the feasibility and effectiveness of this method is proved by the example of emergency material dispatch problem.

Citation: Zixue Guo, Fengxuan Song, Yumeng Zheng, Zefang He. An improved fuzzy linear weighting method of multi-objective programming problems and its application. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020175
References:
[1]

R. E. Bellman and L. A. Zadeh, Decision making in a fuzzy environment, Management Science, 17, (1970), B141–B164. doi: 10.1287/mnsc.17.4.B141.  Google Scholar

[2]

P. ChenS. Zhao and C. Chen, Fuzzy geometric weighting method for solving economic dispatch in power system, Electric Power Science and Engineering, 28 (2012), 1-5.   Google Scholar

[3]

X. Feng, Y. Ruan and W. Zhao, Research on the weight of the assess index system of army equipment support maneuver based on G1 method, Science and Technology Square, (2009), 17–99. Google Scholar

[4]

J. Fu, D. Wang and X. Ma et al., The application research on safety assessment system in drinking water resource, Environmental Monitoring and Assessment, (2013), 109–110. Google Scholar

[5]

L. Guo, Evaluation function method of multi-objective programming, Zhangzhou Institute of Technology, 8 (2006), 12-15.   Google Scholar

[6]

Z. GuoY. Zheng and S. Li, Interval multi-objective programming problems based on fuzzy geometric weighting method, Hebei University (Natural Science edition), 35 (2015), 230-235.   Google Scholar

[7]

S. M. Hosamani, Correlation of domination parameters with physicochemical properties of octane isomers, Applied Mathematics and Nonlinear Sciences, 1 (2016), 345-351.  doi: 10.21042/AMNS.2016.2.00029.  Google Scholar

[8] Y. Hu, Practical Multi-objective Optimization, Shanghai Science Press, 1990.   Google Scholar
[9]

J. Jin and W. J. Mi, An aimms-based decision-making model for optimizing the intelligent stowage of export containers in a single bay, Discrete and Continuous Dynamical Systems Series-S, 12 (2019), 1101-1115.   Google Scholar

[10]

J. Li and L. Yang, Further study on the determination of membership function, Guizhou University of Technology, 33 (2004), 1-4.   Google Scholar

[11]

C. Qiao and G. Zhang, Geometric weighing method of solving multi-objective programming problems, North China Electric Power University, 38 (2011), 107-110.   Google Scholar

[12]

B. Yang and K. Zhang, Multi-Objective Decision Analysis Theory, Method and Application Research, Donghua University Press, 2008. Google Scholar

[13]

D. YuH. Liu and C. Bresser, Peak load management based on hybrid power generation and demand response, Energy, 163 (2018), 969-985.  doi: 10.1016/j.energy.2018.08.177.  Google Scholar

[14]

G. ZhangG. Li and H. Xie, Multi-objective weighted fuzzy nonlinear programming, North China Electric Power University, 31 (2004), 33-35.   Google Scholar

[15]

G. Zhang, G. Li and H. Xie et al. , Weighting fuzzy multi-objective model of economic dispatch, North China Electric Power University, 31(2004), 49-52. Google Scholar

[16]

H. Zhao and S. Luo, Fuzzy multi-objective decision model for selection of logistics modes, Logistics Technology, 32 (2013), 104-107.   Google Scholar

[17]

Z. Zhao, Y. Shao and Y. Bi, Quality evaluation of oil and gas pipeline emergency plan based on G1 weighting method, Oil, Gas Storage and Transportation, (2008), 6–9. Google Scholar

[18]

H. Zuo, Research on solution of fuzzy multi-objective programming problem, North China Electric Power University, (2013). Google Scholar

show all references

References:
[1]

R. E. Bellman and L. A. Zadeh, Decision making in a fuzzy environment, Management Science, 17, (1970), B141–B164. doi: 10.1287/mnsc.17.4.B141.  Google Scholar

[2]

P. ChenS. Zhao and C. Chen, Fuzzy geometric weighting method for solving economic dispatch in power system, Electric Power Science and Engineering, 28 (2012), 1-5.   Google Scholar

[3]

X. Feng, Y. Ruan and W. Zhao, Research on the weight of the assess index system of army equipment support maneuver based on G1 method, Science and Technology Square, (2009), 17–99. Google Scholar

[4]

J. Fu, D. Wang and X. Ma et al., The application research on safety assessment system in drinking water resource, Environmental Monitoring and Assessment, (2013), 109–110. Google Scholar

[5]

L. Guo, Evaluation function method of multi-objective programming, Zhangzhou Institute of Technology, 8 (2006), 12-15.   Google Scholar

[6]

Z. GuoY. Zheng and S. Li, Interval multi-objective programming problems based on fuzzy geometric weighting method, Hebei University (Natural Science edition), 35 (2015), 230-235.   Google Scholar

[7]

S. M. Hosamani, Correlation of domination parameters with physicochemical properties of octane isomers, Applied Mathematics and Nonlinear Sciences, 1 (2016), 345-351.  doi: 10.21042/AMNS.2016.2.00029.  Google Scholar

[8] Y. Hu, Practical Multi-objective Optimization, Shanghai Science Press, 1990.   Google Scholar
[9]

J. Jin and W. J. Mi, An aimms-based decision-making model for optimizing the intelligent stowage of export containers in a single bay, Discrete and Continuous Dynamical Systems Series-S, 12 (2019), 1101-1115.   Google Scholar

[10]

J. Li and L. Yang, Further study on the determination of membership function, Guizhou University of Technology, 33 (2004), 1-4.   Google Scholar

[11]

C. Qiao and G. Zhang, Geometric weighing method of solving multi-objective programming problems, North China Electric Power University, 38 (2011), 107-110.   Google Scholar

[12]

B. Yang and K. Zhang, Multi-Objective Decision Analysis Theory, Method and Application Research, Donghua University Press, 2008. Google Scholar

[13]

D. YuH. Liu and C. Bresser, Peak load management based on hybrid power generation and demand response, Energy, 163 (2018), 969-985.  doi: 10.1016/j.energy.2018.08.177.  Google Scholar

[14]

G. ZhangG. Li and H. Xie, Multi-objective weighted fuzzy nonlinear programming, North China Electric Power University, 31 (2004), 33-35.   Google Scholar

[15]

G. Zhang, G. Li and H. Xie et al. , Weighting fuzzy multi-objective model of economic dispatch, North China Electric Power University, 31(2004), 49-52. Google Scholar

[16]

H. Zhao and S. Luo, Fuzzy multi-objective decision model for selection of logistics modes, Logistics Technology, 32 (2013), 104-107.   Google Scholar

[17]

Z. Zhao, Y. Shao and Y. Bi, Quality evaluation of oil and gas pipeline emergency plan based on G1 weighting method, Oil, Gas Storage and Transportation, (2008), 6–9. Google Scholar

[18]

H. Zuo, Research on solution of fuzzy multi-objective programming problem, North China Electric Power University, (2013). Google Scholar

Figure 1.  The relationship between membership function and objective function
Table 1.  The values of $ r_{i} $
$ r_{i} $ The important degree $ f_{i-1}(x) $ of compared to $ f_{i}(x) $
1.0 Equally important
1.1 Between the equally important and slightly important
1.2 Slightly important
1.3 Between the slightly important and obviously important
1.4 Obviously important
1.5 Between the obviously important and strongly important
1.6 Strongly important
1.7 Between the strongly important and extremely important
1.8 Extremely important
$ r_{i} $ The important degree $ f_{i-1}(x) $ of compared to $ f_{i}(x) $
1.0 Equally important
1.1 Between the equally important and slightly important
1.2 Slightly important
1.3 Between the slightly important and obviously important
1.4 Obviously important
1.5 Between the obviously important and strongly important
1.6 Strongly important
1.7 Between the strongly important and extremely important
1.8 Extremely important
Table 2.  Transportation time and unit transportation cost and other related data
The transportation time and cost Supply
$ D_{1} $ $ D_{2} $ $ D_{3} $ $ s_{i} $
$ S_{1} $ (3, 44) (5, 53) (6, 49) 10
$ S_{2} $ (5, 46) (3, 48) (5, 50) 12
$ S_{3} $ (6, 58) (7, 60) (5, 51) 12
$ S_{4} $ (5, 49) (3, 46) (7, 68) 9
$ S_{5} $ (4, 45) (6, 65) (5, 54) 15
Demand $ d_{j} $ 10 12 10
The transportation time and cost Supply
$ D_{1} $ $ D_{2} $ $ D_{3} $ $ s_{i} $
$ S_{1} $ (3, 44) (5, 53) (6, 49) 10
$ S_{2} $ (5, 46) (3, 48) (5, 50) 12
$ S_{3} $ (6, 58) (7, 60) (5, 51) 12
$ S_{4} $ (5, 49) (3, 46) (7, 68) 9
$ S_{5} $ (4, 45) (6, 65) (5, 54) 15
Demand $ d_{j} $ 10 12 10
Table 3.  The evaluation results of the first group experts
The values of important degree ratio Expert 1 Expert 2 Expert 3 Expert 4
$ r_a^{\left( 4 \right)} $ 1.5 1.6 1.7 1.8
The values of important degree ratio Expert 1 Expert 2 Expert 3 Expert 4
$ r_a^{\left( 4 \right)} $ 1.5 1.6 1.7 1.8
Table 4.  The evaluation results of the second group experts
The values of important degree ratio Expert 1 Expert 2 Expert 3
$ r_a^{\left( 3 \right)} $ 1.4 1.1 1.2
The values of important degree ratio Expert 1 Expert 2 Expert 3
$ r_a^{\left( 3 \right)} $ 1.4 1.1 1.2
Table 5.  The results of fuzzy linear weighting method
Weights $ (w{T},w_{C}) $ Total satisfaction $ \lambda $ Non-inferior solution Function value $ (f_{T},f_{C}) $
(0.55, 0.45) 0.99 $ (X_{ij},Y_{ij},Y_{j}) $ (11, 1976)
Weights $ (w{T},w_{C}) $ Total satisfaction $ \lambda $ Non-inferior solution Function value $ (f_{T},f_{C}) $
(0.55, 0.45) 0.99 $ (X_{ij},Y_{ij},Y_{j}) $ (11, 1976)
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