# American Institute of Mathematical Sciences

## 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:

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##### References:
The relationship between membership function and objective function
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
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
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
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
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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