Recently, various methods have been developed for solving linear programming problems with fuzzy number, such as simplex method and dual simplex method. But their computational complexities are exponential, which is not satisfactory for solving large-scale fuzzy linear programming problems, especially in the engineering field. A new method which can solve large-scale fuzzy number linear programming problems is presented in this paper, which is named a revised interior point method. Its idea is similar to that of interior point method used for solving linear programming problems in crisp environment before, but its feasible direction and step size are chosen by using trapezoidal fuzzy numbers, linear ranking function, fuzzy vector, and their operations, and its end condition is involved in linear ranking function. Their correctness and rationality are proved. Moreover, choice of the initial interior point and some factors influencing the results of this method are also discussed and analyzed. The result of algorithm analysis and example study that shows proper safety factor parameter, accuracy parameter, and initial interior point of this method may reduce iterations and they can be selected easily according to the actual needs. Finally, the method proposed in this paper is an alternative method for solving fuzzy number linear programming problems.
Linear programming is one of the most widely used decision-making tools for solving real-word problems. However, real word situations are characterized by imprecision rather than exactness. Then, fuzzy linear programming (FLP) has been developed to treat uncertainty of optimization problems, such as fuzzy data envelopment analysis and fuzzy network optimization [
Then, many researchers have considered various kinds of FLP problems and have proposed some approaches for solving these problems [
Sensitivity analysis is a basic tool for studying perturbations in optimization problems. There is considerable research on sensitivity analysis for some models of operations research and management science such as linear programming and investment analysis. So, many scholars studied the sensitivity analysis for FVLP [
In a word, existing methods solving FNLP problems are mainly using the concept of comparison of fuzzy numbers and linear ranking function to change the fuzzy number into crisp number, using simplex method and its revised method to solve these FNLP problems. Because the time complexity of simplex methods [
This paper is organized as follows. We demonstrate some preliminaries of fuzzy set theory and the concept of ranking functions in Section
In this section, we review some necessary concepts of fuzzy set theory and the ranking function and then present some definition about fuzzy vectors.
A convex fuzzy set Its membership function is piecewise continuous. There exist three intervals
Let
We denote the set of all trapezoidal fuzzy numbers by
If for any for any
The function
If
If a ranking function
The forms of linear ranking functions on
For any trapezoidal fuzzy number
For any two trapezoidal fuzzy numbers
A
Let
It is quite easy to get the following rules: commutativity: associativity: neutral Element:
Let
It is quite easy to get the following rules: distributivity over fuzzy vectors: distributivity over number:
Let
It is quite easy to obtain
Let
In this section, we recall the definition of FNLP and the fuzzy primal simplex algorithm to FNLP.
An
There is another equivalent form of (
The fuzzy primal simplex algorithm.
The basic feasible solution is given by Let If Pivot on
The idea of this algorithm is to start from a vertex; each step of its iteration is moving to a better vertex until the optimal solution is found or infeasible solution is proved.
In Algorithm
Fortunately, we know that Karmarkar’s interior point method [
In this section, we propose a revised interior-point method to solve FNLP problem.
The basic idea of revised interior point is first starting from an interior point
Combined with the slack variable
In the
Let
From the idea of revised interior point method and the derivation of calculation formula, steps of the revised interior point algorithm to solve model (
Give an initial interior point
Compute
Set the diagonal matrix
Using the vector multiplication of fuzzy vectors (
Compute the vector
Let
Compute the next point:
Using the vector multiplication of fuzzy vectors (
Generally, set
And if
If
If there is the optimal solution of problem (
In this section, first we analyze the algorithm. Then, an example in the practical production is given. At last, we analyze some factors influencing the results of this method through the given example.
The time complexity of simplex methods [ Iterations are rising rapidly as the number of planning variables and constraints increasing. The simplex method is terminated in optimal basis of original and dual programs. Although it has reached optimal solution in the degenerate case, it often needs to iterate the basis many times in order to prove that it is optimal.
As we know, interior point methods (IPMs) are the most effective methods for solving a large-scale linear optimization problem. Since the creative work of Karmarkar [
The relationship among product demand, production capacity, and pure profit.
Product | Daily demand | Manual system | Machine system | ||||
---|---|---|---|---|---|---|---|
Production capacity | Pure profit | Production capacity | Pure profit | ||||
Shift 1 | Shift 2 | Shift 1 | Shift 2 | ||||
1 | 5 | 3 |
|
|
4 |
|
|
2 | 10 | 5 |
|
|
7.5 |
|
|
In Table
In Table
: the output of product 1 in shift 1 produced by manual system; : the output of product 1 in shift 2 produced by manual system; : the output of product 2 in shift 1 produced by manual system; : the output of product 2 in shift 2 produced by manual system; : the output of product 1 in shift 1 produced by machine system; : the output of product 1 in shift 2 produced by machine system; : the output of product 2 in shift 1 produced by machine system; : the output of product 2 in shift 2 produced by machine system.
(ii) Now an FNLP model is established as follows:
These inequalities
(iii) Then, solve the FNLP problem (
Given
Compute
Set the diagonal matrix
Compute
Compute the vector
Let
Compute
Compute
Above all, the number of iteration is 5 and the results are listed in Table
The interior point value and the corresponding objective function value of each iteration.
Interior point value | Objective function value | |
---|---|---|
1 |
|
(81.9560 |
2 |
|
(82.4542 |
3 |
|
(82.6160 |
4 |
|
(82.6559 |
5 |
|
(82.6633 |
The optimal solution is
Factors influencing the results of this method are mainly safety factor parameter Table Table Table Table
The influence of the safety factor parameter
Parameter |
0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 0.95 |
---|---|---|---|---|---|---|---|---|---|---|
Iterations |
75 | 40 | 27 | 20 | 16 | 13 | 11 | 10 | 9 | 9 |
The influence of the accuracy parameter
Parameter |
0.9 | 0.5 | 0.1 | 0.01 | 0.005 | 0.001 |
---|---|---|---|---|---|---|
Iterations |
9 | 10 | 13 | 16 | 17 | 19 |
The influence of the initial interior point
Initial interior point | Iterations |
Initial interior point | Iterations |
---|---|---|---|
|
33 |
|
16 |
|
31 |
|
15 |
|
26 |
|
14 |
|
21 |
|
13 |
|
20 |
|
13 |
|
19 |
|
12 |
|
18 |
|
12 |
|
17 |
|
11 |
|
16 |
The influence of the accuracy parameter
Parameter |
0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 0.95 | |
---|---|---|---|---|---|---|---|---|---|---|---|
Iterations |
|
52 | 29 | 20 | 16 | 13 | 11 | 9 | 8 | 8 | 7 |
|
75 | 40 | 27 | 20 | 16 | 13 | 11 | 10 | 9 | 9 | |
|
97 | 50 | 33 | 25 | 19 | 16 | 13 | 12 | 11 | 11 |
A new interior point method is presented to solve FNLP problems using linear ranking function in this paper. Compared with simplex method or revised simplex algorithm, this method is more outstanding in solving the large scale of the FNLP problem, for it has a polynomial time complexity. And some factors influencing the results of this method are analyzed. The result shows that proper safety factor parameter, accuracy parameter, and initial interior point of this method may reduce iterations and they can be selected easily according to the actual needs. Although a general method to select the initial point has been given in this paper, it is not feasible in some cases. For example, under the condition
The authors thank the anonymous referees for their suggestions and comments to improve an earlier version of this paper. The authors are also grateful for the financial support by Scientific Research Fund of Sichuan Provincial Education Department (no. 11ZA024) and Science Foundation of Southwest Petroleum University of China (no. 2012XJZ031).