Application of Homotopy Perturbation Method for Fuzzy Linear Systems and Comparison with Adomian’s Decomposition Method

Abstract

We present an efficient numerical algorithm for solution of the fuzzy linear systems (FLS) based on He’s homotopy perturbation method (HPM). Moreover, the convergence properties of the proposed method have been analyzed and also comparisons are made between Adomian’s decomposition method (ADM) and the proposed method. The results reveal that our method is effective and simple.

1. Introduction

Consider the following linear system: 𝐴𝑥=𝑏, (1) where 𝑥 denotes a vector in a finite-dimensional space and 𝐴∈𝑅^𝑛×𝑛. Linear systems are of fundamental importance in various fields of science and engineering and there are numerous methods to find a solution for them; see [1–8] and the references therein.

When coefficients of the system are imprecise, accessing the solution with no ambiguity will be difficult. Therefore, fuzzy logic was proposed by Zadeh (1965) and encountered seriously over Aristotle classic logic. Professor Zadeh discussed fuzzy set theory, fuzzy number concept, and arithmetical operation with these numbers in [9–12] and following that many articles and books were published in fuzzy systems.

We refer the reader to [13] for more information on fuzzy numbers and fuzzy arithmetic. Fuzzy systems are used to study a variety of problems including fuzzy metric spaces [14], fuzzy differential equations [15], particle physics [16, 17], game theory [18], optimization [19], and especially fuzzy linear systems [20–24]. Fuzzy number arithmetic is widely applied and useful in computation of linear system whose parameters are all or partially represented by fuzzy numbers. Friedman et al. [20] introduced a general model for solving a fuzzy 𝑛×𝑛 linear system whose coefficient matrix is crisp and the right-hand side column is an arbitrary fuzzy number vector. They used the parametric form of fuzzy numbers and replaced the original fuzzy 𝑛×𝑛 linear system by a crisp 2𝑛×2𝑛 linear system. Furthermore, some iterative methods have been presented for solving such fuzzy linear systems [25–32].

In this paper, we solve fuzzy linear systems (FLS) via an analytical method called homotopy perturbation method (HPM). This method was first proposed by the Chinese mathematician He in 1999 [33] and was further developed and improved by him [34–36]. This author presented a homotopy perturbation technique based on the introduction of homotopy in topology coupled with the traditional perturbation method for the solution of algebraic equations. This technique provides a summation of an infinite series with easily computable terms, which converges rapidly to the solution of the problem. In the literature, various authors have successfully applied this method for any kinds of different problems [37–42]. The HPM has some advantages over routine numerical methods. This method does not involve discretization of the variables and hence is free from rounding off errors and does not require large computer memory or time. In this paper we focus on the use of the homotopy perturbation method for solving the FLS.

2. Fuzzy Linear Systems (FLS)

In this section we provide some basic notations and definitions of fuzzy number and fuzzy linear system.

Definition 1. An arbitrary fuzzy number is represented, in parametric form, by an ordered pair of functions (𝑢(𝑟),𝑢(𝑟)), 0≤𝑟≤1, which satisfy the following requirements (see [20]):(i)𝑢(𝑟) is a bounded monotonic increasing left continuous function over [0,1];(ii)𝑢(𝑟) is a bounded monotonic decreasing left continuous function over [0,1];(iii)𝑢(𝑟)≤𝑢(𝑟), 0≤𝑟≤1. A crisp number 𝛼 can be simply expressed as 𝑢(𝑟)=𝑢(𝑟)=𝛼.
The addition and scalar multiplication of fuzzy numbers 𝑥=(𝑥(𝑟),𝑥(𝑟)) and 𝑦=(𝑦(𝑟),𝑦(𝑟)) can be described as follows:(i)𝑥=𝑦 if and only if 𝑥(𝑟)=𝑦(𝑟) and 𝑥(𝑟)=𝑦(𝑟),(ii)𝑥+𝑦=(𝑥(𝑟)+𝑦(𝑟),𝑥(𝑟)+𝑦(𝑟)), (iii)𝐾𝑥=^{(𝐾𝑥,𝐾𝑥),𝐾≥0,}_{(𝐾𝑥,𝐾𝑥),𝐾<0,}𝐾∈𝑅.

Definition 2. Consider the following 𝑛×𝑛 linear system of equations: 𝑎₁₁𝑥₁+𝑎₁₂𝑥₂+⋯+𝑎_1𝑛𝑥_𝑛=𝑏₁,𝑎₂₁𝑥₁+𝑎₂₂𝑥₂+⋯+𝑎_2𝑛𝑥_𝑛=𝑏₂,⋮𝑎_𝑛1𝑥₁+𝑎_𝑛2𝑥₂+⋯+𝑎_𝑛𝑛𝑥_𝑛=𝑏_𝑛, (2) where the coefficient matrix 𝐴=(𝑎_𝑖𝑗), 1≤𝑖, 𝑗≤𝑛, is a crisp matrix and 𝑏_𝑖∈𝐸¹; 1≤𝑖≤𝑛 is called a fuzzy linear system (FLS).

Definition 3. A fuzzy number vector 𝑋=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇, given by parametric form 𝑥_𝑖=(𝑥_𝑖(𝑟),𝑥_𝑖(𝑟)), 1≤𝑖≤𝑛, 0≤𝑟≤1, is called a solution of the fuzzy linear system (2) if ^𝑛_𝑗=1𝑎_𝑖𝑗𝑥_𝑗=^𝑛_𝑗=1𝑎_𝑖𝑗𝑥_𝑗=𝑏_𝑖,_𝑛𝑗−1𝑎_𝑖𝑗𝑥_𝑗=^𝑛_𝑗=1𝑎_𝑖𝑗𝑥_𝑗=𝑏_𝑖. (3) Friedman et al. [20] in order to solve the system given by (3) have solved a 2𝑛×2𝑛 crisp linear system as 𝑆𝑋=𝐵, (4) where 𝑆=(𝑠_𝑖𝑗) are determined as follows: 𝑎_𝑖𝑗≥0⟶𝑠_𝑖𝑗=𝑎_𝑖𝑗,𝑠_{𝑖+𝑛,𝑗+𝑛}=𝑎_𝑖𝑗,𝑎_𝑖𝑗<0⟶𝑠_{𝑖,𝑗+𝑛}=−𝑎_𝑖𝑗,𝑠_{𝑖+𝑛,𝑗}=−𝑎_𝑖𝑗. (5) And the others are zero. Then refering to [20] we have 𝑠𝑆=¹𝑠₂𝑠₂𝑠₁𝑥,𝑋=−𝑥𝑏,𝐵=−𝑏⟹𝑠₁𝑥−𝑠₂𝑥=𝑏,𝑠₂𝑥−𝑠₁𝑥=−𝑏. (6) Or 𝑠𝑆=¹−𝑠₂−𝑠₂𝑠₁𝑥,𝑋=𝑥𝑏,𝐵=𝑏⟹𝑠₁𝑥−𝑠₂𝑥=𝑏,−𝑠₂𝑥+𝑠₁𝑥=𝑏, (7) where 𝑠₁, 𝑠₂≥0 and 𝐴=𝑠₁−𝑠₂.

3. Homotopy Perturbation Method for Linear Systems

Consider (1), where 𝐴=𝑎_𝑖𝑗,𝑏=𝑏_𝑖,𝑥=𝑥_𝑗,𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑛. (8) Let also 𝐿(𝑢)=𝐴𝑢−𝑏 and 𝐹(𝑢)=𝑢−𝑤₀, where 𝑤₀ is a known vector. Then we define homotopy 𝐻(𝑢,𝑝) as follows: 𝐻𝑢,𝑝=1−𝑝𝐹(𝑢)+𝛼𝑝𝐿(𝑢)=0, (9) where 𝑝∈[0,1] is an embedding parameter and 𝛼 is nonzero auxiliary parameter.

Obviously, we will have 𝐻()(𝑢),𝐻()(𝑢)𝑢,0=𝐹𝑢,1=𝛼𝐿=0−−−−→^𝛼≠0𝐻(𝑢,1)=𝐿(𝑢). (10) According to the HPM, we can first use the embedding parameter 𝑝 as a small parameter and assume that the solution of (1) can be written as a power series in 𝑝: 𝑢=𝑢₀+𝑝𝑢₁+𝑝²𝑢₂+⋯, (11) and the exact solution is obtained as follows: 𝑥=lim_𝑝→1𝑢=lim_𝑝→1𝑢₀+𝑝𝑢₁+𝑝²𝑢₂+𝑝³𝑢₃=+⋯_∞𝑗=0𝑢_𝑗. (12) Putting (11) into (9) and comparing the coefficients of identical degrees of 𝑝 on both sides, we find 𝑝⁰∶𝑢₀=𝑤₀,𝑝¹∶(𝛼𝐴−𝐼)𝑢₀+𝑢₁−𝑤₀𝑢−𝛼𝑏=0,₁=𝛼𝑏−(𝛼𝐴−𝐼)𝑢₀+𝑤₀,𝑝²∶(𝛼𝐴−𝐼)𝑢₁+𝑢₂=0,𝑢₂=−(𝛼𝐴−𝐼)𝑢₁⋮ (13) And in general, 𝑢_𝑛+1=−(𝛼𝐴−𝐼)𝑢_𝑛,𝑛=1,2,…. (14) Taking 𝑢₀=𝑤₀=0 yields. 𝑢₁𝑢=𝛼𝑏,₂=−(𝛼𝐴−𝐼)𝑢₁(),𝑢=−𝛼𝐴−𝐼)(𝛼𝑏₃=−(𝛼𝐴−𝐼)𝑢₂=(𝛼𝐴−𝐼)²()⋮𝑢𝛼𝑏_𝑛+1=−(𝛼𝐴−𝐼)𝑢_𝑛=(−1)^𝑛(𝛼𝐴−𝐼)^𝑛(𝛼𝑏). (15) Therefore the solution can be in the following form: 𝑥=^∞_𝑖=0(−1)^𝑖(𝛼𝐴−𝐼)^𝑖(𝛼𝑏). (16) The convergence of the series equation (16) when 𝛼=1 is proved for diagonally dominant matrix 𝐴 in [41].

4. HPM for the Fuzzy Linear Systems

Asady and Mansouri in [29] proposed a method for finding a fuzzy solution of 𝑛×𝑛 FLS without increasing the order. In this method the original system with a matrix 𝐴 is replaced by two 𝑛×𝑛 crisp linear systems. Furthermore, improvement of computing time compared with other methods by using this method is shown. This method is given by the following.

By noting the structure of 𝑆 we have the following linear system: 𝑠₁𝑥+𝑠₂(−𝑥)=𝑏,𝑠₂𝑥+𝑠₁(−𝑥)=−𝑏. (17) Therefore, by adding and then subtracting the two parts of (17) we obtain the following: 𝐴𝛿=𝑏₁,𝐸𝜎=𝑏₂, (18) where 𝐴=𝑠₁−𝑠₂,𝐸=𝑠₁+𝑠₂,𝛿=𝑥−𝑥,𝜎=𝑥+𝑥,𝑏₁=𝑏+𝑏,𝑏₂=𝑏−𝑏. (19) Thus, by adding and subtracting the two solutions of systems equation (18) we have the following: 𝑥₋=𝛿−𝜎2,𝑥=𝛿+𝜎2. (20) Now, we solve the FLS using HPM.

By using the above model, HPM method for FLS is as follows: 𝛿=^∞_𝑖=0(−1)^𝑖𝛼𝐷⁻¹𝐴−𝐼^𝑖𝛼𝐷⁻¹𝑏₁,𝜎=^∞_𝑖=0(−1)^𝑖𝛼𝐷⁻¹𝐴−𝐼^𝑖𝛼𝐷⁻¹𝑏₂, (21) where 𝐷 is the diagonal part of 𝐴=𝐷−𝐿−𝑈 and −𝐿, −𝑈 are, respectively, strictly lower and upper triangular matrices of 𝐴.

So as iterative method we have Algorithm 1.

Given a stopping criterion with tolerance 𝜀>0,    for 𝑛=1,2,…, until 𝛿𝑛+1−𝛿𝑛𝜎≤𝜀,𝑛+1−𝜎𝑛≤𝜀, do;        𝛿𝑛+1=∑𝑛𝑖=0(−1)𝑖𝛼𝐷−1𝐴−𝐼𝑖𝛼𝐷−1𝑏1,        𝜎𝑛+1=∑𝑛𝑖=0(−1)𝑖𝛼𝐷−1𝐴−𝐼𝑖𝛼𝐷−1𝑏2,          Obtain 𝑥−=𝛿𝑛+1−𝜎𝑛+12,𝛿𝑥=𝑛+1+𝜎𝑛+12.    End for.

Algorithm 1 

HPM method for FLS.

Next, we will present the convergence analysis of this method.

Definition 4 (see [1, 4, 6–8]). (a) A matrix 𝐴=𝑎_𝑖𝑗 is called a 𝑍-matrix if for any 𝑖≠𝑗, 𝑎_𝑖𝑗≤0.
(b) A 𝑍-matrix is a nonsingular 𝑀-matrix, if 𝐴 is nonsingular and if 𝐴⁻¹≥0.
(c) For any matrix 𝐴 the comparison matrix ⟨𝐴⟩=(𝑚_𝑖𝑗)∈𝑅^𝑛×𝑛 is defined by 𝑚_𝑖𝑖=𝑎_𝑖𝑖,𝑚_𝑖𝑗=−𝑎_𝑖𝑗⋅|.,𝑖≠𝑗,1≤𝑖,𝑗≤𝑛|issymbolofabsolutevalue (22)
(d) Matrix 𝐴 is an 𝐻-matrix if and only if ⟨𝐴⟩ is a nonsingular 𝑀-matrix.

Definition 5 (see [1, 4, 6–8]). Let 𝐴 be a real matrix. The splitting 𝐴=𝑀−𝑁 is called;(a)convergent if 𝜌(𝑀⁻¹𝑁)<1;(b)𝑀-splitting if 𝑀 is a nonsingular 𝑀-matrix and 𝑁≥0.

Lemma 6 (see [1]). Let 𝐴=𝑀−𝑁 be an 𝑀-splitting of 𝐴. Then 𝜌(𝑀⁻¹𝑁)<1 if and only if 𝐴 is a nonsingular 𝑀-matrix.

Lemma 7 (see [43]). If 𝐴 is an 𝐻-matrix, then |𝐴⁻¹|≤⟨𝐴⟩⁻¹.

Theorem 8. The series 𝑥=∑^∞_𝑖=0(−1)^𝑖((𝛼𝐷⁻¹𝐴)−𝐼)^𝑖(𝛼𝐷⁻¹𝑏) in HPM method for solving linear system convergence if  0<𝛼≤1 and 𝐴 is an 𝐻-matrix.

Proof. It is well known that the series convergence if and only if 𝜌(𝛼𝐷⁻¹𝐴−𝐼)<1.
Furthermore, we have 𝜌𝛼𝐷⁻¹𝐴−𝐼=𝜌𝛼𝐷⁻¹𝐴−𝛼⁻¹𝐷=𝜌𝛼𝐷⁻¹𝛼−1𝛼𝐷−(𝐿+𝑈). (23) Since 0<𝛼≤1 and 𝐴=𝐷−𝐿−𝑈 is an 𝐻-matrix, we see that 𝐷 is also an 𝐻-matrix, ⟨𝐴⟩=(1/𝛼)|𝐷|_𝑀−(((1−𝛼)/𝛼)|𝐷|+|𝐿|+|𝑈|)_𝑁 is an 𝑀-matrix, and ⟨𝐴⟩=𝑀−𝑁 is 𝑀-splitting; by Definition 5 and Lemma 6 we have 𝜌𝛼|𝐷|⁻¹1−𝛼𝛼|𝐷|+(|𝐿|+|𝑈|)<1. (24) Furthermore, from Lemma 7, 𝐷⁻¹≤|𝐷|⁻¹,𝛼𝐷⁻¹1−𝛼𝛼𝐷⁻¹𝐷+𝐿+𝑈≤𝛼⁻¹1−𝛼𝛼𝐷⁻¹|𝐷|+𝐿+𝑈≤𝛼⁻¹1−𝛼𝛼|𝐷|⁻¹+|𝐿|+|𝑈|. (25) Therefore, 𝜌𝛼𝐷⁻¹𝛼−1𝛼𝐷−(𝐿+𝑈)≤𝜌−𝛼𝐷⁻¹1−𝛼𝛼𝛼|𝐷|𝐷+𝐿+𝑈≤𝜌⁻¹1−𝛼𝛼|𝐷|+(|𝐿|+|𝑈|)<1. (26) And the proof is completed.

Theorem 9. Algorithm 1 for solving the fuzzy linear system converges if  0<𝛼≤1 and 𝐴 is an 𝐻-matrix.

Proof. We need to show that max𝜌𝛼𝐷⁻¹𝐴−𝐼,𝜌𝛼𝐷⁻¹𝐴−𝐼<1. (27) Also it is easy to see that if 𝐴 is an 𝐻-matrix, then |𝐴|  is and 𝐻-matrix too.
Therefore by Theorem 8 the proof is completed.

5. Comparison Results

Adomian’s decomposition method (ADM) was first introduced by G. Adomian in the beginning of 1980s [44, 45] and has been rapidly growing in recent years. In this method the solution of a functional equation is considered as the sum of an infinite series usually converging to an accurate solution. Allahviranloo in [26] applied the Adomian decomposition method to solve the fuzzy linear systems by model of (6) and has shown that this method is equivalent to the Jacobi iterative method. But since Flop count and the computing time in model (18) are shorter than model (6) we present ADM for FLS based on model (18). This algorithm is as Algorithm 2.

Given𝛿0=𝐷𝐴−1𝑏1,𝜎0=𝐷𝐸−1𝑏2, and the stopping criterion with tolerance 𝜀>0,   for 𝑛=1,2,…, until 𝛿𝑛+1−𝛿𝑛𝜎≤𝜀,𝑛+1−𝜎𝑛≤𝜀,  do       𝛿𝑛+1=𝐷𝐴−1𝐿𝐴+𝑈𝐴𝛿𝑛+𝐷𝐴−1𝑏1,       𝜎𝑛+1=𝐷𝐸−1𝐿𝐸+𝑈𝐸𝜎𝑛+𝐷𝐸−1𝑏2,          Obtain 𝑥−=𝛿𝑛+1−𝜎𝑛+12,𝛿𝑥=𝑛+1+𝜎𝑛+12.   End for.

Algorithm 2 

ADM method for FLS.

Now, to show the ability of these algorithms we test the following FLS, using Matlab software.

Example 10. Consider the following 𝑛×𝑛 fuzzy system 2𝑥₁−𝑥₃=(),𝑥2+𝑟,4−𝑟₂+2𝑥₃−𝑥₄=(),𝑥2+𝑟,4−𝑟₃+2𝑥₄−𝑥₅=()⋮𝑥2+𝑟,4−𝑟_𝑛−3+2𝑥_𝑛−2=(),𝑥2+𝑟,4−𝑟_𝑛−2+2𝑥_𝑛−1=(),𝑥2+𝑟,4−𝑟_𝑛−1+2𝑥_𝑛−𝑥₁=(2+𝑟,4−𝑟). (28) The tables show the numerical results of the above example. In Table 1, we reported the number of iterations (ITER) for the HPM and ADM methods with different 𝑛. Also 𝛼=0.9.

Table 1 

Number of iterations of HPM and ADM.

From Table 1, we can see that the number of iterations in HPM algorithm is superior to the number of iterations in ADM algorithm.

Tables 2 and 3 also show comparison of accumulated errors for system (28) obtained using HPM and ADM.

Table 2 

Accumulated error for ‖𝑥₋‖ and 𝑛=100.

Table 3 

Accumulated error for ‖𝑥‖ and 𝑛=100.

From the above tables, we can see that the results of the HPM are close to the exact solution in comparison with ADM which confirm the validity of our method.

6. Conclusions

The homotopy perturbation method (HPM) is a powerful tool which is capable of handling linear/nonlinear equations. In this paper, we proposed homotopy perturbation method and compared it with Adomian’s decomposition method (ADM) for solving fuzzy linear systems (FLS). This method does not require parameter in any equation, the same as the perturbation approach, and also is very simple and straightforward. The numerical results show that the results of the present method are in excellent agreement with those of exact solution and reductions in the size of calculations compared with the ADM.

Acknowledgments

The authors would like to thank all the three reviewers for their valuable suggestions which have led to an improvement in both the quality and clarity of this paper.