Finding solutions of fuzzy polynomial equations systems by an Algebraic method

1 Introduction

Solving fuzzy polynomial equations is one of the most critical problems in the field of uncertainty modelling as it has wide applications in science, engineering, and economics [6].

So far, two different categories of the methods have been developed for solving fuzzy polynomial equations, namely, approximate and exact computations. In the first category, several authors applied Newton’s method and some of its extensions such as the neural nets, evolutionary algorithms, and a few other iterative methods in order to solve fuzzy polynomial equations systems [1, 27]. In this category, many techniques have been proposed to solve fuzzy linear systems [17, 30]. The results of such approaches are difficult to evaluate.

In most cases, solving non-linear systems is fundamental. Although the linearisation can be done, it is only valid for a small region around the linearisation point. The non-linear fuzzy equations of multivariable polynomial systems are important in various areas [5]. Abbasbandy et al. used Newton’s method to solve such systems 1, 2. In 2006, a learning algorithm of crisp weights of a three-layer feed forward fuzzy neural network was introduced to find one of the roots of a fuzzy multivariable polynomial equation 4. A system of fuzzy polynomial equations with real coefficients and fuzzy variables is considered in 5 and only one real root of the system is found by a learning algorithm. In the most of these approaches, choosing a suitable initial point is needed. Also, the existence of a solution for the fuzzy systems is not determined and all solutions of the system cannot be obtained at the same time.

To overcome the challenges mentioned above, another category of methods based on symbolic computation techniques has been developed recently [16, 26]. For instance, in [26], Abbasi Molai et al. used Gröbner bases to solve fuzzy polynomial equations systems. In their approach, a parametric form for the system of fuzzy polynomial equations with respect to parameter of r is obtained. Then, parameter r is considered as a variable and the variety of the crisp form system is computed using Gröbner bases. Lastly, the solutions of the fuzzy system are obtained by isolating the family of elements of the variety where r is arbitrary in [0,1]. Therefore, one might need a quantifier elimination routine to eliminate the extra variable r.

In this paper, we propose a new approach to solving the fuzzy polynomial equations systems. The variety of our crisp system is the set of sought solutions of the fuzzy polynomial equations system. Wu’s algorithm can be then used to solve real polynomial equations systems [32]. In some cases, as in [10, 24], this algorithm is more efficient than the Gröbner bases approach for solving real polynomial equations systems. Characteristic sets of Wu’s algorithm was first introduced by Ritt [28]. Since 1980, Wu Wen-Tsun has considerably improved Ritt’s theory and the method of characteristic sets by removing irreducibility requirements. It has developed efficient algorithms for zero decomposition of arbitrary polynomial systems [31, 33]. Thereafter, Ritt-Wu’s method has been improved and extended by many researchers and successfully was applied to many problems in science and engineering [35]. Some attempts have also been made in order to speed up the computation of characteristic sets [9, 11]. The lasted was presented by Jin Meng et al. [24].

As the characteristic sets have a triangular structure, finding the variety of these sets can be done simply by a forward substitution. In this manner, we try to make a bridge between the solutions of the fuzzy polynomial equations systems and the varieties of the characteristic sets. For this purpose, our focus is on finding the real solutions of the polynomial equations system in n variables{ f 1 ( x 1 , x 2 , … , x n ) = b ˜ 1 ⋮ f s ( x 1 , x 2 , … , x n ) = b ˜ s(1)where x₁, x₂, …, x _n are real variables and all coefficients and right hand values b ˜ 1 , b ˜ 2 , … , b ˜ s are fuzzy numbers. The main idea of the proposed approach is based on converting fuzzy system (1) into a crisp system and getting a polynomial system of 2s equations. This system is converted into a system with one less variable which is called a collected crisp system. Then, by using Wu’s algorithm, the variety of the collected crisp system is found. This variety is composed of the all the solutions of the fuzzy system.

The paper is organized as follows. In Section 2, some basic definitions of fuzzy set theory and fuzzy polynomial systems are recalled. Then Wu’s algorithm is introduced in Section 3. Section 4 focuses on finding real roots of univariate polynomials. The main algorithm to find all solutions of the system of fuzzy polynomials is presented in Section 5. Finally, the proposed method is illustrated by solving some examples in Section 6.

In this section, some necessary definitions and notations of fuzzy set theory are reviewed. Let A ˜ be a subset of universal set X.

Definition 2.1. [22] (Membership function of a fuzzy set) In a fuzzy set each element is mapped to [0, 1] by a membership function μ A ˜ : X ⟶ [ 0 , 1 ] where [0, 1] means real numbers between 0 and 1 (including 0, 1).

Fuzzy sets are a generalization of classical sets. Classical bivalent sets are usually called crisp sets in fuzzy set theory. By Definition 2.1, a fuzzy set is defined by its membership function. The membership function μ A ˜ of fuzzy set A ˜ is said normalized if there exists an element x ∈ X such that μ A ˜ ( x ) = 1. For each real number r in [0, 1], the r-cut set A ˜ r of fuzzy set A ˜ is defined by A ˜ r = { x ∈ X ∣ μ A ˜ ( x ) ≥ r }. Now define the universal set X in the m-dimensional Euclidean Vector space ℝ m. If for each r ∈ [0, 1] the r-cut sets A ˜ r is convex, the fuzzy set A ˜ in origin of these r-cut sets is said convex.

Definition 2.2. [22] A convex, normalized fuzzy set whose membership function defined in ℝ is piecewise continuous is called a fuzzy number.

Note that the r-cuts of a fuzzy number are closed and bounded intervals. For arbitrary fuzzy numbers A ˜ , B ˜ the sum operation is defined by μ A ˜ + B ˜ ( z ) = max z = x + y min ( μ A ˜ ( x ) , μ B ˜ ( y ) ) .

An equivalent definition of fuzzy numbers is also given in [19] as follows:

Definition 2.3. A fuzzy number A ˜ in parametric form is an ordered pair [ A _ , A ¯ ] of functions A _ ( r ) and A ¯ ( r ), r ∈ [0, 1], which satisfy the followingconditions:

A _ ( r ) is a bounded left continuous non decreasing function on [0, 1],

This parametric form of fuzzy numbers so described is accompanied by the fuzzy arithmetic described below. For two arbitrary fuzzy numbers A ˜ , B ˜ given in the parametric form and a real number k, we have:

A ˜ = B ˜ if and only if A _ ( r ) = B _ ( r ) and A ¯ ( r ) = B ¯ ( r ), for each r ∈ [0, 1],

Let m , α , β ∈ ℝ and α, β ≥ 0. The fuzzy number A ˜ with membership function μ A ˜ ( x ) = { x - m α + 1 m - α ≤ x ≤ m m - x β + 1 m ≤ x ≤ m + β 0 otherwise is called a triangular fuzzy number [5]. We represent this fuzzy number by the triplet (m, α, β). The shape of this fuzzy number is shown in Fig. 1. Triangular fuzzy numbers are the most popular fuzzy numbers. These numbers have linear membership functions and their addition is also a triangular fuzzy number. The parametric form of the triangular fuzzy number A ˜ = ( m , α , β ) is given by its r-cut sets: A _ ( r ) = m + α ( r - 1 ) , A ¯ ( r ) = m + β ( 1 - r ) .

In Section 5, we will obtain solutions of some fuzzy polynomial equations systems as the system (1) where coefficients are triangular fuzzy numbers.

Definition 2.4. A vector ( a 1 , a 2 , … , a n ) ∈ ℝ n is called a solution of system (1) if f l ( a 1 , a 2 , … , a n ) = b ˜ l for all 1 ≤ l ≤ s.

3 Wu’s algorithm

In this section, we introduce some notations and give a brief presentation of Wu’s algorithm [20]. Let R = 𝕂 [ x 1 , … , x n ] be the polynomial ring in n variables x₁, …, x _n over a field 𝕂 of characteristic zero. Without loss of generality we may assume that the variables are ordered as follows x₁ < x₂ < ⋯ < x _n . If the variable x _m is selected, then the polynomial f ∈ R can be written as f = I t x m t + I t - 1 x m t - 1 + ⋯ + I 0 where t is the degree of f with respect to x _m , denoted by deg  _{x m}  (f), and I i ∈ 𝕂 [ x 1 , … , x m - 1 , x m + 1 , … , x n ] for 0 ≤ i ≤ t. We denote by lc (f, x _m ) the coefficient I _t of x m t. The class class (f) of f is defined as the greatest subscript c of x appearing in f. The class of a constant polynomial is defined to be zero. The variable x _c and the polynomial lc (f, x _c ) are respectively called the leading variable lv (f) of f and the initial of f, denoted by initial (f). A polynomial g ∈ R is said to be reduced with respect to a non constant polynomial f if deg  _{x c}  (g) < deg  _{x c}  (f) where c = class (f) ≠0. The polynomial g is reduced with respect to F ⊂ R if g is reduced with respect to any f ∈ F. Now define a partial order on polynomials. Let f , g ∈ R. We say that a polynomial g has a higher rank than f, denoted by f < g, if one of the following two conditions holds:

class (f) < class (g)

If class (f) = class (g) = c and deg  _{x c}  (f) = deg  _{x c}  (g) or both polynomials are constant, then f and g are said equivalent and it is denoted by f ∼ g. An ordered polynomial set F = {f₁, …, f _r } is said triangular if either r = 1 or class (f₁) < ⋯ < class (f _r ). The triangular set F is called an ascending set if each f _j is reduced with respect to each f _i for i < j. Now we extend the partial order on polynomials to provide a partial order for ascending sets. Let F = {f₁, …, f _r } and G = {g₁, …, g _k } be two ascending sets. We say that F has a lower rank than G, denoted by F < G, if one of the following two conditions holds:

There exists j ≤ min {r, k} such that f _j  < g _j and f _i  ∼ g _i for each i < j

For incomparable ascending sets we write F ∼ G.

Lemma 3.1. A sequence of ascending sets steadily lower in ordering is finite.

Proof. See [34], Lemma 2.1. □

By Lemma 3.1 ascending sets of lowest rank consisting of polynomials chosen from F exists and is called a basic set of F. It is obvious that two basic sets of a polynomial set have the same cardinality. By proof of Theorem 4.11 in [15], we can give an algorithm for computing a basic set of a polynomial set.

Algorithm 3.2. (Basic Set Algorithm)

Input: F ⊂ R, a non-empty polynomial set

Output: B, a basic set of F

B : =∅

Lemma 3.3. Let B be a basic set of a polynomial set F. If g ∈ R is reduced with respect to F, then a basic set of F ∪ {g} has a lower rank than F.

Proof. See [34], Lemma 2.2.□

Now introduce the interesting division algorithm for multivariable polynomials known as the pseudo-division and used in the Wu’s algorithm.

Proposition 3.4. Let f , g ∈ R and c = class (f). Then there exists an equation I t m g = qf + r , where q , r ∈ R, I _t  = initial (f), m ≥ 0, and either r = 0 or r is reduced with respect to f. In particular, q and r are unique if m is minimal.

Proof. see [12], Section 6.5, Proposition 1 andExercise 3.□

The polynomials q and r in Proposition 3.4 are respectively the pseudo-quotient and the pseudo-remainder prem (g, f) of g on its pseudo-division by f (not unique when m is not minimal). The following algorithm computes the pseudo-remainder prem (g, f) and the pseudo-quotient:

Algorithm 3.5. [12] (Pseudo-division Algorithm)

Input: g , f ∈ R

Output: r, q pseudo-remainder and pseudo-quotient of g on its pseudo-division by f

r : = g, q : =0

Given an ascending set F = {f₁, …, f _r } and g ∈ R. By the following successive pseudo-divisions R = prem ( ⋯ prem ( prem ( g , f r ) , f r - 1 ) , … , f 1 ) , we get the following remainder formulaI 1 s 1 I 2 s 2 ⋯ I r s r g = ∑ q i f i + R(2)where I _i  = initial (f _i ), s _i  ≥ 0, q i ∈ R and R is reduced with respect to all f _j of F. By Proposition 3.4, if smaller s _i are selected, the remainder R is unique and denoted by prem (g, F). For a finite subset G from R we put prem ( G , F ) = { prem ( g , F ) ∣ g ∈ G } .

We denote by the ideal generated by F ⊂ R. The set V ( F ) = { ( a 1 , … , a n ) ∈ 𝕂 n ∣ f ( a 1 , … , a n ) = 0 , ∀ f ∈ F } is the variety defined by F. For a polynomial set G ⊂ R,we define V (F/G) = V (F) ∖ V (G), called a quasi-algebraic variety.

Definition 3.6. An ascending set B in R is called a characteristic set of a non-empty polynomial set F ⊂ R if and prem (F, B) = {0}.

According to procedure expressed in [34] Section 2.2, the following algorithm computes a characteristic set.

Algorithm 3.7. (Characteristic Set Algorithm)

Input: F ⊂ R, a non-empty set

Output: B, a characteristic set of F

S : = F

The main properties of characteristic sets are summarized in the following theorem.

Theorem 3.8. [31] (Wu’s Well-ordering Principle) Let B be a characteristic set of F ⊂ R. Then V ( F ) = V ( B / I B ) ⋃ ∪ b ∈ B V ( F ∪ B ∪ { initial ( b ) } ) where I B = ∏ b ∈ B initial ( b ).

The following corollary about the computation of varieties is the main key in Wu’s Algorithm.

Corollary 3.9. By repeat the Wu’s Well-ordering Principle Theorem, for each F ∪ B ∪ {initial (b)}, b ∈ B, the procedure will end in a finite number of steps. Therefore, V (F) can be obtained as union of some finite numbers of varieties V (B/I _B ).

Proof. According to Lemmas 3.1 and 3.3, the corollary is proved.□

Wu’s algorithm is presented to give all characteristic sets that are required to calculate the variety V (F) by application of Corollary 3.9.

Algorithm 3.10. [32] (Wu’s Algorithm)

Input: F ⊂ R, a non-empty set

Output: Z, a set of characteristic sets such that V (F)  =  ⋃ _B∈ZV (B/J), where J is the product of the initials of polynomials in corresponding B.

Z : = ∅ ,  D : = {F}

By Wu’s algorithm, we can write the variety V (F) as a union of quasi-algebraic varieties of characteristic sets. Therefore, we can find V (F) easily because these sets are easy to solve.

Example 3.11. Apply Wu’s algorithm to F = {xy  +  x + y, xy²  +  x + y} with y < x. Put F′ : = F, then D =∅.First use Characteristic Set algorithm. A basic set of F′ is B = {xy + x + y} and R = y² - y³ is the pseudo-remainder of xy² + x + y with respect to B. By adding R to F′ the new basic set B = {y³ - y², xy + x + y} obtained is a characteristic set. Thus Z : = {B}. We have initial (xy + x + y) = y + 1 and initial (y² - y³) =1, therefore, D : = {F′ ∪ {y + 1}} because B is already added to F′. Now put F′ : = {xy + x + y, xy² + x + y, y + 1}. A basic set of F′ is B = {y + 1}. The pseudo-remainder of xy + x + y with respect to B is 1. Thus, {1} is a characteristic set of F′ and D =∅. Therefore, the output is Z = {{y³ - y², xy + x + y}} and V ( F ) = V ( { y 3 - y 2 , xy + x + y } ) ∖ V ( y + 1 ) = { ( x = 0 , y = 0 ) , ( x = - 1 2 , y = 1 ) } .

As mentioned in the previous section, solving real multivariate polynomial equations systems leads to solving univariate polynomials by Wu’s Well-ordering Principle Theorem. This section presents a summary of the number and isolation of real solutions of univariate polynomials. Let f ( x ) = c 0 x a 0 + c 1 x a 1 + ⋯ + c m x a m be a univariate polynomial where c _i  ≠ 0 are real numbers and a₀ < a₁ < ⋯ < a _m . By fundamental theorem of algebra, the polynomial f has a _m roots, counted with their multiplicities, in the complex field. Descartes’s rule of signs [14] says that the number of positive real roots of f, counted with their multiplicities, is at most equal to the cardinality of the set F = { i ∣ 1 ≤ i ≤ m and c i - 1 . c i < 0 } , and the parity rule says that the difference between the cardinality of F and the number of positive roots is even. We use the notation var (f) for the cardinality of set F associated to polynomial f. If we replace x by -x in f (x), Descartes’s rule gives a bound on the number of negative real roots.

Example 4.1. Let f (x) = -2 + 4x + x² - 3x⁵. The number N of positive real roots is at most var (f) =2. Actually, by the parity rule, there won’t be just one positive root. Then N is 0 or 2. Change x by -x and put g (x) : = f (- x) = -2 - 4x + x² + 3x⁵. The number of positive roots of g is at most var (g) =1. On the other, the number 1 can be reduced to only an even number. Then, by the parity rule, the polynomial f has exactly one negative root.

On the other hand, there is a symbolic method for obtaining information about the real roots of a univariate polynomial in an interval [a, b]. From the univariate polynomial f, construct the following sequence f₀, f₁, … called the Sturm sequence of f: f 0 ( x ) = f ( x ) , f 1 ( x ) = f ′ ( x ) , f i ( x ) = - remain ( f i - 2 , f i - 1 ) for 2 ≤ i where f′ is the derivation of f and remain (g, h) is the remainder of the euclidean division of the polynomial g by the polynomial h.

Theorem 4.2. (Sturm’s Theorem) Assume f _m  (x) is the last non-zero polynomial in the Sturm sequence of f (x). If a < b in ℝ and neither is a zero of f (x), then the number of real zeros of f (x) in the interval [a, b] is the number of sign changes in the sequence f₀ (a) , f₁ (a) , …, f _m  (a) minus the number of sign changes in the sequence f₀ (b) , f₁ (b) , …, f _m  (b).

Proof. See [13]. □

Each zero is ignored when we count the number of sign changes in the sequence of real numbers.

Fabrice Rouillier et al. presented an efficient algorithm to isolate real roots of a univariate polynomial using Descartes’ rule [29]. This algorithm is optimal in terms of memory usage. Also, it is implemented in package RS of computer algebra system Maple. To show how one can use this package, assume we wish to obtain the real solutions of the above example. Related commands are as follows:

with(RS): f(x):=-2+4*x+x^2-3*x^5: rs_isolate(f,scalars=floating_point);

The output is [ { x = - 1.117172099 } , { x = . 4624009258 } , { x = 1 . } ] As we expect there are one negative and two positive real solutions.

5 Proposed algorithm

In this section, we present an algorithm for solving a system of fuzzy polynomial equations. To this end, consider the system of fuzzy polynomial equations (1) where all coefficients are fuzzy numbers. We write the all coefficients in the parametric form. By arithmetic operations for fuzzy numbers and definition of equality of two fuzzy numbers described in Section 2, we obtain a real polynomial system of system (1) as follows:{ f 1 , 1 ( x 1 , x 2 , … , x n , r ) = b 1 ¯ ( r ) f 1 , 2 ( x 1 , x 2 , … , x n , r ) = b 1 _ ( r ) ⋮ f s , 1 ( x 1 , x 2 , … , x n , r ) = b s ¯ ( r ) f s , 2 ( x 1 , x 2 , … , x n , r ) = b s _ ( r )(3)with 2s polynomials and n + 1 variables x₁, …, x _n , r where r ∈ [0, 1]. We call this new system (3) the crisp form of system (1). The next proposition expresses relationship between solutions of a fuzzy polynomial equations system and solutions of its crisp form system.

Proposition 5.1. Let F be a fuzzy polynomial equations system as system (1) and F₁ be its crisp form system. Then { ( a 1 , … , a n ) ∈ ℝ n ∣ ( a 1 , … , a n , r ) ∈ V ( F 1 ) , ∀ r ∈ [ 0 , 1 ] } is the solution set of F where r is the parameter.

Proof. Consider the fuzzy polynomial equations system F as system (1). We write all fuzzy numbers in parametric form and get F′ the parametric form system of F with parameter r. Systems F and F′ have the same solutions when parameter r is arbitrary between 0 and 1. By arithmetic operations for fuzzy numbers and definition of equality of two fuzzy numbers, we get the crisp form F₁ that is a system with real coefficients and variables x₁, …, x _n , r. Then we get solutions of F′ from solutions of F₁ with arbitrary variable r between 0 and 1. Therefore, the set { ( a 1 , … , a n ) ∈ ℝ n ∣ ( a 1 , … , a n , r ) ∈ V ( F 1 ) , ∀ r ∈ [ 0 , 1 ] } is the set of solutions of F′ and consequently also that of F.□

If all coefficients are triangular fuzzy numbers, then the system (3) is linear with respect to r. Therefore, this system can be written as{ h 1 ( x 1 , x 2 , … , x n ) r - g 1 ( x 1 , x 2 , … , x n ) = 0 h 2 ( x 1 , x 2 , … , x n ) r - g 2 ( x 1 , x 2 , … , x n ) = 0 ⋮ h 2 s ( x 1 , x 2 , … , x n ) r - g 2 s ( x 1 , x 2 , … , x n ) = 0(4)where h i , g i ∈ 𝕂 [ x 1 , x 2 , … , x n ]. The polynomial system formed by the polynomials h _i and g _i , 1 ≤ i ≤ 2s, is called the collected crisp form system of the fuzzy system. The polynomials h _i and g _i are called the left collects and the right collects polynomials respectively. The following theorem shows the relation between the solutions of a fuzzy polynomial equations system and the variety of the its collected crisp form system.

Theorem 5.2. Let F be a fuzzy polynomial equations system as system (1). Then, the set of solutions of F equals the variety of the its collected crisp formsystem.

Proof. By proposition 5.1, the set of solutions of F is v = { ( a 1 , … , a n ) ∈ ℝ n ∣ ( a 1 , … , a n , r ) ∈ V ( F 1 ) , ∀ r ∈ [ 0 , 1 ] } where F₁ is the crisp form system of F and r is the parameter. Let v′ = V (h₁, ⋯ , h_2s, g₁, ⋯ , g_2s) where h₁, ⋯ , h_2s and g₁, ⋯ , g_2s are the left collects and the right collects polynomials respectively. Let (a₁, …, a _n ) ∈ v. Then (a₁, …, a _n , 0) ∈ V (F₁). By substituting x _i  = a _i and r = 0 in each element of the crisp form system, we obtain g _i  (a₁, …, a _n ) =0 for 1 ≤ i ≤ 2s. Also, (a₁, …, a _n , 1) ∈ V (F₁). By substituting x _i  = a _i and r = 1, we obtain h i ( a 1 , … , a n ) = - g i ( a 1 , … , a n ) = 0 for 1 ≤ i ≤ 2s. Therefore, v ⊆ v′. Now, Let (a₁, …, a _n ) ∈ v′. Then, (a₁, …, a _n , r) is an element of the variety of the crisp form for each r ∈ ℝ. Thus, v′ ⊆ v and the proof is complete.□

Our main algorithm based on the previous discussions for solving a system of fuzzy polynomial equations is described below. We use Wu’s algorithm and compute the variety of the polynomial set obtained from the collected crisp form system in the ring ℝ [ x 1 , x 2 , … , x n ].

Algorithm 5.3. (Main Algorithm)

Input: F, a fuzzy polynomial equations system

Output: The set of solutions of F

Compute the parametric form of F

The main result of this paper is stated in the following theorem.

Theorem 5.4. Main Algorithm is a correct and finiteness algorithm.

Proof. By Corollary 3.9 and Theorem 5.2, the theorem is proved.□

Consider the fuzzy polynomial equations systems F 1 : { ( 2 , 1 , 1 ) xy + ( 3 , 1 , 1 ) x 2 y 2 + ( 2 , 1 , 1 ) x 3 y 3 = ( 7 , 3 , 3 ) ( 5 , 1 , 1 ) xy + ( 2 , 3 , 1 ) x 2 y 2 + ( 2 , 2 , 1 ) x 3 y 3 = ( 9 , 6 , 3 ) F 2 : { ( 2 , 1 , 1 ) x 2 + ( 3 , 2 , 2 ) y 2 = ( 5 , 3 , 3 ) ( 2 , 3 , 1 ) x + ( 2 , 1 , 2 ) y = ( 0 , 5 , 2 ) F 3 : { ( 0 , 0.5 , 0.5 ) x 5 + ( 0.5 , 0.5 , 0.25 ) y = ( 0.25 , 0.75 , 0.625 ) ( 0.5 , 1 , 0.5 ) x 4 + ( 1.5 , 0.5 , 0.5 ) xy = ( 1.25 , 1.25 , 0.75 ) F 4 : { ( 1 , 1 , 1 ) xy + ( - 2 , 1 , 1 ) x 2 y 2 = ( 14 , 2 , 5 ) ( 5 , 1 , 1 ) xy + ( 2 , 3 , 1 ) x 2 y 2 = ( - 2 , 1 , 1 ) . In [5], an approximate solution with the error value of e ≤ 0.01 was found for each of these systems by the neural nets method where a suitable initial point is required. In that paper, the method produces some approximate solutions for system F₄ and there is no hope to make the measure of error close to zero while it has no any real solution. Also, these systems are considered in [26]. The method presented leads to solve the crisp systems with n + 1 variables where n is the number of variables. Also, the solutions must be isolated with regard to the value of r. Through this method, it is require to solve univariate polynomials of degree 4 to obtain the solutions of system F₃. We solve these examples by our algorithm and find all exact solutions of these systems without any initial point. By our method, the solutions are obtained by computing the variety of the collected crisp form system and there is no need any isolation.

Also, the following system with three variables is solved for more example, F 5 : { ( 1 , 2 , 3 ) x 2 + ( 0 , 1 , 1 ) y 2 = ( 1 , 2 , 2 ) ( 2 , 4 , 3 ) x 2 + z 2 = ( 0 , 2 , 3 ) .

According to the parametric form of triangular fuzzy number introduced in Section 2, the parametric form of this system is as follows: { [ 2 r - 1 , 4 - 3 r ] x 2 + [ r - 1 , 1 - r ] y 2 = [ 2 r - 1 , 3 - 2 r ] [ 4 r - 2 , 5 - 3 r ] x 2 + z 2 = [ 2 ( r - 1 ) , 3 ( 1 - r ) ] . Always, x² and y² are non-negative. Therefore, by arithmetic operations for fuzzy numbers and definition of equality of two fuzzy numbers, the crisp form system is as follows: { ( 2 x 2 + y 2 - 2 ) r - x 2 - y 2 + 1 = 0 ( - 3 x 2 - y 2 + 2 ) r + 4 x 2 + y 2 - 3 = 0 ( 4 x 2 - 2 ) r - 2 x 2 + z 2 + 2 = 0 ( - 3 x 2 + 3 ) r + 5 x 2 + z 2 - 3 = 0 .

Thus, the collected crisp form system will be { 2 x 2 + y 2 - 2 = 0 - 3 x 2 - y 2 + 2 = 0 4 x 2 - 2 = 0 - 3 x 2 + 3 = 0 - x 2 - y 2 + 1 = 0 4 x 2 + y 2 - 3 = 0 - 2 x 2 + z 2 + 2 = 0 5 x 2 + z 2 - 3 = 0 .

We used Epsilon package in Maple created by Dongming Wang for computing characteristic sets. The result of solving these fuzzy polynomial equations systems are placed in Table 1. In the examples we adopt the following notations. If the polynomial set F i ′ is the collected crisp form of fuzzy polynomial equations system F _i , the output of Wu’s algorithm for F i ′ is denoted by Z = [z₁, z₂, …, z _m ]; by Wu’s Well-ordering Principle Theorem, the variety V ( F i ′ ) is the union of sets V (z _j  ∖ I _{z j} ) for j = 1, 2, …, m. The notation V is the variety of F i ′ with regard to the sign of variables. The solution set of F _i is union of sets V which are obtained in all cases. In the sequel, an economic example is expressed as an application of the fuzzy polynomial equations systems.

Example 6.1. The balance between supply and demand determine the market price of a commodity and the quantity of production. Suppose that demand and supply are nonlinear polynomial functions of the price as: { q d + a = b . p 2 , q s + c = d . p 2 where q _s and q _d are the quantity of supply and required demand, respectively, which is required to be equal, p is the price and a, b, c and d are coefficients to be estimated. The coefficients a, b, c and d are represented by fuzzy triangular numbers and q _s , q _d and p are real variables. The fuzzy nonlinear system F : { x + ( - 1 , 1 , 1 ) = ( - 2 , 1 , 1 ) y 2 , x + ( 3 , 1 , 1 ) = ( 2 , 1 , 1 ) y 2 should be solved by imposing the equality between quantity supplied and requested. We solve this system by our algorithm. The crisp form and collected crisp form systems of F are { ( 1 - y 2 ) r + x + 3 y 2 - 2 = 0 , ( y 2 - 1 ) r + x + y 2 = 0 , ( 1 - y 2 ) r + x - y 2 + 2 = 0 , ( y 2 - 1 ) r + x - 3 y 2 + 4 and F ′ : { 1 - y 2 = 0 , y 2 - 1 = 0 , x + 3 y 2 - 2 = 0 , x + y 2 = 0 , x - y 2 + 2 = 0 , x - 3 y 2 + 4 = 0 , respectively. Using Wu’s algorithm, we obtain Z = [{x + 1, y² - 1}] and V = {(x = -1, y = ±1)}. Therefore, all solutions of F are obtained exactly by the presented method. Only one solution has been obtained with the initial point (-0.75, 0.75) and the error value of e ≤ 0.01 in [5].

7 Summary and conclusions

This paper presented an algorithm, based on Wu’s algorithm, to find all solutions of fuzzy polynomial equations systems. This algorithm leads us to solve triangular systems that are easy to solve. The proposed method is independent from a suitable starting point and all solutions can be obtained simultaneously. Also, no isolation with regard to the value of parameter r is needed. Numerical results confirm the competencies of the provided algorithm for obtaining all solutions of fuzzy polynomial equations systems.

Acknowledgments

The authors would like to thank Ali Abbasi Molai, Xiao-Shan Gao and Dongming Wang for their helpful discussions.