Leontief input-output model with trapezoidal fuzzy numbers and Gauss-Seidel algorithm

Transcription

1 Int. J. Process Management and Benchmarking, Vol. x, No. x, xxxx 1 Leontief input-output model with trapezoidal fuzzy numbers and Gauss-Seidel algorithm Charmi Panchal* Laboratory of Applied Mathematics, Lappeenranta University of Technology, P.O. Box 20, FIN Lappeenranta, Finland charmi.panchal@abo.fi *Corresponding author Pasi Luukka School of Business, Lappeenranta University of Technology, P.O. Box 20, FIN Lappeenranta, Finland pasi.luukka@lut.fi Jorma K. Mattila Laboratory of Applied Mathematics, Lappeenranta University of Technology, P.O. Box 20, FIN Lappeenranta, Finland jorma.mattila@lut.fi Abstract: In this work a fuzzy linear system is used to solve Leontief input-output model with fuzzy entries. For solving this model, we assume that the consumption matrix from different sectors of the economy and final demand are known. These assumptions heavily depend on the information obtained from the industries. Hence it is clear that uncertainties are involved in this information. The aim of this work is to model these uncertainties and to address them by fuzzy entries such as fuzzy numbers and LR-type fuzzy numbers (triangular and trapezoidal). Fuzzy linear system has been developed and it is solved using Gauss-Seidel algorithm. Numerical examples show the efficiency of this algorithm. The famous example from Prof. Leontief, where he solved the production levels for US economy in 1958, is also further analysed to demonstrate the efficiency of our approach. Keywords: fuzzy numbers; Leontief input-output model; Gauss-Seidel algorithm; fuzzy linear system. Reference to this paper should be made as follows: Panchal, C., Luukka, P. and Mattila, J.K. (xxxx) Leontief input-output model with trapezoidal fuzzy numbers and Gauss-Seidel algorithm, Int. J. Process Management and Benchmarking, Vol. x, No. x, pp.xxx xxx. Copyright 200x Inderscience Enterprises Ltd.

2 2 C. Panchal et al. Biographical notes: Charmi Panchal is working in Computational Biomodelling Laboratory at Department of Information Technologies, Åbo Akademi Finland. His research interests includes computational and systems biology, especially computational modelling of biochemical systems, simulations of biochemical systems under various perturbations, application of mathematical logic approaches to represent behaviour of biological pathways. Pasi Luukka is working in Applied Mathematics at LUT, Department of Mathematics and Physics (Finland) and in strategic research in School of Business. His research interests include soft computing methods, especially fuzzy decision making, fuzzy data analysis, classification, feature selection and regression analysis. He has published about 30 refereed journal articles, some books chapters and about 50 conference papers. Jorma K. Mattila is from Department of Mathematics and Physics, Lappeenranta University of Technology (Finland). His main works consist of papers on modifiers, Zadeh algebras, relationships between Zadeh algebras and Lukasiewicz many-valued logics, hidden modalities of Lukasiewicz logics, and fuzzy-valued fuzzy logic. 1 Introduction The fuzzy sets are developed in order to handle the situations where final decision is perception-based. It has been investigated since 1960s when it was introduced by Zadeh (1965, 1975) from the University of California, Berkeley. Zadeh (2003) has explained the relationship between fuzzy logic and probability theory. In this work fuzzy number theory is used to solve an economic input-output (I-O) model. To be more exact Leontief I-O model is used. The model was developed by Prof. Wassily Leontief who was awarded the Nobel prize for his work in This economic model is more thoroughly introduced in Lay s (2003) text book. As a brief description, suppose that nation s economy is divided into n sectors those produce goods and services. Let X be a production vector in R n, which gives output of each sector. There is also an another part of economy, which does not produce goods or services but only consumes them. This is called open sector. The values of goods and services demanded from this non-productive part are listed by demand vector. The demand vector d can represent consumer demand, government demand, or other external demand. The data provided to this model can not be assumed to be exact, but estimates and due to this reason they most certainly contain uncertainties. In our work fuzzy number theory is used to represent these uncertainties. In brief instead of regular numbers, fuzzy numbers are used in the Leontief I-O model. In order to solve Leontief I-O model with fuzzy entries we use fuzzy linear systems applied with trapezoidal fuzzy numbers and use numerical well known Gauss-Seidel algorithm to solve this. In order to do this we use fuzzy numbers instead of regular number. This makes it somewhat more complicated compared to solving it in traditional way. Earlier, in Luukka and Mattila (2009), LR-type fuzzy triangle numbers were used in both input coefficients of I-O model as well as in total demand coefficients. Here in our present work, we extend this to cover also fuzzy trapezoidal numbers in both input coefficients as well as total

3 Leontief input-output model with trapezoidal fuzzy numbers 3 demand vector coefficients. Also we compare and study two different ways of analysing the uncertainties in the coefficients. Rest of the paper is outlined in following way. In Section 2 we explain the Leontief I-O model, in Section 3 we present our method how trapezoidal fuzzy numbers can be used to solve Leontief I-O model. In Section 4 we study the practical results and compare three different approaches to solve this and in Section 5 we give discussion. 2 Leontief I-O model The sectors involved in the Leontief model, produce goods to meet consumer demand. They also produce goods to meet intermediate demand of the sectors for their own production. The total production should be such that it should satisfy the final demand d and intermediate demand. Resulting in that total production is intermediate demand added with final demand. I-O model is very convenient tool to study and analyse the national or regional economy (Leontief, 1953). This model is well explained in Jensen (2001). This model is usually represented in a matrix form. There columns represent sectors and rows represent inputs from sectors to other column sectors. Considering following example given in Table 1. Table 1 Leontief s I-O economic model Economic Inputs to Inputs to Inputs to Final Total activities energy services transport demand output Energy Services Transport Labour This tabular representation of model shows the inter-industry relations between the sector in the economy. It shows how output of one industry serves as an input to other industry. From the first row it can be seen that energy inputs 5 to energy itself to produce 90 units of output. Services needs 15 units from energy to produce an output of 80 units. The transport industry needs two units from Energy to produce an output of 30 units. The balance equation for this system is x i = x i1 + x i2 + x i x in + d i (1) Equation (1) shows that, the total production (output) of sector i is x i and it depends on intermediate inputs x i1, x i2,..., x in in sectors 1, 2,....n from sector i, and partly to the final demand d i. Leontief, the innovator of input output analysis, used a special production function which depends linearly on the total output variables x i. This production function uses the intermediate Leontief s coefficients c ij. Using c ij one can manipulate transaction information into the I-O table, which can be presented as follows. x ij = c ij x j. (2) where x ij stands for the use of products from a sector i as an input in a sector j and x j stands for the total production in a sector j. The matrix filled with the entries c ij, is

4 4 C. Panchal et al. called I-O matrix. In order to solve this model the linear algebra is applied. Replacing (2) in the balance equation (1) we get c 11 x 1 + c 12 x c 1j x j + d 1 = x 1 c 21 x 1 + c 22 x c 2j x j + d 2 = x 2.. =. c j1 x 1 + c j2 x c jj x j + d j = x j which can be expressed in matrix formulation as, CX + d = X, from where X can be solve as X = [I C] 1 d. Provided that (I C) is invertible. Here x 1 x j d 1 d j x 2 X =., d = d 2., c 11 c c 1j c 21 c c 2j C =... c j1 c j2... c jj and I is an identity matrix. Matrix C is known as unit consumption matrix and column of C is called unit consumption vector of that sector. Let say I C = A. Entries in matrix A can be understood as a factor inputs per unit of output. The total factor used is the vector d which is given as a vector of final demand. Using this notation we get X = A 1 d. The basic assumption of Leontief s I-O model is that for each sector, there is a unit consumption vector that lists the inputs needed from the other sectors to produce a unit of output. In order to solve the model the matrix C and demand d must be known. 3 Mathematical background In this section the basic concepts of fuzzy sets and logic are provided. Fuzzy arithmetic operations are also briefly explained. After that mathematical background of fuzzy linear system and Gauss-Seidel algorithm is given. 3.1 Fuzzy numbers and arithmetic operations The concept of fuzzy numbers and arithmetic operations were first introduced and investigated by Zadeh (1965) and then later by Mizumoto and Tanaka (1979), Dubois and Prade (1980) and Nahmias (1978). The different approaches to fuzzy numbers and the structure of fuzzy number spaces were provided by Puri and Ralescu (1983), Goetschell and Voxman (1986) and Cong-Xin and Ming (1991, 1992). Generally fuzzy numbers are represented by two end points a 1 and a 3, and a modal point a 2 as A =

5 Leontief input-output model with trapezoidal fuzzy numbers 5 (a 1, a 2, a 3 ). If we are dealing with trapezoidal fuzzy number then a 2 will be an interval. Every fuzzy triangular number has its own membership function which can be defined as follows, µ A (x) = 0 if x < a 1, x a 1 a 2 a 1 if a 1 x a 2, a 3 x a 3 a 2 if a 2 x a 3, 0 if x > a 3. Instead of taking modal point, if modal interval is used then the form of fuzzy interval is A = (a 1, a 2, a 3, a 4 ), where [a 2, a 3 ] is an interval. This fuzzy interval is called fuzzy trapezoidal number. Membership function of a fuzzy trapezoidal number is given in equation (4). µ A (x) = 0 if x < a 1, x a 1 a 2 a 1 if a 1 x a 2, 1 if a 2 x a 3, a 4 x a 4 a 3 if a 3 x a 4, 0 if x a 4. Another way of defining a fuzzy number is LR-type fuzzy number (Miin-Shen et al., 2005), which can be defined as follows. Definition 3.1: A fuzzy number A is of LR-type if there exists reference functions L (for left), R (for right) and scalars l > 0, r > 0 with, (3) (4) µ A (x) = { L((M x)/l) if x M R((x M)/r) if x M, (5) where M is called modal value of A, it is real number. Moreover l and r are called the left and right spreads respectively. Symbolically A is denoted by (M, l, r) and for LR-type function we use L(x) = R(x) = max {1 x, 0},which gives fuzzy triangular numbers (Dubois and Prade, 1980). For short this fuzzy number A can be represented by a modal value M, and the lengths of fuzzy spreads as A = (M, l, r). The LR-type representation of fuzzy trapezoidal number is represented in following way. Definition 3.2: A fuzzy number A is of LR-type if there exists reference functions L (for left), R (for right) and scalars l > 0, r > 0 with, L((M 1 x)/l) if x M 1 µ A (x) = 1 if x [M 1, M 2 ] R((x M 2 )/r) if x M 2, modal interval M is such that (M 1, M 2 ) R, M 1 < M 2, and µ M (x) = 1, x [M 1, M 2 ]. Coefficients l and r are called the left and right spreads respectively. Symbolically A is denoted by (M 1, M 2, l, r) and for LR-type function we use L(x) = R(x) = max {1 x, 0}, which gives fuzzy trapezoidal numbers. (6)

6 6 C. Panchal et al. Similarly short representation is simply A = (M 1, M 2, l, r). Next we shortly view α-cuts of fuzzy numbers of form (3) and (4) Definition 3.3 α-cuts: An α-cuts of a fuzzy set A is a crisp set A α = {x X µ A (x) α} (7) The set A α > = {x X µ A (x) > α} is called the strick α-cut. For a fuzzy triangular number A = (a 1, a 2, a 3 ), an interval A α arbitrary α cut where α [0, 1]. corresponds to an A α = [a α 1, a α 3 ] = [(a 2 a 1 )α + a 1, (a 3 a 2 )α + a 3 ]. (8) In the same way the α-cuts can be calculated for fuzzy trapezoidal numbers. If A = (a 1, a 2, a 3, a 4 ) is fuzzy trapezoidal number then an interval A α corresponds to an arbitrary α cut where αϵ[0, 1] that is calculated as follows. A α = [(a 2 a 1 )α + a 1, (a 4 a 3 )α + a 4 ]. (9) This type of α-cut representation is often used in representation of a fuzzy number in parametric form which is defined as Definition 3.4: A fuzzy number A in parametric form is a pair (A(α), A(α)) which satisfies the following conditions: A(α) is a bounded monotonic increasing left continuous function A(α) is a bounded monotonic decreasing left continuous function A(α) A(α). where α [0, 1] There for example if we are dealing with triangular numbers parametric form of a fuzzy number would be { A(α) = a1 + (a A(α) = 2 a 1 )α A(α) = a 3 (a 3 a 2 )α or if it would be trapezoidal number it would be { A(α) = a1 + (a A(α) = 2 a 1 )α A(α) = a 4 (a 4 a 3 )α Arithmetic operations between LR-type fuzzy triangular numbers Next we view the basic arithmetic operations of LR-type fuzzy numbers [see (5)]. Consider two LR-type fuzzy numbers A 1 = (M 1, l 1, r 1 ) and A 2 = (M 2, l 2, r 2 ). The arithmetic operations to LR-type fuzzy numbers were introduced by Dubois and Prade (1980). The different arithmetic operations are defined as follows.

7 Extended sum Leontief input-output model with trapezoidal fuzzy numbers 7 A 1 A 2 = (M 1 + M 2, l 1 + l 2, r 1 + r 2 ) LR. (10) Extended difference A 1 A 2 = (M 1 M 2, l 1 + r 2, r 1 + l 2 ) LR. (11) Extended product If spreads are not small with respect to M 1 and M 2 then multiplication can be approximated by the following formula, when M 1 > 0 and M 2 > 0. A 1 A 2 = (M 1 M 2, M 1 l 2 + M 2 l 1 l 1 l 2, M 1 r 2 + M 2 r 1 + r 1 r 2 ) LR. (12) When M 1 < 0 and M 2 > 0 A 1 A 2 = (M 1 M 2, M 1 l 2 + M 2 l 1 + l 1 r 2, M 1 r 2 + M 2 r 1 r 1 l 2 ) RL.(13) When M 1 < 0 and M 2 < 0 A 1 A 2 = (M 1 M 2, M 1 r 2 M 2 r 1 l 2 r 1, M 1 l 2 M 2 l 1 + r 2 l l ) RL.(14) More about these product rules can be found in Dubois and Prade (1980) book. Extended quotient The extended quotient can be approximated as follows. ( M1 r 2 M 1 + l 1 M 2 l 2 M 1 + r 1 M 2 A 1 A 2, M 2 M2 2, M2 2 ). (15) LR The approximation for the above equation can also be derive in the same way like equations (12), (13) and (14). It is also good to notice that the production levels can not be negative in Leontief s I-O model with fuzzy entries. That means when we solve AX = Y, where X is the production level vector which can not have negative modal values. Hence in the calculations only the case when M 1 > 0 and M 2 > 0 is considered Arithmetic operations with parametric form of fuzzy triangular numbers Multiplication and division are considered for only positive numbers. In the case of triangular fuzzy numbers A = (a 1, a 2, a 3 ) and B = (b 1, b 2, b 3 ), another way to express multiplication and division can be by using parametric forms of fuzzy numbers A and B which are A(α) and B(α) where { A(α) = a1 + (a A(α) = 2 a 1 )α A(α) = a 3 (a 1 a 2 )α same for B(α). Basic operations for parametric form of fuzzy numbers are given as follows

8 8 C. Panchal et al. Addition: A B = (A(α) + B(α), A(α) + A(α)) (16) Substraction: A B = (A(α) B(α), A(α) A(α)) (17) Let λ be positive scalar constant, then the scalar multiplication is λa(α) = (λa(α), λa(α)) (18) Multiplication is expressed as follows. A B = (AB(α), AB(α)). (19) where AB(α) = min{a(α)b(α), A(α)B(α), A(α)B(α), A(α)B(α)} AB(α) = max{a(α)b(α), A(α)B(α), A(α)B(α), A(α)B(α)}. Division can be expressed as follows. where A B = ( A B ( ), A ) B (+). (20) { A A(α) ( ) = min B A (+) = max B B(α), A(α) B(α), A(α) B(α), A(α) } B(α) { A(α) B(α), A(α) B(α), A(α) B(α), A(α) } B(α) Arithmetic operations between fuzzy trapezoidal numbers Here we next show how the previous operations can be extented to cover also trapezoidal fuzzy numbers. Consider first LR-type trapezoidal fuzzy numbers of form given in definition (3.1) now written shortly as A = (a 1, a 2, a l, a r ) meaning A = (a 1, a 2, a l, a r ) L((a 1 x)/a l ) if x a 1 = 1 if x [a 1, a 2 ] R((x a 2 )/a r ) if x a 2, (21) and similarly B = (b 1, b 2, b l, b r ). The different arithmetic operations can be calculated as follows.

9 Extended sum Leontief input-output model with trapezoidal fuzzy numbers 9 A B = (a 1 + b 1, a 2 + b 2, a l + b l, a r + b r ) LR. (22) Extended difference A B = (a 1 b 2, a 2 b 1, a l + b r, a r + b l ) LR. (23) Extended product Extented product can be approximated by the following formula, when a 1 > 0 and b 1 > 0. A B = (a 1 b 1, a 2 b 2, a 1 b l + b 1 a l a l b l, a 2 b r + b 2 a r + a r b r ) LR. (24) When a 2 < 0 and b 1 > 0 A B = (a 1 b 2, a 2 b 1, a 1 b l + b 2 a l + a l b r, a 2 b r + b 1 a r a r b l ) RL. (25) When a 2 < 0 and b 2 < 0 A B = (a 2 b 2, a 1 b 1, a 2 b l b 2 a r b l a r, a 1 b l, b 1 a l + b r a l ) LR (26) Extended quotient The extended quotient for positive fuzzy numbers can be approximated as follows. A B ( a1 b 2, a 2 b 1, a 1 b r + a l b 2 a l b r b 2, 2 ) a 2 b l + a r b 1 + a r b l b 2. (27) 1 LR This can be straightforwardly derived by using the product (24) and the inverse for LR-type fuzzy trapezoidal number of form ( 1 B 1 =, 1, b r b 2 b 1 b 2, b ) l 2 b 2 1 In similar manner also the other quotients for negative cases can be derived Arithmetic operations with parametric form of fuzzy trapezoidal numbers Extending the formulas from triangular number to trapezoidal number of parametric form is more straightforward than with the LR-type of case. Here simply parametric forms of fuzzy numbers are now given as A(α) = { A(α) = a1 + (a 2 a 1 )α A(α) = a 4 (a 4 a 3 )α

10 10 C. Panchal et al. and B(α) = { B(α) = b1 + (b 2 b 1 )α B(α) = b 4 (b 4 b 3 )α Basic operations for parametric form of fuzzy numbers are given as follows: Addition: A B = (A(α) + B(α), A(α) + A(α)) (28) Substraction: A B = (A(α) B(α), A(α) A(α)) (29) Let λ be positive scalar constant, then the scalar multiplication is λa(α) = (λa(α), λa(α)) (30) Multiplication is expressed as follows. A B = (AB(α), AB(α)). (31) where AB(α) = min{a(α)b(α), A(α)B(α), A(α)B(α), A(α)B(α)} AB(α) = max{a(α)b(α), A(α)B(α), A(α)B(α), A(α)B(α)}. Division can be expressed as follows. A B = ( A B ( ), A ) B (+). (32) where { A A(α) ( ) = min B A (+) = max B B(α), A(α) B(α), A(α) B(α), A(α) } B(α) { A(α) B(α), A(α) B(α), A(α) B(α), A(α) } B(α)

11 Leontief input-output model with trapezoidal fuzzy numbers Fuzzy linear system and Gauss-Seidel algorithm The fuzzy linear system has been studied by many authors (Dehghan et al., 2006; Friedman et al., 1998). Fuzzy models and systems are widely used in many real world engineering applications. For example population models, control chaotic systems (Chang and Zadeh, 1972; Rao and Chen, 1998), economics and finance (Buckley, 1992; Simon and Blume, 1994), etc. In some of these models, the parameters and measurements are represented by fuzzy numbers rather than crisp numbers. The concept of fuzzy numbers and arithmetic operations with these numbers were first investigated by Zadeh (1965). Application of fuzzy linear system for solving Leontief s production model can be found in Luukka and Mattila (2009) and Mattila and Luukka (2009). The Leontief s model is expressed as a n n linear system as written in here. a 11 x 1 + a 12 x a 1n x n = y 1 a 21 x 1 + a 22 x a 2n x n = y 2. a n1 x 1 + a n2 x a nn x n = y n (33) In this article, all the parameters are considered to be fuzzy numbers, that means the coefficient matrix A = (a ij ), 1 i, j n and column vector Y = (y i ), 1 i n, are fuzzy numbers. We may take either of them as a fuzzy numbers. This we called fuzzy linear system.the general fuzzy linear system has also been discussed by Zheng and Wang (2006). Many authors have been worked for solving fuzzy linear system using different approaches (Chi-Tsuen, 2007; Jun-Feng and Wang, 2009; Allahviranloo, 2004; Rao and Chen, 1998; Abbasbandy and Amirfakhrian, 2006). The Gauss-Seidel iteration formula for the crisp(regular) numbers, is as follows. Here k shows the iteration number. i 1 N x k+1 i = y i k+1 a ij x j k a ij x j /a ii. (34) j=1 j=i+1 i = 1, 2,..., N, a ii 0. For solving fuzzy linear system Gauss-Seidel algorithm with following formula is used. x i k+1 = ( y i ( i 1 j=1 a ij x j k+1 ) ( N j=i+1a ij x j k )) a ii (35) where the operations,, and are for fuzzy numbers. This Gauss-Seidel iteration formula is implemented using MAT LAB T M software. In equation (35) fuzzy arithmetic operations were given in general form, but as it is obvious from previous section we can apply different types of arithmetic operations to it. For this algorithm three different approaches are applied and compared in this article 1 LR-type fuzzy triangle numbers are used 2 this is extended to LR-type fuzzy trapezoidal numbers 3 parametric forms of fuzzy trapezoidal numbers with α-cuts are applied.

12 12 C. Panchal et al. The comparison of these three approaches is given in the next section. In this work for the stopping criteria we introduce to apply ( -norm) with fuzzy numbers. This -norm is extended so that it also takes into account also the spread areas. It is applied in the following way. We have x k+1 = (M k+1, l k+1, r k+1 ) and x k = (M k, l k, r k ), as our LR-type fuzzy triangular numbers. Now for x k+1 x k ϵ we now calculate, max{ M k+1 M k, l k+1 l k, r k+1 r k } < ϵ. where ϵ is predefined value. In practice we used ϵ = This norm guarantees that spread areas are also converged. Extenting this criteria to trapezoidal fuzzy numbers is straightforward. For x k+1 = (M1 k+1, M2 k+1, l k+1, r k+1 ) and x k = (M1 k, M2 k, l k, r k ), as our LR-type fuzzy trapezoidal numbers. Now for x k+1 x k ϵ we now calculate, max{ M1 k+1 M1 k, M2 k+1 M2 k l k+1 l k, r k+1 r k } < ϵ. 4 Practical results In the Leontief s economic model when many sectors are involved, the Gauss-Seidel algorithm is the good choice to use. To test this algorithm Leontief s example is used where total 81 sectors of the economy are considered. These 81 sectors are grouped in to seven major sectors which are as follows. 1 non-metal household and personal products 2 final metal products (such as motor vehicles) 3 basic metal products and mining 4 basic nonmetal products and agriculture 5 energy 6 service 7 entertainment and miscellaneous products. The consumption matrix for this economy is as follows C =

13 Leontief input-output model with trapezoidal fuzzy numbers 13 The production level for each sector has to be found to satisfy following final demands. d = Final demand Nonmetal household and personal products Final metal products (such as motor vehicles) Basic metal products and mining Basic nonmetal products and agriculture Energy Service Entertainment and miscellaneous products 74,000 units 56,000 units 10,500 units 25,000 units 17,500 units 196,000 units 5,000 units Here we assume that in all the values of consumption matrix and final demand the uncertainty exists and they are modelled with fuzzy numbers. Fuzzy linear system is developed using C and d, so that instead of crisp numbers fuzzy numbers are used. After that Gauss-Seidel algorithm is used to solved it to find the production levels. This algorithm is applied in these three different approaches as follows. 4.1 Approach 1 In this approach LR-type fuzzy triangular entries are considered with left and right spread values as l = r = That means that for example the entry in C becomes (0.1588, 0.005, 0.005). Same is applicable for other entries in C and d. Fuzzy linear system is being constructed using matrix C and d. After applying Gauss-Seidel algorithm following fuzzy production levels are found. The arithmetic operations mentioned in Section are used in this calculation. X = (99, 575, 6, 150, 6, 150) (97, 703 8, 583, 8, 583) (51, 230, 9, 012, 9, 012) (131, 569, 12, 493, 12, 493) (49, 488, 7, 587, 7, 587) (329, 554, 13, 995, 13, 995) (13, 835, 4, 549, 4, 549) As entries in the data were LR-type fuzzy triangular entries with specific spread values so as the results contain same. To plot them they are formulated in the form A = (a 1, a 2, a 3 ) where a 1 = modal value left spread, a 2 = modal value and a 3 = modal value + right spread. Solved production levels can be seen in Figure Approach 2 Here all the entries in matrix C and vector d are considered as LR-type fuzzy trapezoidal numbers together with left-right spread values as l = r = As a sample the entry understood as (0.1588, , 0.005, 0.005). The entries of demand vector is considered like, e.g., the i th entry is written as (demand(i) 1, demand(i) + 1, l, r). Conversion from LR-type fuzzy number to a fuzzy number of the form A = (a1, a2, a3, a4) (see Section 3.1 and Definition 3.1) can be done easily by writing a 1 = left boundary of the modal interval left spread,

14 C. Panchal et al. 14 [a2, a3 ] = modal interval, a4 = right boundary of the modal interval + right spread. For example formulating (225.92, , 77.83, ) as mentioned in above and it becomes fuzzy trapezoidal number (a1, a2, a3, a4 ) becomes A = (148.09, , , ). Figure 1 Production levels using Approach 1 (see online version for colours) x 10 After that our algorithm is applied to solve the fuzzy linear system. Here the arithmetic operations are used as mentioned in Section The resultant fuzzy production level for each sector, are as follows. These are regular fuzzy trapezoidal numbers. (93, 763, (89, 757, (42, 884, X= (119, 943, (42, 421, (316, 564, (9, 624, 99, 574, 97, 700, 51, 228, 131, 566, 49, 486, 329, 550, 13, 834, 99, 577, 97, 705, 51, 232, 131, 573, 49, 490, 329, 558, 13, 836, Results are plotted as follows in Figure , 268) 107, 042) 61, 036) 145, 163) 57, 743) 344, 782) 18, 785)

15 Leontief input-output model with trapezoidal fuzzy numbers 15 As it is difficult to see the plateau of the trapezoid so let s zoom the circled portion, the clear image look like as follows in Figure 3. Figure 2 Production level using Approach 2 (see online version for colours) x 10 5 Figure 3 Zoomed circled portion of the previous figure (see online version for colours) x 10 5

16 16 C. Panchal et al. 4.3 Approach 3 Like in previous case here also LR-type fuzzy numbers are used but instead of left-right spread values percentage of uncertainty is used, which is taken as 5%. That means the entry logically becomes (0.1588, , * u 100, * u 100 ) and the i th demand(i)*u entry in demand vector is written as (demand(i) -1, demand(i)+1, 100, demand(i)*u 100 ). Before applying the algorithm, we transform the crisp number into trapezoidal fuzzy number. Meaning that for example the entry becomes ( * u u 100, , , * 100 ). So all the entries in matrix C are treated like this. The entries in vector d are understood like, e.g., i th entry in demand is written as (demand(i) 1 demand(i)*u 100, demand(i) 1, demand(i) + 1, demand(i) demand(i)*u 100 ). The fuzzy results produced by Gauss-Seidel algorithm is presented in vector X as follows. The arithmetic operations used are mentioned in Section These results are obtained in the form of LR-type fuzzy trapezoidal numbers. (88447, 99574, 99577, ) (85448, 97700, 97705, ) (42265, 51228, 51232, 62671) X = (108926, , , ) (41825, 49486, 49490, 59132) (288316, , , ) (11415, 13834, 13836, 16941) These trapezoidal numbers are plotted in following Figure 4. Figure 4 Production level using Approach 3 (see online version for colours) x 10 5

17 Leontief input-output model with trapezoidal fuzzy numbers 17 Same convergence criteria is applied in above all approaches and the result X is produced using Gauss-Seidel algorithm. In the Table 2 number of iterations and time taken by the algorithm in different approaches are listed. Table 2 Comparison of the computational time with our three approaches Approach Number TOC after TOC before cputime before cputime after of iteration plotting (in sec.) plotting (in sec.) plotting (in sec.) plotting (in sec.) As can be seen from Table 2 third approach with trapezoidal fuzzy nummber cave the fastest convergence in computational time. 5 Discussion The Leontief closed I-O model with uncertain consumption matrix and with uncertain final demand is considered. In this work fuzzy linear system has been developed, by implementing fuzzy entries in this model. In our given work previously presented (Luukka and Mattila, 2009) work in this area is now extended to cover fuzzy trapezoidal numbers this way creating a new fuzzy linear system problem to be solve. Besides this also computational times are examined with three different approaches. This fuzzy linear system has been solved by applying Gauss-Seidel algorithm. Different possibilities of representing uncertainty and possible results have been analysed. In our work, we tested the algorithm for economic system which involves seven major sectors of economy. It is assumed that uncertainty is involved in all data contained in the model. In the presented algorithm, to solve fuzzy linear system we assumed that both the co-efficient matrix and demand vector are fuzzy trapezoidal numbers. Table 2 represents the number of iterations and approximate time for the algorithm to produce results in different approaches. After getting solution by various approaches from Table 2 it can be seen that third approach seemed to save the computation time as it gives results in few iterations which enables fast computations. It is here therefore consider the most preferred approach. There LR-type fuzzy entries together with percentage of uncertainty, are used in the linear system. The arithmetic operations for solving this system are given in The second preferable choice from our three cases studied here can be given to the 2nd approach. There algorithm works also well and gives result in very short time. In this calculation entries are used same as in Approach 3, but uncertainty is presented in the form of spread values. The percentage of uncertainty is most advisable to use instead of left right spread values since this seems more realistic and also to give suitable spread values for each number can be time consuming and difficult to be gained realistically. Moreover comparing the results in both second and third approaches, one may find that the modal intervals are the same, but in Approach 2 the extreme left and right values differ due to m different uncertainties. In Approach 3 one can easily define all the uncertainty values using percentage of the modal value. This way we get a more realistic estimate since it is not necessarily realistic to set certain constant values as left and right side spreads as modal

18 18 C. Panchal et al. values in consumption matrix differ quite a lot, making situation such that constant uncertainty can be quite small to some values and large to others. This of course effects the production level calculations. The solution in Approach 1 is obtained by using arithmetic operations mentioned as in This was also implemented earlier in Luukka and Mattila (2009). Solution is LR-type fuzzy triangular numbers, with constant left and right spread values. By implementing trapezoidal entries, algorithm seems to work faster than in previous work. That can be seen from Approaches 2 and 3. Moreover by using LR-type fuzzy trapezoidal values, more general form of uncertainty in the data is now allowed to be exist. Because instead of modal value, only approximate modal interval is needed to provide. This enlarges the possibilities to use this approach, i.e., in the case that we would have missing data values these can easily be covered with trapezoidal fuzzy numbers where as it will not be possible by using triangular fuzzy numbers. References Abbasbandy, S. and Amirfakhrian, M. (2006) The nearest trapezoidal form of a generalized left right fuzzy number, International Journal of Approximate Reasoning, Vol. 43, No. 2, pp Allahviranloo, T. (2004) Numerical methods for fuzzy system of linear equations, Applied Mathematics and Computation, Vol. 155, No. 2, pp Buckley, J.J. (1992) Solving fuzzy equations in economics and finance, Fuzzy Sets and Systems, Vol. 48, No. 3, pp Chang, S.L. and Zadeh, L.A. (1972) On fuzzy mapping and control, IEEE Trans., Syst. Man Cyb., Vol. 2, No. 1, pp Chi-Tsuen, Y. (2007) Reduction of fuzzy linear systems of dual equations, Int. J. Fuzzy Syst., Vol. 9, No. 3, pp Cong-Xin, W. and Ming, M. (1991) Embedding problem of fuzzy number space: Part I, Fuzzy Sets and Systems, Vol. 44, No. 1, pp Cong-Xin, W. and Ming, M. (1992) Embedding problem of fuzzy number space: Part III, Fuzzy Sets and Systems, Vol. 46, No. 2, pp Dehghan, M., Hashemi, B. and Ghatee, M. (2006) Solution of the fully fuzzy linear systems using the decomposition procedure, Applied Mathematics and Computation, Vol. 182, No. 2, pp Dubois, D. and Prade, H. (1980) Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York. Friedman, M., Ming, M. and Kandel, A. (1998) Fuzzy linear systems, Fuzzy Sets and System, Vol. 96, No. 2, pp Goetschell, R. and Voxman, W. (1986) Elementary calculus, Fuzzy Sets and Systems, Vol. 18, No. 1, pp Jensen, I. (2001) The Leontief Open Production Model or Input-Output Analysis [online] (accessed 25 July 2012). Jun-Feng, Y. and Wang, K. (2009) Splitting iterative methods for fuzzy system of linear equations, Computational Mathematics and Modeling, Vol. 20, No. 3, pp Lay, D.C. (2003) Linear Algebra and its Applications, Addison Wesley, University of Maryland, College Park.

19 Leontief input-output model with trapezoidal fuzzy numbers 19 Leontief, W.W. (1953) Studies in the Structure of the American Economy, Oxford University Press, New York, NY. Luukka, P. and Mattila, J.K. (2009) Fuzzy linear systems applied to Leontief input-output model, Acta Technica Jaurinensis (International Journal), Vol. 2, No. 2, pp Mattila, J.K. and Luukka, P. (2009) Solving Leontief input-output model with fuzzy entries, FUZZ-IEEE Korea, International Conference on Fuzzy Systems, pp Miin-Shen, Y., Wen-Liang, H. and Chang-Chien, S.J. (2005) On a similarity measure between LR-type fuzzy numbers and its application to database acquisition, International Journal of Intelligent Systems, Vol. 20, No. 10, pp Mizumoto, M. and Tanaka, K. (1979) Some properties of fuzzy numbers, in M.M. Gupta, R.K. Ragade and R.R. Yager (Eds.): Advances in Fuzzy Set Theory and Applications, pp , North-Holland, Amsterdam. Nahmias, S. (1978) Fuzzy variables, Fuzzy Sets and Systems, Vol. 1, No. 2, pp Puri, M.L. and Ralescu, D.A. (1983) Differentials for fuzzy functions, Journal of Mathematical Analysis and Applications, Vol. 91, No. 2, pp Rao, S.S. and Chen, L. (1998) Numerical solution of fuzzy linear equations in engineering analysis, International Journals for Numerical Methods in Engineering, Vol. 42, No. 5, pp Simon, C.P. and Blume, L. (1994) Mathematics for Economists, WW Norton and Company, New York, London. Zadeh, L.A. (1965) Fuzzy sets, Information and Control, Vol. 8, No. 3, pp Zadeh, L.A. (1975) Fuzzy logic and approximate reasoning, Synthese, Vol. 30, Nos. 3 4, pp Zadeh, L.A. (2003) Probability Theory and Fuzzy Logic, Computer Science Division Department of EECS UC Berkeley, Los Alamos National Laboratory, Sponsored by ESA Div. LANL Uncertainty Quantification Working Group (April 24) [online] (accessed 25 July 2012). Zheng, B. and Wang, K. (2006) General fuzzy linear systems, Appl. Math. Comput., Vol. 181, No. 2, pp