An Application of Fuzzy Linear System of Equations in Economic Sciences

Transcription

1 Austalian Jounal of Basic and Applied Sciences, 5(7): 7-14, 2011 ISSN An Application of Fuzzy Linea System of Equations in Economic Sciences 1 S.H. Nassei, 2 M. Abdi and 3 B. Khabii 1 Depatment of Mathematics, Mazandaan Univesity, Babolsa, Ian. 2 Depatment of Accounting, Pasa Pivate Institute, Babolsa, Ian. 3 Depatment of Mathematics, Islamic Azad Univesity, Jouyba Banch, Ian. Abstact: The maket pice of a good and the quantity poduced ae detemined by the equality between supply and demand. Demand is the amount of a good that consumes ae willing and able to buy at a given pice. Supply is the amount of good poduces is willing and able to sell at a given pice. Suppose that demand and supply ae linea functions of the pice: qd a* pb qs c* pd whee is the quantity supplied, which is equied to be equal to, the quantity equested, is the pice and a and c ae cisp eal numbes and b and d ae coefficients to be estimated. Suppose that only some impecise data on the elation between the quantity supplied and equested at a given pice ae available and that the elation can be natually descibed as follows: qd a* p b qs c* p d whee and ae the fuzzy vesion of the paametes and, espectively. As we see fo obtaining a solution, it is necessay to solve a fuzzy vesion of linea system of equations. Hence, in this pape, an application of solving fuzzy linea system of equations in economic is discussed. We fist biefly state some fundamental concepts of fuzzy sets theoy and economic and then give an example to illustate the main idea of this study. Key wods: Demand, equilibium, fuzzy aithmetic, fuzzy linea systems, fuzzy numbes, supply INTRODUCTION In the cuent complex economical system thousands goods and sevices ae exchanged evey day and often maket of these goods and sevices ae intedependent and have intepl ay. I t s necessay to have a mechanism fo egulaize poduction and consumption of the numeous goods and sevices and allocation of economical esouces. This mechanism is needed to oganize the economists decisions in ode to have balance in the whole maket. In the othe wod, this mechanism must be able to answe the questions such as: what should be poduced? How should be poduced? And fo whom should be poduced? The ole of supply and demand mechanism (maket mechanism) in the goods maket and poduction esouces can eply to the economical basic questions (what, how, and fo whom to poduce). In supply and demand mechanism and maket opeation, the pice which will be getting fom intesection and contast of supply and demand will be main facto in allocating goods and esouces. The pices in such mechanism have many data s fo supplies and applicants and poduce equied motivation in ode to making necessay eaction in them. Fo example, when the pice of goods, and o sevices is high by eason of lack of supply (uppe than othe goods), consume will undestand that thee is not enough fom this good, and then consumption level will decease and on the contay, supplies will be encouaged to poduce that good, and poviding fit condition, poduction of this goods will incease in next peiod, as by inceasing supply, the deceased pice will adjust. Coesponding Autho: S.H. Nassei, Depatment of Mathematics, Mazandaan Univesity, Babolsa, Ian 7

2 This pape is oganized in 5 sections. In Sections 2 and 3, we give some fundamental concepts of economic science concening to ou discussion. In Section 4, we fist eview some necessay definitions and concepts of fuzzy sets theoy and then define the fuzzy linea system of equations and discuss on solving these systems. Finally we give an illustative example in Section 5. We conclude in Section Demand: Individual demand means amount good that a buye tends and buys it accoding to its pice and fixed othe elements in a specific ea (Baumol, 1977; Maddala and Mille, 1989). In fact, demand is maximum amount goods ae bought in egad to thei pice by a puchase. Of couse, it s necessay to mention that demand also happens fo sevices such as: tavele s tanspot sevices. It s necessay to mention that equiement and demand diffe with togethe. We need a lot goods and sevices but may not demand. Fo example, a peson may equie an ai plane but he/she doesn t demand. Some ou needs become changed into demand egading to pice and income and etc Effective factos on demand: The amount demand of good is affect on following factos: d Qx F(p x,i,p y,t,a x,e d, ) As, P x is the pice of good x. P y is the pice of othe goods. I is consume s budget o income. T is consume s style o taste and pefeence that can esult fom his/he needs and the souce of these needs can be because of social custom and supeio to all, because of his/he value and belief standads. A x is publicity fo good x. E d is expectation facto of demand pice, while consume s demand is affecting on his/he expectation fo poviding o not good in the futue and also his/he foecast about pice pocess of this good in futue Demand function: d If we fix othe effective factos on demand except the pice of good, we can wite: Q f( p ) It calls demand function. Then demand function is a function that shows elation between good pice and amount demand of same good if the othe factos emain constant. Individual s demand table on good shows the diffeent amount of this good that he/she tends to buy with othe multiple pices on good if the othe factos emain constant. Accoding to the given table, that is Table 2.1 we obtain Fig. 2.1 (demand cuve). Table 2.1 Demand table P X x Q D X x 8

3 Fig. 2.1: Accoding to the above demand table, the equation of peson s demand fo good is Q x =30-5Px (if othe factos emain fix). 3- Supply: Individual supply means amount good that a selle tends and can give it accoding to its pice and fixed of othe factos in a specific peiod. In fact, supply of a good is the exteme of that good that selle ente it into the maket egading to its pice. Of couse, it s necessay to mention that in economy thee is sevices supply, too. Such as, taxi sevices supply Effective factos on supply: Except of pice facto(px) which is the most impotant vaiable affecting on supply, the othe factos such as poduction expense(tc involves poduction qualities pice and..), coelate goods pice(py), technology level o technical knowledge(t), expectations facto of supply pice(es) and ae effective on supply of a good. Geneally, supply has contact with benefit, if benefit inceases, supply also will incease and vice vesa; and also benefit depend on the above factos. Theefoe, we can show geneal fom of individual supply function like this: s Qx F(px,py,TC,T,Es,..) 3.2. Supply function: If we assume (in the above function) othes factos ae fixed except good pice, we can wite: s Qx f(px) This function is called supply function. Then, supply function is a function that shows elation between good pice and supply amount of same good if the othe elements is consideed fix. This function is defined in two ways: (1) Minimum pice that supplie tends to supply the good. (2) Maximum amount that ae supplied in font of each pice. Individual supply table fo good shows diffeent amount of this good that he/she tends to sell by othe pices of good if the othe factos emain fixed (see Table 3.1). Now accoding to Table 3.1 we obtain Fig. 3.1 (supply cuve). 9

4 Table 3.1: Supply table p X Aust. J. Basic & Appl. Sci., 5(7): 7-14, Q S X Fig. 3.1: The supply table concludes that the equation of individual supply to good x is as follows: Q x = px (if othe factos emain constant). Now accoding to above demand and supply functions, balance amount of good by using maket balance tem is as follows: S x= D x : balance condition p 30 5p p 2 x x x and then Qx=30-5(2)=20. The following figue shows the equilibium of the good maket which is concluded fom the contact of the demand and supply functions. Fig. 3.2: 4- Fuzzy linea system of equations: In this section we fist state some necessay backgounds of fuzzy aithmetic which is useful in thoughout ou discussion (taken fom Matinfa, et al., 2008 and Nassei, 2008). Then we define a fuzzy linea system of equations and discuss on solving it. Paametic fom of an abitay fuzzy numbe is defined as following. uu u (), u (),0 A fuzzy numbe u in paametic fom is a pai numbe (, ) of function 1, which satisfies the following equiements: 1. u ( ) is a bounded left semi continuous non-deceasing function ove [0, 1], 10

5 2. u ( ) is a bounded left semi continuous non-inceasing function ove [0, 1], 3.() u u (),01. If in the above definition u () and u () ae linea functions, we have tiangula fuzzy numbe. We show the set of all tiangula fuzzy numbes by F(R). Definition 4.1: The following n n linea system is called a fuzzy linea system (FLS): n j1 a x ( ), x ( ) y ( ), y ( ), i 1,..., n ij j j i j whee the coefficients matix A= [a ij ], 1#i,j#n, is a cisp n n matix and i i i ae given and 1#j#n. x x (), x () j j j (4.1) y y (), y () F( R),1 in, ae unknown tiangula fuzzy numbes and should be detemined, fo all Fo abitay fuzzy numbes x ( x ( ), x ( )), y ( y ( ), y ( )) and eal numbe, we define the addition and the scala multiplication of fuzzy numbes by using the extension pinciple as: 1. x y x() y(), y() x(), 2. kx kx( ), kx( ), k 0, 3. kx kx( ), kx( ), k 0, 4. x y x() y(), x() y() Definition 4.2: Two fuzzy numbes and ae equal and we show it by if and only if x(), y(), y() x(), x y x y fo all 0##1. Definition 4.3: A fuzzy vecto (,..., ) T x x1 x n x x(), x (),1 i1, j i i given by n j1 n j1 is called a solution of the fuzzy linea system (4.1), if satisfy the equations. Now if a ij > 0, we simply get ax() y(), ij j i ax() y(), ij j i Consequently, in ode to solve the system given by Eq. (4.1), one must solve a 2n 2n cisp linea system (4.2) whee the ight-hand side column is the function vecto system ( y,..., y, y,..., y ) T 1 n 1 1. We get the 2n 2n linea 11

6 S11x1 S12 x2... S1nx1 n S1, n1x1... S1,2 nxn y1 Sn1x1Sn2x2... Snnx1 n Sn, n1( x1)... Sn 1,2n( xn) y1 S x S x... S x S ( x )... S ( x ) y 2n 1 2n 2 2 n, n 1n 2 n, n1 1 2 n,2n n n whee S ij ae detemined as follows: aij 0 Sij Sin, jn aij, 1 in,1 j n. aij 0 Sin, j Si, jn aij, (4.4) and any S ij which is not detemined is zeo such that: A=B-C (4.5) Using matix notation we get SX=Y (4.6) whee T X ( x,..., x, x,..., x ), Y ( y,..., y, y,..., y ) Sij 0, 1 i, j 2 n, and n n n n T Theefoe, B CX Y C B X Y Below, we give some theoems and esults ae taken fom Matinfa et al., (2008) and Nassei, (2008). Theoem 4.1: The matix S is nonsingula, if and only if the matix B+C and A=B-C ae both non-singulas. Definition 4.4: Let X {( x( ), x ( )),1 in} u {( u ( ), u ( )),1 in} i i i i ui() min xi(), xi(), xi(1), ui() max xi(), xi(), xi(1), defined denote the unique solution of (4.7). The fuzzy numbe vecto (4.8) is called the fuzzy solution of SX=Y. If x x(), x (),1 i n ae all fuzzy numbes, then u () x(), u () x (), fo alli,1#i#n i i i and u is called a stong fuzzy solution. Othewise, is a weak fuzzy solution. 5- Solving an economic poblem using fuzzy linea system of equations: Suppose that demand and supply ae linea functions of the pice: qd a* pb qs c* pd 12 i i i i

7 whee q s is the quantity supplied, which is equied to be equal to q d, the quantity equested, p is the pice and a and c ae cisp eal numbes and b and d ae coefficients to be estimated. Suppose that only some impecise data on the elation between the quantity supplied and equested at a given pice ae available and that the elation can be natually descibed as follows: qd p 5 qs 3p 13 whee the coefficients b and d ae epesented by tiangula fuzzy numbes. Now if we denote x x 1 and 2 espectively as the fuzzy value of supply (demand) and the fuzzy value of pice, by imposing the equality between quantities supplied and equested, the following fuzzy linea system should be solved: x1 x2 5 x1 3x2 13 If we shift some unknowns to the othe side of the equations, we get: x1x2 5 x1 3x2 13 Now if we conside the tiangula fuzzy numbes 5 and 13 as 5 = (3 + 2, 7-2) and 13 = (9 + 4, 17-4), espectively then the supply (and demand) and the pice will be achieved by solving the following fuzzy linea system. x1x2 (3 2,7 2 ) x1 3 x2 (9 4,17 4 ) So, the extended 44 matix is S And finally we may obtain the solution by using one of the methods in the liteatue of fuzzy linea systems as follows (fo finding some methods see Matinfa et al., (2008) and Nassei, (2008): x x (), x () (6,8 ), x x (), x () (1,3 ) Theefoe, the fuzzy value of supply (demand) and the fuzzy value of pice ae 7 and 2, espectively. Conclusion: We applied a fuzzy vesion of the linea system of equations to obtain the demand (supply) and pice whee the equilibium condition of a good maket in the fuzzy envionment is hold. We guess the application fuzzy linea system of equation will be useful in othe fields of economic sciences too. 13

8 ACKNOWLEDGMENT The fist autho thanks to the Reseach Cente of Algebaic Hype-stuctue and Fuzzy Mathematics and Ianian National Elite Foundation fo thei suppots. REFERENCES Baumol, W.J., Economic Theoy and Opeations Analysis, 4th Edition, Pentice-Hall. Maddala, G.S. and E. Mille, Micoeconomics Theoy and Applications, McGaw-Hill. Matinfa, M., S.H. Nassei and M. Sohabi, Solving fuzzy linea system of equations by using Householde decomposition method, Applied Mathematical Sciences, 52: Nassei, S.H., Solving Fuzzy Linea System of Equations by use of the matix decomposition, Intenational Jounal of Applied Mathematics, 21: