KEYWORDS Fuzzy numbers, trapezoidal fuzzy numbers, fuzzy Vogel s approximation method, fuzzy U-V distribution method, ranking function.

Transcription

1 Applications (IJERA ISSN: Method For Solving The Transportation Problems Using Trapezoridal Numbers Kadhirvel.K, Balamurugan.K Assistant Professor in Mathematics,T.K.Govt.Arts College, Vriddhachalam Associate professor in Mathematics, Govt.Arts.College - Thiruvannamalai. ABSTRACT The fuzz set theor has been applied in man fields such as operation research, management science control theor etc. The fuzz numbers fuzz values are widel used in engineering applications because of their suitabilit for representing uncertain information. In stard fuzz arithmetic operations we have some problem in subtraction multiplication operations. In this paper, we investigate fuzz transportation, Problem with the aid of trapezoidal fuzz numbers. U-V distribution method is proposed to find the optimal solution in terms of fuzz numbers. A new relevant numerical example is also included. KEYWORDS numbers, trapezoidal fuzz numbers, fuzz Vogel s approximation method, fuzz U-V distribution method, ranking function. INTRODUCTION The transportation problem (TP refers to a special class of linear programming problems. In a tpical problem a product is to transported from m sources to n designations their capacities are,. respectivel. In additions there is a penalt associated with transporting unit of product from source i to destination j. This penalt ma be cost or deliver time or safet of deliver etc. A variable represents the unknown quantit to be shipped from some i to destination j. Iserman (1 introduced algorithm for solving this problem which provides effective solutions. Lai Hwang(5, Bellman Zadeh (3 proposed the concept of decision making in fuzz environment. The Ringuest Rinks (2 proposed two iterative algorithms for solving linear, multicriteria transportation problem. Similar solution in Bit A.K (8. In works b S.Chanas D.Kuchta (10 the approach introduced in work (2. In this paper, the fuzz transportation problems using trapezoidal fuzz numbers are discussed. Here after, We have to propose the method of fuzz U-V distribution method to be finding out the optimal solution for the total fuzz transportation minimum cost. 2. FUZZY CONCEPTS L.A.Zadeh advanced the fuzz theor in The theor proposes a mathematical technique for dealing with imprecise concept problems that have man possible solutions. The concept of fuzz mathematical programming on a general level was first proposed b Tanaka et al (1947 in the frame work of the fuzz decision of bellman Zadeh(3 now, we present some necessar definitions Definition:- A real fuzz number is a fuzz subset of the real number R. with membership function satisfing the following conditions. 1. is continuous from R to the closed interval [0,1] 2. is strictl increasing continuous on [ ] 3. is strictl decreasing continuous on [ ] Where are real numbers, the fuzz number denoted b = [ ] is called a trapezoidal fuzz number Definition:- The fuzz number = [ ] is a trapezoidal number, denoted b [ ] its membership function figure. (x 1 is given below the 0 a1 a2 a3 a P a g e

2 Applications (IJERA ISSN: Fig: Membership function of a fuzz number 2.3 Definition:- We define a ranking a function R: F(R R, Which maps each fuzz number into the real line, F(R represents the set of all trapezoidal fuzz numbers, If R be an ranking function, then R ( = ( / Arithmetic Operations:- Let = ( = ( two trapezoidal fuzz numbers them the arithmetic operation on as follows: Additions: + = Subtraction: - = Multiplication:. =[ (, (, (, ( ] Require ment Where if R( >0 The linear programming model representing the fuzz transportation is given b Minimize if R 3. FUZZY TRANSPORTATION PROBLEM Consider transportation with m fuzz origins (rows n fuzz destinations (Columns. Let be the cost of transporting one unit of the product from i th fuzz origin to j th fuzz destination be the quantit of commodit available at fuzz origin i, be the quantit of commodit requirement at fuzz destination j. is quantit transported from i th fuzz origin to j th fuzz destination. The above fuzz transportation problem can be stated in the below tabular form. available Subject to the constraints The given fuzz transportation problem is said to be balanced if if the total fuzz available is equal to the total fuzz requirement. 4. THE COMPUTATIONAL PROCEDURE FOR FUZZY U-V DISTRIBUTION METHODS 4.1. Vogel s Approximathical Methods There are numerous methods for finding the fuzz initial basic feasible solution of a 2155 P a g e

3 Applications (IJERA ISSN: transportation problem. In this paper we are to follow fuzz Vogel s approximation method. Step 1. Find the fuzz penalt cost, namel the fuzz difference between the smallest smallest fuzz costs in each row column. Step 2. Among the fuzz penalties as found in step1 choose the fuzz maximum penalt, b ranking method. If this maximum penalt is more than one, choose an one arbitraril. Step 3. In the selected row or column as b step 2, find out the cell having the least fuzz cost. Allocate to this cell as much as possible depending on the fuzz available fuzz requirements. Step 4. Delete the row or column which is full exhausted. Again compute column row fuzz penalties for the reduced fuzz transportation table then go to step 2, repeat the procedure until all the rim requirement are satisfied. Once the fuzz initial fuzz feasible solution can be computed, the next step in the problem is to determine whether the solution obtained is fuzz optimal or not. optimalit test can be conducted to an fuzz initial basic feasible solution of a fuzz transportation provided such allocations has exact m+n-1 non-negative allocations where m is the number of fuzz origins n is the number of fuzz destinations. Also these allocations must be independent positions, which fuzz optimalit finding procedure is given below U-V distribution method This proposed method is used for finding the optimal basic feasible solution in fuzz transportation problem the following step b step procedure is utilized to find out the same. Step 1. Find out a set of numbers satisfing + for each row column for each occupied cell. To start with we assign a fuzz zero to an row or column having maximum number zero to an row or column having maximum number of allocation, if this maximum numbers of allocation is more than one, choose an one arbitrar. Step 2. For each empt (un occupied cell, we find fuzz sum. Requ irem Step 3. Find out for each empt cell the net evaluation value ent this step gives the optimalit conclusion. Case(i. If all then the solution is fuzz optimal, but an alternative solution exists. Case (ii. If then the solution is fuzz optimal, but an alternative solution exists. Case (iii.if at least one then the solution is not fuzz optimal. In this case we go to next step, to improve the total fuzz transportation minimum cost. Step.4. Select the empt cell having the most negative value of from this cell we draw a closed path drawing horizontal vertical lines with corner cell occupied. Assign Sign + alternatel find the fuzz minimum allocation from the cell having negative sign. This allocation should be added to the allocation having negative sign. This allocation should be added to the allocation having negative sign. Step.5. The above step ield a better solution b making one (or more occupied cell as empt one empt cell as occupied. For this new set of fuzz basic feasible allocation repeat from the step 1, till an fuzz optimal basic feasible solution is obtained. 5. Example: To solve the following fuzz transportation problem stating with the fuzz initial fuzz basic feasible solution obtained b fuzz Vogel s approximation. avail able,16 (8,16,24,3 2 18,2 6,16 0 8,12,16 12, ,0,2,8 8,12 18,2 6 12,1 6 (8,20,38,5 = 2156 P a g e

4 Applications (IJERA ISSN: Solution: The problem is balanced fuzz transportation problem. There exists a fuzz initial basic feasible solution. avail able Requi reme nt 8,12 16 (8,16,24,3 2 10,- 2,6,1 18, ,12 0 8, ,- 2,12, 2 12,16 22,- 6,12, 2 0 2,0,2, 8 8,12 18,26 12,16 (8,20,38,5 Since the number of occupied cell having m+n-1 = 6 are also independent, there exists a non-degenerate fuzz basic feasible solution. Therefore the initial transportation minimum cost is, for each occupied cell. Since 3 rd row has maximum numbers of allocations, we give fuzz number. The remaining numbers can be obtained as given below = + = [-16,-8,0,16] We find, for each empt cell of the sum. Next we find net evaluation is given b Where To find the optimal solution Appling the fuzz U-V distribution methods, we determine a set of numbers that each row column such 2157 P a g e

5 Applications (IJERA ISSN: U-V.Distribution Methods Table 8,12 16 *36,-8,10,46 16 *36,-8,10,46 16 *44,-,52 2,0,2,8 *34,-2,24,4 *28,-2,14,40 10,-2,12,2 (84,32 12,16 10,-2,6,1 8,12 22,-6,12,2 *22,-4,12,38 18, , ,6,18,30 30,-18,-8,12 16,-8,0,16 (0,0,0,0 All the solution is fuzz optimal unique. The optimal solution in terms of trapezoidal fuzz numbers. Hence the total fuzz transportation minimum cost is 6. CONCLUSION We have thus obtained an optimal solution for a fuzz transportation problem using trapezoidal fuzz number. A new approach called fuzz U-V Computational procedure to find the optimal solution is also discussed. The new arithmetic operations of trapezoidal fuzz numbers are emploed to get the fuzz optimal solutions. The same approach of solving the fuzz problems ma also be utilized in future studies of operational research. REFERENCES 1. Isermann.H, The Enumeration of all Efficient Solutions for a Linear Multiple- Objective Transportation Problem Naval Research Logistics Quarterl 26(1979,PP Ringuest J.L.,Rinks.D.B Interactive Solution for the Linear Multi objective Transportation Problem European Journal of operational Research 32(1987, pp, Bellman.R.E,zadech.L.A, Decision Making in a Environment Management Sci.17(1970,PP Chanas.S,Delgado.M,Verdeg.J.L,vila.M.A, Interval Extensions of Classical Transportation Problems Transportation planning technol.17(1993,pp Lai..J.,Hwang.C.L., Mathematical Programming Methods Application,Springer Berlin( Nagoor Gani.A Stephan Dingar.D, A Note on Linear Programming in Environment,Proc.Nat Acda,Sci.India,Sect.A,Vol.79,pt.I( Maleki.H.R., Ranking Functions their Applications to Linear Programming,Far East J.Math Sci.4(2002,PP Bit.A.K, Biswal.M.P,Alam.S.S, Programming Approach to Multicriteria Decision Making Transportation Problem sets sstems 50 (1992 PP Waiel.F, Abd.E.l-Wahed, A Multi Objective Transportation Problem under iness fuzz sets sstems 117(2001 pp Chanas.S Kuchta.D, Integer Transporting Problem, fuzz sets sstems 98 (1998 pp Kasana.H.S,Kumar.K.D, Introductor Operations Research Theor Applications Springer,Berlin ( P a g e