CHAPTER INTRODUCTION 2. Chapter REVIEW OF LITERATURE LINEAR PROGRAMMING PROBLEM TRANSPORTATION PROBLEM 14

Transcription

1 CHAPTER INTRODUCTION REVIEW OF LITERATURE LINEAR PROGRAMMING PROBLEM TRANSPORTATION PROBLEM BASIC OPERATIONS ON FUZZY SETS STATEMENTS USED IN C PROGRAM NOTES AND COMMENTS 24 1

2 CHAPTER - 1 INTRODUCTION 1.1 REVIEW OF LITERATURE M. Sharif Uddin[19], in his paper - Transportation Time Minimization: An Algorithmic Approach, discussed an initial basic feasible solution of transportation problem with equal constraints, in minimization of time, initial basic feasible solution obtained by this is near to optimal solution. Khalid M. Altassan, Mahmoud M. El-Sherbiny and Bokkasam[15], in their paper - Near optimal solution for the step fixed charge Transportation Problem, three formulae are proposed to construct intermediate coefficient matrix as a base for finding an initial solution for SFCTP. Abdul Quddoos, Shakil Javaid, M.M. Khalid[4], in the paper named - A new method for finding an optimal solution for transportation problems, proposed a new method called ASM method for finding an optimal solution for a transportation problem. Paul S. Dwyer s[26], in the paper - The Direct Solution of the Transportation Problem with Reduced Matrices, gives discussion of importance of a direct method in obtaining all the solution of a transportation problem and in obtaining solution of more general problems. Debiprasad Acharya, Manjusri Basu and Atanu Das[10], in their paper - Discounted Generalized Transportation Problem, authors considered cost of transportation (c ij ) depends upon the amount of transported commodity (x ij ). Based on these situation, developed a new algorithm for obtaining the optimum solution of this problem. P. Pandian and G. Natarajan[25], presented a blocking method for finding an optimal solution to bottleneck transportation problems. Then, for finding all efficient solutions of a bottleneck-cost transportation problem, another method namely, blocking zero point method is proposed which is based on zero point method in the paper - A new method for solving Bottleneck Cost Transportation Problems. M. Shanmugasundari, K. Ganeshan[18], in their paper - A Novel Approach for Fuzzy Optimal Solution of Fuzzy Transportation Problem, given a new method for fuzzy optimal solution to the transportation problem with fuzzy parameters is given. 2

3 They developed a fuzzy version of Vogels and MODI algorithms for finding fuzzy basic feasible and fuzzy optimal solution of fuzzy transportation problem without converting them to classical transportation problem. Bodkhe S.G., M. H. Lohgaonkar s[7],in their paper - Fuzzy Goal Programming Model for the unbalanced Transportation Problem with hyperbolic membership function, fuzzy goal programming models can be formulated for unbalanced transportation problems by using the basic notions of fuzzy subsets and those problems can be solved by non-linear membership function. N. Kishore and AnuragJayswal s[23], in their paper - Prioritized goal programming Formulation of an Unbalanced Transportation Problem with Budgetary Constraints: A Fuzzy Approach, introduces fuzziness in prioritized goal programming formulation of an unbalanced transportation problem with budgetary constraints keeping budget on first priority. Shugani Poonam, Abbas S.H., Gupta V.K.[32], in their paper - Fuzzy Transportation Problem of Triangular Numbers with α-cut and ranking Technique., considered transportation problem with fuzzy cost a ij, the objective function is also considered as a fuzzy number, the Robust Ranking method with alpha optimal solution for solving fuzzy transportation problem is used. Fuzzy demand and supply all are in the form of triangular fuzzy numbers. Zrinka Lukac, Dubravko Hunjet, Luka Neralic[37], in their paper named - Solving the Production Transportation Problem in the Petroleum Industry,formulated two new production transportation models are (i) the model of discrete production transportation problem. (ii) themodel by level discrete production transportation problem. The model applied on two sets of available real life data from a petroleum industry. Bablu Jana and Tapan Kumar Roy[[6], gavethe solution procedure of multiobjective fuzzy linear programming problem with mixed constraints and its application in solid transportation problem is presented in their paper- Multi-objective Fuzzy Linear Programming and Its Application in Transportation Model. Chakraborty Ananya, Chakraborty M.[9], has been proposed a method for the minimization of transportation cost as well as time of transportation. When the demand, supply and the transportation cost per unit of quantities are fuzzy. The problem modelled as multi-objective LPP with imprecise parameters. Fuzzy 3

4 parametric programming has been used to handle impreciseness and the resulting multi-objective problem has been solved by prioritized goal programming approach in the paper - Cost-Time Minimization in Transportation Problem with Fuzzy Parameters: A Case Study. W. Ritha and J. MerlineVinotha[35], used fuzzy geometric programing approach to determine the optimal solution of a multi-objective two stage fuzzy transportation problem in which supplies and demands are trapezoidal fuzzy numbers and fuzzy membership of objective function is defined in paper named - Multiobjective Two stage Fuzzy Transportation Problem. Bodkhe S.G., Bajaj V.H. and Dhaigude R.M.[7], used the method for solving bi-objective transportation problem where objectives are considered as fuzzy, the fuzzy programming technique with hyperbolic membership function in Fuzzy programming technique to solve bi-objective transportation problem. V.J. Sudhakar, V Navaneetha Kumar[33], solved multi-objective two stage fuzzy transportation problem in a feasible method. For this solution zero-suffix method is used in which supplies and demands are trapezoidal fuzzy numbers and fuzzy membership of the objective function is defined in the paper - Solving Multiobjective Two Stage Fuzzy Transportation Problem by Zero Suffix Method. Rachida Abounacer, Monia Rekik[28], Jacques Renaud, proposed an adaptive epsilon-constraint method and prove that it generates the set of exact pare to front of a complex three-objective location-transportation problem in paper - An Exact Solution Approach for Multi-Objective Location-Transportation Problem for Disaster Response. Mistuo GEN, Yinzhen LI[22], suggested a new approach which is spanning tree-based genetic algorithm for solving a multi-objective transportation problem is presented in the paper - Solving Multi-Objective Transportation Problem by Spanning Tree-Based Genetic Algorithm. Yousria Abo-Elnaga, Bothina El-Sbky, HanadiZahed[36], used a weighting approach together with an active set strategy and multiplier method to transform (MOT) problem to unconstrained optimization problem and also used a trust-region algorithm to solve it inthe paper - Trust Region Algorithm for Multi-Objective Transportation, Assignment, and Transshipment Problems. 4

5 M.Zangiabadi and H. R. Maleki[20], used a special type of nonlinear (hyperbolic and exponential) membership functions to solve multi-objective transportation problem in the paper - Fuzzy Goal Programming Technique to solve Multiobjective Transportation Problem with some Non-linear Membership functions. V.J. Sudhakar and V. Navaneetha Kumar[34], solved multi-objective two stage fuzzy transportation problem in a feasible method. For obtaining solution zero suffix method is used in which the supplies and demands are trapezoidal fuzzy numbers and fuzzy membership of the objective function is defined inthe paer - A Different Approach for Solving Two Stage Fuzzy Transportation Problems. A.A. Mousa, Hamdy M. Geneedy and Adel Y Elmekawy[1], presented an improved algorithm for solving MOTP, the algorithm is the integration of GA and local search technique which improves the quality of the founded solution inthe paper - Efficient Evolutionary Algorithm for solving Multi-objective Transportation Problem. Deshabrata Roy Mahapatra, Shankar Kumar Roy and Mahendra Prasad Biswal[[11], applied fuzzy programming technique to the objective function and the stochastic method for the randomness of sources and destination parameters in inequality type of constraints of multi-objective stochastic unbalanced transportation problem in the paer - Stochastic Based on Multi-Objective Transportation Problem Involving Normal Randomness, AMO-Advanced Modeling and Optimization. Lohgaonkar M.H., Bajaj V.H., Jadhav V.A. and Patwari M.B.[16], used special type of linear and non-linear membership functions to solve multi-objective multi-index transportation problem which gives an optimal compromise solution inin the paer - Fuzzy Multi-Objective Multi-index Transportation Problem. Sayed A Zaki,Abd Allah A.Mousa,Hamdy M Geneedi,Adel Y. Elmekawy[31], in the paper - EfficientMulti-objective Genetic Algorithm for Solving Transportation, Assignment, and Transshipment Problems, gave the algorithm that maintains a finite-sized archive of non-dominated solutions which gets iteratively updated in the presence of new solutions based on clustering algorithm. K Venkatasubbaiah,S.G. Acharyulu, K V V Chandra Mouli[14], considered fuzzy membership functions and deviation goals taken for each objective function. Fuzzy max-min operator is implemented to show the effectiveness of proposed methodology. Lingo software package is used to solve constrained optimization 5

6 problem in the paper - Fuzzy Goal Programming Methods for Solving Multi-objective Transportation Problems, Global Journal of Research in Engineering. ANDREI JIRNYI[5], presented in the handout - Using the solver add-in in MS-Excel 2007,that how to install the solver add-in. By using this tool, how to perform simple optimizations and solve the equations using Excel. In - Calling the LP_Solve Linear Program Software from R, S-Plus and Excel, Samuel E Buttrey[30], presented a link that allows R, S-Plus and Excel to call the function in the lp_solve system. This link allows Excel users to handle substantially larger problems at no extra cost. P. Lavanya Kumari, K. Vijaya Kumar[24], focuses on the simplest method of solving linear programming problem and transportation problem using solver tool available in MS Excel in - Some aspects of operations research using solver. Michel Berkelaar and others[21], presented some of the R functions that solved general linear/integer problems, assignment problems and transportation problems using the lp_solve version 5.5 in the paper - Interface to LP_Solve V.5.5 to solve linear integer programs. Zulkipli Ghazali, M. Amin Abd Majid and Mohd Shazwani[38], highlighted the case study of four distribution centers to four plants of Malaysian Company using LPP and spreadsheets in the research article - Optimal Solution of Transportation Problem using LPP a case of Malaysian Trading Company. R.E. Bellman and L.A. Zadeh[27], developed a general theory of decision making in fuzzy environment in paper - Decision Making in a Fuzzy Environment. 1.2 LINEAR PROGRAMMING PROBLEM In 1947, George Dantzig and his associates while working in the US department of Air force, observed that a large number of military programming and planning problems could be formulated as maximizing/minimizing a linear form of profit/cost function whose variables were restricted to values satisfying a system of linear 6

7 constraints (a set of linear equations / or inequalities). A linear form is meant a mathematical expression of the type a 1 x 1 +a 2 x 2 + +a n x n, where a 1, a 2,..., a n are constants, and x 1, x 2,... x n are variables. The term programming refers to the process of determining a particular programme or plan of action. So linear programming (L.P.) is one of the most important optimization (maximization / minimization) techniques developed in the field of Operations Research (O.R.). The methods applied for solving a linear programming problem are basically simple. For simple problem, a solution can be obtained by a set of simultaneous equations. However, a unique solution for a set of simultaneous equations in n-variables (x 1, x 2,..., x n ) can be obtained if there are exactly n relations, at least one of them is nonzero. When the number of relations is greater than or less than n, a unique solution does not exist, but a number of trial solutions can be found. In various practical situations the problems are seen in the number of relations is not equal to number of variables and many of the relations are in the form of inequalities ( or ) to maximize (or minimize) a linear function of the variables subject to such conditions. Such problems are known as Linear Programming Problems (LPP). The general LPP calls for optimizing (maximizing or minimizing) a linear function for variables called the objective function subject to a set of linear equations and /or inequalities called the constraints or restrictions Mathematical formulation of the linear programming problem: The procedure for mathematical formulation of LPP consists of the following steps: 1. Write down the decision variables of the problem. 2. Formulate the objective function to be optimized (maximized or minimized) as a linear function of the decision variables. 3. Formulate the other conditions of the problem such as resource limitations, market constraints and inter-relation between variables etc. as a linear equations or inequations in terms of the decision variables. 4. Add the non-negativity constraint from the consideration that negative values of the decision variables do not have any valid physical interpretation. The objective function, the set of constraints and the non-negativity constraint together form a Linear Programming Problem. 7

8 Chapter - 1 General Linear Programming Problem: Definition 1: Let z be a linear function on R n defined by..(1.1) Where c j s are constants. Let (a ij ) be an m x n real matrix and let { b 1, b 2,..., b m } be a set of constants such that (1.2) And 0 1,2,..(1.3) The problem of determining an n tuple (x 1, x 2, x n ) which makes z a minimum (or maximum) and which satisfies (1.2) and (1.3) called the general linear programming problem. Objective function: The linear function. which is to be minimized (or maximized) is called the objective function of the general LPP. Constraints: The inequations (1.2) are called the constraints of the general LPP. Non-negative restrictions: The set of inequations in (1.3 ) is usually known as the non negative restrictions of the general LPP. Definition 2: (Solution) : An n-tuple (x 1, x 2, x n ) of real numbers which satisfies the constraints of a general LPP is called a solution to the general LPP. Definition 3: (Feasible Solution) : Any solution to a general LPP which also satisfies the non negative restrictions of the problem is called a feasible solution to the general LPP. 8

9 Definition 4: (Optimum Solution): Any feasible solution which optimizes (maximizes or minimizes) the objective function of a general LPP is called an optimum solution to the general LPP Some Important Definitions It has not been possible to get the graphical solution to the LP problem of more than two variables. The analytic solution is not possible because the tools of analysis are not well suited to handle inequalities. In such cases a simple and most widely used Simplex Method is adopted which was developed by G. Dantzig in The simplex method provides an algorithm (a rule of procedure usually involving repetitive application of a prescribed operation) which is based on the fundamental theorem of linear programming. Fundamental theorem of linear programming problem states that the collection of all feasible solutions to L.P. problems constitutes a convex set whose extreme points corresponds to the basic feasible solutions. The simplex method provides a systematic algorithm which consists of moving from one basic feasible solution (one vertex) to another in a prescribed manner such that the value of the objective function is improved. The procedure of jumping from vertex to vertex is repeated. If the objective function is improved at each jump, then no basis can ever repeat and there is no need to go back to vertex already covered. Since the number of vertices is finite, the process must lead to the optimal vertex in a finite number of steps. Basic terms involved in Simplex procedure: 1. The Standard Form: The general LPP in the form! "! ". Subject to the constraints: # #... # #, 1, 2,.,,,., 0 9

10 is known as in standard form. The characteristic of this form are: (i) (ii) All the constraints are expressed in the form of equations, except for the non-negative restrictions. The right hand side of each constraint equation in non-negative. 2. Slack variable: A variable added to the left hand side of less than or equal to constraint to convert the constraint into equality. The value of this variable can usually be interpreted as the amount of unused resource. 3. Surplus variable: A variable subtracted from the left hand side of a greater than or equal to constraint to convert the constraint into equality. The value of this variable can usually be interpreted as the amount over and above the required minimum level. 4. Basic solution: For a general linear programming problem with n variable and m constraints, a basic solution may be obtained by setting (n-m) variables equal to zero and solving the constraint equations for the remaining m variables provided the determinant of coefficients of these m variables is nonzero. Such m variables (any of them may be zero) are called basic variables and remaining n-m zero variables are called non-basic variables. The number of basic solutions thus obtained will be at most $! %!!! ' (!)!, Which is number of combinations of n things taken m at a time. 5. Basic feasible solution: A basic feasible solution which also in feasible region (that is it satisfies the non-negativity requirement). A basic feasible solution corresponds to a corner point of the feasible region. Basic feasible solutions are of two types: a) Non-degenerate A non degenerate basic feasible solution is the basic feasible solution which has exactly m positive x i (i=1,2,.m). In other words, all m basic variables are positive, and remaining n-m variables will be zero. b) Degenerate A basic feasible solution is degenerate if one or more basic variables are zero. 10

11 6. Optimum basic feasible solution: A basic feasible solution is said to be optimum if it optimizes (maximizes or minimizes) the objective function. 7. Unbounded solution: If the value of the objective function increased or decreased indefinitely, such solutions are called unbounded solutions The Simplex method (Maximization case): The stepwise procedure of simplex method: Step 1 : Formulation of the mathematical model. (a) Formulate a linear programming model of the real word problem that is obtain a mathematical representation of the problem s objective function and constraints. (b) Express the mathematical model of LP problem in the standard form by adding slack variables in the left hand side of the constraints and assign a zero coefficients to these in the objective function. Step 2: Set up the Initial Solution. The initial basic feasible solution is obtained by assigning zero value to (n-m) decision variables to initiate the solution procedure from the origin.where n = number of variables and m = number of equations. Complete the initial simplex table by adding two rows C j (cost coefficients in the objective function for the corresponding variables)and Z j - C j (net evaluations), where *, -# #, j=1, 2,,n. Step 3: Test the solution for optimality. Examine the index row or net evaluation row of the simplex table (a) If all * (, 0, then we get an optimum solution (b) If at least one * (,. 0, proceed onto the next step (by putting these information into the following table called as initial simplex table). Step 4: If there are more than one negative * (, then choose most negative of them. Corresponding variable is an entering variable and the column is pivot or key column. Step 5: Then compute minimum ratios using formula 11

12 . /. 0-0, 0 1 0, " 56! " " "3" 9 : 5" Step 6: The variable corresponding to the minimum ratio is outgoing or living variable and corresponding row is key row. Key element: The element at the intersection of incoming vector and outgoing vector is called key element. It is always a non zero positive number. Step 7: Convert the key element to unity by dividing its row element by the key element and other elements in its row to zero by using the relation ;"< "5"!"3 58 "5"!"3 ( "2 "5" ="> <? "2 "5"!"3 ="> "5"!"3 Step 8: Go to step 3 and repeat the computation procedure until the optimal solution is obtain. 12

13 Flow Chart 1 Reformulate the given LPP as a standard maximization LPP Obtain an initial basic feasible solution to the problem Compute the net evaluationsand set-up the starting simplex table Update the simplex table by appropriate operations (pivoting) Examine the row of net evaluations Design a new table. Remove the leaving variable from the basis and introduce the entering one Is there any negative net evaluations? YES NO An optimum solution has been attained the current basic solution is optimal Choose the smallest ratio. The basic variable corresponding to it leaves the basis Choose the most negative net evaluation. The nonbasic variable corresponding to it enters the basis Select the positive and divide the corresponding values of the current basic variables by them NO Are all the constraint coefficients of the entering variable negative? YES There exists an unbounded solution to the problem 13

14 1.3 TRANSPORTATION PROBLEM The Transportation Problem is one of the sub-classes of L.P.Ps in which the objective is to transport various quantities of a single homogeneous commodity, that are initially stored at various origins, to different destinations in such a way that the total transportation cost is minimum. To achieve this objective we must know the amount and location of available supplies and quantities demanded. In addition, we must know the costs that result from transporting one unit of commodity from various origins to various destinations. A single objective transportation problem can be stated mathematically as a LPP as below: Subject to the constraints:! " A A # # #B B A # #, 1, 2,,! B A # #B, 1, 2,, # 0 C 55 8 Where, a i = quantity of commodity available at origin i, b j = quantity of commodity needed at destination j, c ij = cost of transporting one unit of commodity from origin i to destination j, and x i j = quantity transported from origin i to destination j. 14

15 Definitions:- Solution The real numbers, x ij ( i = 1, 2,.m, j = 1, 2,.n ) is called a solution of transportation problem if it satisfies the constraints of the transportation problem. Feasible Solution A set of non-negative individual allocations (x ij 0) which simultaneously removes deficiencies is called a feasible solution. Basic Feasible Solution A feasible solution to a m-origin, n-destination problem is said to be basic if the number of positive allocations are m + n 1, i.e., one less than the sum of rows and columns. If the number of allocations in a basic feasible solution are less than m + n 1 it is called degenerate Basic Feasible Solution (otherwise, non-degenerate Basic Feasible Solution). Optimal Solutions A feasible solution (not necessarily basic) is said to be optimal if it minimizes the total transportation cost IBFS of Transportation Problem by North West Corner Rule (NWCR): Step 1: Starting with the cell at the upper left (north-west) corner of the transportation matrix, we allocate as much as possible so that either the capacity of the first row is exhausted or the destination requirement of the first column is satisfied, i.e., x 11 = min, (a 1, b 1 ). Step 2: If b 1 1 a 1, we move down vertically to the second row and make the second allocation of magnitude x 21 = min. (a 2, b 1 x 11 ) in the cell (2, 1). If b 1. a 1, we move right horizontally to the second column and make the second allocation of magnitude x 12 = min. (a 1 x 11, b 2 ) in the cell (1, 2). If b 1 = a 1, there is a tie for second allocation. One can make the second allocation of magnitude. x 12 = min. (a 1 a 1, b 2 ) = 0 in the cell (1, 2). or x 21 = min. (a 2,b 1 b 1 ) = 0 in the cell (2, 1). 15

16 Step 3: Repeat steps 1 and 2 moving down towards the lower right corner of the transportation table until all the rim requirements are satisfied IBFS of Transportation Problem by Least Cost or Matrix Minima Method Step 1. Determine the smallest cost in the cost matrix of the transportation table. Let it be c ij. Allocate x ij = min (a i, b j ) in the cell (i,j). Step 2. If x ij = a i, cross off the ith row of the transportation table and decrease b j by a i. Go to step 3. If x ij = b j, cross off the jth column of the transportation table and decrease a i by b j Go to step 3. If x ij = a i = b j, cross off the ith row as well as jth column of the transportation table. Go to step 3. Step 3. Repeats step 1 and step 2 for the resulting reduced transportation table until the rim requirements are satisfied. Whenever minimum cost is not unique, make an arbitrary choice among the minimum IBFS of Transportation Problem by Vogel s Approximation Method Step 1. For each row of the transportation table identify the smallest and next to smallest cost. Determine the difference between them for each row called row penalty. Write them alongside the transportation table against the respective row. Similarly compute penalties. Step 2. Identify the row or column with the largest penalty among all the rows or columns. If a tie occurs, break tie arbitrarily. Let the greatest difference corresponds to ith row and let c ij be the smallest cost in the ith row. Allocate the maximum feasible amount x ij = min (a i, b j ) in the ( i, j ) th cell and cross out the i th row or j th column in usual manner. Step 3. Recompute the column and row penalties for the reduced transportation table and go to step 2. Repeat the procedure until all the rim requirements are satisfied. 16

17 1.3.4 Method of obtaining Optimal Solution of transportation problem by UV Method or MODI (Modified distribution) method Step 1. Construct a transportation table entering the origin capacities a i, the destination requirements b j and the cost c ij, check that the given problem is balanced one. Step 2. Determine an Initial Basic Feasible Solution by using any one of the three methods. Step 3. For all the basic variables or allocating cells, compute the numbers u i, i = 1, 2,.m and v j, j = 1, 2,.n. by using relation u i + v j = c ij starting initially with any u i or v j = 0 ( Generally we take u i = 0). Step 4. Compute the net evaluations d ij, d ij = c ij (u i + v j )., for non-allocating cells or non-basic variables. Step 5. Examine the sign of each d ij if all d ij 0, we reach at the optimal solution. If at least one is negative then go to the next step. Step 6. Select the variable which is most minimum among the negative net evaluation. Assign a positive and negative sign to the extreme cells after forming the loop starting from new basic cell. Step 7. Observe the allocation at a negative sign and choose the most minimum allocation. Let it be x, subtract x from all allocations where the sign is negative and add to x where the sign is positive and form a new transportation table and get new solution. Step 8. Return to step 3 and repeat the procedure until an optimal solution has been obtained. 17

18 Flow chart 2 Construct a transportation table for the problem. Determine an initial basic feasible solution to the problem by any method for initial solution. Select a maximum value of Θ so that one of the basic cells become empty maintaining feasibility of new solution. Compute cell-evaluations: d ij = c ij (u i + v j ) for empty (non-basic) cells. Identify a loop which starts and ends at this cell and connecting some or all basic cells. Make ±Θ adjustment in the cells at the corners of this loop maintaining feasibility. Is there any negative cell-evaluation? YES Select the largest negative cell-evaluation for indicating the cell entering the solution Yes (basis). NO The solution under test is optimal. 1.4 BASIC OPERATIONS ON FUZZY SETS Consider the fuzzy sets DE and FG in the universe U, DE HI0,J K G')LM, J K G') N O0,1P, FG HI0, J - G')LM, J - G') N O0,1P. The operations with DE 8 FG are introduced via operations on their membership function J K G') 8 J - G'). 18

19 1. Equality The fuzzy sets DE and F, Q DE = FG, if and only if for every N R, J K G') J - G'). 2. Inclusion The fuzzy set DE is included in the fuzzy set F Q denoted by DE S FG if for every N R, J K G') J - G'), then DE is called a sub set of FG. 3. Proper subset The fuzzy set DE is called a proper subset of the fuzzy set F Q denoted by DE T FG when DEthe subset of isfg and DE U FG i.e. J K G') J - G') for every N R J K G') T J - G') for at least one N R 4. Intersection The intersection of the fuzzy set DE and F Q denoted by DE V FG is defined by J K GWJ - G')! 'J K G'),J - G')) If., min', ). For instancemin'5, 7) 5 5. Union The union of the fuzzy DE and F Q denoted by DE R FG is defined by J K G R J - G')!'J K G'), J - G')) If., max', ).For instancemin'5, 7) Fuzzy Relations Consider the Cartesian product D? F _', >) N D, > N Fa, Where D? F are the subset of the universal sets R and R respectively. A fuzzy relation on D? F denoted by b or b', >) is defined as the set 19

20 b _', >), Jb', >) ', >) N D? F, Jb', >) N O0, 1Pa Where Jb', >)function in two variables is called membership function. It gives the degree of membership of the ordered pair (, y) in D? F, a real number in the interval [0, 1]. The degree of membership indicates the degree to which x is in relation with y. 7. Operations on Fuzzy Relations c"3 b 8 b " 3< "53 2 D? F b _', >), J ', >)a, ',>) N D? F, b _', >), J ',>)a,', >) N D? F, 8. Equality b b C 8 5> C C ":"> 7 ', >) N D? F Jb ', >) Jb ', >) 9. Inclusion dc C ":"> 7 ', >) N D? F Jb ', >) Jb ', >) 34" "53 b 568"8 b b 2 59" 34 b,8"3"8 > b S b C b S b C C 3 5"23 7 ', >) Jb ', >). Jb ', >) Then we have proper inclusion b T b 10. Intersection e4" 3"2"3 Jb 8 Jb 8"3"8 > Jb V Jb 2 8"C "8 > Jb V Jb ', >)! _Jb ', >),Jb ', >)a', >) N D? F. 20

21 11. Union e4" 6 Jb 8 Jb 8"3"8 > Jb R Jb 2 8"C "8 > Jb R Jb ', >)!_Jb ',>), Jb ', >)a', >) N D? F. The operations of intersection and union on fuzzy relations are illustrated in following Example. 12. Illustration c"3 C6> 2"3 DE _,, f a 8 F Q _>,>, > f a " 343,3" "3<"" DE 8 FG 2 9 :" 2 C55<: x 1 x 2 x 3 b f y 1 y 2 y 3 b > > > f y 1 y 2 y 3 b V b f y 1 y 2 y 3 b R b f

22 1.5 STATEMENTS USED IN C PROGRAM 1. Scanf: By using C library function scanf input data can be entered into the computer from a standard input device. In general terms scanf function is written as: scanf(control string, argument 1, argument 2, argument n) where control string refers to a string that contains formatting information and argument 1, argument 2, argument n are arguments that represent the individual input data items.( Actually the arguments represent pointers that indicate the addresses of the data items within the computer s memory.) 2. Printf: The printf function can be used to output any combination of numerical values, single characters, and strings. The printf function moves data from the computer s memory to the standard output device. Whereas the scanf function enters data from the standard input device and stores it in computer s memory. In general terms, printf function is written as : printf (control string, argument 1, argument 2, argument n) where control string refers to a string that contains formatting information and argument 1, argument 2, argument n are arguments that represent the individual output data items. The arguments in printf function do not represent memory addresses and therefore they do not preceded by ampersands (&). 3. While statement: The general form (Syntax) of the statement is, While(expression)statement The included statement executed repeatedly, as long as the value of the expression is not zero. The statement can be simple or compound, though it is typically a compound statement. It must include some feature which eventually alters the value of expression. Thus providing stopping condition for a loop. In practice the included expression is usually a logical expression that is either true or false. Thus the statement executed as long as the logical expression is true. 4. For statement : The general form (syntax) of for statement is, For(expression 1; expression 2; expression 3) 22

23 Statement Where statement 1 is used to initialize some parameter that controls the looping action expression 2 is a condition that must be satisfied for the loop to continue execution and expression 3 is used to alter the value of the parameter initially assigned by expression 1. Expression 1 : assignment expression Expression 2 : logical expression Expression 3 unary or an assignment expression. When for statement is executed expression 2 is evaluated and tested before each pass through the loop and expression 3 is evaluated at the end of each pass. Thus the for statement is equivalent to Expression 1; While(expression 2){ Statement; Expression 3;} The looping action will continue as long as the value of expression 2 is not zero, that is as long as the logical condition represented by expression 2 is true. Nested loops Loops can be nested (embedded) within one another. The inner and outer lops need not be generated by the same type of control structure. 5. The if else statement: The if else statement is used to carry out a logical test and then take one of two possible actions, depending on the outcome of the test (i.e. whether the outcome is true or false). The else portion of the if else statement is optional. The general form is, if (expression) statement The expression must be placed in parentheses as shown. In this form, the statement will be executed only if the expression has non zero value (i.e. the expression is true). If the expression has a value zero (i.e. if the expression is false) then statement will be ignored. The statement can be simple or compound. 23

24 6. The goto statement: The goto statement is used to alter the normal sequence of program execution by transferring control to some other part of the program. General form of goto statement is, Goto label; Where label is an identifier used to label the target statement to which control will be transferred. Control may be transferred to any other statement within the program. The target statement must be labeled and label must be followed by a colon. Thus the target statement will appear as Label: statement Each labeled statement within the program must have a unique label that is no two statements can have the same label. 1.6 NOTES AND COMMENTS The classical transportation problem is one of the many well-structured problems in operations research that has been extensively studied in literature. The transportation problem is one of the subclasses of linear programming problems for which simple and practical procedures have been developed. As well as many of the softwares are developed for solving these problems. There are different types of transportation problems and the simplest of them that is now standard in the literature was first presented by Hitchcock(1941), along with a constructive solution and, later independently, by Koopman(1947). Kantorovich (1942) published a paper on a continuous version of the problem and later with Gavurin, an applied study of the capacitated transportation problem. Many times not all problems are of single objective, therefore many researchers [1,3,11,14,16,20,28,29,31,36] studied the multiobjective transportation problem. In the present work we also studied the multiobjective transportation problem and tried to develop the method to find the solution of multi-objective transportation problem. As well as we have developed the C- programme to find the solution of the developed method and one more method[3] in the literature. 24