EFFECT OF GRAPHICAL METHOD FOR SOLVING MATHEMATICAL PROGRAMMING PROBLEM

Transcription

1 DODIL INTERNTIONL UNIVERSITY JOURNL O SCIENCE ND TECHNOLOGY VOLUME 5 ISSUE JNURY 9 EECT O GRPHICL METHOD OR SOLVING MTHEMTICL PROGRMMING PROLEM imal Chandra Das Department of Tetile Engineering Daffodil International University Dhaka bcdas@daffodilvarsityedubd bstract: In this paper a computer implementation on the effect of graphical method for solving mathematical programming problem using MTL programming has been developed To take any decision for programming problems we use most modern scientific method based on computer implementation Here it has been shown that by graphical method using MTL programming from all kinds of programming problem we can determine a particular plan of action from amongst several alternatives in very short time Keywords: Mathematical programming objective function feasible-region constraints optimal solution Introduction Mathematical programming problem deals with the optimization (maimization/ minimization) of a function of several variables subject to a set of constraints (inequalities or equations) imposed on the values of variables or decision making optimization plays the central role Optimization is the synonym of the word maimization/minimization It means choosing the best In our time to take any decision we use most modern scientific and methods based on computer implementations Modern optimization theory based on computing and we can select the best alternative value of the objective function []ut the modern game theory dynamic programming problem integer programming problem also part of the optimization theory having wide range of application in modern science economics and management In the present work I tried to compare the solution of Mathematical programming problem by Graphical solution method and others rather than its theoretic descriptions s we know that not like linear programming problem where multidimensional problems have a great deal of applications non-linear programming problem mostly considered only in two variables Therefore for nonlinear programming problems we have a opportunity to plot the graph in two dimension and get a concrete graph of the solution space which will be a step ahead in its solutions We have arranged the materials of the paper in the following way: irst I discuss about Mathematical Programming (MP) problem In second step we discuss graphical method for solving mathematical programming problem and taking different kinds of numerical eamples we try to solve them by graphical method inally we compare the solutions by graphical method and others or problem so consider we use MTL programming to graph the constraints for obtaining feasible region lso we plot the objective functions for determining optimum points and compare the solution thus obtained with eact solutions Mathematical Programming Problems The general Mathematical programming (MP) problems in n-dimensional Euclidean space R n can be stated as follows: Maimize f() subject to g i ( ) i m () () h j ( ) s j Where ( ) p () T n is the vector of unknown decision variables and f ( ) g( ) ( i m) h j () ( j p) are the real valued functions The function f() is known as objective function and inequalities

2 DS: EECT O GRPHICL METHOD OR SOLVING MTHEMTICL PROGRMMING PROLEM () equation () and the restriction () are referred to as the constraints We have started the MP as maimization one This has been done without any loss of generality since a minimization problem can always be converted in to a maimization problem using the identity min f() -ma (- f()) () ie the minimization of f() is equivalent to the maimization of (-f()) The set S is normally taken as a connected subset of R n Here the set S is taken as the entire space R n The set X { s gi () i m j p} is known to as the feasible reason feasible set or constraint set of the program MP and any point is a feasible solution or feasible point of the program MP which satisfies all the constrains of MP If the constraint set is empty (ie φ ) then there is no feasible solution; in this case the program MP is inconsistent and it was developed by [] feasible point is known as a global optimal solution to the program MP if f ( ) f ( ) y [] Graphical Solution Method The graphical (or geometrical) method for solving Mathematical Programming problem is based on a well define set of logical steps ollowing this systematic procedure the given Programming problem can be easily solved with a minimum amount of computational effort and which has been introduced by [] We know that simple method is the well-studied and widely useful method for solving linear programming problem while for the class of non-linear programming such type of universal method does not eist Programming problems involving only two variables can easily solved graphically s we will observe that from the characteristics of the curve we can achieve more information We shall now several such graphical eamples to illustrate more vividly the differences between linear and non-linear programming problems Consider the following linear programming problems Maimize z 5 Subject to ig Optimal solution by graphical method The graphical solution is show in ig The region of feasible solution is shaded Note that the optimal does occur at an etreme point In this case the values of the variables that yield the maimum value of the objective function are unique and are the point of intersection of the lines 6 5 so that the optimal values of the variables and are The maimum value of the objective function is z 5 which was by [5] Now consider a non-linear programming problem which differs from the linear programming problem only in that the objective function: z (5) ( 5 ) ( ) Imagine that it is desired to minimize the objective function Observe that here we have a separable objective function The graphical solution of this problem is given in ig ig Optimal solution by graphical method The region representing the feasible solution is of course precisely the same as that for

3 DODIL INTERNTIONL UNIVERSITY JOURNL O SCIENCE ND TECHNOLOGY VOLUME 5 ISSUE JNURY the linear programming problem of ig Here however the curves of constant z are ellipse with centers at the point (5 ) The optimal solution is that point at which an ellipse is tangent to one side of the conve set If the optimal values of the variables are and and the minimum value of the objective function is z * then from ig - 6 and ( 5) ( ) z urthermore the slope of the curve z 5 evaluated at ( ) ( ) ( ) 6 5 must be since this is the slope of Thus we have the additional equation ( 5) We have obtained three equations involving and z * The unique solution is 5 5 and z * 5 Now the point which yields the optimal value of the objective function lies on the boundary of the conve set of feasible solutions but it is not an etreme point of this set Consequently any computational procedure for solving problems of this type cannot be one which eamines only the etreme points of the conve set of feasible solutions y a slight modification of the objective function studied above the minimum value of the objective function can be made to occur at an interior point of the conve set of feasible solutions Suppose for eample that the objective function is - ( ) ( ) z the same as that considered above This case is illustrated graphically in ig The optimal values of and z are * * and z * Thus it is not even necessary that the optimizing point lie on the boundaries Note that in this case the minimum of the objective function in the presence of the constraints and non-negativity restrictions is the same as the minimum in the absence of any constraints or non-negativity restrictions In such situations we say that the constraints and non-negativity restrictions are inactive since the same optimum is obtained whether or not the constraints and non-negativity restrictions are included Each of the eamples presented thus far the property that a local optimum was a global optimum and was introduced by [5] s a final eample I shall eamine an integer linear programming problem Let us solve the problem intigers Ma z 5 The situation is illustrated geometrically in igthe shaded region would be the conve set of feasible solutions in the absence of the integrality requirements When the j are required to be integers there only four feasible solutions which are represented by circles in ig If we solve the problem as a linear programming problem ignoring the integrality C O ig Optimal solution by graphical method ig Optimal solution by graphical method and that the conve set of feasible solutions is requirements the optimal solutions is 75 and z * 75 However it is clear that when it is required that the j be integers the optimal solution is * *

4 DS: EECT O GRPHICL METHOD OR SOLVING MTHEMTICL PROGRMMING PROLEM and z * 5 Note that this is not the solution that would be obtained by the solving the linear programming problem and rounding the results to the nearest integers which satisfy the constraints (this would give ) and z However in the case of a NLP problem the optimal solution may or may not occur at one of the etreme points of the solution space generated by the constraints and the objective function of the given problem Graphical solution algorithm: The solution NLP problem by graphical method in general involves the following steps: Step : construct the graph of the given NLP problem Step : Identify the conve region (solution space) generated by the objective function and constraints of the given problem Step : Determine the point in the conve region at which the objective function is optimum maimum or minimum) Step : Interpret the optimum solution so obtained Which has been introduced by [] Solution of Various Kinds of Problems by Graphical Solution Method Problem with objective function linear constraints non-linear Maimize Z Subject to the constraints Let us solve the problem by graphical method: or this first we are tracing the graph of the constraints of the problem considering inequalities as equations in the first quadrant (since ) We get the following shaded region as opportunity set OCD D C O ig 5 Optimal solution by graphical method The point which maimizes the value z and lies in the conve region OCD have to find The desired point is obtained by moving parallel to k for some k so long as k touches the etreme boundary point of the conve region ccording to this rule we see that the point C ( ) gives the maimum value of Z Hence we can find the optimal solution at this point by [6] z Ma 6 at Problem with objective function linear constraints non-linearlinear Maimize Z slt Let us solve the above problem by graphical method: or this we see that our objective function is linear and constraints are non-linear and linear Constraints one is a circle of radius with center ( ) and constraints two is a straight line In this case tracing the graph of the constraints of the problem in the first quadrant we get the following shaded region as opportunity set C O ig 6 Optimal solution by graphical method Considering the inequalities to equalities (6) (7 ) Solving (6) and (7 ) We get ( ) ( ) 5 5 The etreme points of the conve region are / 5 / 5 and C() O( ) () ( )

5 DODIL INTERNTIONL UNIVERSITY JOURNL O SCIENCE ND TECHNOLOGY VOLUME 5 ISSUE JNURY y moving according to the above rule we see that the line k touches ( / 5 / 5) the etreme point of the conve region Hence the required solution of the given problem is Z Ma at 5 5 Problem with objective function nonlinear constraints linear Minimize Z subject to the constraint s: 5 Our objective function is non-linear which is a circle with origin as center and constraints are linear The problem of minimizing Z is equivalent to minimizing the radius of a circle with origin as centre such that it touches the conve region bounded by the given constraints irst we contracts the graph of the constraints by MTL programming [9] in the st quadrant since ig 7 Optimal solution by graphical method Since and 5 the desire point must be some where in the unbounded conve region C The desire point will be that point of the region at which a side of the conve region is tangent to the circle Differentiating the equation of the circle d d d d Considering the inequalities to equalities C () 5 and Differentiating we get d d and d d d d d d Now from () and (9) we get and This shows that the circle has a tangent to it- (i) the line at the point () (ii) the line 5 at the point () ut from the graph we see that the point () does not lie in the conve region and hence is to be discarded Thus our require point is () Minimum Z (9) at thepoint ( ) 5 Comparison of Solution by Graphical Method and Others Let us consider the problem Maimize Z Subject to the constraints: irst I want to solve above problem by graphical solution method The given problem can be rewriting as: Maimize Z ( ) Subject to the constraints We observe that our objective function is a parabola with verte at ( -/) and constraints are linear To solve the problem graphically first we construct the graph of the constraint in the first quadrant since and by considering the inequation to equation Here we contract the graph of our problem by MTL programming [9] ccording to our previous graphical method our desire point is at (/ 5/)

6 DS: EECT O GRPHICL METHOD OR SOLVING MTHEMTICL PROGRMMING PROLEM ig Optimum solution by graphical method Hence we get the maimum value of the objective function at this point Therefore Z ma at Let us solve the above problem by using [7] Kuhn-Tucker Conditions The Lagrangian function of the given problem is ( ) ( ) y Kuhn-Tucker conditions we obtain () ( ) ( ) ( ) ( ) with ) ( ) ( d c b a Now there arise the following cases: Case (i) : Let in this case we get from and which is impossible and this solution is to be discarded and it has been introduced by []Case (ii): Let In this case we get from ) ( () lso from and If we take then If we consider then Now putting the value of in () we get 5 ( ) 5 for this solution / / / / Thus all the Kuhn-Tucker necessary conditions are at the point (/ 5/) Hence the optimum (maimum) solution to the given NLP problem is ma Z 5 at 6 97 Let us solve the problem by eale s method Maimize ( ) f Subject to the constraints: Introducing a slack variables s the constraint becomes s since there is only one constraint let s be a basic variable Thus we have by [] ( ) ( ) s N with s Epressing the basic and the objective function in terms of non-basic N we have s- - and f - We evaluated the partial derivatives of f wrto non-basic variables at N we get ( ) N N N f f since both the partial derivatives are positive the current solution can be improved s

7 DODIL INTERNTIONL UNIVERSITY JOURNL O SCIENCE ND TECHNOLOGY VOLUME 5 ISSUE JNURY 5 f gives the most positive value will enter the basis Now to determine the leaving basic variable we compute the ratios: α ho ko α min min α hk kk α α min α since the minimum occurs for s will α leave the basis and it was introduced by [] Thus epressing the new basic variable as well as the objective function f in terms of the new non-basic variables ( and s) we have: s s and f 6 s we again evaluate the partial derivates of f w r to the non-basic variables: f f s N N since the partial derivatives are not all negative the current solution is not optimal clearly will enter the basis or the net Criterion we compute the ratios min α / α / since the minimum of these ratios correspond to non-basic variables can be removed Thus we introduce a free variable u as an additional non-basic variable defined by u f Note that now the basis has two basic variables and (just entered) That is we have ( s u ) and ( ) N Epressing the basic in terms of non-basic N we have u 5 and ( ) u s The objective function epressing in terms of N is 5 f u u 97 s u 6 Now s δf δu N u N u u s u f f since for all j in N and u u the current solution is optimal Hence the optimal basic feasible solution to the given problem is: 5 Z * ; 97 6 Similarly we can find that by Wolfe s algorithm the optimal point is at (/ 5/) which was introduced by [] Thus for the optimal solution for the given QP problem is MaZ at ( ) 6 Therefore the solution obtained by graphical solution method Kuhn-Tucker conditions eale s method and Wolf s algorithm are same The computational cost is that by the graphical solution method using MTL Programming it will take very short time to determine the plan of action and the solution obtained by graphical method is more effective than any other methods we considered

8 6 DS: EECT O GRPHICL METHOD OR SOLVING MTHEMTICL PROGRMMING PROLEM 6 Conclusion This paper has been presented a direct fast and accurate way for determining an optimum schedule (such as maimizing profit or minimizing cost) The graphical method gives a physical picture of certain geometrical characteristics of programming problems y using MTL programming graphical solution can help us to take any decision or determining a particular plan of action from amongst several alternatives in very short moment ll kinds of programming problem can be solved by graphical method The limitation is that programming involving more than two variables ie for -D problems can not be solved by this method Non-linear programming problem mostly considered only in two variables Therefore from the above discussion we can say that graphical method is the best to take any decision for modern game theory dynamic programming problem science economics and management from amongst several alternatives References [] Greig D M: Optimization Lougman- Group United New York (9) [] Keak N K: Mathematical Programming with usiness pplication Mc Graw-Hill ook Company New York [] G R Walsh: Methods of optimization Jon Wiley and sons ltd 975 Rev 95 [] Gupta P K Man Mohan: Linear programming and theory of Games Sultan Chand & sons New Delhi [5] M S azaraa & C M Shetty: Non-linear programming theory and algorithms [6] G Hadley: Non-linear and dynamic programming [7] dadie J: On the Kuhn-Tucker Theory in Non- Linear Programming J dadie (Ed) 967b [] Kanti Sawrup Gupta P K Man Monhan Operation Research Sultan Chandra & Sons New Delhi India (99) [9] pplied Optimization with MTL Programming Venkataraman (Chepter-) [] Yeol Je Cho Daya Ram Sahu Jong Soo Jung pproimation of ied Points of symptotically Pseudocontractive Mapping in ranch Spaces Southwest Journal of Pure and pplied Mathematics Vol Issue Pub July [] Mittal Sethi Linear Programming Pragati Prakashan Meerut India 997 [] D Geion Evans On the K-theory of higher rank graph C*-algebras New York Journal of Mathematics Vol Pub January [] Hildreth C: Quadratic programming procedure Naval Research Logistics Quarterly [] imal Chandra Das: Comparative Study of the Methods of Solving Non-linear Programming Problems Daffodil Int Univ Jour of Sc and Tech Vol Issue January 9 imal Chandra Das has completed M Sc in pure Mathematics and Sc (Hons) in Mathematics from Chittagong University Now he is working as a Lecturer under the Department of Tetile Engineering in Daffodil International University His area of research is Operation Research