Linear Programming Introduction to linear programming And Problem formulation
Defnition And Characteristics O Linear Programming Linear Linear Progr Programm amming ing is that that branch branch of mathem mathemati atical cal progr programm amming ing which which is designed designed to solve optimization optimization problems where where all the constraint constraints s as will as the object objective ives s are expr express essed ed as Linear Linear functi function on .It was develop developed ed by George George . !enting !enting in "#$%. Its earlier application application was solely related related to the activities activities of the second& 'orld 'ar. (owever soon its importance was recognized and it came to occupy a prominent place in the industry and trade. Linear Programming is a techni) techni)ue ue for ma*ing ma*ing decisio decisions ns under certain certainty ty i.e.+ when when all the courses courses of options options available to an organisat organisation ion are *nown , the objective objective of the -rm along with its constraints are )uanti-ed. hat course of action is chosen out of all possible alternatives which yields the optimal results. Linear Programming can also be used as a veri-cation and chec*ing mechanism to ascertain the accuracy and and the the relia reliabi bilit lity y of the the decis decisio ions ns whic which h are are ta* ta*en sole solely ly on the the basi basis s of manager/s experience0without the aid of a mathematical model. Some o the defnitions o Linear Programming are as ollows: 1Linear Programming is a method of planning and operation involved in the construction of a model of a real0life situation having the following elements2 3a4 5ariables 5ariables which denote the available choices and 3b4 the relat related ed mathem mathemati atical cal expre expressi ssions ons which which relat relate e the variab variables les to the controlling conditions6 re7ect clearly the criteria to be employed for measuring the bene-ts bene-ts 7owing 7owing out of each each course course of action action and provi providin ding g an accura accurate te measurement of the organization&s objective. he method maybe so devised/ as to ensu ensurre the the sele select ctio ion n of the the best best alte altern rnat ativ ive e out out of a lar large numb number er of alternative available to the organization. Linear Programming is the analysis of problems in which a Linear function of a number of variables is to be optimized 3maxim 3maximize ized d or minimized minimized44 when whose whose variable variables s are subjec subjectt to a number number of constraints in the mathematical near ine)ualities. 8rom the above de-nitions6 it is clear that2 3i4 Linear Linear Program Programming ming is an optimiza optimization tion techni)ue6 techni)ue6 where where the underlying underlying objective is either to maximize the pro-ts or to minim is the 9osts. 3ii4 It deals with with the problem problem of allocation of -nite -nite limited resour resources ces amongst amongst di:erent competing activities in the most optimal manner. manner. 3iil4 It generates solutions solutions based on the the feature feature and and characteristics characteristics of the actual problem or situation. (ence (ence the scope of linear programming is very wide as it -nds -nds applic applicati ation on in such such divers diverse e -elds -elds as mar*e mar*etin ting6 g6 produ producti ction6 on6 -nance -nance , personnel etc. 3iv4 Linear Linear Programming Programming has be0en highly highly successful successful in solving solving the following types of problems 2 3a4 Product0mix problems 3b4 Investment planning problems 3c4 lending strategy formulations and 3d4 ;ar*eting , !istribution management. 3v4
3b4 Presence of constraints which limit the extent to which the objective can be pursued=achieved. 3c4 Availability of alternatives i.e.+ di:erent courses of action to choose from6 and 3d4 he objectives and constraints can be expressed in the form of linear relation. 35I4 >egardless of the size or complexity6 all LP problems ta*e the same form i.e. allocating scarce resources among various compete ting alternatives. Irrespective of the manner in which one de-nes Linear Programming6 a problem must have certain basic characteristics before this techni)ue can be utilized to -nd the optimal values. The characteristics or the basic assumptions o linear programming are as ollows: 1 Decision or Acti!it" #ariables $ Their %nter&'elationship he decision or activity variables refer to any activity which is in competition with other variables for limited resources.
- Presence o Di.erent Alternati!es !i:erent courses of action or alternatives should be available to a decision ma*er6 who is re)uired to ma*e the decision which is the most e:ective or the optimal. 8or example6 many grades of raw material may be available6 the& same raw material can be purchased from di:erent supplier6 the -nished goods can be sold to various mar*ets6 production can be done with the help of di:erent machines. / 0on&0egati!e 'estrictions @ince the negative values of 3any4 physical )uantity has no meaning6 therefore all the variables must assume non0negative values. If some of the variables is unrestricted in sign6 the non0negativity restriction can be enforced by the help of certain mathematical tools without altering the original informati#n contained in the problem. Linearit" Criterion he relationship among the various decision variables must be directly proportional Le.+ oth the objective and the constraint6B must be expressed in terms of linear e)uations or ine)ualities. 8or example. if00Cne of the factor inputs 3resources li*e material6 labour6 plant capacity etc.4 9reases6 then it should result in a proportionate manner in the -nal output. hese linear e)uations and in e)uations can graphically be presented as a straight line. 2 Additi!el" It is assumed that the total pro-tability and the total amount of each resource utilized would be exactly e)ual to the sum of the respective
individual amounts. hus the function or the activities must be additive 0 and the interaction among the activities of the resources does not exist. D. ;utually
Ad!antages o Linear Programming 1 Scientifc Approach to Problem Sol!ing Linear Programming is the application of scienti-c approach to problem solving. (ence it results in a better and true picture of the problems0which can then be minutely analysed and solutions ascertained. ( 5!aluation o All Possible Alternati!es ;ost of the problems faced by the present organisations are highly complicated 0 which can not be solved by the traditional approach to decision ma*ing. he techni)ue of Linear Programming ensures that&ll possible solutions are generated 0 out of which the optimal solution can be selected. + 6elps in 'e&5!aluation Linear Programming can also be used in .re0 evaluation of a basic plan for changing conditions. @hould the conditions change while the plan is carried out only partially6 these conditions can be accurately determined with the help of Linear Programming so as to adjust the remainder of the plan for best results. - 7ualit" o Decision Linear Programming provides practical and better )uality of decisions& that re7ect very precisely the limitations of the system i.e.+ the various restrictions under which the system must operate for the solution to be optimal. If it becomes necessary to deviate from the optimal path6 Linear Programming can )uite easily evaluate the associated costs or penalty. / )ocus on 8re"&Areas (ighlighting of grey areas or bottlenec*s in the production process is the most signi-cant merit of Linear Programming. !uring the periods of bottlenec*s6 imbalances occur in the production department. @ome of the machines remain idle for long periods of time6 while the other machines are unable to:ee the demand even at the pea* performance level. )le9ibilit" Linear Programming is an adaptive , 7exible mathematical techni)ue and hence can be utilized in analyzing a variety of multi0dimensional problems )uite successfully. 2 Creation o %normation ase y evaluating the various possible alternatives in the light of the prevailing constraints6 Linear Programming models provide an important database from which the allocation of precious resources can be don rationally and judiciously.
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ation o )actors o Production Linear Programming helps in optimal utilization of various existing factors of production such as installed capacity6 labour and raw materials etc. Limitations o Linear Programming Although Linear Programming is a highly successful having wide applications in business and trade for solving optimization/ problems6 yet it has certain demerits or defects. @ome of the important0limitations in the application of Linear Programming are as follows2 1 Linear 'elationship Linear Programming models can be successfully applied only in those situations where a given problem can clearly be represented in the form of linear relationship between di:erent decision variables. (ence it is based on the implicit assumption that the objective as well as all the constraints or the limiting factors can be stated in term of linear expressions 0 which may not always hold good in real life situations. In practical business problems6 many objective function , constraints cannot be expressed linearly. ;ost of Fhe business problems can be expressed )uite easily in the form of a )uadratic e)uation 3having a power 4 rather than in the terms of linear e)uation. Linear Programming fails to operate and provide optimal solutions in all such cases. e.g. A problem capable of being expressed in the form of2 axHbxHc ? where a J ? cannot be solved with the help of Linear Programming techni)ues.
( Constant #alue o ob*ecti!e $ Constraint 5?uations efore a Linear Programming techni)ue could be applied to a given situation6 the values or the coeEcients of the objective function as well as the constraint e)uations must be completely *nown. 8urther6 Linear Programming assumes these values to be constant over a period of time. In other words6 if the values were to change during the period of study6 the techni)ue of LP would loose its e:ectiveness and may fail to provide optimal solutions to the problem. (owever6 in real life practical situations often it is not possible to determine the coeEcients of objective function and the constraints e)uations with absolute certainty. hese variables in fact may6 lie on a probability distribution curve and hence at best6 only the Ii*elil"ood of their occurrence can be predicted. ;over over6 often the value&s change due to extremely as well as internal factors during the period of study. !ue to this6 the actual applicability of Linear Programming tools may be restricted. + 0o Scope or )ractional #alue Solutions here is absolutely no certainty that the solution to a LP problem can always be )uanti-ed as an integer )uite often6 Linear Programming may give fractional0varied answers6 which are rounded o: to the next integer. (ence6 the solution would not be the optimal one. 8or example6 in -nding out /the pamper of men and machines re)uired to perform a particular job6 a fractional Larson0integer solution would be meaningless. - Degree Comple9it" ;any large0scale real life practical problems can not be solved by employing Linear Programming techni)ues even with the help of a computer due to highly complex and Lengthy calculations. Assumptions and approximations are re)uired to be made so that Be6 given problem can be bro*en down into several smaller problems and6 then solve separately. (ence6 the validity of the -nal result6 in all such cases6 may be doubtful2
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