Operational Research1

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LINEAR PROGRAMMING PROJECT



V.PAVITHRAV.PAVITHRASUKANYAH.V.KSUKANYAH.V.KRIZWANA SULTANARIZWANA SULTANASHILPA JAINSHILPA JAIN



INTRODUCTION

Modern technological advance growth of scientific techniques

Operations Research (O .R.) recent addition

to scientific tools O.R. new outlook to many conventional

management problems Seeks the determination of best (optimum )

course of action of a decision problem underthe limiting factor of limited resources



Operational Research can be considered asbeing the application of scientific method byinter-disciplinary teams to problemsinvolving the control of organized

systems so as to provide solutionswhich best serve the purposes of theorganization as a whole .

WHAT IS OR?WHAT IS OR?



CHARACTERISTIC NATURE OF OR

InterInter--disciplinary team approachdisciplinary team approach Systems approach Systems approach Helpful in improving the quality of solution Helpful in improving the quality of solution Scientific method Scientific method Goal oriented optimum solution Goal oriented optimum solution Use of models Use of models

Require willing executives Require willing executives Reduces complexity Reduces complexity



PHASES TO OR

Judgment phase Determination of the problem

Establishment of the objectives and values Determination of suitable measures of effectivenessResearch phase

Observation and data collection Formulation of hypothesis and models Observation and experimentation to test the hypothesis Prediction of various results, generalization, considerationof alternative method

Action phase Implementation of the tested results of the model



METHODOLOGY OF ORMETHODOLOGY OF OR

Formulating the problem Constructing the model

Deriving the solution Analytical method Numerical method Simulation method

Testing the validity Controlling the solution Implementing the result



PROBLEMS IN OR AllocationAllocation Replacement Replacement Sequencing Sequencing Routing Routing Inventory Inventory

Queuing Queuing Competitive Competitive Search Search



OR TECHINIQESOR TECHINIQES Linear programming

Waiting line or queuing theory

Inventory control / planning Game theory Decision theory Network analysis

Program Evaluation and ReviewTechnique

Critical Path Method (CPM ) etc . Simulation Integrated production models



SIGNIFICANCE OF ORSIGNIFICANCE OF OR

Provides a tool for scientific analysis Provides solution for various business problems Enables proper deployment of resources

Helps in minimizing waiting and servicing costs Enables the management to decide when to buyand how much to buy? Assists in choosing an optimum strategy Renders great help in optimum resource allocation Facilitates the process of decision making Management can know the reactions of the integrated

business systems . Helps a lot in the preparation of future managers .



LIMITATIONS OF ORLIMITATIONS OF OR

The inherent limitations concerning mathematicalexpressions

High costs are involved in the use of O .R. techniques O.R. does not take into consideration the intangible

factors O.R. is only a tool of analysis and not the complete

decision-making process Other limitations

Bias Inadequate objective functions Internal resistance Competence Reliability of the prepared solution



INTRODUCTION TO LINEAR PROGRAMMING

Today many of the resources needed asinputs to operations are in limited supply .

Operations managers must understand theimpact of this situation on meeting theirobjectives .Linear programming (LP ) is one way thatoperations managers can determine howbest to allocate their scarce resources .



Linear programming

We use graphs as useful modeling abstractions tohelp us develop computational solutions for a widevariety of problemsA linear program is simply another modelingabstraction (tool in your toolbox )Developing routines that solve general linearprograms allows us to embed them in sophisticatedalgorithmic solutions to difficult problems (e .g. like

we did for TSP )The cutting edge algorithmic solutions to manyproblems use the ideas from mathematicalprogramming, linear programming forming thefoundation .



BASIC CONCEPT OF LP PROGRAM

Objective functionConstraintsOptimizationSolution of lpp .Feasible solution

Optimal solution



LP PROBLEMS IN OM: PRODUCT MIX

Objective

To select the mix of products or servicesthat results in maximum profits for the planningperiod

Decision VariablesHow much to produce and market of each

product or service for the planning period

ConstraintsMaximum amount of each product orservice demanded; Minimum amount of productor service policy will allow; Maximum amount of resources available



Objective function:the linear functions which is to be optimized i .e

maximized or minimized this may be expressed inlinear expression .

Solution of Lpp:The set of all the values of the variable

x1,x2 xn which satisy the constraints is calledthe solution of Lpp .

Feasible solution:The set of all the values of the variable

x1,x2 xn which satisy the constraints and alsothe non negative conditions is called the feasiblesolution of lpp .



Recognizing LP Problems

Characteristics of LP Problems in OMA well-defined single objective must bestated .

There must be alternative courses of action .The total achievement of the objectivemust be constrained by scarce resources

or other restraints .The objective and each of theconstraints must be expressed as linearmathematical functions .





Linear Programming

An optimization problem is said to be a linearprogram if it satisfied the following properties:

There is a unique objective function .Whenever a decision variable appears in

either the objective function or one of theconstraint functions, it must appear onlyas a power term with an exponent of 1,possibly multiplied by a constant .



LP Problems in General

Units of each term in a constraint mustbe the same as the RHS

Units of each term in the objectivefunction must be the same as ZUnits between constraints do not haveto be the sameLP problem can have a mixture of constraint types



No term in the objective function or in anyof the constraints can contain products of the decision variables .

The coefficients of the decision variables inthe objective function and each constraintare constant .

The decision variables are permitted toassume fractional as well as integer values



Examples of lpp

We are already familiar with the graphical representation of equations and inequations . here we describe the application of linear equations and inequations in solving different kinds of

problems . The examples are stated below .Example 1:Find two positive numbers such that whosesum is atleast 15 and whose difference is at themost 7 such that the product is maximum .

Step1:we have to choose the positive two

numbers . Let the 2 positive numbers be x and y . this x and y are decision variables .



Step 2:our objective is to minimize the product x ,yLet z=xy we have to maximize z

Step3:we have the following conditions on the variables as x

and y .

step 4:x+y>=15x-y<=7x,y>0

as the linear constraints .the mathemetical constraint of

this equation is to maximize the objective function z=xy .



PROBLEMS 1. A producer wants to maximise revenues producing two goods x

1and x2 in the market . Market prices of goods are 10 and 5respectively . Production of x 1and x2 requires 25 and 10 unitsof skilled labour and total endowment of skilled labour is1000 .

Similarly production of x1 and x2 also requires 20 and 50 unitsof unskilled labour and whose total endowment is 1500 . Howmuch should this firm produce x 1and x 2 in order to maximisethe total revenue.

Max R =10x 1 + 5x 2Subject to:

Skilled labour constraint: 25 x1 +10x 2<=1000Unskilled labour constraint: 20x 1 +50x 2 <=1500Non-negativity constraints: x 1 ,x2 >=0



SOLUTION

Max R =10x 1 + 5x 2

Subj ect to:Skilled labour constraint: 25 x1 +10x 2<=1000Unskilled labour constraint: 20x 1 +50x 2 <=1500Non-negativity constraints: x 1 ,x2 >=0





Define the objectiveMaximize total weekly profitDefine the decision variables

x1 = number of Deluxe frames

produced weeklyx2 = number of Professional framesproduced weekly

Write the mathematical objectivefunction

Max Z = 10x 1 + 15x 2



Write a one- or two-word description of

each constraintAluminum availableSteel available

Write the right-hand side of eachconstraint

10080

Write <, =, > for each constraint< 100< 80



Write all the decision variables on theleft-hand side of each constraint

x1 x2 < 100

x1 x2 < 80Write the coefficient for each decisionin each constraint

+ 2x1 + 4x2 < 100+ 3x1 + 2x2 < 80



LP in Final FormMax Z = 10x 1 + 15x2

Subject To

2x1 + 4x2 < 100 ( aluminumconstraint )3x1 + 2x2 < 80 ( steel constraint )

x1 , x2 > 0 (non-negativityconstraints



Example:graphical method

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So far we find an optimal point by searchingamong feasible intersection points .

The search can be improved by starting withan initial feasible point and moving to a

better solution until an optimal one is found .The simplex method incorporates both

optimality and feasibility tests to find theoptimal solution(s ) if one exists



An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best value

found so far .A feasi b ility test determines whether the

proposed intersection point is feasible .The decision and slack variables are

separatedinto two nonoverlapping sets, which we callthe independent and dependent sets



THE SIMPLEX METHOD

Transform Linear Program into a systemof linear equations using slack variables :

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THE SIMPLEX METHOD



Start from the vertex (x=0 , y=0 )Move to the next vertex that increases profitas much as possible .

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At (0,0 ), P = 0Increasing x can increase P the most (x coefficient

has larger magnitude than they coefficient )Compute check ratios to find pivot row (smallest

ratio )



Basic Idea: Start from a vertex (x=0, y=0 )Move to next vertex that increases

profit as much as possible

At (0,0 ), P = 0

Increasing x can increase P the most (x coefficient haslarger magnitude than the y coefficient )Compute check ratios to findpivot row (smallest ratio )Pivot around the element inboth pivot column and row



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Operational Research1