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Transcript of Operational Research1
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LINEAR PROGRAMMING PROJECT
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V.PAVITHRAV.PAVITHRASUKANYAH.V.KSUKANYAH.V.KRIZWANA SULTANARIZWANA SULTANASHILPA JAINSHILPA JAIN
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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
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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?
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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
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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
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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
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PROBLEMS IN OR AllocationAllocation Replacement Replacement Sequencing Sequencing Routing Routing Inventory Inventory
Queuing Queuing Competitive Competitive Search Search
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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
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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 .
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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
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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 .
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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 .
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BASIC CONCEPT OF LP PROGRAM
Objective functionConstraintsOptimizationSolution of lpp .Feasible solution
Optimal solution
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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
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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 .
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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 .
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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 .
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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
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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
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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 .
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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 .
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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
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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
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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
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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
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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
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LP in Final FormMax Z = 10x 1 + 15x2
Subject To
2x1 + 4x2 < 100 ( aluminumconstraint )3x1 + 2x2 < 80 ( steel constraint )
x1 , x2 > 0 (non-negativityconstraints
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Example:graphical method
0,
152
203 :subject to
920 max,
u
e
e
y x
y x
y x
y x y x
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0,
152
203 :sub ect to
920 max,
u
e
e
y x
y x
y x
y x y x
x
20
0
Example:graphical method
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0,
152
203 :sub ect to
920 max,
u
e
e
y x
y x
y x
y x y x
x
y
20
15
0
(5,5)
Example:graphical method
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Example:graphical method
0,
152
203 :subject to
920 max,
u
e
e
y x
y x
y x
y x y x
x
y
20
15
0
(5,5)
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x
20
15
0
(5,5)
0, 152
203
:sub ect to
920 max,
u
e
e
y x y x
y x
y x y x
Example:graphical method
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0,
152
203 :subject to
920 max,
u
e
e
y x
y x
y x
y x y x
x
20
15
0
Example:graphical method
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0,
152
203 :subject to
920 max,
u
e
e
y x
y x
y x
y x y x
x
20
15
0
(5,5)
Example:graphical method
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Example:graphical method
0,
152
203 :subject to
920 max,
u
e
e
y x
y x
y x
y x y x
x
y
20
15
0
(5,5)
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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
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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
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THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables :
0,,,
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s s y x
P y x
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¼¼¼
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THE SIMPLEX METHOD
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Start from the vertex (x=0 , y=0 )Move to the next vertex that increases profitas much as possible .
¼¼
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½
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THE SIMPLEX METHOD
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¼¼¼
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0100920
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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 )
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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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¼¼¼
½
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0100920
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x y s1 s2 P RHS
Pivoting means solve for that variable,Then substitute into the other equations
3
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3
1
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s y x
s y x
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Pivoting means solve for that variable,Then substitute into the other equations
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! s y x
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x y s1 s2 P RHS
3
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3
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! s y x
Pivoting means solve for that variable,Then substitute into the other equations
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THE END