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Operations Research
Prof G R C Nair
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Introduction
What is OR / MS ? Why OR / MS ?
Applications
What is a Model ?
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Types of Models
Physical / Mathematical
Deterministic / ProbabilisticAllocation / Optimization / Inventory
/ Sequencing / Queuing /
Replacement / Investment /Competitive / Net working
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Define the problem
Identify Measurable Objectives
Identify the Constraints and Variables
Collect Data
Establish inter relationships
Make a Model
Solve Test / Analyse / Modify
Implement
Steps in O.R
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Linear Programming
Linear programming is an Optimization Model.
It is a deterministic model.All the relationships
/ variations are to be straight line or linear.
Objective Maximization / Minimization.
Constraints Limitations.
There is a Graphic Solution when the
variables involved are just two. If there are
more variables, Simplex Algorithm is used.
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Formulation of
L.P Model
1. Maximization Case
A manufacturer produces two products Chairand
Table each requiring processing by two sections,carpentry and painting.Carpentry section has 32hours and painting section has has 34 hours oftime available per week.A chair needs 3 hours incarpentry and 1 hour in painting section,while atable needs 2 and 4 hours respectively.The profitmargin from the sale of a chair is Rs 5 and table isRs 6. Find the product mix for the maximumprofit.Formulate it as a LP model.
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Let the number of chairs and tables made be x&y.
Objective.Maximize Profit, subject to limitations
of processing capacity.
Profit Z = 5x+ 6y.
So Maximize Profit, Z = 5x + 6y,
Subject to constraints,
1.Carpentry capacity,
3x+2y < 32
2. Painting capacity,X+4y < 34
3.Nonnegative constraints,
X > 0, Y > 0
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2. Minimization case.Vitamins A and B are required for broiler
chicken at the rate of40 and 50 units per
day.T
here are 2 feedsF
1 andF
2 whichhave vitamin A at the rate of 2 and 4 units
and vitamin B at the rate of 3 and 2 units
per Kg of feed. The cost of feed F1 and
F2 are Rs 3 and 2.5 respectively perKg.Minimize the cost of feed. Formulate it
as a LP model.
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Let the optimum quantity of feed F1 and F2 be x
and y.
Objective function
Minimize cost Z=3x+2.5y,
Subject to constraints,
1. 2x+4y > 40 ( Mini requirement of Vitamin A )
2. 3x+2y > 50 (Mini requirement of Vitamin B)3. Non negative constraints
x > 0 , y > 0
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Graphic Solution. Maximization case
Draw the constraint lines3x+2y=32 at its limit
when x = 0, y = 16, when y=0, x = 10.667.
Draw a line joining points (0,16) and(10.667,0).
x+4y = 34 at its limit
When x=0, y = 8.5, when y = 0, x = 34Draw a line joining (0,8.5) and (34,0).
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Feasible area
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Within these two lines, above X axis and
right of Y axis is the feasible area. Any point
here, by its coordinates, gives the possiblecombination of chairs and tables that can be
made. Optimum will be at any corner, which
can be found out graphically or by solvingthe equations of the lines concerned.
Here, x = 6 and y = 7. Profit = 72.
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Graphic Solution. Minimization Case.
Draw the constraint lines
1. 2x+4y = 40.Put x = 0, y = 10, y = 0, x =20.
Points (0,10) and (20,0) define this line
2. 3x+2y = 50Put x = 0, y = 25, y = 0, x = 16.67
Points (0,25) and (16.67,0) define this line.
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Feasible area
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The feasible area is above these lines.
(why?) Any point here, by its coordinates,
shows a possible combination of the twofeeds that satisfy the vitamin requirements.
Optimum will be at any corner, which can be
found out graphically or by solving the
equations of the lines concerned.
Here, x = 15 and y = 2.5 Cost= 51.25
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1.More constraints. We can have any number of
constraints. It may further limit the feasible area.
2.Redundant Constraint - A constraint whichdoes not affect in any way the feasible area.It
does not supply any new information.
3. > and < constraints together. Mark feasiblearea appropriately.
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Formulation Examples
2007 Terminal (Make-up) Q 4
2007 Terminal Q 4
2009 Terminal Q 7 (HW)
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TransportationModel
Allocation and Optimizing Technique.
Special case of L.P model.
The total Supply and total Demandmust be equal or else made equal.
Basically the algorithm is for minimization
case. (Algorithm for solving this model are VAM,
MODI, NWCR, Stepping Stone etc).
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Problem 2
Formulate a transportation model for the least costtransportation plan satisfying the followingrequirements.
Availability of commodity
Delhi-100, Mumbai- 25, Bangalore 75
Requirement at Jaipur-80, Calcutta-30, Kanpur-90
Cost of transportation:
From Delhi - to Jaipur, Calcutta, and Kanpur
(Rs 000s / Unit) - 5,10,2 From Mumbai - 3,7,5
From Bangalore - 6, 8, 4.
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Minimize the cost of transportation
Pay offM
atrix Rs 000s
To
FromJaipur Calcutta Kanpur Total
Available
Delhi 100
Mumbai 25
Bangalore 75
Total
Required80 30 90 200
200
5 10 2
86
573
4
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Special Cases
For case of maximisation, subtract all profits
from the highest value in profit matrix.
This gives the opportunity cost of not makingthe most efficient allocation. This opportunity
cost is to be minimized by the method.
When the total supply and total demand are not
the same, introduce a dummy row or dummycolumn with zero cost for the balance quantity.
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Assignment Model
Special case of LP and transportation models.
Here, part allocation and part fulfilling not possible
-one to one allocation only.
Basicaly for minimising.Men to Jobs, Funds to Projects, boy to girl etc.
Minimising case - Cost, Time, Loss.
Condition The number of rows and columns should
be the same. ie, the matrix should be square or elsemade square by adding dummy.
(Algorithm for solving this model is known as theHungarian technique / Floods technique).
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Problem 1.Assigning Projects to Competing Contractors.
4 contractors have quoted for all the 4 projects to be
executed. But the management wants to give one
project only to each contractor for speedy completionof all the projects and to avoid putting all eggs in one
basket.The rate quoted by the contractors for these
projects are different as given in the pay off
matrix.The objective is to assign the projects to thevarious contractors such that the total cost is the
minimum. Formulate an Assignment Model.
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Pay off MatrixBids in Rs Crores
Projects Contractors
L&T Gammons IVR CL GMR
A 48 48 50 44
B 56 60 60 68
C 96 94 90 85
D 42 44 54 46
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Special Cases
Maximization case - convert it to minimization
case by subtracting all values in the given pay
off matrix from the highest value in the table.
This gives the opportunity cost of not making themost efficient allocation.This opportunity cost is to be
minimized.
Unbalanced Assignment -If the number of rows
and columns differs, then the matrix has to be
made square by adding a dummy row or a
dummy column as the case may be, with zero
cost.
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General Remarks
L.P is the most general form of all these
three models.
Transportation Model is a special Case ofL.P
Assignment Model is a special Case of
Transportation Model.
Hence all the above can be solved by L.P.
