01 Intro Duc
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AAOC C222: OPTIMISATION
Text Book:
Operations Research: An Introduction
By Hamdy A.Taha (Pearson Education)7th Edition
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Reference Books:
1. Hadley, G: Linear Programming,
Addison Wesley2. Pant, J.C: Optimization,
Jain Brothers
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3. Hillier & Lieberman
Introduction to OperationsResearch, Tata McGraw-Hill
4. Bazaraa, Jarvis & SheraliLinear Programming and Network
Flows, John Wiley
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http://discovery.bits-
pilani.ac.in/discipline/math/msr/index.html
You may view my lecture slides in the
following site.
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The formal activities of Operations Research
(OR) were initiated in England during World
War II when a team of British scientists set
out to make decisions regarding the best
utilization of war material. Following the
end of the war, the ideas advanced in
military operations were adapted to improve
efficiency and productivity in the civiliansector. Today, OR is a dominant and
indispensable decision making tool.
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Example: The Burroughs garment
company manufactures men's shirts
and womens blouses for WalmarkDiscount stores. Walmark will accept
all the production supplied byBurroughs. The production process
includes cutting, sewing and
packaging. Burroughs employs 25workers in the cutting department, 35
in the sewing department and 5 in the
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packaging department. The factory works
one 8-hour shift, 5 days a week. Thefollowing table gives the time
requirements and the profits per unit for
the two garments:
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Garment Cutting Sewing Packaging Unitprofit($)
Shirts 20 70 12 8.00
Blouses 60 60 4 12.00
Minutes per unit
Determine the optimal weeklyproduction schedule for Burroughs.
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Solution: Assume that Burroughs
producesx1
shirts andx2
blouses per
week.8x1 + 12x2
Time spent on cutting =
Profit got =
Time spent on sewing = 70x1
+ 60x2
mts
Time spent on packaging =12x1 + 4x2 mts
20x1 + 60x2mts
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The objective is to findx1,x2 so as to
maximize the profitz= 8x1 + 12x2
satisfying the constraints:
20x1 + 60x2 25 40 60
70x1 + 60x2 35 40 60
12x1 + 4x2 5 40 60
x1,x2 0, integers
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This is a typical optimization problem.
Any values ofx1, x2 that satisfy all
the constraints of the model is called
a feasible solution. We areinterested in finding the optimum
feasible solution that gives the
maximum profit while satisfying allthe constraints.
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More generally, an optimization
problem looks as follows:
Determine the decision variables
x1,x2, ,xn so as to optimize anobjectivefunction f(x1,x2, ,xn)
satisfying the constraints
gi (x1,x2, ,xn) bi (i=1, 2, , m).
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Linear Programming Problems(LPP)
An optimization problem is called aLinear Programming Problem (LPP) when
the objective function and all the
constraints are linear functions of thedecision variables, x1, x2, , xn. We also
include the non-negativity restrictions,
namely xj 0 for all j=1, 2, , n.Thus a typical LPP is of the form:
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Optimize (i.e. Maximize or Minimize)
z = c1x1 + c2x2+ + cnxnsubject to the constraints:
a11x1 + a12x2+ + a1nxn b1
a21x1 + a22 x2+ + a2nxn b2
. . .
am1x1 + am2x2+ + amnxnbm
x1,x2, ,xn 0
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A LPP satisfies the two properties:
Proportionality and additivity
Proportionality means the contributionsof each decision variable in the
objective function and its requirements
in the constraints are directlyproportional to the value of the variable.
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Additivity stipulates that the total
contributions of all the variables in theobjective function and their
requirements in the constraints are the
direct sum of the individualcontributions or requirements of each
variable.
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We shall first look at formulation of
some LPPs,
Graphically solve some LPPs
involving two decision variables
Study some mathematical
preliminaries regarding the solutions
of LPPs
Finally look at the Simplex method
of solving a LPP
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Wild West produces two types of cowboy hats.
Type I hat requires twice as much labor as a
Type II. If all the available labor time is
dedicated to Type II alone, the company can
produce a total of 400 Type II hats a day. Therespective market limits for the two types of
hats are 150 and 200 hats per day. The profit is
$8 per Type I hat and $5 per Type II hat.Formulate the problem as an LPP so as to
maximize the profit.
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Solution: Assume that Wild West producesx1
Type I hats andx2 Type II hats per day.
8x1 + 5x2
Labour Time spent is (2x1 + x2) c minutes
Per day Profit got =
Assume the time spent in producing onetype II hat is c minutes.
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The objective is to findx1,x2 so as to
maximise the profitz= 8x1 + 5x2
satisfying the constraints:
(2x1 +x2 ) c 400 c
x1 150
x2 200
x1,x2 0, integers
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That is: The objective is to findx1,x2 so
as tomaximise the profitz= 8x1 + 5x2
satisfying the constraints:
2x1 +x2 400
x1 150
x2 200
x1,x2 0, integers
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Feed Mix problem: The manager of a milk
diary decides that each cow should get at least
15, 20 and 24 units of nutrients A, B and Crespectively. Two varieties of feed are
available. In feed of variety 1(variety 2) the
contents of the nutrients A, B and C arerespectively 1(3), 2(2), 3(2) units per kg. The
costs of varieties 1 and 2 are respectively
Rs. 2 and Rs. 3 per kg. How much of feed of
each variety should be purchased to feed a cow
daily so that the expenditure is least?
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Trim Loss problem: A company has to
manufacture the circular tops of cans. Two
sizes, one of diameter 10 cm and the other
of diameter 20 cm are required. They are to
be cut from metal sheets of dimensions 20
cm by 50 cm. The requirement of smaller
size is 20,000 and of larger size is 15,000.
The problem is : how to cut the tops from
the metal sheets so that the number of
sheets used is a minimum. Formulate the
problem as a LPP.
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A sheet can be cut into one of the following
three patterns:
Pattern I
Pattern II
Pattern III
10
20
20
10
10
10
20
10
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Pattern I: cut into 10 pieces of size 10 by 10
so as to make 10 tops of size 1
Pattern II: cut into 2 pieces of size 20 by 20
and 2 pieces of size 10 by 10 so as to make
2 tops of size 2and 2 tops of size 1
Pattern III: cut into 1 piece of size 20 by 20
and 6 pieces of size 10 by 10 so as to make1 top of size 2 and 6 tops of size 1
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So assume thatx1 sheets are cut according to
pattern I,x2
according to pattern II,x3
according to pattern III
The problem is to
Minimizez=x1 +x2 +x3
Subject to 10x1 + 2x2 + 6x3 20,000
2x2 + x3 15,000
x1,x2,x3 0, integers
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A Post Office requires different number of
full-time employees on different days of the
week. The number of employees required oneach day is given in the table below. Union
rules say that each full-time employee must
receive two days off after working for fiveconsecutive days. The Post Office wants to
meet its requirements using only full-time
employees. Formulate the above problem asa LPP so as to minimize the number of full-
time employees hired.
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Requirements of full-time employees
day-wise
Day No. of full-time
employees required
1 - Monday 10
2 - Tuesday 6
3 - Wednesday 8
4 - Thursday 125 - Friday 7
6 - Saturday 9
7 - Sunday 4
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Solution: Letxibe the number of full-time
employees employed at the beginning of day
i (i = 1, 2, , 7). Thus our problem is to findxi so as to
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Minimize 1 2 3 4 5 6 7z x x x x x x x
Subject to1 4 5 6 7 10 (Mon)x x x x x
1 2 5 6 7 6 (Tue)x x x x x
1 2 3 6 7 8 (Wed)x x x x x
1 2 3 4 7 12 (Thu)x x x x x
1 2 3 4 5 7 (Fri)x x x x x
2 3 4 5 6 9 (Sat)x x x x x
3 4 5 6 7 4 (Sun)x x x x x
xi 0.
integers
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BITS wants to host a Seminar for five
days. For the delegates there is anarrangement of dinner every day. The
requirement of napkins during the 5
days is as follows:
Day 1 2 3 4 5
NapkinsNeeded
80 50 100 80 150
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Institute does not have any napkins in the
beginning. After 5 days, the Institute has no
more use of napkins. A new napkin costsRs. 2.00. The washing charges for a used one
are Rs. 0.50. A napkin given for washing after
dinner is returned the third day before dinner.The Institute decides to accumulate the used
napkins and send them for washing just in time
to be used when they return. How shall theInstitute meet the requirements so that the total
cost is minimized ? Formulate as a LPP.
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Solution Letxj be the number of napkins
purchased on day j, j=1,2,..,5
Letyj be the number of napkins given for
washing after dinner on day j, j=1,2,3
Thus we must have
Also we have
y1 80,y2 (80 y1) + 50
y3 (80 y1) + (50y2) + 100
x1 = 80,x2 = 50,x3 +y1 = 100,x4 +y2 = 80
x5 +y3 = 150
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Thus we have to Minimize
z= 2(x1+x2+x3+x4+x5)+0.5(y1+y2+y3)Subject to
x1 = 80,x2 = 50,x3 +y1 =100,
x4 +y2 = 80,x5 +y3 = 150,
y1 80,y1+y2 130,y1+y2+y3 230,
all variables 0, integers
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There are many Software packages
available to solve LPP and related problems.
Your book contains a CD having the
package TORA probably developed by
the author.There is also Microsofts Excel Solver.
Normally this would not have been loaded;
you mut check whether it is loaded.
There is also a commercial package
LINGO
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Dr. J C Pants book contains in the end a
C code for solving some of the LPPproblems (of course developed by some of
your seniors).
You may yourself develop programs tosolve LPP problems.