AAOC C222: OPTIMISATION

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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
• Hadley, G Linear Programming, Addison Wesley
• Pant, J.C Optimization, Jain
  Brothers

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3. Hillier Lieberman Introduction to
Operations Research, Tata McGraw-Hill 4. Bazaraa,
Jarvis Sherali Linear Programming and
Network Flows, John Wiley
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You may view my lecture slides in the following
site.
http//discovery.bits-pilani.ac.in/discipline/math
/msr/index.html
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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
civilian sector. 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 Walmark Discount stores. Walmark will accept
all the production supplied by Burroughs. The
production process includes cutting, sewing and
packaging. Burroughs employs 25 workers 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. The following table
gives the time requirements and the profits per
unit for the two garments
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Minutes per unit
Determine the optimal weekly production schedule
for Burroughs.
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Solution Assume that Burroughs produces x1
shirts and x2 blouses per week.
8 x1 12 x2
Profit got
20 x1 60 x2 mts
Time spent on cutting
70 x1 60 x2 mts
Time spent on sewing
Time spent on packaging
12 x1 4 x2 mts
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The objective is to find x1, x2 so as to
maximize the profit z 8 x1 12 x2
satisfying the constraints
20 x1 60 x2 25 ? 40 ? 60 70 x1 60 x2
35 ? 40 ? 60 12 x1 4 x2 5 ? 40 ? 60
x1, x2 0, integers
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This is a typical optimization problem.
Any values of x1, x2 that satisfy all the
constraints of the model is called a feasible
solution. We are interested in finding the
optimum feasible solution that gives the maximum
profit while satisfying all the constraints.
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More generally, an optimization problem looks as
follows Determine the decision variables x1,
x2, , xn so as to optimize an objective function
f (x1, x2, , xn) satisfying the constraints
gi (x1, x2, , xn) bi (i1, 2, , m).
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Linear Programming Problems(LPP)

• An optimization problem is called a Linear
  Programming Problem (LPP) when the objective
  function and all the constraints are linear
  functions of the decision variables, x1, x2, ,
  xn. We also include the non-negativity
  restrictions, namely xj 0 for all j1, 2, ,
  n. Thus a typical LPP is of the form

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• Optimize (i.e. Maximize or Minimize)
• z c1 x1 c2 x2 cn xn
• subject to the constraints
• a11 x1 a12 x2 a1n xn b1
• a21 x1 a22 x2 a2n xn b2
• . . .
• am1 x1 am2 x2 amn xn bm
• x1, x2, , xn ? 0

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A LPP satisfies the two properties Proportionalit
y and additivity
Proportionality means the contributions of each
decision variable in the objective function and
its requirements in the constraints are directly
proportional to the value of the variable.
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Additivity stipulates that the total
contributions of all the variables in the
objective function and their requirements in the
constraints are the direct sum of the individual
contributions 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. The respective 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 produces x1 Type
I hats and x2 Type II hats per day.
8 x1 5 x2
Per day Profit got
Assume the time spent in producing one type II
hat is c minutes.
Labour Time spent is
(2 x1 x2) c minutes
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The objective is to find x1, x2 so as to
maximise the profit z 8 x1 5 x2
satisfying the constraints
(2 x1 x2 ) c 400 c x1
150 x2 200 x1, x2
0, integers
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That is The objective is to find x1, x2 so as to
maximise the profit z 8 x1 5 x2
satisfying the constraints
2 x1 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 C
respectively. Two varieties of feed are
available. In feed of variety 1(variety 2) the
contents of the nutrients A, B and C are
respectively 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
10
10
Pattern I
Pattern II
20
10
10
20
10
Pattern III
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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 2 and 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
make 1 top of size 2 and 6 tops of size 1
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So assume that x1 sheets are cut according to
pattern I, x2 according to pattern II, x3
according to pattern III
The problem is to Minimize z x1 x2
x3 Subject to 10 x1 2 x2 6 x3 20,000
2 x2 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 on each
day is given in the table below. Union rules say
that each full-time employee must receive two
days off after working for five consecutive days.
The Post Office wants to meet its requirements
using only full-time employees. Formulate the
above problem as a LPP so as to minimize the
number of full-time employees hired.
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Requirements of full-time employees day-wise
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Solution Let xi be the number of full-time
employees employed at the beginning of day i (i
1, 2, , 7). Thus our problem is to find xi so
as to
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Minimize
Subject to
xi ? 0. integers
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BITS wants to host a Seminar for five days. For
the delegates there is an arrangement of dinner
every day. The requirement of napkins during the
5 days is as follows
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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 costs Rs.
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 the Institute meet
the requirements so that the total cost is
minimized ? Formulate as a LPP.
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Solution Let xj be the number of napkins
purchased on day j, j1,2,..,5 Let yj be the
number of napkins given for washing after dinner
on day j, j1,2,3 Thus we must have
x1 80, x2 50, x3 y1 100, x4 y2 80 x5
y3 150
Also we have y1 80, y2 (80 y1) 50
y3 (80 y1) (50 y2) 100
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Thus we have to Minimize
z 2(x1x2x3x4x5)0.5(y1y2y3) Subject to
x1 80, x2 50, x3 y1 100,
x4 y2 80, x5 y3 150,
y1 80, y1y2 130, y1y2y3
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 LPP problems (of
  course developed by some of your seniors).
• You may yourself develop programs to solve LPP
  problems.