AAOC C222: OPTIMISATION

Text Book:

Operations Research: An Introduction

By Hamdy A.Taha (Pearson Education)

7th Edition

Reference Books:

1. Hadley, G: Linear Programming,

Addison Wesley

2. Pant, J.C: Optimization,

Jain Brothers

3. Hillier & Lieberman

Introduction to Operations

Research, Tata McGraw-Hill

4. Bazaraa, Jarvis & Sherali

Linear Programming and Network

Flows, John Wiley

http://discovery.bits-

pilani.ac.in/discipline/math/msr/index.

html

You may view my lecture slides in the

following site.

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.

Example: The Burroughs garment

company manufactures men's shirts

and women’s 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

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:

Garment Cutting Sewing Packaging Unit

profit($)

Shirts 20 70 12 8.00

Blouses 60 60 4 12.00

Minutes per unit

Determine the optimal weekly

production schedule for Burroughs.

Solution: Assume that Burroughs

produces x1 shirts and x2 blouses per

week. 8 x1 + 12 x2

Time spent on cutting =

Profit got =

Time spent on sewing = 70 x1 + 60 x2 mts

Time spent on packaging = 12 x1 + 4 x2 mts

20 x1 + 60 x2 mts

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

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.

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 (i=1, 2, …, m).

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 j=1, 2, …, n.

Thus a typical LPP is of the form:

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

A LPP satisfies the two properties:

Proportionality 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.

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.

• 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

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.

Solution: Assume that Wild West produces x1

Type I hats and x2 Type II hats per day.

8 x1 + 5 x2

Labour Time spent is (2 x1 + x2) c minutes

Per day Profit got =

Assume the time spent in producing one

type II hat is c minutes.

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

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

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?

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.

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

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

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

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.

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 12

5 - Friday 7

6 - Saturday 9

7 - Sunday 4

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

Minimize 1 2 3 4 5 6 7z x x x x x x x

Subject to

1 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

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:

Day 1 2 3 4 5

Napkins

Needed

80 50 100 80 150

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.

Solution Let xj be the number of napkins

purchased on day j, j=1,2,..,5

Let yj 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) + (50 – y2) + 100

x1 = 80, x2 = 50, x3 + y1 = 100, x4 + y2 = 80

x5 + y3 = 150

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

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 Microsoft’s Excel Solver.

Normally this would not have been loaded;

you mut check whether it is loaded.

• There is also a commercial package

“LINGO”

• Dr. J C Pant’s 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.