Operations Research Project.docx

7/28/2019 Operations Research Project.docx

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A Report on

OPERATIONS RESEARCH TERM PROJECT

Preparedfor:

Dr. AKM Kais Bin Zaman

Assistant ProfessorDeptartment of Industrial and Production Engineering

Preparedby:

Md. Golam Kibria, Student No.: 0908013Md. Rassel Sarker, Student No.: 0908014

Level: 03, Term: 01

Dept. of Industrial and Production Engineering

Submitted on: 20. 06. 2013

BangladeshUniversityofEngineeringandTechnology,Dhaka


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ACKNOWLEDGEMENT

Our project report would be incomplete without thanking some people. We would like to

acknowledge the people who helped and supported us throughout the project work.

First and foremost we would like to convey sincere gratitude to our course teacher, Dr. AKM

Kais Bin Zaman, Assistant Professor, Department of IPE, BUET. He has helped us in

developing and clarifying our concepts on the course and this project by providing us his

valued teaching. While working for this project, his teaching became an integrated part of our

thoughts and ideas.

We would also like to express our gratitude and sincere thanks to Mr. Ramiz Uddin, owner of

the retail shop under study. He provided us most of the required data needed for the project

work. Thanks also go to the employees of the shop who helped us by all means whilecollecting data for the study.

Finally, we would like to thank our classmates for their generous support and encouragement.


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Contents


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List of Illustrations


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ABSTRACT

Operations Research (OR) is a quantitative analysis of complex management problems. From

this analysis, management can make an objective decision.

The main objective of this project is to apply different class concepts to a real life problem.

For completing the project work, we have to handle various phases of an OR study. This will

help us in defining and solving more complex practical optimization problems in the future.

This project study is irrevocably important. Engineering is not all about reading books and

learning techniques. It is valueless if we do not implement it in our day-to-day life. This

project report is an outcome of practical implementation of Operations Research that we have

learned in the classes during the course.

The method we utilized for the purpose of the project study is called Integer Linear

Programming.

INTRODUCTION

Operations Research (OR) emerges During World War II. It began as an interdisciplinary

activity in the military to solve complex problems. It has grown in the last 50 years to a fully

developed engineering discipline. OR is considered as a platform of established mathematical

models. OR has achieved characteristics from mathematics, engineering, business, computer

science, economics, and statistics. Applications of OR is seen in business, industry,

government, and military.

The optimization problem we have solved is faced by the owner of Boimela and Photocopy,

a retail shop of books, photocopy and other study materials. Mr. Ramiz Uddin is the owner

of the shop. It is situated near the Mirpur1 bus stand in Dhaka.

Boimela and Photocopy is a renowned shop established in 1995 as a small retail shop of

books and other educational accessories. It has been fulfilling a variety of demand of the

general people. The shop owner first installed a photocopier in 2003. The shop was then only


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150 square feet. At present, the size of the shop is 500 square feet and there are 4 copiers

(two Toshiba 2860 and two Toshiba 3560). There are six workers that include four copiers

operators. The shop opens at 9 am and closes at 8 pm daily.

PROBLEM DESCRIPTION

Mr. Ramiz Uddin, the owner of Boimela & Photocopy, thinks that he is not currently

utilizing all his resources and hence his current yearly profit is not maximized. He has a lot of

empty space in his shop. Since the demand for photocopy is very high at present, he thinks he

should install more copiers in the shop to maximize his profit. He wants to buy some

reconditioned copiers. There are six types of copiers he is considering to buy. Each of the

type has its own privilege and expense.

It is not clear to him how many of each type of copiers would be most profitable.

We begin by having discussions with Mr. Ramiz Uddin to identify the objective, variables,

constraints and parameters of the problem. These discussions led to developing the following

definition of the problem:

Determine how many of each type of copiers should be bought in order to maximize Mr.

Ramiz Uddins total yearly profit, subject to the restrictions imposed by the limited space for

new machines, owners allocated budget for buying new copiers, current demand, and

photocopy quality that the owner intends to serve the customers to make them pleased as well

as to keep the fame of his shop.


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Methodology Used for Solving the Problem

Since the number of copiers cannot be fractions, the methodology to solve the problem is

Integer Programming. To solve the problem we have used Microsoft Words Excel Solver.

Integer Programming (IP):

An integer programming (IP) problem is a mathematical programming problem in which

some or all of the variables are restricted to assume only integer values. If all variables take

integer values, then the problem is called a pure IP. On the other hand, if both integer and

continuous variables coexist, the problem is called a mixed integer program (MIP). Those

variables whose values can only be either 0 or 1 are called binary variables.Consequently, IP

problems that contain only binary variables sometimes are called binary integer programming

(BIP)problems.An IP problem can be either linear or non-linear.

Our project is to solve a pure Integer Linear Programming. With the use of the matrix

notation, an IP problem can be represented as below.

minimize Z = CX

subject to,

AX bX0 and integer

Here,

X = the vector of integer variables, C = the coefficient vector of the objective function, A= the coefficient matrix of the constraints, b = right-hand-side vector of the constraints,

All elements ofd, B, and b are known constants. Some constraints may be equations and of

the type ..


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Numerical Calculation

DataCollection:


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Table 2: Paper Cost

Type of Papers Price (Tk / 500 pages)

A4 Offset 420

A4 Normal 210

Legal Normal 250

Table 3: Photocopy Price

Type of Papers Single Side (Tk per piece) Both Sides (Tk per piece)

A4 Offset 2 2.5

A4 Normal 1.5 2.0

Legal Normal 1.5 2.0

Table 4: Rating for image quality compare to original copy (on a scale of 100)

Excellent 90100

Very good 8089

Good 7579

Moderate 7074

Bad 6069

Very bad 059

Table 5: Other Expenses

Electricity (Tk per copier in a month) Rent of Shop (Tk per month)

1000 25000


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Other Information:

1. Available space for copiers: 200 square feet.2. Available money for buying new copiers: 1000000 Tk.3. Current demand: 20000 copies per day.4. Amount of money that the owner is willing to spend for maintenance of the machines:

10000 Tk per month.

5. Maximum expected paper waste or copy waste:A4 size offset: 80 copies per day.

A4 size Normal: 150 copies per day.

Legal size Normal: 150 copies per day.

6. Image quality the owner wishes to provide: above 80 on a scale of 100.


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Calculation of Profit Per Year from Various Copy Machines

Copiers Model Toshiba 2860:

Revenue from Toshiba 2860

Revenue from number of copies:

Type of paper No. of copies per

day, n

Price per copy,

p (Tk.)

Total income per day,

np (Tk.)

A4 size Offset (Both Sides) 100 2.5 250

A4 size Offset (Single Sides) 100 2 200

A4 size Normal (Both Sides) 300 2 600

A4 size Normal (Single Sides) 200 1.5 300

Legal size Normal (Both Sides) 175 2 350

Legal size Normal (Single Sides) 50 1.5 75

Total 1775

Total income per day = 1775 Tk

So, Total income per year = 1775 30 12 Tk

= 639000 Tk

Revenue from Salvage Value: Time value of money should be considered here.

Buying Cost: 90,000 TK

Life: 6 yrs

Salvage: 10,000 TK

Interest Rate: 5% (Assumed)

Revenue per year from salvage value = 10000 (A|F, 5%, 6) Tk

= 10000 0.147 Tk

= 1470 Tk


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Total Revenue from Toshiba 2860 per year = (639000 + 1470) Tk

= 640470 Tk

Expenses incurred from Toshiba 2860

Machine Cost = 90000 Tk

Life: 6 years

Machine cost per year = 90000 (A|P, 5%, 6) Tk

= 90000 0.197 Tk

= 17730 Tk

Maintenance cost per month = 1000 Tk

Maintenance cost per year = 1000 12 Tk

= 12000 Tk

Operators salary per month = 10000 Tk

Operators salary per year = 10000 12 Tk

= 120000 Tk

Ink price per packet (190 gm) = 225 Tk

No. of copies can be done using one packet = 6000

Capacities (Copies per day) = 1500

Ink cost per year = 123015006000

225 Tk

= 20250 Tk

Paper Cost

Type of Paper Cost per 500

pages (Tk)

Cost per unit

page (Tk)

No. of paper

used per day

Total cost per

day (Tk)

Total cost per

year (Tk)

A4 size offset 420 0.84 200 168 60480

A4 size normal 210 0.42 500 210 75600

A4 size legal 250 0.5 225 112.5 40500

Total 176580

Paper wastage

Type of Paper Cost per 500

pages (Tk)

Cost per unit

page (Tk)

No. of paper

wasted per day

Total

wastage

Total wastage

cost per year


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cost per day

(Tk)

(Tk)

A4 size offset 420 0.84 8 6.72 2419

A4 size normal 210 0.42 15 6.3 2268A4 size legal 250 0.5 15 7.5 2700

Total 7387

Electricity Cost:

Electricity cost per month = 1000 Tk

Electricity cost per year = 12000 Tk

Total Rent for 500 square feet area per month = 25000 Tk

Space occupied by Toshiba 2860 = 25 square feet

Rent expense for Toshiba 2860 per month = 25500

25000 Tk

= 1250 Tk

Rent expense for Toshiba 2860 per year = 1250 12 Tk

= 15000 Tk


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Total Expense for Toshiba 2860

Type of expense Expense (Tk)

Machine cost 17730

Maintenance cost 12000

Operator expense 120000

Ink cost 20250

Paper cost 176580

Paper wastage expense 7387

Electricity cost 12000

Rent expense 15000

Total expense 380947

Profit from Toshiba 2860 per year = Total revenue - Total expense

= (640470 - 380947) Tk

= 259523 Tk

Like Toshiba 2860 we have evaluated total revenue, total expense and profit per year for

other copiers model. They are given in the following sections.

Copiers Model - Toshiba 3560:


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Total Revenue from Toshiba 3560 = 769005 Tk

Total Expense for Toshiba 3560:


Machine cost 21670



Ink cost 24300

Paper cost 211896



Rent expense 15000



Total Revenue for Toshiba 2060 = 512376 Tk



Machine cost 12805



Ink cost 19440

Paper cost 141264



Rent expense 15000



Total Revenue from Toshiba 4560 = 1066005 Tk



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Machine cost 23640


Operator expense 144000Ink cost 27000

Paper cost 293544



Rent expense 15000


Copiers ModelCanon 1215:

Total Revenue from Canon 1215 = 339135 Tk

Total Expense for Canon 1215


Machine cost 6895



Ink cost 14400

Paper cost 92952



Rent expense 15000


Copiers ModeleStudio 4560:

Total Expense for Toshiba eStudio 4560 = 1198140 Tk

Total Expense for Toshiba eStudio 4560


Machine cost 25610


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Ink cost 39200

Paper cost 330840Paper wastage expense 6091


Rent expense 15000


Summary of per year revenue, expense and profit from different copiers calculated

using collected data:

Type of copiers Revenue per year

(Tk)

Expense per year

(Tk)

Profit per year (Tk)

Toshiba 2860 640470 380947 259523

Toshiba 3560 769005 451409 317596

Toshiba 2060 512376 338989 173387

Toshiba 4560 1066005 563453 502552

Canon 1215 339135 254609 84526

Toshiba eStudio 456 1198140 728741 469399


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Formulation as a Linear Integer Programming Problem:

To formulate the mathematical (linear programming) model for this problem, let

X1 = number of copiers of the type Toshiba 2860.




X5 = number of copiers of the type Canon 1215.

X6 = number of copiers of the type Toshiba eStudio 456.

Z = Total profit per year.

Thus, X1, X2, X3, X4, X5 and X6 are the decision variable for the model.

Table 5: Calculated Profit for different type of copier

Name of Machines Profit

(Tk per year)

Toshiba 2860 259523

Toshiba 3560 317596

Toshiba 2060 173387

Toshiba 4560 502552

Canon 1215 84526

Toshiba eStudio 456 469399

The objective is to choose the values of X1, X2, X3, X4, X5 and X6 so as to maximize Z =

259523 X1 + 317596 X2 + 173387 X3 + 502552 X4 + 84526 X5 + 469399 X6, subject to the

restrictions imposed on their values by the limited space in the shop.


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(1)Each of the copiers occupies 25 square feet and available space for new copiers is 200square feet. This restriction is expressed mathematically by the inequality 25 X1 + 25

X2+ 25 X3+ 25 X4+25 X5+ 25 X6 200.

(2)From Table 1 we find the market price of different reconditioned copiers. Since theowner allocated 1000000 Tk. for new copiers, another restriction can be

mathematically expressed by the inequality 90000 X1 + 110000 X2 + 65000 X3 +

120000 X4+ 35000 X5+ 130000 X6 1000000.

(3)The capacity of each copier is tabulated in table 1. Since the owner thinks the currentdemand is 20000 copies per day, this restriction can be mathematically expressed by

another inequality 1500 X1 + 1800 X2 + 1200 X3 + 2500 X4 + 800 X5 + 2800 X6

20000.

(4)The maintenance cost for each of the copiers is given in table 1. The owner does notwish to spend more than 10000 Tk. per month for the maintenance. This restriction is

expressed mathematically by the inequality 1000 X1 + 1200 X2 + 1000 X3 + 1200 X4

+1100 X5+ 1000 X6 10000.

(5)Paper wastage for each type of paper in each machine is given in table 7. The owner isnot willing to waste more than 80 A4 size offset paper, 150 A4 size normal paper and

150 legal size normal paper per day. These restrictions can be mathematically

expressed by the inequalities:

(i) 8 X1 + 8 X2+ 5 X3+ 12 X4+ 5 X5+ 8 X6 80,

(ii) 15 X1 + 20 X2+ 15 X3+ 20 X4+ 8 X5+ 10 X6 150

(iii) 15 X1 + 15 X2+ 15 X3+ 20 X4+ 8 X5+ 12 X6 150

(6)Rating for image quality compare to original copy is given in table 5. The ownerwishes to provide a minimum image quality of 80 on a scale of 100. This image

quality restriction can be expressed mathematically by the following inequality:

80907085758580

654321

654321

XXXXXX

XXXXXX


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The final form of this equation yields:

0 X1 + 5 X2- 5 X3+ 5 X4 - 10 X5+ 10 X6 0

To summarize, in a mathematical language of linear integer programming, the problem is to

choose values of X1, X2, X3, X4, X5 and X6 so as to

Maximize Z = 259523 X1 + 317596 X2 + 173387 X3 + 502552 X4 + 84526 X5 + 469399 X6,

subject to the constraints

25 X1 + 25 X2+ 25 X3+ 25 X4+25 X5+ 25 X6 200

90000 X1 + 110000 X2+ 65000 X3+ 120000 X4+ 35000 X5+ 130000 X6 1000000

1500 X1 + 1800 X2+ 1200 X3+ 2500 X4+ 800 X5+ 2800 X6 20000

1000 X1 + 1200 X2+ 1000 X3+ 1200 X4+1100 X5+ 1000 X6 10000

8 X1 + 8 X2+ 5 X3+ 12 X4+ 5 X5+ 8 X6 80

15 X1 + 20 X2+ 15 X3+ 20 X4+ 8 X5+ 10 X6 150

15 X1

+ 15 X2+ 15 X

3+ 20 X

4+ 8 X

5+ 12 X

6 150

0 X1 + 5 X2- 5 X3+ 5 X4 - 10 X5+ 10 X6 0

Xj 0, for j = 1, 2. . . 6.

And Xj is integer, for j = 1, 2. . . 6.


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Solution:

The mathematical model formulated in the previous section has been solved using the Excel

Solver. The optimal solution which maximizes the objective function is

(X1, X2, X3, X4, X5, X6) = (0, 2, 0, 4, 0, 2)

The corresponding value of Z = 3584198 Tk

The screenshot of the Excel Solver is attached here.

Figure 1: Solution and optimized objective function value obtained using the Excel Solver


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Conclusion

During our project work we first defined the problem, and then collected data. To represent

the problem, we formulated a mathematical model. Finally, we developed a computer-based

procedure, i.e. the Excel Solver, to solve the problem from the developed model.

The most challenging and yet most interesting phase of this OR study was the mathematical

formulation of the real-life system. After identifying the problem we detected the parameters

and the variables which are involved in this problem. To keep the model as simple as possible

we selected those variables which seemed most influential. Then we stated verbal

relationship among these variables based on collected data.

After completing this project work, we have gathered a great deal of knowledge which, we

believe, we will be able to implement very efficiently in the future. Since OR has its

applications in defense, industry and in all public system, the importance of having a clear

knowledge about Operations Research is beyond description.

References

Documents

Operations Research Project.docx