Sains (DFX Global)

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BJQP 2023 MANAGEMENT SCIENCE

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1.0 QUESTION 1

Describe how linear programming is related to your chosen area of study.

Background of the Company (DFX GLOBAL RESOURCES)

For this Management Science Course, we are required to do assignment on how linear

programming is related to our chosen area of study. Our group consists of 6 members and

we chose marketing as our study field. We conducted an interview with the DFX GLOBAL

RESOURCES¶s Manager and Marketing Coordinator, Mr. Mohamad Rosli Bin Abdullah.

DFX GLOBAL RESOURCES was founded on February 1, 2008. Kind of business

is a partnership. Registration number of companies is JM0506982-H. DFX GLOBAL

RESOURCES located at No. 9 Tingkat 1, Jalan Lada Hitam, Taman Sri Tengah, 86000

Kluang, Johor and certified under the Business Registration Act 1956. This company led by

two partners, where one of them has extensive experience of over 15 years experience in

Information Technology. Meanwhile, his partner was more than 5 years experience in

management and business administration.

DFX GLOBAL RESOURCES is also controlled by the management board of highly

qualified and trained manpower. The company always ready to carry out any work entrusted

to the well and put the best quality service and customer satisfaction.

DFX GLOBAL RESOURCES was established in 2008 as the ICT-based creative

solutions in Kluang, Johor. Previously known as the DFX GLOBAL RESOURCES, aims to

improve competitiveness by providing assistance to partner them to better exploit

opportunities in ICT atmosphere. In February 2008, while retaining the status of the

Bumiputra, the team of experienced personnel recruiting and diversity efforts only focus on




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providing communications consulting and media service company, primarily through the

use of ICT and multimedia systems.

High expertise in providing advice in all aspects related to the Software

Development technology and management and Technical Training is a priority in the DFX

GLOBAL RESOURCES. Effective advisory provides opportunities for the DFX GLOBAL

RESOURCES to provide and supply services from foundation up to the supply of trained

teachers and high quality.

DFX GLOBAL RESOURCES also carry a variety of business services according to

the current market. Its main activities are focused on Information Technology (IT). Among

the supplies, to rent and sell equipment related to IT such as computers, scanners, printers

and so forth. The company also provides computer repair services to foreign companies and

individuals as well as offering audio (PA) systems, video and photography for a variety of

receptions, such as weddings, engagements and so forth.

With the concept of IT, DFX GLOBAL RESOUCES also provides software

development services (Desktop & Web Applications) and making web pages and E-

commerce. Offers made to all individuals and companies who are interested. The company

is also available to supply product such as clothing and souvenirs by the demand and

preferences of customers at present.

In an effort to improve the quality and service, DFX GLOBAL in December 2009

has established two branches, each of which is located in Kluang, Johor and Bangi,

Selangor. In addition, DFX GLOBAL RESOURCES has also widened the scope of services

such as supply of stationary, teaching aids, food and beverage, advertising, agricultural

services, cleaning, construction work and hire a tent (canopy).




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Linear Programming

Linear programming (LP) is a mathematical procedure for determining optimal

allocation of scarce resources. LP is a procedure that has found practical application in

almost all facets of business, from advertising to production planning. Transportation,

distribution, and aggregate production planning problems are the most typical objects of LP

analysis.

Linear programming deals with a class of programming problems where both the

objective function to be optimized is linear and all relations among the variables

corresponding to resources are linear. This problem was first formulated and solved in the

late 1940¶s. Rarely has a new mathematical technique found such a wide range of practical

business, commerce and industrial applications and simultaneously received so thorough a

theoretical development, in such a short period of time. Today, this theory is being

successfully applied to problems of capital budgeting, conversation of resources, games of

strategy, economic growth prediction, and transportation systems.

In very recent times, linear programming theory has also helped company to resolve

and unify many outstanding applications. It is important for the reader to appreciate, at the

outset that the ³programming´ in Linear Programming is of a different flavor than the

³programming´ in Computer Programming.

Any LP problem consists of an objective function and a set of constraints. In the most

cases, constraints come from the environment in which you work to achieve your objective.

When we want to achieve the desirable objective, we will realize that the environment is

setting some constraint (i.e., the difficulties, restrictions) in fulfilling your desire or

objective.




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A function is a thing that does something. For example, a coffee machine is a function

that transforms the coffee beans into powder. The (objective) functions maps and translates

the input domain (called the feasible region) into output range, with the two end-values

called the maximum and minimum values.

When we formulated a decision making problem as a linear program, we must check

the following conditions:

y The objective function must be linear. That is, check if all variables have power of 1

and they are added or subtracted (not divided or multiplied).

y The objective must be either maximization or minimization of a linear function. The

objective must represent the goal of the decision maker.

y The constraints must also be linear. Moreover, the constraint must be of the following

forms (�, �, or =, that is, the LP constraints are always closed).

For most LP problems one can think of two important classes of objects: The first is

limited resources as land, plant capacity, or sales force size; the second is activities such as

produce low carbon steel, produce stainless steel and produce high carbon steel. Each

activity consumes or possibly contributes additional amounts of the resources. There must

be an objective function. The problem is to the best combination of activity levels, which do

not use more resources than are actually available. Many managers are faced with this task

every day. Fortunately, when a well-formulated model is input, linear programming

software helps to determine the best combination.




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Linear programming can be used to determine the proper mix of media to use in an

advertising campaign. In this case, it can be used to determine the proper mix of channel to

use in a promotion activity in DFX GLOBAL RESOURCES. Suppose that the available

media are internet (website of the company and Facebook), salespersons (Sales and Marketing

Coordinator and Officers) and promotion by DFX GLOBAL (indirect agents).

The problem is to determine how many advertisements to place in each medium. Of

course, the cost of placing an advertisement depends on the medium chosen. Since each

medium may provide a different degree of exposure of the target population, there may be a

lower bound on the total exposure from the campaign. Also, each medium may have a

different efficiency rating in producing desirable results; there may thus be a lower bound on

efficiency. In addition, there may be limits on the availability of each medium for advertising.




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2.0 QUESTION 2

Briefly and analytically introduce a problem related to your area of study. Clearly

highlight the problem you want to solve. Make sure your decision variables are more

than 3.

The DFX GLOBAL RESOURCES has assigned the sales and marketing coordinator to

determine the types and amount of advertising it should invest in for its company. The four

types of advertising and promotion are by the sales person, internet, promotion by DFX

GLOBAL RESOURCES and media massa. Promotion by sales persons is use brochures and

talks to consumers, internet is absolutely use the website, promotion by the company itself

use campaign or open the kiosk stall and lastly is promotion with media massa with using

the electronic media and print media like radio, television or pamphlets.

DFX GLOBAL RESOURCES has a yearly promotion budget of RM100, 000. The

company desires to know the number of each type of promotion it should invest more, in

order to maximize exposure. It is estimated that each advertising or promotion will reach the

following potential audience and cost the following amount:

Exposure (people per

advertisement)

Cost (RM per

advertisement)

Sales persons 10 000 8 000

Internet 6 000 5 000

Kiosk stall

promotion

7 000 5 000

Media massa 5 000 4 000




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The company must consider the following resource constraints:

1) The sales persons have time available for 6 promotion talks.

2) The website has page available for 3 advertisements.

3) The kiosk stall promotion has events available for 5 advertisements.

4) The media massa promotion available for 4 advertisements.

5) The sales and marketing has time and workers available to handle no more than a total

of 15 advertisements.

Decision Variables

This model consists of 4 decision variables that represent the number of each type of

advertisement:

X1: number of sales persons advertisements

X2: number of internet advertisements

X3: number of kiosk stall has event promotion

X4: number of promotion by media massa




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The Objective Function

The objective of this problem is to maximize audience exposure. The objective audience

exposure is determined by summing the audience exposure gained from each type of

advertising:

Maximize Z: 10 000 X1 + 6 000 X2 + 7 000 X3 + 5 000 X4

Where Z = total level of audiences exposure

10 000 X1 = estimated number of people reached by sales persons advertisement.

6 000 X2 = estimated number of people reached by internet advertisements.

7 000 X3 = estimated number of people reached by kiosk stall event promotion by DFX

GLOBAL RESOURCES advertisements.

5 000 X4 = estimated number of people reached by media massa advertisements.




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Model Constraints

The first constraints in this model reflects the limited budget of RM 100 000 allocated for

advertisements.

8 000 X1 + 5 000 X2 + 5 000 X3 + 4 000 X4 � 100 000

Where:

8 000 X1 = amount spent for sales persons advertising.

5 000 X2 = amount spent for internet advertising.

5 000 X3 = amount spent for kiosk stall promotion advertising by DFX GLOBAL

RESOURCES.

4 000 X4 = amount spent for media massa advertising.

The other 3 constraints:

X1 < 6

X2 < 3

X3 < 5

X4 < 4

The final constraint specifies that the total number of advertisement must not exceed 15

because of limitations of the sales and marketing department:

X1 + X2 + X3 + X4 � 15 advertisements




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Model Summary

The complete linear programming model for this problem is summarized as:

Maximize Z: 10 000 X1 + 6 000 X2 + 7 000 X3 + 5 000 X4

Subject to:

8 000 X1 + 5 000 X2 + 5 000 X3 + 4 000 X4 � 100 000

X1 < 6

X2 < 3

X3 < 5

X4 < 4

X1 + X2 + X3 + X4 � 15

X1, X2, X3, X4 � 0




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3.0 QUESTION 3

Analyze your data by using QM for Windows or Excel QM. Show your solution.

We analyze our data and solve the linear problem using QM for Windows.

BJQP2023-LINEAR PROGRAMMING APPLICATION

X1 X2 X3 X4 RHS Equation form

Maximize 100 000 6 000 7 000 5 000 Max 100,000X1 + 6,000X2 + 7,000X3 +

5,000X4

Constraint

1

8 000 5 000 5 000 4 000 < = 100 000 8,000X1+5,000X2+5,000X3+4,000X4

<= 100,000

Constraint

2

1 0 0 0 < = 6 X1 < = 6

Constraint

3

0 1 0 0 < = 3 X2 < = 3

Constraint

4

0 0 1 0 < = 5 X3 < = 5

Constraint

5

0 0 0 1 < = 4 X4 < = 4

Constraint

6

1 1 1 1 < = 15 X1 + X2 + X3 + X4 < = 15

LINEAR PROGRAMMING RESULTS

Variable Status Value

X1 Basic 6

X2 Basic 3

X3 Basic 5

X4 Basic 1

Slack 1 Basic 8 000

Slack 2 NONBasic

Slack 3 NONBasic

Slack 4 NONBasic

Slack 5 Basic 3

Slack 6 NONBasic

Optimal value (Z) 118 000




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RAGING

Variable Value Reduced

Cost

Original Val Lower

Bound

Upper

Bound

X1 6 0 10000 5000 InfinityX2 3 0 6000 5000 Infinity

X3 5 0 7000 5000 Infinity

X4 1 0 5000 0 6000

Constraint Dual Value Slack/Surplus Original Val Lower Bound Upper Bound

Constraint 1 0 8000 100000 92000 Infinity

Constraint 2 5000 0 6 3 7



Constraint 5 0 3 4 1 Infinity

Constraint 6 5000 0 15 14 17

ORIGINAL PROBLEM W/ANSWERS

X1 X2 X3 X4 RHS Dual

Maximize 10000 6000 7000 5000 < = Max 10000X1 + 6000X2 +

7000X3 + 5000X4

Constraint 1 8000 5000 5000 4000 < = 100000 0

Constraint 2 1 0 0 0 < = 6 5000

Constraint 3 0 1 0 0 < = 3 1000

Constraint 4 0 0 1 0 < = 5 2000

Constraint 5 0 0 0 1 < = 4 0

Constraint 6 1 1 1 1 < = 15 5000

Solution-> 6 3 5 1 Optimal Z-> 118000 X1 + X2 + X3 + X4 <= 15




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ITERATIONS

Cj Basic

variables

Qty 10000

X1

6000

X2

7000

X3

5000

X4

0

Slack

1

0

Slack

2

0

Slack

3

0

slack

4

0

slack

5

0

slack

6

Iteration 1

0 Slack 1 100000 8000 5000 5000 4000 1 0 0 0 0 0

0 Slack 2 6 1 0 0 0 0 1 0 0 0 0

0 Slack 3 3 0 1 0 0 0 0 1 0 0 0

0 slack 4 5 0 0 1 0 0 0 0 1 0 0

0 Slack 5 4 0 0 0 1 0 0 0 0 1 0

0 Slack 6 15 1 1 1 1 0 0 0 0 0 1

zj 0 0 0 0 0 0 0 0 0 0 0

cj-zj 10000 6000 7000 5000 0 0 0 0 0 0

Iteration 2

0 Slack 1 52000 0 5000 5000 4000 1 -8000 0 0 0 0

10000 X1 6 1 0 0 0 0 1 0 0 0 0

0 Slack 3 3 0 1 0 0 0 0 1 0 0 0

0 Slack 4 5 0 0 1 0 0 0 0 1 0 0

0 Slack 5 4 0 0 0 1 0 0 0 0 1 0

0 Slack 6 9 0 1 1 1 0 -1 0 0 0 1

zj 60000 10000 0 0 0 0 10000 0 0 0 0

cj-zj 0 6000 7000 5000 0 0 0 0 0

Iteration 3

0 Slack 1 27000 0 5000 0 4000 1 -8000 0 -5000 0 0

10000 X1 6 1 0 0 0 0 1 0 0 0 0

0 Slack 3 3 0 1 0 0 0 0 1 0 0 0

7000 X3 5 0 0 1 0 0 0 0 1 0 0

0 Slack 5 4 0 0 0 1 0 0 0 0 1 0

0 Slack 6 4 0 1 0 1 0 -1 0 -1 0 1

zj 95000 10000 0 7000 0 0 10000 0 7000 0 0

cj-zj 0 6000 0 5000 0 0 -7000 0 0

Iteration 4

0 Slack 1 12000 0 0 0 4000 1 -8000 -5000 -5000 0 0

10000 X1 6 1 0 0 0 0 1 0 0 0 0

6000 X2 3 0 1 0 0 0 0 1 0 0 0

7000 X3 5 0 0 1 0 0 0 0 1 0 0

0 Slack 5 4 0 0 0 1 0 0 0 0 1 0

0 Slack 6 1 0 0 0 1 0 -1 -1 -1 0 1

zj 113000 10000 6000 7000 0 0 10000 6000 7000 0 0

cj-zj 0 0 0 5000 0 -6000 -7000 0 0

Iteration 5

0 Slack 1 8000 0 0 0 0 1 -4000 -1000 -1000 0 -4000

10000 X1 6 1 0 0 0 0 1 0 0 0 0

6000 X2 3 0 1 0 0 0 0 1 0 0 07000 X3 5 0 0 1 0 0 0 0 1 0 0

0 slack 5 3 0 0 0 0 0 1 1 1 1 -1

5000 X4 1 0 0 0 1 0 -1 -1 -1 0 1

zj 118000 10000 6000 7000 5000 0 5000 1000 2000 0 5000

cj-zj 0 0 0 0 0 -5000 -1000 -2000 0 -5000




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ORIGINAL PROBLEM

DUAL PROBLEM

Constraint

1

Constraint

2

Constraint

3

Constraint

4

Constraint

5

Constraint

6

Minimize 100 000 6 3 5 4 15

X1 8 000 1 0 0 0 1 >= 100 000

X2 5 000 0 1 0 0 1 >= 6 000

X3 5 000 0 0 1 0 1 >= 7 000

X4 4 000 0 0 0 1 1 >= 5 000

Maximize X1 X2 X3 X4

Constraint 1 8 000 5 000 5 000 4 000 < = 100 000

Constraint 2 1 0 0 0 < = 6

Constraint 3 0 1 0 0 < = 3

Constraint 4 0 0 1 0 < = 5

Constraint 5 0 0 0 1 < = 4

Constraint 6 1 1 1 1 < = 15




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4.0 QUESTION 4

Discuss and conclude your findings.

Computer software packages are built to handle linear programs involving large numbers of

variables and constraints. In our research, we will make solution which is to maximize

audience exposure. Based on the marketing department at DFX GLOBAL RESOURCES,

there are 4 decision variables that represent the number of each type of advertisement:

X1: number of sales persons advertisements

X2: number of internet advertisements

X3: number of kiosk stalls promotion

X4: number of promotion by media massa

Refer to the information above, there are the management scientist solutions for the

modified DFX GLOBAL RESOURCES problem. The solution was shown at optimal value

in problem and results. It is shown that the optimal solution is 118 000. Besides that, the

result also showed the optimal solution for every variable. The solution of variables was

shown in the raging column at value. We can see that the optimal solution of audience

exposure is 6 sales person advertisements, 3 internet advertisements, 5 kiosk stall promotion

by DFX GLOBAL RESOURCES advertisements and 1 promotion advertisements by media

massa.




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Then we analyze the information contained in Reduced Cost Column. Recall that the

reduced costs indicate how much each objective function coefficient would have improved

before the corresponding decision variable could assume a positive value in the optimal

solution. As the results shown at the answer in question 3, all the decision variables which

are sales person¶s advertisements, internet advertisements, kiosk stall promotion by DFX

GLOBAL RESOURCES advertisements and promotion advertisement by media massa are

zero in the reducing cost. So there will be no resolve on decision variables.

By referring to the answer, there also shown the dual prices of RM5 000 for

constraints 2, RM1 000 for constraints 3, RM2 000 for constraints 4 and RM5 000 for

constraints 6. Respectively, indicating that the four constraints are binding in optimal

solutions.

Next, each additional audience exposure in constraint 2 which is time available for

sales person talk would increase the value optimal solution by RM5 000. The same goes to

exposure in constraints 3 by pages available for advertisement would increase the value

optimal solution at RM1 000. Audience exposure in constraint 4 by kiosk stall promotion

has events available for advertisement increase the value optimal solution at RM2 000.

Lastly, in constraint 6, DFX GLOBAL RESOURCES will increase the value optimal in total

of RM5 000 in terms of time and staff available for handling advertisement.




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Because of slack in RM8 000 in the maximize audience exposure as constraint 1 and

RM3 in advertising promotion by media massa, management might use these unused

audience exposure and diversify it in other media of promotion. So, perhaps DFX GLOBAL

RESOURCES can take any advertisement, staff or sales person in that roles to help other

advertisement such as action to create a superb pages for promotion or making their own

event to interact audience to their products.

As the conclusion, DFX GLOBAL RESOURCES should manage the advertisement

by the best solution which is listed as below:

Decision variables Optimal solution for DFX GLOBAL RESOURCES

Sales persons 6 promotion talks to promoting

Internet 3 pages of advertisement

Kiosk stall promotion 5 event

Media massa 1 advertisement

Total exposure 118 000 people

Sains (DFX Global)

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