Linear Programming Applications

1. LINEAR PROGRAMMING
APPLICATIONS
Kashif Latif
Sumbal Babar

2. What is LP Applications
Most successful quantitative approach to
decision making, also have been reported
almost every industry. Application includes
 Production Scheduling
 Media Selection
 Financial Planning
 Capital Budgeting
 Transportation
 Distribution System Design
 Staffing

3. What we’ll Cover
 Marketing Applications
 Financial Applications
 Operations Management Applications

4. What is Marketing
Marketing is
communicating the value
of a product, service or
brand to customers, for
the purpose of
promoting or selling that
product, service, or
brand.

5. Marketing Applications
 Media Selection
 Marketing Research

6. Media Selection
Help marketing managers to allocate a fixed
advertising budget to various advertising media.
Media includes
 Newspapers
 Magazines
 Radio
 Television
 Direct Mail

7. Objective
Objective of Media Selections includes
 Maximize Reach
 Frequency
 Quality of Exposure

8. Restrictions
 Company Policy
 Contract Requirements
 Media Availability

9. Relax-and-Enjoy Lake Development
Corporation
Advertising Media
No. of
Potential
Customer
Reached
Cost ($) per
Advertiseme
nt
Maximum
Time
Available
per Month
Exposure
Quality
Units
Daytime TV (1 min),
station WKLA
1000 1500 15 65
Evening TV (30 sec),
station WKLA
2000 3000 10 90
Daily Newspaper (full
page), The Morning
Journal
1500 400 25 40
Sunday Newspaper
magazine (1/2 page color),
The Sunday Press
2500 1000 4 60
Radio, 8:00 AM or 5:00 PM
news (30 sec), station
KNOP
300 100 30 20

10. Decision Variables
DTV = number of times daytime TV is used
ETV = number of time evening TV is used
DN = number of time daily newspaper is used
SN = number of time Sunday newspaper is
used
R = number of times radio is used

11. Objective Function
With the objective of maximizing the total
exposure quality units for the overall media
selection plan, the objective function becomes
 Max 65DTV+90ETV+40DN+60SN+20R

12. Formulate Constraints
DTV ≤ 15
ETV ≤ 10
DN ≤ 25
SN ≤ 4
R ≤ 30
Availability of
Media

13. Continue…
1500DTV + 3000ETV + 400DN + 1000SN + 100R ≤ 30000
Budget
DTV + ETV ≥ 10
1500DTV + 3000ETV ≤ 18000
Television Restrictions
1000DTV + 2000ETV + 1500DN + 2500SN + 300R ≥ 50000
Customer Reached
DTV, ETV, DN, SN, R ≥ 0

14. Decision Variables
Decision
Variables
DTV ETV DNP SNP R
Adds 10 0 25 2 30

15. Advertising Plan
Media Frequency Budget ($)
Daytime TV 10 15,000
Daily Newspaper 25 10,000
Sunday Newspaper 2 2,000
Radio 30 3,000
Total 30,000

16. Results
Exposure Quality Units = 2,370
Total Customers Reached = 61,500

17. Marketing Research
A research to learn about
 Consumer Characteristics
 Attitudes
 Preferences

18. Marketing Research Firms
Specialized in marketing research for client
organization. Services they offer includes:
 Designing the Study
 Conducting Market Surveys
 Analyzing the Data Collected
 Providing Summary Reports &
Recommendations

19. Market Survey, Inc.
1. Interview at least 400 households with children.
2. Interview at least 400 households without
children.
3. The total number of households interviewed
during the evening must be at least as great as
the number of households interviewed during the
day.
4. At least 40% of the interviews for households
with children must be conducted during the
evening.
5. At least 60% of the interviews for households
without children must be conducted during the

20. Previous Cost Estimations
Interview Cost
Household Day Evening
Children $20 $25
No Children $18 $20

21. Decision Variables
DC = the number of daytime interviews of
households with children
EC = the number of evening interviews of
households with children
DNC = the number of daytime interviews of
households without children
ENC = the number of evening interviews of
households without children

22. Objective Function
Using previous cost estimation, the object
function would be
Min 20DC + 25EC + 18DNC + 20ENC

23. Formulate Constraints
1. DC + EC + DNC + ENC = 1000
2. DC + EC ≥ 400
3. DNC + ENC ≥ 400
4. EC + ENC ≥ DC + DNC
The usual format for linear programming model
formulation places all decision variables on the left side
of the inequality and a constant (possibly zero) on the
right side. Thus, we rewrite this constraint as
4. – DC + EC – DNC + ENC ≥ 0

24. Formulate Constraints
5. EC ≥ 0.4(DC + EC) or -0.4DC + 0.6EC ≥ 0
6. ENC ≥ 0.6(DNC + ENC) or -0.6DNC + 0.4ENC
≥ 0
Nonnegativity Requirements
DC, EC, DNC, ENC ≥ 0

25. Interview Schedule
Number of Interviews
Household Day Evening Totals
Children 240 160 400
No Children 240 360 600
Totals 480 520 1000

26. Financial Application
In finance, linear programming can be applied in
problem situations involving:
 Capital Budgeting
 Make-or-Buy Decisions
 Asset Allocation
 Portfolio Selection
 Financial Planning, and many more.

27. Financial Application Problems
 Portfolio Selection
 Financial Planning

28. Portfolio Selection
Portfolio selection
problems involve situations
in which a financial
manager must select
specific investments for
example stocks and bonds
from a variety of
investment alternatives.

29. Objective Function
The objective function for portfolio
selection problems usually is maximization
of expected return or minimization of risk.

30. Constraints
The constraints usually reflect restrictions on the
type of
 Permissible Investments
 State Laws
 Company Policy
 Maximum Permissible Risk, and so on.

31. Welte Mutual Funds, Inc.
Projected Rate of Return
Investment (%)
Atlantic Oil 7.3
Pacific Oil 10.3
Midwest Steel 6.4
Huber Steel 7.5
Government Bonds 4.5

32. Decision Variables
A = dollars invested in Atlantic Oil
P = dollars invested in Pacific Oil
M = dollars invested in Midwest Steel
H = dollars invested in Huber Steel
G = dollars invested in government bonds

33. Objective Function
Objective function for maximizing the total return
for the portfolio is
Max 0.073A 0.103P 0.064M 0.075H 0.045G

34. Linear Programming Model
1. A + P + M + H + G = 100,000
2. A + P ≤ 50,000
3. M + H ≤ 50,000
4. -0.25M - 0.25H + G ≥ 0
5. -0.6A + 0.4P ≤ 0
A, P, M, H, G ≥ 0

35. Optimal Portfolio Selection
Investment Amount ($)
Expected Annual
Return ($)
Atlantic Oil 20,000 1,460
Pacific Oil 30,000 3,090
Huber Steel 40,000 3,000
Government Bonds 10,000 450
Totals 100,000 8000
Expected Annual Return of $8000
Overall Rate of Return = 8%

36. Financial Planning
Financial Planning is an ongoing process to help
you make sensible decisions about money that
can help you achieve your goals in life.

37. Hewlitt Corporation
Year 1 2 3 4 5 6 7 8
Cash Requirements 430 210 222 231 240 195 225 255
 The cash requirements (in thousands of dollars) are
due at the beginning of each year.

38. Government Bonds
Investments
Bond Price ($) Rate (%) Years to Maturity
1 1150 8.875 5
2 1000 5.500 6
3 1350 11.750 7

39. Decision Variables
F = total dollars required to meet the retirement plan’s
eight- year obligation
B1 = units of bond 1 purchased at the beginning of year 1
B2 = units of bond 2 purchased at the beginning of year 1
B3 = units of bond 3 purchased at the beginning of year 1
Si = amount placed in savings at the beginning of year i for
I = 1, . . . , 8

40. Objective Function
The objective function is to minimize the total
dollars needed to meet the retirement plan’s
eight-year obligation, or
Min F

41. General Form of Constraint
Funds available
at the beginning
of the year
-
Funds invested
in bonds and
placed in
savings
=
Cash obligation
for the current
year

42. Constraint of Each Year
F - 1.15B1 - 1B2- 1.35B3 - S1 = 430
Year 1
0.08875B1+0.055B2+0.1175B3+1.04S1-S2 = 210 Year
2
0.08875B1+0.055B2+0.1175B3+1.04S2-S3 = 222 Year
3
0.08875B1+0.055B2+0.1175B3+1.04S3-S4 = 231 Year
4
0.08875B1+0.055B2+0.1175B3+1.04S4-S5 = 240 Year
5
1.08875B1+0.055B2+0.1175B3+1.04S5-S6 = 195 Year
6

43. Optimal Solution
Bond Units Purchased Investment Amount
1 B1=144.988 $1150(144.988)=$166,736
2 B2=187.856 $1000(187.856)=$187,856
3 B3=228.188 $1350(228.188)=$308,054

44. Operation Management
Applications
Managing and directing the physical and/or
technical functions of a firm or organization,
particularly those relating to
 Development
 Production
 Manufacturing

45. What we’ll cover
 Make-or-Buy Decision
 Production Scheduling
 Workforce Assignment
 Blending Problems

46. Make-or-Buy Decision
It determine how much of each of several
component parts a company should
manufacture and how much it should purchase
from an outside supplier.

47. Production Scheduling
Establish an efficient low-cost production
schedule for one or more products over several
time periods (weeks or months)

48. Advantages
 Can help to smooth the demand signal
 Protects lead time and helps book future
deliveries
 Acts as a single communication tool to the
business
 Helps the Supply chain prioritize requirement
 Helps stabilize production

49. Disadvantages
 Complexity
 Cost
 Can be Skewed
 Lack of Flexibility

50. Workforce Assignment
Workforce assignment problems frequently
occur when production managers must make
decisions involving staffing requirements for a
given planning period.

51. Blending Problems
Blending problems arise whenever a manager
must decide how to blend two or more resources
to produce one or more products.