1. Managerial Decision Modeling with Spreadsheets
Chapter 3
Linear Programming Modeling Applications: With Computer Analyses in Excel

2. Learning Objectives
Model wide variety of linear programming (LP) problems.
Understand major business application areas for LP problems: manufacturing, marketing, labor scheduling, blending, transportation, finance, and multi-period planning.
Gain experience in setting up and solving LP problems using Excel’s Solver.

3. 3.1 Introduction
Delta Airlines is example of use of LP model for solving real-world scheduling problems resulting in significant cost reductions for company.
Similar examples exist in other managerial decision making areas, such as:
production mix, labor scheduling, job assignment,
production scheduling, marketing research,
media selection, shipping and transportation,
ingredient mix, and financial portfolio selection.
Purpose is to show how one can use LP to modeling for decision-making in these areas.

4. 3.2 Marketing Application
Media Selection -
Win Big Gambling Club promotes gambling junkets from a large Midwestern city to casinos in the Bahamas.
Club has budgeted up to $8,000 per week for local advertising.
Money is to be allocated among four promotional media:
TV spots,
Newspaper ads, and
Two types of radio advertisements.
Win Big’s goal - reach largest possible high-potential audience through various media.

5. Audience Reached Per Ad
Media Selection Data
Audience Reached Per Ad
Cost
Per Ad(s)
Maximum Ads Per Week
TV spot (1minute)
5,000
800 12
Daily newspaper (full page)
8,500
925 5
Radio spot (30 sec, prime time)
2,400
290 25
Radio spot (1 min, afternoon)
2,800
380 20
Contract arrangements require at least five radio spots be placed each week.
Management insists no more than $1,800 be spent on radio advertising each week.

6. Media Selection Data LP Formulation
Objective: maximize audience coverage =
5000T N P +2800A
T = number of 1-minute TV spots taken each week.
N = number of full-page daily newspaper ads taken each week.
P = number of 30-second prime-time radio spots taken each week.
A = number of 1-minute afternoon radio spots taken each week.

7. Media Selection Data LP Formulation
Objective: maximize audience coverage =
5000 T N P A
Subject to

8. Marketing Research Problem
Management Sciences Associates (MSA) handles consumer surveys. MSA has to determine, for a client, that it must fulfill several requirements in order to draw statistically valid conclusions on sensitive issue of new U.S. immigration laws:
1. Survey at least 2,300 U.S. households.
2. Survey at least 1,000 households whose heads
are 30 years of age or younger.

9. Marketing Research Problem
3. Survey at least 600 households whose heads are
between 31 and 50 years of age.
4. Ensure that at least 15% of those surveyed live
in a state that borders on Mexico.
5. Ensure that no more than 20% of those surveyed
who are 51 years of age or over live in a state
that borders on Mexico.

10. MSA’s Goal: Meet Sampling Requirements With Minimum Cost
Objective: minimize total interview costs =
$7.50 B1 + $6.80 B2 + $5.50 B3 +
$6.90 N1 + $7.25 N2 + $6.10 N3
B1 = number 30 years or younger and live in border state.
B2 = number years and live in border state.
B3 = number 51 years or older and live in border state.
N1 = number 30 years or younger and do not live in border
state.
N2 = number years and do not live in border state.
N3 = number 51 years or older and do not live in border state.

11. MSA’s Goal LP Formulation
Objective: minimize total interview costs =
$7.50 B1 + $6.80 B2 + $5.50 B3 +
$6.90 N1 + $7.25 N2 + $6.10 N3
Subject to

12. Rewriting Last Two Constraints
B1 + B2 + B3  0.15(B1 + B2 + B3 + N1 + N2 + N3)
Rewritten as:
 B1 + B2 + B (B1 + B2 + B3 + N1 + N2 + N3)  0
Simplifies to:
0.85B B B N N N3  0
And
B3 ≤ 0.2(B3 + N3)
0.8B N3 < 0

13. Optimal Solution to MSA’s Marketing Research Problem
Optimal solution shows that it costs $15,166 and requires one to survey households as follows:
 State borders Mexico and years =
State borders Mexico and  51 years =
State not borders Mexico and  30 years = 1,000
State not borders Mexico and  51 years =

14. 3.3 Manufacturing Applications
Production Mix Problem
Fifth Avenue Industries
Nationally known menswear manufacturer.
Produces four varieties of neckties.
All-silk tie.
All-polyester tie.
Two different polyester and cotton blends.
Has fixed contracts with major department stores.
Table 3.1 summarizes contract demand for products.

15. 3.3 Manufacturing Applications
Fifth Avenue Industries
Table 3.1 Data for Fifth Avenue Industries
Tie
Price
Monthly Contract
Monthly Demand
Material Required
Material
Silk
6.70
6,000
7,000
0.125
Polyester
3.55
10,000
14,000
0.08
Poly-Cotton
Blend 1
4.31
13,000
16,000
0.10
50%-50%
Blend 2
4.81
8,500
30%-70%

16. Profit Per Unit Fifth Avenue Industries For each all-silk tie -
Cost per tie = yards of silk x $21 per yard = $2.625.
Revenue per tie = $6.70 selling price per silk tie.
Profit per tie = Revenue per tie - Cost per tie =
$ $ = $4.075.
Profit for other three products -
Profit per all-polyester tie = $3.07.
Profit per Blend - 1 poly-cotton tie = $3.56.
Profit per Blend - 2 poly-cotton tie = $4.00.

17. Objective Function Objective: maximize profit menswear ties.
Fifth Avenue Industries
Objective: maximize profit menswear ties.
$4.075 S + $3.07 P + $3.56 B $4.00 B2
Where:
S = number of all-silk ties produced per month.
P = number of polyester ties.
B1 = number of Blend - 1 poly-cotton ties.
B2 = number of Blend - 2 poly-cotton ties.

18. Objective Function and Constraints
Objective: maximize profit =
$4.075 S + $3.07 P + $3.56 B1 + $4.00 B2
Subject to
(Yards of silk)
(Yards of polyester)
(Yards of cotton)
(Contract minimum for all silk)
(Market maximum)
(Contract minimum for all polyester)

19. Objective Function and Constraints
Objective: maximize profit =
$4.075 S + $3.07 P + $3.56 B1 + $4.00 B2
Subject to
Constraints - Continued
(Contract minimum Blend 1)
(Market maximum)
(Contract minimum Blend 2)

20. 3.4 Employee Scheduling Application
Time period
# of Tellers Required
9 a.m. – 10a.m. 10
10 a.m.–11a.m. 12
11 a.m. – noon 14
Noon – 1 p.m. 16
1 p.m. – 2 p.m. 18
2 p.m. – 3 p.m. 17
3 p.m. – 4 p.m. 15
4 p.m. – 5 p.m.
Labor Planning Problem
Hong Kong Bank now employs 12 full-time tellers. Part-time employee (four hours per day) are available.
Tellers requirements:

21. Employee Scheduling Application
Hong Kong Bank
Labor Constraints:
Full-timers work from 9 A.M. to 5 P.M.
Allowed 1 hour for lunch.
Half of full-timers eat at 11 A.M. and other half at noon.
Full-timers thus provide 35 hours per week of productive labor time.
Part-time hours limited to a maximum of 50% of day’s total requirement.
Costs:
Part-timers earn $4 per hour (or $16 per day) on average.
Full-timers earn $50 per day in salary and benefits, on average.

22. Employee Scheduling Application
Hong Kong Bank
Decision Variables:
F = full-time tellers
P1 = part-timers starting at 9 A.M. (leaving at 1 P.M.)
P2 = part-timers starting at 10 A.M. (leaving at 2 P.M.)
P3 = part-timers starting at 11 A.M. (leaving at 3 P.M.)
P4 = part-timers starting at noon (leaving at 4 P.M.)
P5 = part-timers starting at 1 P.M. (leaving at 5 P.M.)

23. Hong Kong Bank LP Formulation
Objective: minimize total daily labor cost
$50 F + $16 ( P1 + P2 + P3 + P4 )
Subject to
(9 A.M A.M. needs)
(10 A.M A.M. needs)
(11 A.M. - noon needs)
(noon - 1 P.M. needs)
(1 P.M. - 2 P.M. needs)
(2 P.M. - 3 P.M. needs)
(3 P.M. - 4 P.M. needs)
(4 P.M. - 5 P.M. needs)
(full-time tellers available)

24. Hong Kong Bank LP Formulation
Constraints (Continued):
Part-time worker hours cannot exceed 50% total hours required each day, which is sum of tellers needed each hour.
Simplifying yields,

25. Hong Kong Bank Solution
Excel entries for model reveal optimal solution.
Employ 10 full-time tellers.
7 part-time tellers at 10 A.M.
2 part-time tellers at 11 A.M.
5 part-time tellers at noon.
Total cost of $724 per day.
There are several alternate optimal solutions.

26. Hong Kong Bank Solution
There are several alternate optimal solutions.
In practice sequence in which constraints are listed in model may affect specific solution found.
One alternate solution.
Employ 10 full-time tellers.
6 part-time tellers at 9 A.M.
1 part-time teller at 10 A.M.
2 part-time teller at 11 A.M.
5 part-time tellers at noon.
Total cost of this policy is also $724.

27. 3.5 Financial Applications
Portfolio Selection
International City Trust (ICT) invests in short-term trade credits, corporate bonds, gold stocks, and construction loans.
ICT has $5 million available for immediate investment and wishes to do two things:
maximize interest earned on investments made over next six months and
satisfy diversification requirements as set by board of directors.

28. Portfolio Specification
International City Trust
Investment Possibilities:
Investment
Interest Earned (%)
Maximum Investment ($millions)
Trade credit 7 1
Corporate Bonds 11
2.5
Gold stocks 19
1.5
Construction Loan 15
1.8
Board specifies at least 55% of funds invested must be in gold stocks and construction loans.
No less than 15% be invested in trade credit.

29. Investment Formulation
International City Trust
Decision Variables:
T = dollars invested in trade credit
B = dollars invested in corporate bonds
G = dollars invested in gold stocks
C = dollars invested in construction loans

30. Investment Formulation
International City Trust
Objective: maximize investment interest dollars earned.
0.07 T B G C
Subject to

31. Rewriting Last Two Constraints
G + C > 0.55(T + B + G +C )
Rewritten as:
-0.55T B G C  0 Gold stock
And
T > 0.15 (T + B + G +C )
0.85T B G C  Trade credit

32. 3.6 Transportation Applications
Truck Loading Problem
Truck loading problem involves deciding which items to load on a truck so as to maximize value of a load shipped.
Consider Goodman Shipping.
One truck with a capacity of 10,000 pounds is next to be loaded.
Several other items are awaiting shipment.
Each items awaiting shipment has associated dollar value and weight.
Objective - maximize total value of items loaded on truck without exceeding truck’s weight capacity.

33. Transportation Applications
Goodman Shipping Items Awaiting Shipment:
Item
Total Value ($)
Weight (Pounds) 1
22,500
7,500 2
24,000 3
8,000
3,000 4
9,500
3,500 5
11,500
4,000 6
9,750

34. Goodman Shipping LP Formulation
Objective: maximize load value =
$22,500 P1+ $24,000 P2 + $8,000 P3 + $9,500 P4 + $11,500 P5 + $9,7500 P6
Subject to
7,500 P1 + 7,500 P2+ 3,000 P3+ 3,500 P4 + 4,000 P5 + 3,000 P < 10,000
P1 < 1
P2 < 1
P3 < 1
P4 < 1
P5 < 1
P6 < 1
P1, P2, P3, P4, P5, P6 > 0
Where: Pi is proportion of each item i loaded on truck.

35. Goodman Shipping Problem Using Pounds – Not Proportions
Formulate alternate model for problem.
Decision variables in model are weights in pounds shipped, rather than proportion.
Layout for model is identical to model shown previously.
Solution to model shows maximum load value is $31,500.
Load value achieved by shipping 2,500 pounds (0.33 of 7,500 pounds available item 1) and 7,500 pounds (all 7,500 pounds available item 2).

36. 3.7 Ingredient Blending Applications
Diet Problems
Diet problem involves specifying a food or food ingredient combination that satisfies stated nutritional requirements at minimum cost.
Whole Food Nutrition Center uses three bulk grains to blend natural cereal that sells by the pound.
Each 2-ounce serving of cereal, when taken with 1.2 cup of whole milk, meets an average adult’s minimum daily requirement for protein, riboflavin, phosphorus, and magnesium.

37. 3.7 Ingredient Blending Applications
Whole Food Nutrition Center
Diet Problems
Minimum adult daily requirement:
Protein 3 units.
Riboflavin 2 units.
Phosphorus 1 unit.
Magnesium unit.
Select blend of grains to meet USRDA at minimum cost.

38. Whole Food’s Natural Cereal Requirements
Grain
Cost per pound (cents)
Protein
(unit/lb)
Riboflavin
Phosphorus
Magnesium A 33 22 16 8 5 B 47 28 14 7 C 38 21 25 9 6
Decision Variables:
A = pounds of grain in one 2-ounce cereal serving.
B = pounds of grain in one 2-ounce cereal serving.
C = pounds of grain in one 2-ounce cereal serving.

39. Whole Food’s LP Formulation
Objective: minimize total cost of mixing 2-ounce serving =
$0.33 A + $0.47 B + $0.38 C
Subject to
(Protein units)
(Riboflavin units)
(Phosphorous units)
(Magnesium units)
(Total mix 2 ounces or pound)

40. Ingredient Blending Applications
Ingredient Mix and Blending Problems
Blending problems arise when decision must be made regarding blending of two or more products to produce one or more products. Resources contain one or more essential ingredients that must be blended so each final product contains specific percentages of each ingredient.
Example -
Deals with application frequently seen in petroleum industry.
Blending crude oils to produce refinable gasoline.

41. Blending Problem Example
Low Knock Oil Company
Low Knock Oil Company produces two grades of cut-rate gasoline for industrial distribution.
Regular.
Economy.
Produced by refining a blend of two types of crude oil.
Type X100.
Type X220.

42. Blending Problem Example
Weekly demand for Regular at least 25,000 barrels.
Weekly demand for Economy grade at least 32,000 barrels.
At least 45% of each barrel of regular must be ingredient A.
At most 50% of each barrel of economy should contain ingredient B.

43. Blending Problem Example
Decision Variables:
R1 = barrels crude oil X100 blended to produce refined Regular.
E1 = barrels crude oil X100 blended to produce refined
Economy.
R2 = barrels crude oil X220 blended to produce refined Regular.
E2 = barrels crude oil X100 blended to produce refined

44. Blending Problem Example
Low Knock Oil Company
Product Type Ingredients and Costs
Crude Oil Type
Ingredient A (%)
Ingredient B (%)
Cost/Barrel ($)
X100 35 55
30.00
X220 60 25
34.80

45. Low Knock Oil Company LP Formulation
Two More Requirements -
FIRST requirement:
At least 45% of each barrel of Regular must be ingredient A.
(R1+R2) is amount of crude blended to produce refined Regular gasoline demanded.
0.45 (R1+R2) is minimum amount of A required.
0.35 R R2 is amount of A in regular gas.
0.35 R R2 > 0.45 (R1+R2)
Recalculate as:
-0.10 R R2 > 0

46. Low Knock Oil Company LP Formulation
Two More Requirements -
SECOND requirement:
At most 50% of each barrel of economy must be ingredient B.
(E1 + E2) is amount of crude blended to produce refined Economy gasoline demanded.
0.50 (E1+E2) is maximum amount of ingredient B allowed.
0.55 E E2 = amount of ingredient B in Economy gas
0.55E1+0.25E2 < 0.50(E1+E2)
Recalculating yields:
0.05E1 – 0.25E2 < 0

47. LP Problem Formulation
Low Knock Oil Company
Objective: minimize cost =
$30 R1 + $30 E1 + $34.80 R2 + $34.80 E2
Subject to
(Demand for Regular)
(Demand for Economy)
(Ingredient A in Regular)
(Ingredient B in Economy)

48. Summary Continued discussion of LP models.
More experience in formulating and solving problems from variety of disciplines and applications:
Marketing, manufacturing, employee scheduling,
Finance, transportation, ingredient blending.
Illustrated setup and solution of models using Excel’s Solver add-in.