Managerial decision modeling with spreadsheets by stair render chapter 03

Chapter 3:
Linear Programming
Modeling Applications

© 2007 Pearson Education


Linear Programming (LP) Can Be Used
for Many Managerial Decisions:
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Product mix
Make-buy
Media selection
Marketing research
Portfolio selection
Shipping & transportation
Multiperiod scheduling


For a particular application we begin with
the problem scenario and data, then:

1)
2)

Define the decision variables
Formulate the LP model using the decision variables

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3)
4)

Write the objective function equation
Write each of the constraint equations
Implement the model in Excel
Solve with Excel’s Solver


Product Mix Problem:
Fifth Avenue Industries
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Produce 4 types of men's ties
Use 3 materials (limited resources)

Decision: How many of each type of tie to
make per month?
Objective: Maximize profit


Resource Data

Material
Silk

Yards available
Cost per yard
per month
$20
1,000

Polyester

$6

2,000

Cotton

$9

1,250

Labor cost is $0.75 per tie


Product Data
Type of Tie
Silk
Selling Price

Polyester Blend 1 Blend 2

$6.70

$3.55

$4.31

$4.81

Monthly
Minimum

6,000

10,000

13,000

6,000

Monthly
Maximum

7,000

14,000

16,000

8,500

0.125

0.08

0.10

0.10

(per tie)

Total material
(yards per tie)


Material Requirements
(yards per tie)
Type of Tie
Material

Silk

Blend 1
Polyester
(50/50)

Blend 2
(30/70)

Silk

0.125

0

0

0

Polyester

0

0.08

0.05

0.03

Cotton

0

0

0.05

0.07

0.125

0.08

0.10

0.10

Total yards


Decision Variables
S = number of silk ties to make per month
P = number of polyester ties to make per

month

B1 = number of poly-cotton blend 1 ties to

make per month

B2 = number of poly-cotton blend 2 ties to

make per month


Profit Per Tie Calculation
Profit per tie =
(Selling price) – (material cost) –(labor cost)

Silk Tie

Profit = $6.70 – (0.125 yds)($20/yd) - $0.75
= $3.45 per tie


Objective Function (in $ of profit)
Max 3.45S + 2.32P + 2.81B1 + 3.25B2
Subject to the constraints:

Material Limitations (in yards)
0.125S

< 1,000 (silk)

0.08P + 0.05B1 + 0.03B2 < 2,000 (poly)
0.05B1 + 0.07B2

< 1,250 (cotton)


Min and Max Number of Ties to Make
6,000 < S < 7,000
10,000 < P < 14,000
13,000 < B1 < 16,000
6,000 < B2 < 8,500

Finally nonnegativity S, P, B1, B2 > 0

Go to file 3-1.xls

Media Selection Problem:
Win Big Gambling Club
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Promote gambling trips to the Bahamas
Budget: $8,000 per week for advertising
Use 4 types of advertising

Decision: How many ads of each type?

Objective: Maximize audience reached


Data
Advertising Options
Radio

Radio

TV Spot

Newspaper

(prime time)

(afternoon)

Audience

Reached
(per ad)

5,000

8,500

2,400

2,800

Cost
(per ad)

$800

$925

$290

$380

Max Ads
Per week

12

5

25

20


Other Restrictions
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Have at least 5 radio spots per week
Spend no more than $1800 on radio

Decision Variables
T = number of TV spots per week
N = number of newspaper ads per week
P = number of prime time radio spots per week
A = number of afternoon radio spots per week


Objective Function

(in num. audience reached)

Max 5000T + 8500N + 2400P + 2800A

Subject to the constraints:
Budget is $8000
800T + 925N + 290P + 380A < 8000
At Least 5 Radio Spots per Week
P+A>5

No More Than $1800 per Week for Radio
290P + 380A < 1800

Max Number of Ads per Week
T < 12

P < 25

N< 5

A < 20

Finally nonnegativity

T, N, P, A > 0
Go to file 3-3.xls


Portfolio Selection:
International City Trust
Has $5 million to invest among 6 investments

Decision: How much to invest in each of 6

Objective: Maximize interest earned

investment options?

Data
Interest
Rate

Risk Score

Trade credits

7%

1.7

Corp. bonds

10%

1.2

Gold stocks

19%

3.7

Platinum stocks

12%

2.4

Mortgage securities

8%

2.0

Construction loans

14%

2.9

Investment


Constraints
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Invest up to $ 5 million

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No more than 25% into any one investment

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At least 30% into precious metals

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At least 45% into trade credits and corporate bonds

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Limit overall risk to no more than 2.0


Decision Variables
T = $ invested in trade credit
B = $ invested in corporate bonds
G = $ invested gold stocks
P = $ invested in platinum stocks
M = $ invested in mortgage securities
C = $ invested in construction loans


Objective Function (in $ of interest earned)
Max 0.07T + 0.10B + 0.19G + 0.12P
+ 0.08M + 0.14C

Subject to the constraints:

Invest Up To $5 Million
T + B + G + P + M + C < 5,000,000


No More Than 25% Into Any One Investment
T < 0.25 (T + B + G + P + M + C)
B < 0.25 (T + B + G + P + M + C)
G < 0.25 (T + B + G + P + M + C)

P < 0.25 (T + B + G + P + M + C)
M < 0.25 (T + B + G + P + M + C)
C < 0.25 (T + B + G + P + M + C)


At Least 30% Into Precious Metals
G + P > 0.30 (T + B + G + P + M + C)

At Least 45% Into
Trade Credits And Corporate Bonds
T + B > 0.45 (T + B + G + P + M + C)


Limit Overall Risk To No More Than 2.0

Use a weighted average to calculate portfolio risk
1.7T + 1.2B + 3.7G + 2.4P + 2.0M + 2.9C < 2.0
T+B+G+P+M+C
OR
1.7T + 1.2B + 3.7G + 2.4P + 2.0M + 2.9C <
2.0 (T + B + G + P + M + C)
finally nonnegativity: T, B, G, P, M, C > 0
Go to file 3-5.xls


Labor Planning:
Hong Kong Bank
Number of tellers needed varies by time of day

Decision: How many tellers should begin work at various times of the day?

Objective: Minimize personnel cost