Managerial Decision Modeling with Spreadsheets Chapter 3 Linear Programming Modeling (Selected) Applications: With Computer Analyses in Excel

Managerial Decision Modeling Managerial Decision Modeling with Spreadsheetswith Spreadsheets

Chapter 3Linear Programming Modeling (Selected)

Applications: With Computer Analyses in Excel

Learning ObjectivesLearning Objectives

1. Model wide variety of linear programming (LP) problems.

2. Understand major business application areas for LP problems: manufacturing, marketing, labor scheduling, blending, transportation, finance, and multi-period planning.

3. Gain experience in setting up and solving LP problems using Excel’s Solver.

Production Mix ProblemProduction 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.


Material Cost per yard Material available per month (yards)

Silk $20 1,000

Polyestar $6 3,000

Cotton $9 1,600

Fifth Avenue uses a standard labor cost of $0.75 per tie (for any variety)


Production Mix Problem Fifth Avenue Industries

Each all-silk tie requires -

• Cost per tie = 0.125 yards of silk x $20 per yard = $2.5

• Revenue per tie = $6.70 selling price per silk tie.

• Profit per tie = Revenue per tie - Cost per tie =

$6.70 - $2.5 – 0.75 = $3.45.

Profit for other three products -

• Profit per all-polyester tie = $2.32.

• Profit per Blend - 1 poly-cotton tie = $2.81

• profit per Blend - 2 poly-cotton tie = $3.25


Fifth Avenue Industries

Objective: maximize profit menswear ties.

$3.45 S + $2.32P + $2.81 B1 + $3.25 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.

Production Mix ProblemProduction Mix ProblemObjective: maximize profit =

$3.45S + $2.32 P + $2.81 B1 + $3.25 B2

Subject to:(Yards of silk)

(Yards of polyester)

(Yards of cotton)

(Contract minimum for all silk)

(Contract minimum)

(Contract minimum for all polyester)

(Contract maximum)

Production Mix ProblemProduction Mix ProblemObjective: maximize profit =

$4.075 S + $3.07 P + $3.56 B1 + $4.00 B2

Subject to

Constraints - Continued(Contract minimum Blend 1)

(Contract maximum)

(Contract minimum Blend 2)

(Contract maximum)

Media SelectionMedia 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.

Media SelectionMedia Selection

• 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.


Objective: maximize audience coverage= 5000T + 8500N + 2400P +2800AT = 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.


Objective: maximize audience coverage =

5000 T + 8500 N + 2400 P +2800 A

Subject to

12

5

25

20

800 925 290 380 $8,000

5

290 380 $1,800

T

N

P

A

T N P A

P A

P A


The optimal solution found to be:

T = 1.97 television spots.

N = 5.00 newspaper ads.

P = 6.21 30-second prime time radio spots.

A = 0.00 1-minute afternoon radio spots.

This produces an audience exposure of 67,240 contacts.

Marketing Research ProblemMarketing 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.

Marketing Research ProblemMarketing Research Problem3. 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 at least 50 % of those surveyed who are 30 years of age

or younger live in a state that does not border Mexico *(missing)

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.


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 31-50 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 31-50 years and do not live in border state.

N3 = number 51 years or older and do not live in border state.

Marketing Research ProblemMarketing Research Problem 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

1 2 3 1 2 3

1 1

2 2

1 2 3 1 2 3 1 2 3

3 3 3

1 2 3 1 2 3

2,300

1,000

600

0.15

0.2

, , , , , 0

B B B N N N

B N

B N

B B B B B B N N N

B B N

B B B N N N


B1 + B2 + B3 0.15(B1 + B2 + B3 + N1 + N2 + N3)

Rewritten as:

B1 + B2 + B3 - 0.15(B1 + B2 + B3 + N1 + N2 + N3) 0

Simplifies to:

0.85B1 + 0.85B2 + 0.85B3 - 0.15N1 - 0.15N2 - 0.15N3 0

And

B3 ≤0.2(B3 + N3)

Rewritten as:

0.8B3 - 0.2N3 < 0


Optimal solution shows that it costs $15,166 and requires one to survey households as follows:

State borders Mexico and 31-50 years = 600

State borders Mexico and 51 years = 140

State not borders Mexico and 30 years = 1,000

State not borders Mexico and 51 years = 560

Employee Scheduling ApplicationEmployee Scheduling Application Hong Kong Bank now employs 12 full-time tellers. Part-

time employees (four hours per day) are available. They can start work anytime between 9 A.M. and 1 P.M.

Tellers requirements:

Employee Scheduling ApplicationEmployee Scheduling ApplicationHong 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 $7 per hour (or $28 per day) on average.

• Full-timers earn $90 per day in salary and benefits, on average.

Employee Scheduling ApplicationEmployee Scheduling Application Hong Kong Bank

Decision Variables:

F = number of. full-time tellers

P1 = part-timers starting at 9 A.M. (leaving at 1 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.)

Employee Scheduling ApplicationEmployee Scheduling ApplicationObjective: minimize total daily labor cost

$90 F + $28 ( P1 + P2 + P3 + P4 + P5 )

Subject to1

1 2

1 2 3

1 2 3 4

2 3 4 5

3 4 5

4 5

5

10

12

0.5 14

0.5 16

18

17

15

10

12

F P

F P P

F P P P

F P P P P

F P P P P

F P P P

F P P

F P

F

(9 A.M. - 10 A.M. needs)

(10 A.M. - 11 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)

Employee Scheduling ApplicationEmployee Scheduling Application

Constraints (Continued):• Part-time worker hours cannot exceed 50% total hours

required each day, which is sum of tellers needed each hour.

Simplifying yields,

1 2 3 4 54( ) 0.50(10 12 14 16 18 17 15 10)P P P P P

1 2 3 4 5

1 2 3 4 5

4 4 4 4 4 0.050(112)

, , , , , 0

P P P P P

F P P P P P


• 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 $1,292 per day.

• There are several alternate optimal solutions.


• 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 $1,292.

Ingredient Blending ApplicationsIngredient 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.


Whole Food Nutrition Center

Diet Problems

• Minimum adult daily requirement:

– Protein 3 units.

– Riboflavin 2 units.

– Phosphorus 1 unit.

– Magnesium 0.425 unit.

• Select blend of grains to meet USRDA at

minimum cost.


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.


Objective: minimize total cost of mixing 2-ounce serving =

$0.33 A + $0.47 B + $0.38 CSubject to

22 28 21 3

16 14 25 2

8 7 9 1

5 0 6 0.425

0.125

, , 0

A B C

A B C

A B C

A B C

A B C

A B C

(Protein units)

(Riboflavin units)

(Phosphorous units)

(Magnesium units)

(Total mix 2 ounces or 0.125 pound)

Multi-period ApplicationsMulti-period Applications

• Most challenging application of LP is modeling multi-period scenarios.

• Situations where decision maker has to determine optimal decisions for several periods (weeks, months, etc.).

• These problems especially difficult because decision choices in later periods are directly dependent on decisions made in earlier periods.

Multi-period ApplicationsMulti-period Applications

Production SchedulingGreenberg Motors, Inc., manufactures two different electrical motors for sale under contract to Drexel Corp.Model GM3A is found in many Drexel foodprocessors, and model GM3B is used in assemblyof blenders. • Production planning must consider four factors:

– Desirability of producing same number of each motor each month.

– Necessity to reduce inventory carrying, or holding, costs. – Warehouse limitations cannot be exceeded.– Company’s no-layoff policy.

Multi-period ApplicationsMulti-period Applications: Production : Production SchedulingScheduling

Greenberg Motors, Inc.• Scheduled Orders.

• Decision Variables.

PAi = number of model GM3A motors produced in

month i (i =1,2,3,4 for January–April).

PBi = number of model GM3B motors produced in

month i (i=1,2,3,4 for January–April).


Greenberg Motors, Inc.

• Associated Costs.

• Production costs now -

– GM3A is $10.

– GM3B is $6.

• Prices will increase in March to $11 and $6.60,

respectively.

• Production Cost =

$10 PA1 + $10 PA2 + $11 PA3 + $11 PA4 + $6 PB1 +

$6 PB2 + $6.60 PB3 + $6.60 PB4


Greenberg Motors, Inc.IAi = level of on-hand inventory for GM3A at end of

month i (i=1,2,3,4 for January–April)

PBi = level of on-hand inventory for GM3B at end of

month i (i=1,2,3,4 for January–April)

• Carrying Cost

– GM3A is $0.18 per month.

– GM3B is $0.13 per month.

Cost of carrying inventory =

$0.18 IA1 + $0.18 IA2 + $0.18 IA3 + $0.18 IA4 +

$0.13 IB1 + $0.13 IB2 + $0.13 IB3 + $0.13 IB4



Objective: minimize total costs = period production costs

+ period inventory costs =

$10 PA1 + $10 PA2 + $11 PA3 + $11 PA4

+ $6 PB1 + $6 PB2 + $6.60 PB3 + $6.60 PB4

+ $0.18 IA1 + $0.18 IA2 + $0.18 IA3

+ $0.18 IA4 + $0.13 IB1 + $0.13 IB2 + $0.13 IB3

+ $0.13 IB4

Multi-period ApplicationsMulti-period Applications:Production :Production SchedulingScheduling

• Inventory at end of month is:

• January’s demand for GM3As is 800 and for GM3Bs is 1,000, write relation as:

IA1 = 0 + PA1 - 800IB1 = 0 + PB1 - 1000

• Rewrite as:

PA1 - IA1 = 800

PB1 - IB1 = 1000



• If Greenberg wants additional 450 GM3As and 300 GM3Bs at end of April, add constraints:

2 1 2

2 1 2

3 2 3

3 2 3

4 3 4

4 3 4

700

1,200

1,000

1,400

1,100

1,400

A A A

B B B

A A A

B B B

A A A

B B B

P I I

P I I

P I I

P I I

P I I

P I I

February GM3A demand

February GM3B demand

March GM3A demand

March GM3B demand

April GM3A demand

April GM3B demand

4 450AI 4 300BI and


• Warehouse space constraints:

1 1

2 2

3 3

4 4

3,300

3,300

3,300

3,300

A B

A B

A B

A B

I I

I I

I I

I I


• Labor Constraints:

1 1

1 1

2 2

2 2

3 3

3 3

4 4

4 4

1.3 0.9 2,240

1.3 0.9 2,560

1.3 0.9 2,240

1.3 0.9 2,560

1.3 0.9 2,240

1.3 0.9 2,560

1.3 0.9 2,240

1.3 0.9 2,560

A B

A B

A B

A B

A B

A B

A B

A B

P P

P P

P P

P P

P P

P P

P P

P P

(January min. hours)

(January max. hours)

(February min. hours)

(February max. hours)

(March min. hours)

(March max. hours)

(April min. hours)

(April max. hours)


SummarySummary• 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, and

– Multi-period planning.

• Illustrated setup and solution of models using Excel’s

Solver add-in.

Documents

Managerial Decision Modeling with Spreadsheets Chapter 3 Linear Programming Modeling (Selected) Applications: With Computer Analyses in Excel