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Chapter 4: Linear Programming Applications

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Chapter 4:Linear ProgrammingApplications

- Marketing Application
- Media Selection

- Financial Application
- Portfolio Selection
- Financial Planning

- Product Management Application
- Product Scheduling

- Data Envelopment Analysis
- Revenue Management

LP Modeling Application

For a particular application we begin with

the problem scenario and data, then:

- Define the decision variables
- Formulate the LP model using the decision variables
- Write the objective function equation
- Write each of the constraint equations
- Implement the Model using QM or MS

- Helps marketing manager to allocate the advertising budget to various advertising media
- News Paper
- TV
- Internet
- Magazine
- Radio

- A Construction Company wants to advertise his new project and hired an advertising company.
- The advertising budget for first month campaign is $30,000
- Other Restrictions:
- At least 10 television commercial must be used
- At least 50,000 potential customer must be reached
- No more than $18000 may be spent on TV advertisement

- Need to recommend an advertising selection media plan

PLAN DECISION CRETERIA

EXPOSURE QUALITY

It is a measure of the relative value of advertisement in each of media. It is measured in term of an exposure quality unit.

Potential customers Reached

We can use the graph of an LP to see what happens when:

- An OFC changes, or
- A RHS changes
Recall the Flair Furniture problem

DTV : # of Day time TV is used

ETV: # of times evening TV is used

DN: # of times daily news paper used

SN: # of time Sunday news paper is used

R: # of time Radio is used

Advertising plan with DTV =65 DTV Quality unit

Advertising plan with ETV =90 DTV Quality unit

Advertising plan with DN =40 DTV Quality unit

Objective Function ????

- Max 65DTV + 90ETV + 40DN + 60SN + 20R (Exposure quality )
- Constraints
- Availability of Media
- Budget Constraint
- Television Restriction

- Availability of Media
- DTV <=15
- ETV <=10
- DN<=25
- SN<=4
- R<=30

- Budget constraints
- 1500DTV +3000ETV +400DN +1000SN +100R <=30,000

- Television Restriction
- DTV +ETV >=10
- 1500DTV +3000ETV<=18000
- 1000DTV+2000ETV+1500DN +2500SN +300R >=50,000

OBJ FUNCTION Value: 2370 (Exposure Quality unit)

Decision variable

Potential customers

????

dtvetvdnsn rRHS dual

Maximize6590406020

Constraint 110000<=150

Constraint 201000<=100

Constraint 300100<=2516

Constraint 400010<=40

Constraint 500001<=3014

Constraint 6150030004001000100<=300000.06

Constraint 711000>=10-25

Constraint 815003000000<=180000

Constraint 91000200015002500300>=500000

Solution->100251.99999930$2,370.

- Dual Price for constraint 3 is 16 ????
- (DN >=25) exposure quality unit ????
- Dual price for constraint 5 is 14
- (R <=30) exposure quality unit ????
- Dual price for constraint 6 is 0.060
- 1500DTV +3000ETV +400DN +1000SN +100R <=30,000 exposure quality unit ????

- Dual price for constraint 7 is -25
- DTV +ETV >=10 ???

Reducing the TV commercial by 1 will increase the quality unit by 25 this means

The reducing the requirement having at least 10 TV commercial should be reduced

Portfolio Selection

1.A company wants to invest $100,000 either in oil, steel or govt industry with following guidelines:

2.Neither industry (oil or steel ) should receive more than $50,000

3.Govt bonds should be at least 25% of the steel industry investment

4.The investment in pacific oil cannot be more than 60% of total oil industry.

What portfolio recommendations investments and amount should be made for available $100,000

Decision Variables

A = $ invested in Atlantic Oil

P= $ invested in Pacific Oil

M= $ invested in Midwest Steel

H = $ invested in Huber Steel

G = $ invested in govt bonds

Objective function ????

Max 0.073A + 0.103P + 0.064M + 0.075H + 0.045G

1.A+P+M+H+G=100000

2.A+P <=50,000, M+H <= 50,000

3. G>=0.25(M + H) or G -0.25M -0.25 H>=0

4. P<=0.60(A+P) or -0.60A +0.40P<=0

Objective Function=8000

Overall Return ????

Dual price for constraint 3 is zero increase in steel industry maximum will not improve the optimal solution hence it is not binding constraint.,

Others are binding constraint as dual prices are zero

For constrain 1 0.069 value of optimal solution will increase by 0.069 if one more dollar is invested.

A negative value for constrain 4 is -0.024 which mean optimal solution get worse by 0.024 if one unit on RHS of constrain is increased. What does this mean

If one more dollar is invested in govt bonds the total return will decrease by $0.024 Why???

Marginal Return by constraint 1 is 6.9%

Average Return is 8%

Rate of return on govt bond is 4.5%/

Associated reduced cost for M=0.011 tells

Obj function coefficient of for midwest steel should be increase by 0.011 before considering it to be advisable alternative.

With such increase 0.064 +0.011 =0.075 making this as desirable as Huber steel investment.

- It is an application of the linear programming model used to measure the relative efficiency of the operating units with same goal and objectives.
- Fast Food Chain
- Target inefficient outlets that should be targeted for further study
- Relative efficiency of the Hospital, banks ,courts and so on

General Hospital; University Hospital

County Hospital; State Hospital

Input Measure

# of full time equivalent (FTE) nonphysician personnel

Amount spent on supplies

# of bed-days available

Output Measures

Patient-days of service under Medicare

Patient-days of service notunder Medicare

# of nurses trained

# of interns trained

ANNUAL SERVICES PROVIDED BY FOUR HOSPITALS

Construct a hypothetical composite Hospital

Output & inputs of composite hospital is determined by computing the average weight of corresponding output & input of four hospitals.

Constraint Requirement

All output of the Composite hospital should be greater than or equal to outputs of County Hospital

If composite output produce same or more output with relatively less input as compared to county hospital than composite hospital is more efficient and county hospital will be considered as inefficient.

Wg= weight applied to inputs and output for general hospital

Wu = weight applied to input & output for University Hospital

Wc=weight applied to input & output for County Hospital

Ws = weight applied to input and outputs for state hospital

Constraint 1

Wg+ wu + wc + ws=1

Output of Composite Hospital

Medicare: 48.14wg + 34.62wu + 36.72wc+ 33.16ws

Non-Medicare:43.10wg+27.11wu+45.98wc+54.46ws

Nurses:253wg+148wu+175wc+160ws

Interns:41wg+27wu+23wc+84ws

Constraint 2:

Output for Composite Hospital >=Output for County Hospital

Medicare: 48.14wg + 34.62wu + 36.72wc+ 33.16ws >=36.72

Non-Medicare:43.10wg+27.11wu+45.98wc+54.46ws>=45.98

Nurses:253wg+148wu+175wc+160ws >=175

Interns:41wg+27wu+23wc+84ws >=23

Constraint 3

Input for composite Hospital <=Resource available to Composite Hospital

FTE:285.20wg+162.30wu+275.70wc+210.40ws

Sup:123.80wg+128.70wu+348.50wc+154.10ws

Bed-dys:106.72wg+64.21wu+104.10wc+104.04ws

We need a value for RHS:

%tage of input values for county Hospital.

E= Fraction of County Hospital ‘s input available to composite hospital

Resources to Composite Hospital= E*Resources to County Hospital

If E=1 then ???

If E> 1 then Composite Hospital would acquire more resources than county

If E <1 ….

FTE:285.20wg+162.30wu+275.70wc+210ws<=275.70E

SUP:123.80wg+128.70wu+348.50wc+154.10ws<=348.50E

Beddays:106.72wg+64.21wu+104.10wc+104.04ws<=104.10E

If E=1 composite hospital=county hospital there is no evidence county hospital is inefficient

If E <1 composite hospital require less input to obtain output achieved by county hospital hence county hospital is more inefficient,.

Min E

Wg+wu+wc+ws=1

48.14wg + 34.62wu + 36.72wc+ 33.16ws >=36.72

43.10wg+27.11wu+45.98wc+54.46ws>=45.98

253wg+148wu+175wc+160ws >=175

41wg+27wu+23wc+84ws >=23

285.20wg+162.30wu+275.70wc+210.40ws-275.70E <=0

123.80wg+128.70wu+348.50wc+154.10ws-348.50E <=0

106.72wg+64.21wu+104.10wc+104.04ws-104.10E <=0

Composite Hospital as much of as each output as County Hospital (constrain 2-5) but provides 1.6 more trained nurses and 37 more interim. Contraint 6 and 7 are for input which means that Composite hospital used less than 90.5 of resources of FTE and supplies

E=0.905

Efficiency score of County Hospital is 0.905

Composite hospital need 90.5% of resources to produce the same output of County Hospital hence it is efficient than county hospital. and county hospital is relatively inefficient

Wg=0.212;Wu=0.26;Ws=0.527.