CHAPTER 4:LINEAR PROGRAMMINGAPPLICATIONS

LINEAR PROGRAMMING (LP) CAN BE USED FOR MANY MANAGERIAL DECISIONS:

Marketing Application Media Selection

Financial Application Portfolio Selection Financial Planning

Product Management Application Product Scheduling

Data Envelopment Analysis Revenue Management

LP Modeling ApplicationFor a particular application we begin

withthe problem scenario and data, then:1) Define the decision variables2) 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

MEDIA SELECTION APPLICATION Helps marketing manager to allocate

the advertising budget to various advertising media News Paper TV Internet Magazine Radio

MEDIA SELECTION 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

MEDIA SELECTIONPLAN DECISION CRETERIA

EXPOSURE QUALITYIt 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

MEDIA SELECTIONWe can use the graph of an LP to see

what happens when:

1. An OFC changes, or 2. A RHS changes

Recall the Flair Furniture problem

ADVERTISING MEDIA

# OF CUSTOMER REACHED

COST PER ADVERTISMENT

MAX TIME AVAIALBLE PER MONTH

EXPOSURE QUALITY UNITS

DAY TIME TV(1 MIN)

1000 1500 15 65

EVENING TV (30 SEC)

2000 3000 10 90

DAILY NEWS PAPER

1500 400 25 40

SUNDAY NEWS PAPER

2500 1000 4 60

RADIO 8 AM TO 5 PM NEWS 30 SEC

300 100 30 20

DECISION VARIABLES 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 ????

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

OPTIMAL SOLUTION OBJ FUNCTION Value: 2370 (Exposure

Quality unit) Decision variable Potential customers ????

MEDIA FREQUENCY

DTV 10ETV 0DN 25SN 2RADIO 30

dtv etv dn sn r RHS dual Maximize 65 90 40 60 20 Constraint 1 1 0 0 0 0 <= 15 0 Constraint 2 0 1 0 0 0 <= 10 0 Constraint 3 0 0 1 0 0 <= 25 16 Constraint 4 0 0 0 1 0 <= 4 0 Constraint 5 0 0 0 0 1 <= 30 14 Constraint 6 1500 3000 400 1000 100 <= 30000

0.06 Constraint 7 1 1 0 0 0 >= 10 -25 Constraint 8 1500 3000 0 0 0 <= 18000 0 Constraint 9 1000 2000 1500 2500 300 >= 50000

0 Solution-> 10 0 25 1.999999 30 $2,370.

DISCUSSION 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

BLENDING PROBLEMFrederick's Feed Company receives four raw grains from which it blends its dry pet food. The pet food advertises that each 8-ounce can meets the minimum daily requirements for vitamin C, protein and iron. The cost of each raw grain as well as the vitamin C, protein, and iron units per pound of each grain are summarized on the next slide.Frederick's is interested in producing the 8-ounce mixture at minimum cost while meeting the minimum daily requirements of 6 units of vitamin C, 5 units of protein, and 5 units of iron.

BLENDING PROBLEM

Vitamin C Protein Iron Grain Units/lb Units/lb Units/lb

Cost/lb 1 9 12 0 .75

2 16 10 14 .90 3 8 10 15 .804 10 8 7 .70

BLENDING PROBLEM Define the constraints

Total weight of the mix is 8-ounces (.5 pounds): (1) x1 + x2 + x3 + x4 = .5

Total amount of Vitamin C in the mix is at least 6 units: (2) 9x1 + 16x2 + 8x3 + 10x4 > 6

Total amount of protein in the mix is at least 5 units: (3) 12x1 + 10x2 + 10x3 + 8x4 > 5

Total amount of iron in the mix is at least 5 units: (4) 14x2 + 15x3 + 7x4 > 5

Nonnegativity of variables: xj > 0 for all j

OBJECTIVE FUNCTION VALUE = 0.406 VARIABLE VALUE REDUCED COSTS

X1 0.099 0.000 X2 0.213 0.000 X3 0.088 0.000 X4 0.099 0.000

Thus, the optimal blend is about .10 lb. of grain 1, .21 lb.of grain 2, .09 lb. of grain 3, and .10 lb. of grain 4. Themixture costs Frederick’s 40.6 cents.

BLENDING PROBLEM

FINANCIAL APPLICATION S 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 ????

Investment Projected Rate of Return %

Atlantic oil 7.3%Pacific oil 10.3%Midwest steel 6.4%Huber Steel 7.5%Govt Bonds 4.5%

CONSTRAINTS & OBJ 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

SOLUTION Objective Function=8000

Variable Value Reduced Cost

A 20000

0.00

P 30000

0.00

M 0.00 0.011H 4000

00.00

G 10000

0.00

Constraint

Slack/surplus

Dual price

1 0 0.0692 0 0.0223 10000 0.004 0 -0.0245 0 0.030Investment

Amount Expected Annual Return

A $20,000 $1460P $30,000 $3090H 40,000 $3000G $10,000 $450Total $100000 $8000

Overall Return ????

DISCUSSION 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

DISCUSSION 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%/

DISCUSSION 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.

DATA ENVELOPMENT ANALYSIS 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

EVUALATING PERFORMANCE OF HOSPITAL 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 RESOURCE CONSUMED BY 4 HOSPITAL

Input Measure General University County StateFTE 285.50 162.30 275.70 210.40Supply Expense

123.80 128.70 348.50 154.10

Bed-days available

106.72 64.21 104.10 104.04

ANNUAL SERVICES PROVIDED BY FOUR HOSPITALSOutput Measure

General University County State

Medicare patient days

48.14 34.62 36.72 33.16

Non-Medicare patient days

43.10 27.11 45.98 56.46

Nurses Trained

253 148 175 160

Interns trained

41 27 23 84

RELATIVE EFFICIENCY OF COUNTY HOSPITAL

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

OUTPUT CONSTRAINTS 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

OUTPUT CONSTRAINTS 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.

INPUT CONSTRAINTS 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 ….

INPUT CONSTRAINTS FTE:285.20wg+162.30wu+275.70wc+210ws<=27

5.70E SUP:123.80wg+128.70wu+348.50wc+154.10ws<

=348.50E Beddays:106.72wg+64.21wu+104.10wc+104.04w

s<=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,.

MODEL 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

OPTIMAL SOLUTIONVariable Valu

eReduced cost

E 0.905

0

WG 0.212

0

WU 0.260

0

WC 0.00 0.95WS 0.52

70

Constraint

Slack/Surplus

Dual Price

1 0 0.2392 0 -0.0143 0 -0.0144 1.615 0.05 37.027 0.06 35.824 0.07 174.42

20.0

8 0.00 0.010

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

DISCUSSION 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.