Chapter 4: Linear Programming Applications. Marketing Application Media Selection Financial Application Portfolio Selection Financial Planning Product Management Application Product Scheduling Data Envelopment Analysis Revenue Management.
Chapter 4:Linear ProgrammingApplications
LP Modeling Application
For a particular application we begin with
the problem scenario and data, then:
PLAN DECISION CRETERIA
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:
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 ????
OBJ FUNCTION Value: 2370 (Exposure Quality unit)
dtvetvdnsn rRHS dual
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
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
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
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
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.
General Hospital; University Hospital
County Hospital; State Hospital
# of full time equivalent (FTE) nonphysician personnel
Amount spent on supplies
# of bed-days available
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.
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
Wg+ wu + wc + ws=1
Output of Composite Hospital
Medicare: 48.14wg + 34.62wu + 36.72wc+ 33.16ws
Output for Composite Hospital >=Output for County Hospital
Medicare: 48.14wg + 34.62wu + 36.72wc+ 33.16ws >=36.72
Input for composite Hospital <=Resource available to Composite Hospital
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 ….
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,.
48.14wg + 34.62wu + 36.72wc+ 33.16ws >=36.72
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
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