Robert DelgadoChris Mui

Amanda Smith

Presented to: Dr. Sima ParisayDue: October 20th, 2011

California State Polytechnic University, Pomona

AgendaProblem StatementSummary of ProblemFormulation of the ProblemSolution using WinQSBReport to Manager

Sensitivity Analysis 1 Basic Variable in O.F. 1 RHS Binding Constraint

Questions/Comments

Problem StatementChandler Oil Company

Problem #5 on Page 92 of Operations Research Applications and Algorithms textbook

Problem Statement5,000

Barrels of Oil 1

10,000 Barrels of

Oil 2

Quality -10

Quality -5

Quality -8Sell:$25/barrel

Demand: 5 barrels/$1 Adv.

Quality -6Sell: $20/barrel

Demand: 10 barrels/$1 Adv.

Summary of the ProblemChandler Oil Company - Oil Information

Oil # of barrels Oil QualityOil 1 5000 10Oil 2 10000 5

Chandler Oil Company - Products Made from Oil

Product (Blend)

Avg. Quality Level

Demand Created per $1 spent on

Advertising

Selling Price per

BarrelGas 8 5 $25 Heating Oil 6 10 $20

Formulation of the ProblemHow much money should be spent in

advertising each one of their products?

How should they blend each type of product from the available oil?

Formulation of the Problem- Step 11) Define Decision Variables

ai = dollars spent daily on

advertising blend i

(i = 1 ,2)xij = barrels of oil i used

daily to produce blend j

(i = 1,2 ; j = 1,2)Sign Restrictions:

ai > 0

xij > 0

 

Variable Name Givenx11 Oil 1 for Gasx12 Oil 1 for Heating Oilx21 Oil 2 for Gas x22 Oil 2 for Heating Oila1 Advertising $ Gasa2 Advertising $ Heating Oil

Formulation of the Problem- Step 1Avg.

Quality Level

Demand of Barrels

Created per $1 spent on Advertising

Selling Price per

BarrelProduct (Blend) Decision

Variables

8 5 $25 Gas x11 x21

6 10 $20 Heating Oil x12 x22

OIL 1 2 # of barrels 5000 10000 Oil Quality 10 5

Formulation of the Problem- Step 1The definition of the decision variables implies:

x11 + x12 = barrels of oil 1 used daily

x11 + x21 = barrels of gas produced daily

x21 + x22 = barrels of oil 2 used daily

x12 + x22 = barrels of heating oil produced daily

 

Formulation of the Problem- Step 22) Provide explanatory information and assumptions

Gas and heating oil cannot be stored, so it must be sold on the day it is produced

Formulation of the Problem- Step 33) Formulate Objective Function (O.F)   Profit = Revenue – Cost

Daily Revenues from Blend Sales (Sales of Gas

and Heating Oil)

= $25(x11 + x21) + $20 (x12 + x22)

  Daily Advertising Cost

= a1 + a2

Daily Profit = Daily Revenues from Blend Sales - Daily Advertising Cost

Daily Profit = [$25(x11 + x21) + $20 (x12 + x22)] – [a1 + a2]

Simplify

Zmax = 25x11 + 25x21 + 20x12 + 20x22 –a1 – a2

GasGas Heating OilHeating Oil

Variable Name Given

x11 Oil 1 for Gas

x12 Oil 1 for Heating Oil

x21 Oil 2 for Gas

x22 Oil 2 for Heating Oil

a1 Advertising $ Gas

a2

Advertising $ Heating Oil

Formulation of the Problem- Step 44.) Formulate Constraints

Constraint 1: Maximum of 5,000 barrels of oil 1 are available for production.

Constraint 2: Maximum of 10,000 barrels of oil 2 are available for production.

Constraint 3: Gasoline must have an average quality level of at least 8.

Constraint 4: Heating oil must have an average quality level of at least 6.

Constraint 5: Demand of gas is increased by 5 barrels for every dollar spent on advertising.

Constraint 6: Demand of heating oil is increased by 10 barrels for every dollar spent on advertising.

Formulation of the Problem- Step 4

Description  Equation Type

Max Profit  Zmax = 25x11 + 25x21 + 20x12 + 20x22 –a1 – a2 Objective Function

Oil 1 Avail. x11 + x12 < 5000 Constraint

Oil 2 Avail.  x21 + x22 < 10,000 Constraint

Gas Quality 2x11 – 3x21 > 0 Constraint

H. Quality Constraint

Demand Gas  x11 + x21 = 5a1 Constraint

Demand H. x12 + x22 = 10a2 Constraint

Explanation for Constraint 3Gasoline must have an average quality level of

at least 8.

Quality of Oil 1 x Total Barrels of Oil 1 Used for gas

Quality of Oil 2 x Total Barrels of Oil 2 Used for gas

Total Barrels of Oil used for Gas * Same idea is applied to

Constraint 4

Explanation for Constraint 3UnitsUsing example of 10x11 in Numerator

Using example of x11 in

Denominator -In the numerator we have quality as units

-In the denominator we have barrels as units

-This means we have quality/barrel in our fraction or “quality per barrel” which is what we are looking for in Constraint 3 on the LHS

* Same idea is applied to Constraint 4

Number of barrels of oil 1 for Gas

Explanation for Constraint 3Gasoline must have an average quality level of

at least 8Simplify so we have a linear equation and not a

fraction1.) Multiply both sides by x11 + x21

2.) Distribute

3.) Get variables on one side 4.)Now you have simplified version* Same idea is applied to Constraint 4

1)1)

2)2)

3)3)

4)4)

Explanation for Constraint 5DEMAND GAS

Equation x11 + x21 = 5a1

Equation Supply of Gas (oil 1 + oil 2) = Demand of Gas (5 barrels for every dollar spent in advertising)

UNITSEquation Barrels = x • Equation Barrels = Barrels• PURPOSE: To show we do having matching units

on both sides of equation.• This method can be applied for constraint 6

Formulation of the Problem- Step 4Create Equality Constraints by Defining:Dr. Parisay’s note: change a1 and a2 names as excess variables to a3 and a4

Description Standard LP Form Equation Type

Max Profit  Zmax = 25x11 + 25x21 + 20x12 + 20x22 –a1 – a2 Objective Function

Oil 1 Avail. x11 + x12 + S1 = 5000 Constraint

Oil 2 Avail.  x21 + x22 + S2 =10,000 Constraint

Gas Quality 2x11 – 3x21 - e1 + a1 = 0 Constraint

H. Quality 4x12 – x22 - e2 + a2 = 0 Constraint

Demand Gas  x11 + x21 - 5a1 = 0 Constraint

Demand H. x12 + x22 - 10a2 = 0 Constraint

Slack Variables

ExcessVariabl

es

Artificial

Variables

Solution using WinQSB: Input

Solution using WinQSB: Output

Report to Manager

To maximize its profit to $323,000 for the current production of gasoline and heating the company should:Produce 5,000 barrels of gasoline by mixing 3,000 barrels of oil 1

with 2,000 barrels of oil 2Produce 10,000 barrels of heating oil by mixing 2,000 barrels of oil 1

with 8,000 barrels of oil 2 Able to meet exact quality requirements

Report to Manager

Oil 1 for Gas- Min $16.83

Oil 1 for Gas- Max $83.17

Oil 2 for Gas- Min $18.88

Oil 2 for Gas- Max $112.25

Report to Manager

Oil 1 for H- Min $0

Oil 1 for H- Max $28.17

Oil 2 for H- Min $5.46

Oil 2 for H- Max $26.13

Report to Manager

We must pay $1000 in advertisement for gas and $1000 in advertisement for heating oil to generate the demand for the 5,000 barrels of gasoline and 10,000 barrels of heating oil

Report to Manager

Optimal if the range of oil 1 usage is from 2,500-15,000 barrels

Optimal if the range of oil 2 usage is from 3,333-20,000 barrels

Sensitivity Analysis of OF CoefficientOil 1 for Gas (Basic Variable)

MOTIVATION: has the highest unit profit of $25 c(j) and the highest allowable max c(j) (taking into account correlation)

Parsiay’s note: table presentation is not helpful.

Sensitivity Analysis of OF Coefficient

This is the current solution. Unit profit is $25 and our max profit is $323,000.

This point shows that when unit profit is increased to $83.17 our max profit will be $497,500.

This point shows a unit cost value outside the allowable max c(j) range.

This flat line shows that the coefficients for x11 on this line will yield the same max profit.

Sensitivity Analysis of RHS Constraint (Non Binding)

Oil 1 AvailableMOTIVATION: Has the highest shadow price of

$29.70Oil 1 Availability:

max RHS of 15,000 barrels Shadow price of $29.70 15,000 x $29.70 = $445,500 increase in profit.

Oil 2 Availability Max RHS of 20,000 barrels Only other constraint with a high shadow price of $17.45 20,000 x $17.45 = $349,000 increase in profit.

Better

Oil 1 Available (table presentation is not helpful in here)

Sensitivity Analysis of RHS Constraint (Non Binding)

This is the current solution. Barrels of oil 1 used is 5,000 and our max profit is $323,000.

If we increase barrels of oil 1 to 15,000 our max profit will be $620,000.

If we can only obtain 2,500 barrels of oil 1, our max profit will be $248,750.

Sensitivity Analysis of RHS Constraint (Non Binding)

Sensitivity Analysis of RHS Constraint (Binding)

Parisay: I explained in Word File to skip this discussion MOTIVATION: Sensitivity analysis on Demand Gas because it has the highest shadow

price of $.20 between the two binding constraints available

COMPARE:

- Shadow Price of Demand Gas x Max RHS = Amount of Increased Profit Due to Demand Gas

$.20 x 5,000 = $1000

 - Shadow Price of Demand H. x Max RHS = Amount of Increased Profit Due to Demand H.

$.10 x 10,000 = $1000

Sensitivity Analysis of RHS Constraint (Binding)Parisay: It is better to use graph not table.

If demand for gas is equal to 5000 we will have a profit maximization of $323,000

Once our demand goes over 5000 our profit will reduce because we cannot meet demand

When gas demand equals 8333 barrels our profit will reduce to $208,333 because more money has to be spent in advertising to create that demand

Any demand above 8333 barrels is infeasible.

QUESTIONSand

COMMENTS