Robert Delgado Chris Mui Amanda Smith Presented to: Dr. Sima Parisay Due: October 20 th, 2011 California State Polytechnic University, Pomona

Agenda Problem Statement Summary of Problem Formulation of the Problem Solution using WinQSB Report to Manager Sensitivity Analysis 1 Basic Variable in O.F. 1 RHS Binding Constraint Questions/Comments

Problem Statement Chandler Oil Company Problem #5 on Page 92 of Operations Research Applications and Algorithms textbook

Problem Statement 5,000 Barrels of Oil 1 10,000 Barrels of Oil 2 Quality -10 Quality -5 Quality -8 Sell:$25/barrel Demand: 5 barrels/$1 Adv. Quality -6 Sell: $20/barrel Demand: 10 barrels/$1 Adv.

Summary of the Problem Chandler Oil Company - Oil Information Oil# of barrelsOil Quality Oil Oil Chandler Oil Company - Products Made from Oil Product (Blend) Avg. Quality Level Demand Created per $1 spent on Advertising Selling Price per Barrel Gas85$25 Heating Oil610$20

Formulation of the Problem How 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 1 1) Define Decision Variables a i = dollars spent daily on advertising blend i (i = 1,2) x ij = barrels of oil i used daily to produce blend j (i = 1,2 ; j = 1,2) Sign Restrictions: a i > 0 x ij > 0 VariableName Given x 11 Oil 1 for Gas x 12 Oil 1 for Heating Oil x 21 Oil 2 for Gas x 22 Oil 2 for Heating Oil a1a1 Advertising $ Gas a2a2 Advertising $ Heating Oil

Formulation of the Problem- Step 1 Avg. Quality Level Demand of Barrels Created per $1 spent on Advertising Selling Price per Barrel Product (Blend) Decision Variables 85$25 Gas x 11 x $20 Heating Oil x 12 x 22 OIL12 # of barrels Oil Quality 105

Formulation of the Problem- Step 1 The definition of the decision variables implies: x 11 + x 12 = barrels of oil 1 used daily x 11 + x 21 = barrels of gas produced daily x 21 + x 22 = barrels of oil 2 used daily x 12 + x 22 = barrels of heating oil produced daily

Formulation of the Problem- Step 2 2) 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 3 3) Formulate Objective Function (O.F) Profit = Revenue – Cost  Daily Revenues from Blend Sales (Sales of Gas and Heating Oil) = $25(x 11 + x 21 ) + $20 (x 12 + x 22 )  Daily Advertising Cost = a 1 + a 2  Daily Profit = Daily Revenues from Blend Sales - Daily Advertising Cost Daily Profit = [$25(x 11 + x 21 ) + $20 (x 12 + x 22 )] – [a 1 + a 2 ]  Simplify Z max = 25x x x x 22 –a 1 – a 2 Gas Heating Oil VariableName Given x 11 Oil 1 for Gas x 12 Oil 1 for Heating Oil x 21 Oil 2 for Gas x 22 Oil 2 for Heating Oil a1a1 Advertising $ Gas a2a2 Advertising $ Heating Oil

Formulation of the Problem- Step 4 4.) 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 EquationType Max Profit Z max = 25x x x x 22 –a 1 – a 2 Objective Function Oil 1 Avail.x 11 + x 12 < 5000Constraint Oil 2 Avail. x 21 + x 22 < 10,000Constraint Gas Quality2x 11 – 3x 21 > 0Constraint H. QualityConstraint Demand Gas x 11 + x 21 = 5a 1 Constraint Demand H.x 12 + x 22 = 10a 2 Constraint

Explanation for Constraint 3 Gasoline 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 3 Units Using example of 10x 11 in Numerator Using example of x 11 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 3 Gasoline must have an average quality level of at least 8 Simplify so we have a linear equation and not a fraction 1.) Multiply both sides by x 11 + x 21 2.) Distribute 3.) Get variables on one side 4.)Now you have simplified version * Same idea is applied to Constraint 4 1) 2) 3) 4)

Explanation for Constraint 5 DEMAND GAS Equation  x 11 + x 21 = 5a 1 Equation  Supply of Gas (oil 1 + oil 2) = Demand of Gas (5 barrels for every dollar spent in advertising) UNITS Equation  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 4 Create Equality Constraints by Defining: Dr. Parisay’s note: change a1 and a2 names as excess variables to a3 and a4 DescriptionStandard LP Form EquationType Max Profit Z max = 25x x x x 22 –a 1 – a 2 Objective Function Oil 1 Avail.x 11 + x 12 + S 1 = 5000Constraint Oil 2 Avail. x 21 + x 22 + S 2 =10,000Constraint Gas Quality2x 11 – 3x 21 - e 1 + a 1 = 0Constraint H. Quality4x 12 – x 22 - e 2 + a 2 = 0Constraint Demand Gas x 11 + x a 1 = 0Constraint Demand H.x 12 + x a 2 = 0Constraint Slack Variables Excess Variables 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 2 Produce 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 Coefficient Oil 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 x 11 on this line will yield the same max profit.

Sensitivity Analysis of RHS Constraint (Non Binding) Oil 1 Available MOTIVATION: Has the highest shadow price of $29.70 Oil 1 Availability: max RHS of 15,000 barrels Shadow price of $ ,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 $ ,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 = $ 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.

QUESTIONS and COMMENTS