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IE 416 Operations Research I Extra Credit Project. 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

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## IE 416 Operations Research I Extra Credit Project

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**IE 416 Operations Research I Extra Credit Project**Robert Delgado Chris Mui Amanda Smith Presented to: Dr. Sima Parisay Due: October 20th, 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**Quality -8 Sell:$25/barrel Demand: 5 barrels/$1 Adv. 5,000 Barrels of Oil 1 Quality -10 10,000 Barrels of Oil 2 Quality -6 Sell: $20/barrel Demand: 10 barrels/$1 Adv. Quality -5**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 • 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**Formulation of the Problem- Step 1**• The 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 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(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 Gas 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.**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 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 3**• Gasoline must have an average quality level of at least 8 • Simplify so we have a linear equation and not a fraction 1) 1.) 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 2) 3) 4)**Explanation for Constraint 5**• DEMAND GAS • Equationx11 + x21= 5a1 • 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 4Create Equality**Constraints by Defining:Dr. Parisay’s note: change a1 and a2 names as excess variables to a3 and a4 Slack Variables Excess Variables Artificial Variables**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 2 for Gas- Min $18.88 Oil 1 for Gas- Max $83.17 Oil 2 for Gas- Max $112.25**Report to Manager**Oil 2 for H- Max $26.13 Oil 1 for H- Max $28.17 Oil 1 for H- Min $0 Oil 2 for H- Min $5.46**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 point shows a unit cost value outside the allowable max c(j) range. This point shows that when unit profit is increased to $83.17 our max profit will be $497,500. This flat line shows that the coefficients for x11 on this line will yield the same max profit. This is the current solution. Unit profit is $25 and our max profit is $323,000.**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 $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**Sensitivity Analysis of RHS Constraint (Non Binding)**• Oil 1 Available (table presentation is not helpful in here)**Sensitivity Analysis of RHS Constraint (Non Binding)**If we can only obtain 2,500 barrels of oil 1, our max profit will be $248,750. If we increase barrels of oil 1 to 15,000 our max profit will be $620,000. This is the current solution. Barrels of oil 1 used is 5,000 and our max profit is $323,000.**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.

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