IE 416 Operations Research I Extra Credit Project

1. 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

2. 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

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

4. 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

5. Summary of the Problem

6. 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?

7. 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

8. Formulation of the Problem- Step 1

9. 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

10. 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

11. 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

12. 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.

13. Formulation of the Problem- Step 4

14. 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

15. 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

16. 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)

17. Explanation for Constraint 5 • DEMAND GAS • Equationx11 + 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

18. 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

19. Solution using WinQSB: Input

20. Solution using WinQSB: Output

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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.

27. 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.

28. 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

29. Sensitivity Analysis of RHS Constraint (Non Binding) • Oil 1 Available (table presentation is not helpful in here)

30. 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.

31. 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

32. 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.

33. QUESTIONSand COMMENTS