Case 11-2: California Arabian Oil Company, Inc

2

AMARCO, Inc.William D. WhislerCalifornia State University, Hayward

1. (a) There are two groups of variables: the amount of the four feedstocks (alkylate, catalytic cracked gasoline, straight run gasoline, and isopentane) used in producing the three types of gasoline (A, B, and C) and the amounts of the four feedstocks left over after production. Twelve variables are in the first group and four in the second, making a total of 16 variables.

Xi,j = amount of feedstock i used in gasoline j, in gallonsXi = amount of feedstock i left over after production, in gallons

for i = feedstock A (Alkylate), CC (Catalytic Cracked gasoline), SR (Straight Run gasoline), I (Isopentane), and j = gasoline A, B, C. All of these variables must be nonnegative.

With the feedstocks available, the objective function maximizes the revenue from the gasoline produced and the feedstocks left over after production.

where vi is the value of feedstock i, Xi.

There are five groups of conditions: (i) Reid Vapor Pressure, (ii) feedstock availability, (iii) minimum demand, (iv) octane level, and (v) the marketing condition. The first, third, and fourth groups have three restrictions each, the second has four, and there is one marketing condition. Thus, the total number of constraints is 14.

(i) Reid Vapor Pressure

for j = gasoline A, B, C, where ri is the Reid Vapor Pressure of feedstock i and 7 is the maximum Reid Vapor Pressure. This gives three constraints

-2XA,j + XCC,j - 3XSR,j + 13XI,j < 0

for j = gasoline A, B, C.

(ii) The feedstock availability conditions are

for i = A, CC, SR, I where Ai are the input stream availabilities from the case.(iii) The minimum demand requirements are

where Dj is the demand for gasoline j = gasoline A, B, C.

(iv) The octane number constraints are

for j = gasoline A, B, C, where oni is the octane number of feedstock i and ONj is the minimum octane number for gasoline j. This gives three constraints

14XA,A + 3XCC,A 6XSR,A + 15XI,A > 0 16.5XA,B + 2XCC,B 4XSR,B + 17XI,B > 0 7.5XA,C 7XCC,C 13XSR,C + 8XI,C > 0

(v) The marketing condition that the amount of avaiation gas A produced must be at least as great as the amount of gas B is

(b) The spreadsheet summarizing the formulation is given below.

2. The solution summary table that follows gives the solutions, obtained by Excels Solver, to all parts of the case. All answers have been rounded off to the nearest whole number.

Below is the Excel spreadsheet used to find the solution for Question 2. The formula =SUMPRODUCT(E5:T5,$E$22:$T$22) is entered in cell Z5 and copied down to cells Z6:Z19. The Solver dialog boxes are shown immediately after the spreadsheet. Cell Z5 contains the value of the objective function for the solution, $692,645 and the values of the variables are in cells E22:T22.

3. (a) If the price of aviation gas doubles the coefficients in the objective function in cells E5:P5 double as shown below. This leads to a new solution and an increase in the objective function to $1,538,308 as shown in the solution summary table.

(b) Doubling all the coefficients in the objective function, as shown below in cells E5:T5, yields the identical solution as for the original case, except that the revenue doubles to $1,632,400.

4. If supplies decrease, the right-hand sides of constraints 4, 5, and 6 in cells V9:V11 change to the values shown below. The solution summary table shows that the solution is infeasible. Consequently AMARCO will not be able to meet the minimum demands.

5. The solution to Question 2 has 6,400 barrels per day of alkylate, zero catalytic cracked gasoline, zero straight run gasoline, and 7,600 barrels per day of isopentane left over. Thus, additional quantities of alkylate and isopentane will increase revenues only by their values, 17, and 14, respectively. This can be seen also from the shadow prices in Excels sensitivity report below, in cells E32 and E35. The other two feedstocks increase revenues by more that their value because the shadow prices are 16.4 for catalytic cracked gasoline (cell E33) and 17.2 for straight run gasoline (cell E34). The allowable increase column gives the largest increase in the feedstock amounts for which the above answers remain valid: 925.7 barrels for catalytic cracked gasoline (cell G33) and 694.3 barrels for straight run gasoline (cell G34). There are no upper limits for the other two feedstocks.

6. From the above portion of Excels Sensitivity Report, 1,000 more barrels than the minimum demand for gasoline A are produced (cell D36 minus cell F36) and gasoline B and C production is equal to their minimum demand (cells D37:D38 are equal to cells F37:F38). Thus, with the economy weakening, a decrease in the minimum demand for gasoline A does not affect the solution, whereas, decreases in the minimum demands for gasoline B or C do affect the solution. The decreases in revenues are given by the shadow prices in cells E37:E38. The shadow price for aviation gasoline B is $2.20 per barrel and is valid until demand decreases by 456.3 barrels (cell H37) and the shadow price for gasoline C is $0.10 and it is valid until demand decreases by 1,786.2 barrels (cell H38). That is, each barrel per day decrease in the minimum demand for B and C increase revenue by $2.20 and $0.10, respectively.

7. If aviation gasoline prices drop then the coefficients in cells E5:P5 of the objective function change as shown below. From the solution summary table the revenue drops to $626,200.

8. If the octane constraints are dropped the last three constraints (11, 12, and 13 in rows 16, 17, and 18) are deleted. The optimal solution does not change.

9. The new Environmental Protection Agency regulation that decreases RVP to 6 changes the coefficients in the first three constraints as shown below. This has no effect on the optimal solution.

10. Adding the new marketing constraint, that the production of aviation gasoline A plus B must be at least as great as that of C will not change the solution. This can be seen by noting that the solution from Question 2 produces 13,000 barrels per day of A, 13,000 barrels per day of B, and 12,000 barrels per day of C. Consequently, the new marketing condition already is satisfied by the current solution. Adding a new constraint and re-solving yields the same result.