The Sebastopol Refinery processes two different kinds of crude oil, Venezuelan and Saudi, to
produce two general classes of products, Light and Heavy. Either crude oil can be processed by
either of two modes of processing, Short or Regular. The processing cost and amounts of Heavy
and Light produced depend upon the mode of processing used and the type of crude oil used.
Costs vary, both across crude oils and across processing modes. The relevant characteristics are
summarized in the table below. For example, the short process converts each unit of Venezuelan
crude to 0.45 units of Light product, 0.52 units of Heavy product, and 0.03 units of waste.
Short Process Regular Process
Venezuela Saud Venezuela Saud
Light prod 0.45 0.60 0.49 0.68
Heavy prod 0.52 0.36 0.50 0.32
Unused prod 0.03 0.04 0.01 0.00
Saudi crude costs $20 per unit, whereas Venezuelan crude is only $19 per unit. The short
process costs $2.50 per unit processed, while the regular process costs $2.10 per unit. Sebastopol
can process 10,000 units of crude per week at the regular rate. When the refinery is running the
Short process for the full week, it can process 13,000 units per week.
The refinery may run any combination of short and regular processes in a given week.
The respective market values of Light and Heavy products are $27 and $25 per unit.
Formulate the problem of deciding how much of which crudes to buy and which processes to run
as an LP. What are the optimal purchasing and operating decisions?
Why did the oil refinery break up with its significant other? Because they had too many crude differences!
Now let's get to the problem at hand. We need to formulate an LP to determine the optimal purchasing and operating decisions for the refinery. Here's how we can do it:
Let:
- VL = Number of units of Venezuelan crude oil to purchase
- SL = Number of units of Saudi crude oil to purchase
- VS = Number of units of Venezuelan crude oil to process using the Short process
- RS = Number of units of Saudi crude oil to process using the Short process
- VR = Number of units of Venezuelan crude oil to process using the Regular process
- RR = Number of units of Saudi crude oil to process using the Regular process
We need to maximize the total profit, which is the sum of the profits from the Light and Heavy products. The profit from each product is the market value minus the cost. The objective function is:
Maximize: 27(VS + RS + VR + RR) + 25((0.45VS + 0.52RS + 0.49VR + 0.50RR) + (0.60VS + 0.36RS + 0.68VR + 0.32RR))
Subject to the following constraints:
1. The total amount of Venezuelan crude oil processed cannot exceed the available amount: VS + VR <= VL
2. The total amount of Saudi crude oil processed cannot exceed the available amount: RS + RR <= SL
3. The total amount of crude oil processed cannot exceed the refinery's capacity:
VS + RS + VR + RR <= 10,000 (for regular process)
VS + RS + VR + RR <= 13,000 (for short process)
There are also non-negativity constraints for all decision variables:
VS, RS, VR, RR >= 0
Solve this LP, and you'll find the optimal values for the purchasing and operating decisions. Good luck, and may your profits flow like oil!
To formulate the problem as a linear program (LP), let's define the decision variables:
Let V = the number of units of Venezuelan crude oil to buy
Let S = the number of units of Saudi crude oil to buy
Let X = the number of units of crude oil processed using the Short process
Let R = the number of units of crude oil processed using the Regular process
The objective is to maximize the total profit. The profit is calculated by subtracting the total cost from the total revenue.
The revenue is the product of the market values of Light and Heavy products and the respective amounts produced.
The cost is the sum of the cost of purchasing the crude oil and the cost of processing it.
Maximize:
27 * (0.45X + 0.60R) + 25 * (0.52X + 0.36R) - 19V - 20S - 2.50X - 2.10R
Subject to the following constraints:
- The refinery can process a maximum of 10,000 units of crude oil using the Regular process:
R <= 10,000
- The refinery can process a maximum of 13,000 units of crude oil using the Short process:
X <= 13,000
- The total amount of crude oil processed cannot exceed the sum of the crude oil bought:
X + R <= V + S
- The amount of Venezuelan crude oil bought cannot exceed the refinery's maximum capacity for processing using the Regular process:
R <= 10,000
- The amount of Saudi crude oil bought cannot exceed the refinery's maximum capacity for processing using the Regular process:
R <= 10,000
- All decision variables must be non-negative:
V, S, X, R >= 0
Solving this LP will give the optimal values of V, S, X, and R, which represent the optimal purchasing and operating decisions.
To formulate the problem as a Linear Programming (LP) problem, we need to define the decision variables, objective function, and constraints.
Decision Variables:
Let's denote the amount of Venezuelan crude oil processed using the short process as V_S and using the regular process as V_R.
Similarly, let's denote the amount of Saudi crude oil processed using the short process as S_S and using the regular process as S_R.
Objective Function:
The objective is to maximize the profit from the production of Light and Heavy products. The profit can be calculated as the revenue from selling the products minus the cost of purchasing and processing the crude oil. The revenue is the market value of Light product multiplied by the amount produced, plus the market value of Heavy product multiplied by the amount produced.
Let's denote the profit as P.
P = (0.45V_S + 0.60S_S + 0.49V_R + 0.68S_R) * $27 + (0.52V_S + 0.36S_S + 0.50V_R + 0.32S_R) * $25 - ($19(V_S + V_R) + $20(S_S + S_R) + $2.50(V_S + S_S) + $2.10(V_R + S_R))
Constraints:
1. The amount of Venezuelan crude oil processed cannot exceed the available amount (10,000 units per week).
V_S + V_R <= 10,000
2. The amount of Saudi crude oil processed cannot exceed the available amount (10,000 units per week).
S_S + S_R <= 10,000
3. The short process for a full week can process 13,000 units in total.
V_S + S_S <= 13,000
Non-negativity constraints:
V_S >= 0, V_R >= 0, S_S >= 0, S_R >= 0
Now, we can solve the LP problem using an optimization solver to find the optimal purchasing and operating decisions, i.e., the values of V_S, V_R, S_S, and S_R that maximize the profit (P) while satisfying all the constraints.