In the highly competitive world of oil refineries, it is always a challenge to maximize profits while maintaining operational excellence. Manufacturing facilities need to carefully balance various factors such as availability, energy consumption, and market demand. Fortunately, advanced optimization techniques such as linear programming (LP) models have emerged as incredibly valuable tools for achieving this delicate balance. In this article, we will discuss how LP models can be used to maximize profit.

Understanding Linear Programming: Linear programming is a mathematical optimization technique for solving complex linear relationship problems. It uses mathematical models to optimize refinery operations and product allocation. The model represents the production, constraints, and objectives of the refinery as a system of linear equations and inequities. Through the use of variables and optimization algorithms, the LP model identifies optimal inputs, outputs, and operating conditions that maximize profits.

Let’s consider an example of a crude unit in a refinery and see how an LP model can be used to maximize profit:

**Objective Function**: Define an objective function to maximize or minimize the output variable. In our case, the objective is to maximize profit. Profit is calculated as the difference between the revenue generated from the sale of refined products and the cost of crude oil processing. The objective function can be defined as:

Maximize: Profit = Revenue — Cost

**Decision Variables**: The decision variables represent the quantities of inputs and outputs in the crude unit that can be adjusted to optimize profitability. In this example, the decision variables may include:

- Crude oil input quantities: The amounts of different crude oil types (e.g., Crude A, Crude B, etc.) to be processed in the crude unit.
- Factors affecting the cost of processing crude like energy requirement
- Quantity of the crude processing can also depend on various factors like asphaltene presence, sulfur content, viscosity, etc.
- Product yields: The production quantities of various refined products, such as gasoline, diesel, jet fuel, and others.
- Prices of the various products

**Constraints**: Constraints are limitations imposed on the decision variables for the refinery’s operations. These constraints ensure that the LP model generates feasible solutions. In the case of the crude unit, some common constraints are:

- Crude oil availability: The total amount of each crude oil type available for processing.
- Processing capacity: The maximum processing capacity of the crude unit, which limits the total amount of crude oil that can be processed.
- Product quality specifications: The required quality standards for each refined product.
- Yield constraints: The minimum and maximum yield percentages for each refined product.

**Linear Relationships**: The relationships between the decision variables, constraints, and objective function need to be expressed as linear equations or inequalities. For instance:

- The revenue generated from each refined product is calculated as the product of its yield and market price.
- The cost of crude oil processing depends on the quantities of each crude oil type processed and their associated costs, further, the quantity processed is dependent on the various decision variables mentioned above.
- The yield percentages of refined products are determined by the processing conditions and the characteristics of the crude oil inputs.

**Solving the Model**: With the objective function, decision variables, constraints, and linear relationships defined, an LP solver can be employed to solve the model. The solver uses optimization algorithms like the Simplex Method or interior point methods to iteratively adjust the decision variables, satisfying the constraints while maximizing the objective function (profit). The result is an optimal solution that provides the quantities of each crude oil type to be processed and the corresponding production quantities of refined products that maximize profit.

**Note**: The LP model may not perform well when the number of decision variables is very high and the relation between the decision variables and the objective function is highly non-linear. The same model can be used for column or process optimization

When non-linearity exists between the decision variables and the objective function and the objective function is not known, We can use ML models like random forest regression to identify the relationship between the objective function and the decision variables using the historical data and then use the optimization method to minimize or maximize the objective function with constraints.

For step-by-step development of the optimization model, you can connect with me on LinkedIn

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