# Example of a Product Mix Problem in Linear Programming

One of the classic applications of Linear Programming models is the product mix problem. If the quality of a product that is processed through the mixture of certain inputs can be approximated reasonably through a proportion, then a linear model may be useful. The example below shows this situation:

An oil refinery produces two types of unleaded gasoline: regular and extra, which sells to distributors in \$12 and \$14 per barrel, respectively. Both types are made from a national refined oil and an imported refined oil stock, that the company owns (ie by blending them), which must comply with the specifications contained in the following table:

The inventory characteristics of the refined oils are:

We need to formulate and solve a linear programming model to maximize the weekly income of the refinery, satisfying the requirements previously detailed.

Decision Variables:

• Xnr: Barrels of national oil used in the production of regular gasoline
• Xne: Barrels of national oil used in the production of extra gasoline
• Xir: Barrels of imported oil used in the production of regular gasoline
• Xie: Barrels of imported oil used in the production of extra gasoline

Objective Function: Maximize the weekly income received by the refinery for production of regular and premium gasoline.

Max 12*(Xnr + Xir) + 14*(Xne + Xie)

Constraints:

Steam Pressure: The weighted average of the steam pressure of the different types of oils which make up the mixture should not exceed 23 units (for each type of gas).

• (25Xnr + 15Xir) / (Xnr + Xir) <= 23
• (25Xne + 15Xie) / (Xne + Xie) <= 23

Minimum octane rating: The weighted average of the different types octane petroleum included in the mixture must be at least 88 and 93 units for regular and extra, gasoline respectively.

• (87Xnr + 98Xir) / (Xnr + Xir) >= 88
• (87Xne + 98Xie) / (Xne + Xie) >= 93

Minimum and Maximum Demand: For each weekly gasoline must produce a quantity of barrels between the minimum and the maximum must be produced weekly for each gasoline type.

• 50,000 <= Xnr + Xir <= 100,000
• 5,000 <= Xne + Xie <= 20,000

Inventory: The availability of national and imported barrels of oil must be respected for the production of regular and premium gasoline.

• Xnr + Xne <= 40,000
• Xir + Xie <= 60,000

Non Negativity: The decision variables must naturally take greater than or equal to zero values.

• Xnr, Xne, Xir, Xie >= 0

The following optimal solution and optimal value result from implementing the above model of optimization with Excel Solver:

30,909.09 barrels of national oil should be targeted to the production of regular gasoline,  9,090.91 barrels of national oil for the production of extra gasoline, 49,090.91 barrels of imported oil for the production of regular gasoline and 10,909.09 barrels of imported for the production of extra gasoline. The previous production policy allows to generate a weekly income of MUS\$1,240.

One recommendation in the computational task is to rewrite the constraints including equivalent proportions, to avoid divisions between changing cells (decision variables) and also denominators that initially adopt a value equal to zero. For example this constraint: (25Xnr + 15Xir) / (Xnr + Xir) <= 23 can be represented in analogous manner as follows: (25Xnr + 15Xir) -23 (Xnr + Xir) <= 0. Thus, it may be corroborated, for example, that in the optimal solution the steam pressure reached by the production of barrels for regular gasoline is: (25*30.909,09 + 15*49.090,91) / (30.909,09 + 49.090,91)=18,8636 (approx) that is less than or equal to the limit of 23 units.

Do you want to have the Excel file with the solution of this problem with Solver? Recommend us on Facebook, Google or Twitter using the social network tool at the bottom of this article and then download the file.