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.

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Greate info on this site! Helped me a lot! Thanx.