5. This example illustrates gasoline blending problems faced in a petroleum refinery. We need to blend gasoline from three ingredients: butane, catalytic reformate, and heavy naphtha. Four characteristics of the resultant gasoline and its inputs are im- portant: cost, octane number, vapor pressure and volatility. These characteristics are summarized in the following table. COMMODITY CAT. CHARACTERISTIC BUTANE REFORMATE HEAVY NAPHTHA REGULAR GASOLINE (B) (CR) (HN) (RG) $/Unit Octane 7 18 12 120 100 74 94 Vapor Pressure Volatility 60 2 4 > 11 105 3 12 17 In this planning period there are only 1,000 units of butane available. the produc- tion of gasoline competes with heavy fuel (HF) for production facilities. The sum of gasoline plus heavy fuel production this planning period cannot exceed 12,000 units. Any blending problems associated with production of heavy fuel are disre- garded. For this example heavy fuel is treated as a simple product. At the highest level this is a product mix problem involving the two competing products, gasoline and heavy fuel. The profit contribution of heavy fuel is $3 per unit, including the cost of all ingre- dients. The profit contribution of regular gasoline is $18 per unit exclusive of the cost of its ingredients. Formulate a linear program for determining the optimal method for blending reg- ular gas and the optimal production levels for regular gas and heavy fuel. Note: A slight simplification assumed in this example is that the interaction between ingredients is linear: e.g., if a "fifty/fifty" mixture of B and CR is made, then its octane will be 0.5 x 120 +0.5 x 100 = 110 and its volatility will be 0.5 x 105+0.5 x 354. In reality this linearity is violated slightly, especially if gasoline containing tetraethyl lead is blended.