Example 15.2 Blending Oil Products at Chandler Oil Blending Models

Objective To use LP to find the best way to blend crude oils to achieve quality requirements and maximize profits.

Background Information Chandler Oil has 5,000 barrels of crude oil 1 and 10,000 barrels of crude oil 2 available. Chandler sells gasoline and heating oil. These products are produced by blending the two crude oils. Each barrel of crude oil 1 has a “quality level” of 10 and each barrel of crude oil 2 has a quality level of 5. Gasoline must have an average quality level of at least 8, whereas heating oil must have an average quality level of at least 6.

Background Information Gasoline sells for $25 per barrel and heating oil sells for $20 per barrel. The advertising cost to sell one barrel of gasoline is $0.20 and the advertising cost to sell one barrel of heating oil is $0.10. We assume that demand for heating oil and gasoline is unlimited, so that all of Chandler’s production can be sold. Chandler wants to maximize its profit.

Solution To model Chandler’s problem, we must keep track of the following: the number of barrels of gasoline and heating oil produced (the outputs) the number of barrels of each crude oil (the inputs) used to produce each output the quality levels of the inputs used to make the outputs the total profit earned

BLENDING.XLS The spreadsheet model for this problem appears on the next slide and can be developed by following these steps: Inputs. Enter the monetary inputs, the quality characteristics, and the crude availabilities in the various shaded ranges. Inputs blended in each output. The quantities Chandler must choose to specify any solution are the barrels of each input used to produce each output. Therefore, enter any trial values for these values in the BlendPlan range. This range will be the changing cell range.

Developing the Model Inputs used and outputs sold. Calculate the amount of crude oils 1 and 2 used in the Used range by summing across the rows of the BlendPlan range. Then calculate the amount of gasoline and heating oil sold in row 19 by summing down the columns of the BlendPlan range. Quality constraints. Keeping track of the quality level of gasoline and heating oil is the trickiest part of this model. Begin by calculating for each product the number of “quality points” (QP) in the inputs used to produce the output: QP in gasoline = 10(Oil 1 in gasoline) + 5(Oil 2 in gasoline) QP in heating oil = 10(Oil 1 in heating oil) + 5(Oil 2 in heating oil)

Developing the Model -- continued Quality constraints continued. For the gasoline product to have a quality level of at least 8 we must have QP in gasoline >= 8 (Gasoline sold) For the heating oil product to have a quality level of at least 6 we must have QP in heating oil >= 6 (Heating oil sold) These inequalities are operationalized in the spreadsheet in rows 23-25. First, we determine the QP for gasoline in cell B23 with the formula =SUMPRODUCT(B17:B18,$B$8:$B$9) and copy this to cell C23 to generate the QP for heating oil. Then calculate the required QP for gasoline in cell B25 with the formula =B13*B19 and copy this to cell C25 for heating oil.

Developing the Model -- continued Profit. Calculate the total revenue from both products in cell B28 with the formula =SUMPRODUCT(B4:C4, $B$19:$C$19) Then copy this formula to cell B29 to calculate the advertising cost. Finally, calculate the profit in the Profit cell with the formula =B28-B29

Using the Solver Here is the solver dialog box for this blending model. We maximize profit subject to the quality constraints and using no more of the inputs than are available.

Solution -- continued The optimal solution implies that Chandler should make 5,000 barrels of gasoline with 3,000 barrels of crude oil 2. It should also make 10,000 barrels of heating oil with 2,000 barrels of crude oil 1 and 8,000 barrels of crude oil 2. With this blend Chandler will earn a profit of $323,000.

Modeling Issues We could have used this (QP in gasoline)/(Gasoline sold) >= 8 instead of the inequality used for gasoline. However, the preference is towards the previous inequality for two reasons: gasoline sold could conceivably be zero and that would require division by zero which would lead to an error in Excel and because the inequality is technically nonlinear, we prefer to keep the problem linear. The lesson is to “clear denominators” in blending problem constraints.

Modeling Issues -- continued We assume that the quality of a mixture is a linear function of the fraction of inputs used in the mixture. If the quality level of the output is not a linear function of the fraction of each input used in the mixture, then we have a nonlinear problem. Then we do not have a LP model. In reality, a company using a blending model would run the model periodically and set production on the basis of the current inventory of inputs and the current demand forecasts. The forecasts and the input levels would be updated and the model would be run again to determine the next day’s production.

Sensitivity Analysis One possible sensitivity analysis is to see how a change in the price per barrel of gasoline changes Chandler’s optimal product mix.

Sensitivity Analysis -- continued The input mix remains the same until the product reaches between $55 and $65. At this point the optimal solution is to produce all gasoline (approximately 8,333 barrels) and no heating oil. Of course, as the gasoline price increases, so does the optimal profit - even when the blending plan stays the same.