EXAMPLE
4.4 B LENDING
AT
C HANDLER O IL
C
handler Oil has 5000 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 together. Each barrel of crude oil 1 has a “quality level” of 10 and each barrel
of crude oil 2 has a quality level of 5.6 Gasoline must have an average quality level of at
least 8, whereas heating oil must have an average quality level of at least 6. Gasoline sells
for $75 per barrel, and heating oil sells for $60 per barrel. 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 revenue from selling gasoline and heating oil.
Objective To develop an LP spreadsheet model for finding the revenue-maximizing plan
that meets quality constraints and stays within limits on crude oil availabilities.
WHERE DO THE NUMBERS COME FROM?
Most of the inputs for this problem should be easy to obtain.
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The selling prices for outputs are dictated by market pressures.
The availabilities of inputs are based on crude supplies from the suppliers.
The quality levels of crude oils are known from chemical analysis, whereas the
required quality levels for outputs are specified by Chandler, probably in response to
competitive or regulatory pressures.
Solution
In typical blending
problems, the correct
decision variables are
the amounts of each
input blended into
each output.
The variables and constraints required for this blending model are listed in Table 4.7. The
key is the selection of the appropriate decision variables. Many students, when asked what
decision variables should be used, specify the amounts of the two crude oils used and the
amounts of the two products produced. However, this is not enough. The problem is that
this information doesn’t tell Chandler how to make the outputs from the inputs. The company instead requires a blending plan: how much of each input to use in the production of
a barrel of each output. Once you understand that this blending plan is the basic decision,
all other output variables follow in a straightforward manner.
Table 4.7 Variables and Constraints for Blending Model
Input variables
Decision variables (changing cells)
Objective cell
Other calculated variables
Constraints
Unit selling prices, availabilities of inputs, quality levels
of inputs, required quality levels of outputs
Barrels of each input used to produce each output
Revenue from selling gasoline and heating oil
Barrels of inputs used, barrels of outputs produced (and
sold), quality obtained and quality required for outputs
Barrels of inputs used … Barrels available
Quality of outputs obtained Ú Quality required
A secondary, but very important, issue in typical blending models is how to implement
the quality constraints. (The constraints here are in terms of quality. In other blending
To avoid being overly technical, we use the generic term quality level. In real oil blending, qualities of interest
might be octane rating, viscosity, and others.
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164 Chapter 4 Linear Programming Models