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In writing the mathematical formulation of a linear program, we need to define three things: the variables, the objective function, and the constraints. Sometimes the formulation will be obvious and other times constructing the model will be more challenging. Lets start with an example. |
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The Problem ChemCo produces two fertilizers, FastGro and ReallyFastGro. Each is made of a mix of two growth agents. FastGro is a 50/50 mix of the two, and it sells for $13 per pound. ReallyFastGro is a 25/75 mix (i.e., its 25% Agent 1 and 75% Agent 2), and sells for $15/pound. Agent 1 costs $2/pound and Agent 2 costs $3/pound. ChemCo can purchase up to 250 pounds of Agent 1 and up to 350 pounds of Agent 2 each day. What is ChemCo's optimal production strategy? i.e., How much of each product should it produce in order to maximize its profits? We start by defining the variables.
In defining the variables, we need to ask ourselves what it is that we wish the model to determine. In this case, we need to know how much of each product to produce. We will use the units that are specified; namely, pounds and days. These observations lead us to believe the following might be appropriate variable definitions: |
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= number of pounds of FastGro to produce each day. |
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= number of pounds of ReallyFastGro to produce each day. |
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