Blending

# Blending - EMSE 154-254 Blending Blending of Input...

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EMSE 154-254 Blending Instructor: Hernan Abeledo Source: Optimization Modeling with LINGO 1 Blending of Input Materials In a blending problem, there are: 1) Two or more input commodities (raw materials); 2) One or more qualities associated with each input commodity; 3) One or more output products to be produced by blending the input commodities, so certain output quality requirements are satisfied. Often, a good approximation is that the quality of the finished product is the weighted average of the qualities of the products going into the blend. Some examples are:

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EMSE 154-254 Blending Instructor: Hernan Abeledo Source: Optimization Modeling with LINGO 2 Blending models are used most frequently in the following industries: 1) Feed and food (e.g., the blending of cattle feed, hotdogs, etc.); 2) Metals industry (e.g., the blending of specialty steels and nonferrous alloys 3) Petroleum industry (e.g., the blending of gasolines of specified octanes and volatility). Structure of Blending Problems Suppose we must produce a batch of cattle feed having a protein content of at least 15%. Mixing corn (which is 6% protein) and soybean meal (which is 35% protein) produces this feed. In words, the protein constraint is: Let C = # bushels of corn in the mix, and let S = # bushels of soybean meal, then we have: This constraint is not linear. However, if we multiply both sides by C + S , we get: 0.06 C + 0.35 S 0.15 ( C + S )
EMSE 154-254 Blending Instructor: Hernan Abeledo Source: Optimization Modeling with LINGO 3 Constraints on additional characteristics (i.e., fat, carbohydrates and even such slightly nonlinear things as color, taste, and texture) can be handled in similar fashion. The distinctive feature of a blending problem is that the crucial constraints, when written in intuitive form, are ratios of linear expressions. They can be converted to linear form by multiplying through by the denominator. Formulations are simpler when the batch size (output volume) is specified beforehand. The first example below illustrates this case. The second example considers the slightly more complicated scenario where the batch size is a decision variable. Example: The Pittsburgh Steel Company Blending Problem The Pittsburgh Steel (PS) Co. has been contracted to produce a new type of very high carbon steel which has the following quality requirements:

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EMSE 154-254 Blending Instructor: Hernan Abeledo Source: Optimization Modeling with LINGO 4 PS has the following materials available for mixing in a batch: Chemical content of a blend is the weighted average of the chemical content of its components. A one-ton (2000-lb.) batch must be blended. What mix will minimize the cost? An experienced steel man looks at the problem and claims that the least cost mix will not use
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## This note was uploaded on 01/02/2010 for the course EMSE 254 taught by Professor Hernanabeledo during the Fall '08 term at GWU.

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Blending - EMSE 154-254 Blending Blending of Input...

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