Our simple blending problem was concerned with making a single product, HFO, from various components. In practice, blenders have to make more than one product from a given pool of materials. A complication arises when some materials are available in different quantities at different prices. In particular, the manager may assign a lower price to the use of existing stocks of materials than to the use of material that has to be newly purchased. Multi-mix blending problems are defined by the following data:
The natural formulation for this problem is to write x Then we must
It
is of some interest to note that this problem has an apparently simpler
formulation in terms of a single type of variable. If z
This alternative formulation omits one set of material balance constraints. But it has more decision variables, and, more seriously, more non-zero coefficients than the recommended formulation. One consequence of this is that the model makes decisions about the source s of material r used in making product p which are entirely arbitrary (and will actually reflect the naming convention used in the matrix). If calculations are made of the cost of making the individual products, these will show some products to be more profitable than others simply because they have used the high-cost source of material r rather than the low-cost source. This allocation has been entirely arbitrary but it could lead to erroneous decisions as to which plants are profitable and which loss-making. This illustrates the importance of building a model which reflects the data which are available. More generally, it shows that one should not believe something must be true just because the computer says so. |

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