We shall now consider the structure of our simple oil blending model. What are the classes of object with which we are dealing and whose behaviour is essentially similar? They are the: - components: LGO, HGO, WAXD, ARES and VRES
- qualities: SG, VBI, Sulphur
We might also consider that there was a class of objects: - products: HFO
because a similar matrix would exist for other products. We shall disregard this for now. We shall use individual lower case letters as subscripts to denote these classes of object. To be specific we shall use: - r for components;
- q for qualities
- the cost of component r, COST
_{r} - the value of quality q of component r, QUAL
_{qr} - the maximum value of quality q in the blend, QMAX
_{q} - the demand for product, DEMAND
- the availability of component r, AVAIL
_{r}
What are our decision variables? These are: - the quantity of component r to use in the blend, x
_{r}.
To complete our armoury of notation we shall use the symbols Σ
Then we can write our problem informally as
We can write this more formally as:
If we are being formal we should also constrain each x
although in practice this is assumed by convention.
This mathematical formulation may appear more verbose than our simple matrix. Indeed, it occupies much more space when presented with the data as an input file for an algebraic modelling language such as XPRESS. But this is an unfair comparison. Our mathematical formulation remains the same whether we are working with 5 or 50 components and 3 or 30 qualities. It also holds true for blends of any product and for making batches of the product today, tomorrow and at any time. More important than this, it provides a much better way of - the minimum value of quality q in the blend, QMIN
_{q}
and the minimum quality constraints:
It is now a trivial piece of manipulation to write the quality constraints as:
Now
that we are being precise we should record the units in which we
measure the data items, decision variables and constraints. Let us say
that we are working in tonnes, i.e. AVAIL What
should we do about qualities? What are the units of quality q? Specific
gravity is the dimensionless ratio of the density of the material
divided by the density of water. VBI is a numerical index which is
derived by taking the logarithm of the logarithm of a number measured
in centiStokes, which has dimensions Mass Length In
order to avoid getting bogged down in technical details, when dealing
with qualities we shall use the term "units of quality q" as the
equivalent of $ for costs. Then the units of QUAL But what are the units of the quality constraints? More seriously, are the quality constraints correct? In Blending by Weight and by Volume we saw that some qualities blended by weight while others blended by volume. VBI and Sulphur content are examples of qualities which blend by weight, so the quality constraints we have defined are correct for those qualities. But specific gravity blends by volume, so the constraints which we require are
where v Now v
or, cancelling out the ρ
These
are the form of quality constraint which we require for qualities which
blend by volume. The fact that we happened to be considering specific
gravity is irrelevant to the appearance of SG The advantages of an algebraic notation should now have become more clear. It would have been far harder to have developed these constraints correctly if we had been working with specific instances of the constraints rather than algebra. The notation is a prop to our thoughts. |

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