3.8 A Processing Problem

The Problem

Many industrial plants contain a number of processing units, each of which can be operated in one or more modes. Each mode uses one or more types of raw material and produces one or more products. (In practice the modes may not really exist as distinct entities; they may be convenient abstractions for the purposes of modelling.) To illustrate the types of constraint arising in such models we consider a simple problem defined by the following data:

Subscripts

h Material

i Unit

m Mode of operation

Sets

SHB Set of materials that can be bought, e.g. crude oils

SHS Set of materials that can be sold

Constants

YIELDhim Net production of material h from unit i in mode m (tonnes/day).

DAYS        Duration of time period (days)

DEMMAXh Maximum demand for material h (h є SHS) (tonnes)

DEMMINh Minimum demand for material h (h є SHS) (tonnes)

BUYPh Buying price for material h (h є SHB) ($/tonne)

SELLPh Selling price for material h (h є SHB) ($/tonne)

Note that YIELDhim is negative if the process uses material h as an input, and positive if the process produces material h as an output.


The Formulation

The natural formulation is then to use three sets of variables

bh Quantity of material h bought (h є SHB) ($/tonne)

sh Quantity of material h sold (h є SHS) (tonnes)

xim Period for which unit i operates in mode m (days)

Then the problem is to

maximize profit ($)

Σ h є SHS SELLPh sh - Σ h є SHB BUYPh bh

subject to

1. the capacity constraints (days)

Σm xim ≤ DAYS for all i

2. the material balance constraints (tonnes)

bh + Σi Σm YIELDhim xim - Σh = 0 for all h

(where the term bh exists only if h є SHB and sh exists only if h є SHS),

3. the demand constraints (tonnes)

DEMMINh ≤ sh ≤ DEMMAXh for all h є SHS


Commentary

There are a number of points to note about this formulation.

It is simplified by using the general notion of material rather than the more specialized notions of input material (e.g. crude oil), intermediate material and finished product.

No data have been provided for costs or capacities of transporting material within the plant. So these activities are assumed to be effectively unlimited and to have a negligible cost. Similarly the operating costs of the units are not specified, so they are assumed to be independent of how the units are used.

If one wanted to limit the amount of material that can be transmitted from some unit, say X, to some other unit, say Y, then the simplest approach is to define the pipework connecting these units as another unit, say P, with a specified capacity. One must then also define the output material from P as mathematically different from the input material, even though they are physically the same. The unit P then converts input material into output material, and Y can only use output material from P.

It will also be noted that there is a slack on the capacity constraints, indicating that the units do not have to be used all the time. But there is no slack on the material balance constraints. This is important because some by-products may be surplus to requirements and have to be processed to turn them into something else that can be sold. Some products may have a negative selling price, indicating that one has to pay someone to dispose of them.

Note particularly that the processing variables are xim and not xhim: the amounts of material either used or produced are proportional to the overall level of each production activity and are not independent variables. This is different from blending problems, although one can think of blending problems in this way, with each mode making the product from a single material and with quality constraints restricting the choice of combinations of modes. In practice a model may contain processing activities of the type considered in this example and also blending activities.

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