3.7 A Multi-Mix Blending Problem

The Problem

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:

Subscripts

p Product

q Quality

r Material

s Source

Sets

SQL Set of qualities with a lower limit

SQU Set of qualities with an upper limit

SRp Set of materials that can be used to make product p

SPr Set of products in which material r can be used.

Constants

AVAILrs       Availability of material r from source s (tonnes)

DEMANDp  Demand for product (tonnes)

COSTrs       Price of raw material ($/tonne)

QMAXpq       Maximum permitted amount of q in product p (units of quality q)

QMINpq       Minimum permitted amount of q in product p (units of quality q)

QUALqr       Amount of q per tonne of r (units of quality q).

The Natural Formulation

The natural formulation for this problem is to write xrs for the amount of raw material purchased (tonnes) and ypr for the amount of raw material used (tonnes).

Then we must

minimize costs ($)

Σ r Σ s COSTrs xrs

subject to

1. quality constraints (units of quality q * tonnes)

Σ r є SRp ( QUALqr - QMAXpq ) ypr ≤ 0 for all p, q є SQU

Σ r є SRp ( QUALqr - QMINpq ) ypr ≥ 0 for all p, q є SQL

2. the availability constraints (tonnes)

xrs ≤ AVAILrs for all r, s

3. the material balance constraints (tonnes)

Σ r xrs - Σ p є SPr ypr = 0 for all r

Σ r є SRp ypr = DEMANDp for all p

together with the constraints that all variables are nonnegative.

An Alternative Formulation

It is of some interest to note that this problem has an apparently simpler formulation in terms of a single type of variable. If zprs is the number of tonnes of material r bought from source s and used to make product p, the problem is to

minimize

Σp Σr Σs COSTrs zprs

subject to the constraints

Σ r є SRp Σ s ( QUALqr - QMAXpq ) zprs ≤ 0 for all p, q є SQU

Σ r є SRp Σ s ( QUALqr - QMINpq ) zprs ≥ 0 for all p, q є SQL

Σ p zprs ≤ AVAILrs for all r, s

Σ r є SRp zprs = DEMANDp for all p

Why the Natural Formulation is Better

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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