




Supply chain planning for a manufacturer of polyester yarn. Please see the
"Changeovers" topic for details. Once the production plan has been developed and agreed,
a detailed schedule must be created for the near term. This is used to directly control
operations, steering towards planned production quantities whilst dealing with production upsets and
unexpected orders as smoothly as possible.


Clearly the schedule must reflect the production plan, which is the embodiment of a
"contract" between sales and operations, as closely as possible. It
must also obey restrictions that allow only certain groups of products to be made together
on a line or a group and avoid expensive changeovers as far as possible. This
problem does not fall within the scope of standard scheduling algorithms, so we use
mixed integer programming.
The timescale is shorter than for the production plan – typically six weeks rather
than one year – but a daily resolution is needed to match production quantities
accurately. The problem is therefore more complex than the production planning model.

There are no interactions between lines once the target production quantities have been
defined, so we can solve for one line at a time. Given the number of lines, each problem
has to be solved in a few minutes at the most, which is still challenging.
We adopt a threestage approach for each line:

Decide which products are to be made on the line in each day


Decide how many positions make each product in each day


Decide which product is made on each position in each day.



In the first stage, the model is similar to the main production planning model (but for
one line only). Individual positions are not recognised but the number of positions
making each product is calculated as a continuous value. One zeroone variable for
each product and day represents the decision to make the product at all, and these are used to
enforce the complex constraints on concurrent production.

Only those zeroone decisions from the first stage are kept to use in the second stage
model. This model can then be simplified by leaving out the concurrent production
constraints. The number of positions making each product is defined to be an integral
quantity, and the model essentially balances the estimated changeover costs against meeting
production targets from the plan.



Having decided how many positions to use for a product in each day, production quantities
are determined, and the third stage model no longer needs to be concerned with targets.
Instead products are assigned to specific spinnerets simply to minimise changeovers. (A
small arbitrary bias is introduced to reduce redundancy.)

This threestage approach is not only faster than a combined model, but finds better solutions
in any reasonable amount of time. Stages 2 and 3 in particular solve very quickly, to a value
close or equal to the linear relaxation.
