A production model and maintenance planning model for the process industry

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total number of variables = j(i+1)t!+(ij+2i+j)t + 2i
number of constraints = j(t!)²+2jt!+(ij+i+5j)t+i+2j

Table 8: Model Complexity (RT expressed in periods)

Fortunately, in the chemical industry, there is hardly ever a need for a planning horizon of over
20 periods, except perhaps in the pharmaceutical branch, where smaller time buckets could be
useful because of the smaller batches and larger variety of products. Even then the model could
still be used with a rolling horizon. Ten products on three production lines is also the maximum
to be scheduled in most chemical companies. If more products are manufactured, grouping could
be used.

For the sake of completeness, it is worth pointing out that the model can also be solved using one
of the standard algorithms for solving mixed binary linear programming problems. For a review
of some of these algorithms we refer to Shapiro (1979). Other computationally efficient
commercial software include IBM Mathematical Programming system Extended/370 Program
Reference Manual (1979), or IBM Optimization Subroutine Library (OSL) (1990), which have
already been extensively applied in a study of large scale binary linear programming problems
(Crowder et al. (1983)). MPSARX, developed by Van Roy and Wolsey (1987), is a state-of-the-
art Mathematical Programming system (MPS) that can be implemented for solving our model.

For additional references, see also Van Roy (1983, 1989), Van Roy and Wolsey (1983),
Mikhalevich et al. (1983), Jackson and O'Neil (1983), Côté and Laughton (1984), Glover (1984),
and Jeroslow (1984). Finally, for a review of the performance evaluation literature of mixed
binary programming algorithms, we refer to von Randow (1985, pp. 198 and 199).

Obviously, for large scale problems the model may be hard to solve to optimality with a simple
branching policy or standard software. In such cases, one may be forced to stop the procedure
early, or to develop an appropriate heuristic. Solution techniques such as Simulated Annealing or
Tabu search may be used as well. A direction for further research therefore involves improving
the computational efficiency for large scale problems, which the authors are currently pursuing.

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