Thursday, December 3, 2015

Optimality of a small MINLP model

When I read a paper I often try quickly to implement the example and see if I can solve it. Here is a small MINLP model related to RRAP (Reliability-Redundancy Allocation Problem).

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It is very similar to the approach in http://yetanothermathprogrammingconsultant.blogspot.com/2015/11/linearization-by-enumeration-of-small.html. That is we want to maximize the reliability of a system by adding redundant components subject to cost, weight and volume constraints. The special thing here is that we not only decide on the redundancy (variable n) but also choose components along a cost-reliability curve (see equation cdef). The result is a small MINLP model. In the paper a PSO (Particle Swarm Optimization) algorithm is developed to solve model. However we can also just throw this model into a MINLP solver. The results are:

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With the MINLP solver BONMIN I get:

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BONMIN solves this is 4 seconds with a solution that is slightly better. The integer configuration (redundancy) is the same, but the cost/reliability trade-off is slightly different.

In general I would say that meta-heuristics are more appropriate when standard off-the-shelve solvers are running out of steam. This can happen when the problem becomes too big. That is certainly not the case here. Also I think one should never mention “optimal” in connection with a result found with a (meta-)heuristic. These methods do not give any guarantee about optimality.

Reference

HARISH GARG, S. P. SHARMA
RELIABILITY–REDUNDANCY ALLOCATION PROBLEM OF PHARMACEUTICAL PLANT
Journal of Engineering Science and Technology Vol. 8, No. 2 (2013) 190 - 198