Lee, B. H. and Deininger, R. A. (1992) “Optimal Locations of Monitoring Stations in Water Distribution Systems”, Journal of Environmental Engineering, 118(1) pp. 4-16
The EPA requires that all water distribution authorities to test the water quality in their system. Lee and Deininger have done an important work in contributing to the efficiency and effectiveness of this testing by providing and demonstrating an integer programming approach to testing the water quality. This assists the entire nation as the improved efficiency saves taxpayers' and water users' money.
To set up the integer program, testing nodes need to be defined and assumptions need to be made. When a water sample is taken at a single tap, the quality is thought to represent the quality at the closest node. d_i is the demand at that node, and D is the total demand. With the single sample, the quality of the fraction d_i/D is known. Also, the nodes nearby may be inferred depending on the hydraulics. If all the water at the sampled node came from an upstream node, then the water quality of this upstream node would also be known. Since water distribution systems are complex and the water in one node likely came from a combination of other nodes, the authors set up a method for deciding when the water quality of an upstream node can be assumed to be represented by a downstream node.
I'm not entirely convinced that these assumptions can be made. The weakness of the article is that they make assumptions without any argument attached as to why this is acceptable. But perhaps these assumptions are intuitive to most reading the article, and I'm not familiar enough with the water quality testing industry to know this. For example, the authors make an assumption that if atleast 50% of the sampled water has passed through a node, the sample is considered representative of that node (p. 8). I'd be interested in hearing others' opinions on whether they think this is an intuitively accurate assumption.
I'm sure that since 1992 many more complex models have been presented for how to effeciently and effectively monitor water quality across a large system, but for 1992 standards integer programming was an excellent and quick solution (17-27s to solve!). Nowadays integer programming is rare because engineers don't need to simplify to that level simply to save computing time. Integer programming is only used when it really is the most accurate model of a system. I'm trying to think of real life applications where it would be used today, but I can't think of any. I'd be interested in hearing if anybody else can think of some applications.
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Michelle,
ReplyDeleteI had similar reflections from this article. I think the notion of upstream nodes presumably having better quality than their downstream counterparts is a valid assumption. It was the other assumptions of the article that I had trouble with, which are discussed in my blog. I think integer programming is still a legitimate optimization method, especially in situations where decisions are "yes/no" or 0/1. Nowadays, with quad-core processors being available to the general public, I think we try and make our models as complex and inclusive as possible in order to take advantage of available computational power. In situations where a somewhat simpler solution is appropriate, however, I feel integer programming still has a useful place.