# Fuzzy mathematical programming vs stochastic programming

 1 Dear all As we know, there are some approaches to tackle uncertainty in mathematical programming amongst with stochastic programming, robust optimization, and fuzzy mathematical programming. To the best of my knowledge, SP is used when there are enough data to estimate the probability distribution function, RO is used when we just know the interval that data varies (continuous form), and FMP is used when data are vague, ambiguous, or when data are not sufficient to estimate PDF of data. There are a lot of problem in operations management in which there is no historical data, such supply chain network design, location of warehouses and so on. In this situation, it seems more appropriate to use FMP while in most prestigious journals such as Management science, Operations Research, etc, SP and RO are more acceptable while these approaches increase the complexity of model and in some cases make them nonlinear. Can anyone tell me why SP and RO are more desirable to model uncertainty in these journals and North of America? Thanks asked 01 Oct '16, 16:06 Amin-Sh 61●1●6 accept rate: 0%

2 Answers:
 4 Disclaimer: Similar to Paul, the following is based on my own thoughts and discussions with colleagues, hence not supported with hard data. Also, I'm no expert in fuzzy theory, but I've relatively more experience with RO and SP. Originally, fuzzy theory was proposed to deal with ambiguous data. For example, how you define coldness or hotness of water is ambiguous. However, most of data we use in our optimization models are by nature crisp (e.g., price, time, demand, etc). Perhaps you don't know what the future demand would be, however the demand itself is crisp. In other words, the fact that you cannot measure the future demand, does not make it ambiguous. It's just unknown. Following the previous point, FMP community has abused FMP to a great extent. I've seen papers published in good journals that are absolutely outrageous. For example in some scheduling studies, the authors have considered due dates to be fuzzy parameters. This means that even customers are not sure which date they need their order. AFAIK, FMP models, at least basic ones, have no structural property that models the uncertainty. For example, some FMP models are built on the idea of $$\alpha$$-cut that is very similar to the expected value models (i.e., nominal models in RO terminology), and we already know they are not as good as stochastic models. Recently, I've seen efforts to extend FMP models to consider notions of robustness and chance constraints (e.g., possibilistic models). However, I don't know about their rigor. Fuzzy data is based on experts' opinion. I don't know how you can derive a membership function from the expert's mind. However, asking people for scenarios and intervals seems more natural. answered 05 Oct '16, 08:20 Ehsan ♦ 4.8k●3●11●22 accept rate: 16% Thank you dear Ehsan. The first reason was so surprising for me. (07 Oct '16, 06:20) Amin-Sh
 2 Disclaimer: The following is purely personal opinion, not supported with hard data. One problem I see with publishing applications of FMP is that it is simply not as widely known as SP and RO (despite the fact that I think FMP predates RO as a methodology). Reviewers who do not know much about SP or RO (which describes me) have at least heard of them and perhaps know (a) the basics and (b) when/why to apply them. That brings me to the second problem: when (and, perhaps more importantly, why) to apply each method. As you noted, SP applies when (a) parameters are random and (b) you can model their distribution(s) with some plausible accuracy. When parameters are random, or at least "uncertain", and you can't be very specific about what they are likely to be, robust optimization appeals. To me (and I'm not the definitive source here), FMP applies when the data is fairly concrete but decision makers will differ in their interpretations of certain things (such as what constitutes a "defective part", or a "serious delay", or a "good solution"). FMP lets you try to quantify what I would call "perceptions" or "opinions" by the decision makers. To me, that's a "soft OR" concept. In the UK (and perhaps Europe more generally?), "soft OR" is a staple of OR programs; in the US, we seem to be fixated with "hard OR" ("hard" meaning algorithmic and mathematically rigorous, not necessarily NP-screwy). So, again, it's harder for FMP to get traction here. I haven't paid much attention to FMP in quite a long time, but at least back when I did there was one other issue that might inhibit its publication. Probabilities can be estimated in a somewhat rigorous manner, and RO bounds can be derived fairly rigorously (I think). I don't know whether there are comparably rigorous methods for deriving fuzzy set membership functions. Without them, FMP may look too arbitrary to some reviewers. answered 01 Oct '16, 16:34 Paul Rubin ♦♦ 14.6k●4●12 accept rate: 19% Paul, In fairness, you were answering a question about journals. Nevertheless, in the real world, not everyone is fixated on proofs and rigor. There are exceptions though, such as the other day when I briefed the CEO of a major corporation on an initiative to cut costs 20% "Today I will be proving theorem 2.9.3. WLOG we can assume the conditions in Lemma 1.2, thereby reducing the problem to the usual situation covered in theorem 1.5.1. The really exciting part is that this new theorem has important applications, such as proving theorem 2.9.4." For some reason, I didn't win the job. (01 Oct '16, 17:50) Mark L Stone Mark: Indeed! I take it you didn't bring an "academic speak to English" translator with you? On a related note, I used to teach integer programming to (mostly) doctoral students in business. Every so often, I'd point out that most decision makers aren't interested in waiting three days for your solver to squeeze out the last 1% gap in a model that somewhat vaguely approximates their reality. (01 Oct '16, 18:14) Paul Rubin ♦♦ 1 Sorry, just a little attempt at humor. I was remembering the course description for a probability special topics course at a major west coast university in the heart of silicon valley a third of a century ago. It was something like this "The theory of xyz will be developed in detail. But this is not just a theoretical course. We will also cover important applications, such as using the theory of xyz to provide a new proof of theorem abc." As an O.R. guy, I remember thinking at the time that that's not what the typical businessperson would consider to be an application. (02 Oct '16, 08:52) Mark L Stone 1 I feel you. I spent 33 years on the faculty of a business school. You might think that B-school research (and courses) would actually be applied (or at least applicable) in the real world. I used to think that myself. ;-) (02 Oct '16, 11:46) Paul Rubin ♦♦ Thank Dear Paul and Mark. Although there are some approaches to estimate probability distribution of data sets, but these approaches need sufficient data to estimate pdf of data appropriately. In case that we want to design a supply chain network, there is no historical data to estimate pdf.In this situation, it might be more realistic to use fuzzy numbers (for example, pessimistic, moderate, optimistic). (07 Oct '16, 05:20) Amin-Sh I agree that FMP is not rich mathematically compared with SP or RO, but I think for some decision-making problems in MS/OM, FMP matches the problem characteristic in comparison with SP. Moreover, counterpart problem of a fuzzy MIP problem is still MIP, while SP might change it to NLMIP and FMP does not usually make a tractable problem to an intractable one. (07 Oct '16, 05:20) Amin-Sh I still think that even if you cannot build sophisticated SP or RO models due to lack of historical data, basic yet relevant SP and RO models based on experts' opinion are better suited to address the issue of uncertainty. (07 Oct '16, 06:02) Ehsan ♦ showing 5 of 7 show 2 more comments
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Asked: 01 Oct '16, 16:06

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Last updated: 08 Oct '16, 15:44

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