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I'm currently the 2nd-year MS student in Applied Mathematics. As I'm planning for which classes to take the next semester, I would love to hear your feedback about my selection. The reason is because I'm almost completing the M.S program (this one is my second-to-last semester) before I apply to the PhD programs in Operation Research and/or Applied Math next Fall. As you probably realize, I want to hedge my bet by applying for both OR and Applied Math programs, most of which allow to specialize in interdisciplinary fields like OR or Finance, not necessarily Physics or Biology. To prepare for an intense competition in top PhD programs in OR/Applied Math, I think I would need to display stronger interest in OR, as I'm only taking 2 OR Grad courses in Linear and Nonlinear Programming (I got both As in them, and love NLP's materials quite a lot. LP is also fine, but it gave me impression that there is not much to do heavy research in this topic). I currently wonder if the best way to show a true interest in OR is to either take more OR grad courses, or to do some research with my OR professor(s) who taught me in the two OR courses above. The reason for my concern is that I'm afraid I overkilled with math courses (I still find my knowledge in Math is so little, considering how steep the learning curve is for PDEs, but my time is running out).

My main interests in OR include 3 areas: optimal dynamic pricing for manufactured products and its impacts on a business's top + bottom line, derivatives pricing /asset allocation/portfolio optimization, and optimal routing/production scheduling in manufacturing and logistics.

Now, the major problem is that I can only take at most $8$ more courses, and $2$ of them I have decided to be on Measure Theory and Stochastic Differential Equations (book by M. Steele) , I still have 2 more classes to choose for next semester, and below is my potential course to take (ranked from most to least important):

Integer Programming - Applied Integer Programming - Modeling and Solution by Chen

Dynammic Programming - Models + Application book by Eric Denardo

Regression and Time Series - Econometrics Model + Economics Forecast book by Pindyck

Numerical Solutions of Differential Equations (purely Finite Difference method) - A first course in Numerical Analysis of DEs (Cambridge Series Textbook)

Complex Analysis I - no book decided yet

My current temporary pick is the first two on the list, but that might be completely wrong. Needless to say, Integer programming is quite useful when applying for OR/Applied Math programs. But this is just my thought, and I don't really have that much experience, so please help offer your insight/suggestion/new courses proposal for me.

I sincerely appreciate your time and help.

asked 18 Dec '15, 03:42

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ghjk
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edited 18 Dec '15, 18:04


Disclaimer: I've studied IE, not applied math. Therefore, my two cents come from my personal experience, research, and contact with other OR researchers.

If you want to work on FE, I think three courses are very important for research, namely NLP, Stochastics (including two separate courses on stochastic processes and stochastic/robust optimization), and forecasting. In this category, I think you could take the regression and time series course as the other two are almost covered (except for the stochastic programming course). As for your OM-related interests, I think your current choices are fine and better suited.

Finally, I think you could skip some of the more important courses at MSc level and take them at PhD level, specially if your target universities for PhD are better in those courses than your current university. That also applies to courses that are required courses in your desired PhD programs and you cannot skip them even if you have already pass them at the MSc level.

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answered 20 Dec '15, 01:01

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Ehsan ♦
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@Ehsan: Thank you very much for your thoughtful comments. I sincerely appreciate it. Just want to clarify that all my courses taken are at the PhD-level, some are for advanced grad students as well. So it looks like I should take Time Series + Numerical in your opinion? One thing that concerns me is that when I read some papers on different numerical methods/Monte-carlo simulation to price American/Asian/exotic/barrier /etc options, many of them use Fourier Transform and its inverse, along with SDEs, Integration and Probability. So should I take Complex Analysis or Statistical Inference?

(20 Dec '15, 01:15) ghjk

a. What I meant by MSc and PhD level was in fact referring to your MSc and PhD studies, not the course level.

b. If you are more interested in FE than OM, Time Series and Numerical DE seem better choices.

c. Research papers are usually good indicators of which courses you need to take (unless there are some kinds of prerequisite courses that you should take before those main courses).

(20 Dec '15, 03:09) Ehsan ♦

d. I'm not sure about Complex analysis as I'm not familiar with its applications. But Statistical Inference (provided it covers probability, hypothesis testing, interval estimation, etc.) is a must if you have no prior experience with it (usually it's a prerequisite for courses on Econometrics).

e. I think it would be a good idea to take a look at the book "Optimization Methods in Finance" (its TOC is available from here). It would give you a good idea on which OR techniques are more suited to your interests.

(20 Dec '15, 03:10) Ehsan ♦

Excellent thought. Thank you for your reference. Do you also know of any books where I can learn in-depth OR's techniques in OM? I will be able to take 2 courses though. I'm actually equally interested in OM as well (I don't want to be too narrow-minded before getting to grad school), particularly the areas of designing optimal product-pricing/optimal production scheduling scheme/optimal routing in logistics, what is another course besides among the above list I should take besides the Time Series? It seems to me the choice is between Dynamic Programming vs Statistical Inference in this case?

(20 Dec '15, 03:24) ghjk
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  1. I'm not aware of a similar book for OM. However, if you read any OM book, you would get an idea what OR techniques are required. However, you should consult separate books for getting to know those techniques.

  2. If you want to keep your options open with regard to OM, I strongly suggest to take the IP course as it has many applications in OM (in particular routing and scheduling in which you're interested) as well as FE (e.g., portfolio optimization and ALM). You should take Statistical Inference in case you haven't passed any similar course (if its syllabus is what I've already guessed).

(20 Dec '15, 03:49) Ehsan ♦

Thank you for your clear advice. I would look up myself about the OM book to get an overall picture of the common techniques frequently used in that area (optimization is surely one of them, which is nice for me). Statistical Inference would cover these topics: "limiting distributions and stochastic convergence, sufficient statistics, exponential families, statistical decision theory and optimality for point estimation, Bayesian methods, maximum likelihood, asymptotic results, interval estimation, optimal tests of statistical hypotheses, and likelihood ratio tests." Is it sufficiently good?

(20 Dec '15, 04:35) ghjk
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  1. Please note that many OM books are targeted towards MBA students. Therefore they usually touch the surface of the relevant OR aspects. So make sure to identify OM topics and then check books the cover those topics in detail. For example, see books on routing, scheduling, inventory, revenue management, facility location, etc.

  2. The syllabus seems like a good and advanced one. Feel free to take that or IP, whichever you feel is more suited to your interests.

(20 Dec '15, 05:10) Ehsan ♦

Thanks for your confirmation. So Statistical Inference and IP are EQUALLY important (of course, besides LP and NLP, which I already took and did well in them) if I want to be able to do significant research in OM? Or is one more important than the other? Btw, I also saw that some professors in OR use queuing theory to set up the models for problems in OM. I can also take that course this semester, if I forgo either Numerical DE or IP/Time Series. Any thought?

(20 Dec '15, 13:10) ghjk
  1. Probability and statistics are essential tools for every OR researcher. However, you could do OR research without taking advanced courses on them. If you have taken enough fundamental courses on probability and statistics, I'd say go with IP. However if not, take SI as it is useful for both OM and FE research.

  2. Queueing theory is a good and essential course for OM. However, I suggest that you take a stochastic processes course which covers the necessary foundations of studying stochastic systems such as queueing systems. That course would be more useful for your FE interest as well.

(21 Dec '15, 02:57) Ehsan ♦

@Ehsan: Excellent thought!! You really know quite a lot about OR and FE. But it turns out that the IP course conflicts with the SDE course, and I already committed to SDE as it's quite an important course for FE. But now I'm having a hard time to decide between Numerical DEs + Time Series, or Numerical DEs+ DP/IP, or Queuing Theory + DP? May you please give your thought on this, if you were me (I also listed my courses taken so far in the answer below)? I also wonder why IP is more applicable than DP, as the book "Optimization methods in Finance" devotes an entire chapter to DP but not IP?

(21 Dec '15, 21:15) ghjk
  1. You should prioritize your interests regarding FE and OM. Otherwise, you would never choose a course with which you would be satisfied. In my opinion, take Numerical DEs + Time Series for FE. For OM, take Queuing Theory + IP.

  2. DP is an important methodology applicable to a wide range of problems, while IP is useful to deal with large scale discrete problems, specially in network design, transportation, production planning, routing, etc.

  3. Talk to your advisor and fellow students. There are some hidden aspects of courses that are not shown in names of courses (e.g., course quality).

(22 Dec '15, 02:02) Ehsan ♦
showing 5 of 11 show 6 more comments

@Ehsan: My current choice is Numerical DEs + Dynamic Programming, although I'm still not sure whether Time Series or Queuing Theory would be better choice than DP? I'm more interested in Financial Engineering than Operation Management, but I don't want to keep my focus too narrow to avoid not having enough professors who match my interest at the schools I applied for (like Columbia, Cornell, UT-Austin's IROM, UMD's AMSC, UPenn's ACMS, etc.). Below is the syllabus for each course:

  • Time Series and Regression: Mathematics of regression, exponential smoothing, time series, and forecasting

Book: Econometric Models + Economic Forecast - Pindyck

  • Dynamic Programming: (Deterministic DP) Dynamic Programming Networks and the Principle of Optimality. Formulating dynamic programming recursions, Shortest Path Algorithms, Critical Path Method, Resource Allocation (including Investments). Knapsack Problems, Production Control, Capacity Expansion, and Equipment Replacement. Infinite Horizon Optimization including Equipment Replacement over an Unbounded Horizon. Infinite Decision Trees and Dynamic Programming Networks. (Stochastic DP) Stochastic Shortest Path Problems with examples in Inventory Control. Markov Decision Processes, value and policy iteration for discounted cost criteria. MDP with examples in Equipment Replacement and inventory problems. Semi-Markov Decision Process.

Book: Dynamic Programming: Models and Applications by Eric Denardo

  • Queuing Theory: Introduction to queueing theory Review of Poisson processes, Markov chains, Little’s law, Simulation of queueing models, Fluid models, Simple Markovian queues, Advanced Markovian queues, Queueing networks, Models with general distributions.

Book: Fundamentals of Queueing Theory by John Shortle

I'm sorry if I bore you with my questions, but since I only have 2 semesters to go before grad application to top OR/Applied Math, it's very important for me to take a right course to show the admission that I have substantial background for doing research/passing the qual.

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answered 23 Dec '15, 23:53

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ghjk
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edited 25 Dec '15, 23:52

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If you want to keep your options open, take DP as it's a core OR technique. Forecasting (TS and Reg.) could be not necessary for an optimization-focused research in either FE or OM. The same goes for QS in OM. Please note that both TS and QS are useful for practice and you should eventually know them if you want to be successful in these fields.

(25 Dec '15, 15:53) Ehsan ♦

@Ehsan: thanks a lot! I decided to go ahead with Numerical DEs + DP. Hopefully I won't regret later on. Btw, what is QS? Did you mean "Queuing Theory?"

(25 Dec '15, 23:53) ghjk
1

I meant queueing theory, but I mistyped. QS stands for Queueing Systems, which was the name of a similar course I took when I was a graduate student. Sorry for the confusion.

Good luck.

(26 Dec '15, 07:55) Ehsan ♦

Nobody minds giving me some advice on which course to take? It's quite a hard choice to choose between Complex Analysis, Regression and Time series, and Integer programming. Anyone has experienced this situation (professors @Andy, @Matthew, @Mark, @Michael, may you please give your thoughts here?) A little bit more about my previous courses in Math and OR: I took Advanced Linear Algebra, Fourier Analysis, Linear Analysis, Nonlinear Functional Analysis, Numerical Analysis, PDE + ODE, Topology, Linear & Nonlinear Programming, and received A/A+ in all of them. Hopefully this information is somewhat helpful to you.

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answered 19 Dec '15, 16:34

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ghjk
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edited 22 Dec '15, 21:01

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