The problem of unit in the relationship between backorder and waiting time

The formula for the relationship between backorder and waiting time is
V[B]=lambda E[BW] + lambda^2 V[BW]

I think there is exact same formula in queueing theory about the number in the queue and waiting time. But the puzzle to me is the match of unit. Suppose the unit of B, lambda, BW are item, item/day, day. V[B]=item^2, lambda E[BW] = item and lambda^2 V[BW]=item^2.

If it was me to derive this formula, I would throw it into garbage after this check. But sometimes, we cannot trust common sense.

The difference and similarity between Poisson and BM

The more I learn about BM, the more I feel there are strong connection between Poisson and BM. Last week, I ask a question. If we discrete Poisson process to small time interval, it behaves exactly as random walk with probablity lamda*h up by one and 0 otherwise. And counterpart of BM in discrete world is also random work with probability 0.5 up by 1 and probability 0.5 down by 1. Then I ask a question, what if we change the measure just like that we can change the BM with drift to BM without drift. Answer is obvious no, there are several differences.

• discrete BM, up and down probability is fixed. the only scale factor is step size. But the scale factor of poisson process is probability up and step size is fixed
• when we change the measure, we require that two probability measuresare equivalent which means the null space is the same. The null space of discrete one step of randome walke is R/{-1,1}. But poisson is R/{0,1}.

But when I ask Professor what is discrete case of BM with drift. Is it with the symmetric step size but different probability or different step size but the same probability or both or does not matter. I have not get answer yet.

IP homework is way too hard

The second set of homework takes me two days with 16+ hours to finish and without leaving a logic hole.
The last two logic gaps I filled just now are
1. LP relaxation of TSP formulation with subtour elimination constrainted is strictly contained in TSP formulation with subtour elimination constrainted replaced by $u_i-u_j-nx_{ij} \leq n-1$. This invovle how to scale up the problem then scale down.
2. Find as many affine independent points as possible for $P={x \in R^+ | \sum^n_{i=1} x_{ij}=1 for i=1,...n and \sum^n_{j=1} x_{ij}=1 for j=1,...n}$. It would be easy to see the rules by beginning with n=3 and 4.

In homework 1 of IP class, the following question is challenging.
1. show that $max{x_1 − \sqrt(2) x_2 | 1 \leq x1 \leq \sqrt(2) x_2 and x_1, x_2 is integer}$ is feasible and bounded, but has no optimal solution. I use the contradiction to show that there is no optimal solution.

uniqueness of the fixed point

In section 2.4.3 (Univalent mapping arguement) from 'game theory in SC analysis' by Cachon and Netessine, they claim that if best response function is one-to-one, then there is at most one fixed point(uniqueness of equilibrium). But why? It is possible that we can have x and y where x not equal to y such that x=f(x) and y=f(y). For example, f(x_1,x_2)=(x_2,x_1) is univalent mapping. But when x_1 = x_2=a in R, f(a,a)=(a,a). We can construct a trival game, player x and y write down a number, if x=y, then they both get 10, if not then their payoff is 10-(x-y)^2. In this game, the best response is f(x_1,x_2)=(x_2,x_1) .

Pros and cons of staying in a small university

The most advantage is that the relationship between prof. and students are close. But sometimes you will be shocked. There is only one student registering real analysis class, so that the class is cancelled. It is hard to imagine that this could happen to the basic graduate math courses in those bigger universities.

Very limited courses are offered in Lehigh. After my three years in Lehigh, there is no open course about heuristics, the stochastic process class only using introductory level of book 'Introduction to Probability Models' by Ross. It takes me waiting for 2 years to have IP open again. And there are many other regrets about courses in Lehigh.

More guns can be added into my powder magazine

Today, my wife gets and accepts her first offer. Now I can continue to my hobby to buy books. The books I plan to buy recently will be
Stochastic Calculus and Financial Applications
Nonlinear Programming: Theory and Algorithms
Theory and Practice of Revenue Management
An Annotated Timeline of Operations Research: An Informal History

Another new book in two days

Today my wife's sister comes to my home with book 'Real and Complex Analysis' I ordered from dangdang.com. It only cost me 39RMB (5 dollar).

Why Toy's R doesn't have sample for people to try

I also drop at Toy's R today due 25% gift certificate again. I try to get some gifts for my relatives and friends children. But there is no sample for most items for people to try except PS2 and Xbox 360. This setting loses the advantage to have a real store rather than online store only. Finally I get two idog mini which I already know what it is. But if I can try some other stuffs, I may buy more stuff. It seems that they don't know how to model customer behaviors

A new book comes to my book shelf

Today, I use 25% gift certificate of Border to get 'Game theory' by Fudenberg and Tirole. It is must have book for people doing game theory in SCM.

Crazy Tuesday and Thursday Next Semester

Finally I get my schedule for next semester. All the courses and meeting are concentrated in Tuesday and Thursday.
10:45-12:00 Advanced Stochastic Process II
12:10-1:00 IP Seminar (Th)
1:10:2:25 Real Analysis II
2:35-3:50 Financial Calculus II
4:00-5:00 Weekly Meeting with Larry(T)
7:00-8:15 Integer Programming
After crazy day, I will play tennis on Tuesday and Ping Pong on Thursday.

Wiki for Econ. Job Market

Is anything similar in IE or B-school.

Assignment of interview time slot

One of my friends from Econ. will graduate this year. He gets about 30 interviews during a conference at Chicago. Each interview lasts about half an hour. It is interesting to see how interviewees and interviewers agree on the time slot especially when there is no centralized system.

To queue or not to queue

It is not about the book To queue or not to queue. It is about my experience today.

Today, I sit in the Wendy's from 10:30am to 2:50pm to read and think. When I realized that I soon need to have lunch, there is waiting line. Then it becomes an optimization problem. My hungry degree increases by time. I don't want to waiting in the line waste my time not doing meaningful things. The length of waiting line is changing. What should I do?!

It turns out that I was poor to make that decision. I waited until that I cannot stand more and joined a long line. What is the form of best strategy under this situation? What if I have perfect information or not? Is any similiar situation in the real business situation?

Martingale and NP Complete problem

Last semester, I have two courses related to martingale. But how can it be used in inventory related problems. So far my limited literature reading hasn't touched any paper using martingale properties. I guess the reason is that it is not easy to find a martingale in inventory problem.

Today when I read 'The logic of Logistics', I found an application to use martingale difference sequence to develop average case analysis for Bin pack problem. Although it is not close related to inventory problems I am interests. It is a good start.

The reference is
Martingale inequalities and NP-complete problems

Wrong impression: any NP complete problem has no finite worst case bound for any heuristic

In network and graph class, we learn that there is no finite worst case bound for TSP. But I thought there is no finite worst case bound for any NP complete problem. Because of this wrong understanding, I argued with a speaker during Informs about his speech about bound of one multi-echolen problem. Of course, he cannot agree on my wrong opinion.

I realize this understand is wrong until today when I happen to read the bounds on Bin-Packing problem.

live as a pig at the end of 2006

Maybe florida is too warm. I caught the cold right back to Bethlehem. Now everyday, I sleep, eat, sleep. Today, I finally try to do something different and played monopoly. As usually, I defeat computer players with my perfect preformance in stock market. If it were real money, oh...

Hope tomorrow 2007, I can recover and go back to normal status.

Two OR applications in Florida Trip

The first one is queueing theory.
When I was in Magic Kingdom, I found an interesting application of queueing theor - Fast Pass. The procedure is the following. 1) When you see a waiting line is quite long at one attraction say Space Mountain, you can insert your ticket in one machine to get a fast pass which indicates you return during the specific time usually one or two hours laters. 2) Then when you return after playing other stuffs, you can go through the fast pass line.
There are several key points. 1) Once you get your fast pass for one attraction, you cannot get fast pass for another attraction until your current fast pass expire. For example, if I get fast pass for Space Mountain from 1pm to 2pm. I can only get fast pass for Splash Mountain after 2pm. 2) The number of fast pass in one attraction must be limited. Since the available time for the fast pass is different at different attraction, the hotter the attraction is, the longer time difference between available time of fast pass and current will be. In the last attraction-Jungle Cruise we go at 5pm, its fast pass is unavailable. We need to wait 40mins to play.
The decision to make for the visitor. 1) What is the sequence of attraction I should follow. 2) Which attraction should I get the fast pass.
The decision to make for the Theme Park. 1) How much fast passes to available at each attraction and each time periods.
Game theory might not be suitable in this case. Probably, something like priority queue plus abandon and other customer behavior can be used to formulate the problem.

My experience of fast pass in Unversal Studio at CA is different. We got the fast pass after we experienced a half hour technique difficulty at Jurassic Park. Then we can get into fast pass line for every attraction. But actually, you need to spend 30$ to get one fast pass. So the decision for the theme park is how much fast pass to offer and how much to charge.

The second one is network flow. It is very wise decision to buy a GPS just before this trip. We don't need to print every map. And we can avoid the traffic jam in the high way by going through the local. It was very crucial when we tried to catch up our flight back. And the calculation of route is very fast, it is much faster than O(A) I learned last semester. But actually it can only calculate the route from two points. If I want to go several points a day, I need to manually choose the sequence of points I need to goal. If GPS can provide the function of solve small TSP problem such as 5 points, it will be wonderful. Since real map satisfy triangle inequality, there is heuristic to get solution within 1.5 time of optimal solution. So the calculation time won't increase a lot.

Final week is over

This should be my second last final week. Now I am enjoying the final quarter of game between Rocket and Laker, two of my favorite NBA teams. As usual, I don't have mood to study once there is two or three weeks to final week. This year this mood is especially strong cause Rocket comes strong this season. After I finish the trip to Florida, I should use my time sheet again to do my time management which I haven't done after INFORMS.

Textbook: Convex Analysis and Optimization

Today I want to search how to prove a function is unimodal. I get the following book, Convex Analysis and Optimization by Dimitri P. Bertsekas. What I like Dimitri P. Bertsekas most is that he always post his class slide accompanying his book. Another example is his book Dynamic Programming and Optimal Control.

How to Bring Our Schools Out of the 20th Century

Time shows the views how American people would like education to be in the 21th century. Let's compare to Chinese education I received

• Knowing more about the world. (China: Almost all high school will teach English and we learn history of world. USA: where fewer than half of high school students are enrolled in a foreign-language class and where the social-studies curriculum tends to fixate on U.S. history.)
• Thinking outside the box. (This is weakness of Chinese education. But it seems with no child left behind(NCLB), American goes to Asian type education more and more.)
• Becoming smarter about new sources of information. (This one is hard to tell. Only thing I can say is that more knowledge I have, the smarter I can deal with new information)
• Developing good people skills. (This is also a weakness of Chinese people. There is a saying that 'three monk don't have water'. And I found the counterpart in English:Everybody's business is nobody's business. So it is also a problem in USA. I think it is due to how people get measured. Maybe in USA, there is fairer measure of team work than China. For example, almost every teacher ask students to put down name of people who help their homework)

One surprising thing is to see the following question for a second grade student: How many ways can you combine nickels, dimes and pennies to get 20¢? This question can be solved by recursion. It would be interesting to see how teachers tell students the way to solve the problem.