私立中原大學八十九學年度博士班招生考試命題紙

系所組別:工業工程學系 科目：工業工程基礎科目 考試時間：6月14日第1節

壹. 工程統計
1-1. There are four kinds of stocks A, B, C, and D. The game rule is that if you buy stock A, then you have to buy stock B; if you stock C, then you have to buy stock D. Let random variable Xi denote the (random) profit earned if you buy stock i and sell it one year later, i = A, B, C, and D. Let = (XA, XB, XC, XD)'and the mean vector and covariance matrix of Σ are
 =  and Σ =.(unit: thousand dollars)
The market prices per share are 20, 5, 200, and 50 thousand dollars for stocks A, B, C, and D, respectively. Assume that you want to spend 1,000 thousand dollars to buy these four kinds of stocks; however, you would buy at most 20 shares of stock A.
(1). Specify which variables of XA, XB, XC, and XD are independent of each other? (5%)
(2). How many shares of stocks A, B, C, and D should you buy so that the mean total profit earned is maximized? (8%)
(3). Compute the standard deviation of the profit earned. (5%)
(4). Ifis a normal random vector, what is the distribution of the profit earned (based on your answer in (2))? Note: you need to write down the values of all distribution parameters. (4%)
1-2. At Department of Industrial Engineering there are two professors teaching Engineering Statistics, Mark and John. Most college students think that John's teaching is better than Mark's. The classes that Mark and John taught last semester have the following grade statistics:
　

Mark's class

John's class

Number of students in class

75

62

Average grade of all students in class

68

60

Sample standard deviation of class grades

15

7

We want to use the above statistics to test hypothesis that John's teaching is better than Mark's.

(1). Define a performance measure for evaluating the teaching quality. (3%)
(2). Test hypothesis that John’s teaching is better than Mark's, using the performance measure defined in (1). Let α = 0.05, where α is the probability of Type I error. Note: you need to write down the hypotheses and define all symbols you used. No credit will be given if you do not define your own symbols. (10%)
(3). What assumptions are required for the testing procedure in (2) to be valid? (3%)
1-3. Suppose that by any time t the number of people that have arrived at a train depot is a Poisson random variable with mean λt. If the initial train arrives at the depot at a time (independent of when the passengers arrive) that is uniformly distributed over (0, T), what is the mean and variance of the number of passengers that enter the train? (12%)
Reference numbers:
1. Let Z be a standard normal random variable.
P{ Z > 1.645} = 0.05, P{ Z > 1.96} = 0.025
2. Let T(v) has a student-t distribution with v degrees of freedom.
P{ T(61) > 1.67}= 0.05, P{ T(61) > 2.0}= 0.025,
P{ T(74) > 1.666}= 0.05, P{ T(74) > 1.993}= 0.025,
P{ T(135) > 1.6562}= 0.05, P{ T(135) > 1.978}= 0.025,
P{ T(136) > 1.6561}= 0.05, P{ T(136) > 1.977}= 0.025.
3. Let F(v1, v2) has an F distribution with v1 and v2 degrees of freedom.
P{ F(74, 61) > 1.506}= 0.05, P{ F(74, 61) > 1.631}= 0.025,
P{ F(61, 74) > 1.493}= 0.05, P{ F(61, 74) > 1.613}= 0.025.

貳. 作業研究
2-1. Consider the following linear programming problem:

s.t.
    
where is , is , is , is . Suppose that rank is .
(a). Write down the dual of this problem(5%)
(b). What is the weak duality theorem? What is the strong duality theorem? (5%)
(c). What is the complement slackness theorem? (5%)
(d). Prove that if both the primal problem and the dual problem have feasible solution then both must have optimal solution. (5%)
(e). Prove that is a convex set. (5%)
2-2. Consider a gas station has only one gasoline pump. This station can contain only 4 cars. Let arrival process of the cars to this station be a Poisson process with rate cars per hour. Furthermore, an arriving car will drive away with probability when it finds cars already in this station, . The time required to service a car has an exponential distribution with a mean 4 minutes.
(a). Construct the transition rate diagram for this queueing system. (5%)
(b). Write down the balance equations and find the limiting probabilities. (5%)
(c). Find the expected number of cars in system. (5%)
(d). What is the actual (effective) arrival rate to this system? (5%)
(e). Use Little's formula to find the expected waiting time in system for any car.(5%)

參 計算機概論
3-1. Please write down the complete terminology of the following abbreviation.（2% for each）
1. ODBC：
2. COM：
3. OLE：
4. IDL：
5. CORBA：
6. OODB：
7. ASP：
8. EBCDIC：
9. FAT：
10. DSS：
3-2. Please explain the difference between the bridge, the router and the gateway in the computer network. (9 %)
3-3. A programmer writes the following C code：
char *ps;
for(int i = 0; i <1000; i++)
ps = malloc((i+1)*sizeof(char));
free(ps);
What kind of program the programmer may encounter in the program containing the above code? (5 %) 　
3-4. Define an NP-hard and an NP- complete problem. (6 %)
3-5. Please define the Data Warehouse, and explain the main difference between Data Warehouse and Database briefly. (10 %)

肆 案例分析
Most workers spend a major portion of their time in a small work area, called the workspace. The workspace should be designed in such a way that employees will be able to perform their jobs effectively. The design specifications of the workspace in relationship with worker’s physical characteristics and job requirements have significant impact on their productivity, and physical and mental wellbeing. Based on your understanding of two quite different assembly lines, the automobile assembly and the notebook computer assembly, answer the following questions with regard to the optimal design of the job and the workplace for efficiency, safety, and comfort.
4-1. What are the major differences between the two assembly lines in terms of their assembly operations for the operator. (10%)
4-2. Perform an analysis as to whether seated workspace or standing workspace should be utilized for each of the two assembly lines, clearly listing your design criteria. (10%)
4-3. Traditionally, an automobile assembly line produces one car model only. That is, each line is designed to assemble all parts for the same car model. What are the possible advantages and disadvantages for the operator when two or three car models are mixed in the same assembly line? That is, the operator may be assembling a car model in a minute and another different model in the next. (10%)
4-4. Job rotation and job enlargement have both been successfully employed in the automobile assembly. What are the implications for the two design methods? (10%)
4-5. Discuss the limitations and possibilities to employ job enlargement in the notebook assembly line. (10%)

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