ISOM2700 Practice Set 2 sol PDF

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Professor Dongwook Shin ISOM2700

Fall 2020 Operations Management

Practice Questions Set #2 (Solutions)

1. (Advanced Problem: Little’s law) Mt. Kinley is a strategy consulting firm that divides its consultants into three classes: associates, managers, and partners. The firm has been stable in size for the last 20 years, ignoring growth opportunities in the 90s, but also not suffering from a need to downsize in the recession at the beginning of the 21st century. Specifically, there have been – and are expected to be – 200 associates, 60 managers, and 20 partners. The work environment at Mt. Kinley is rather competitive. After four years of working as an associate, a consultant goes “either up or out”; that is, becomes a manager or is dismissed from the company. Similarly, after six years, a manager either becomes a partner or is dismissed. The company recruits MBAs as associate consultants; no hires are made at the manager or partner level. A partner stays with the company for another 10 years (a total of 20 years with the company).

a. How many new MBA graduates does Mt. Kinley have to hire every year? We can use Little’s law to find the flow rate for associate consultants: Inventory=Flow Rate*Flow Time; 200 consultants=Flow Rate* 4 years; thus, the flow rate is 50 consultants per year, which need to be recruited to keep the firm in its current size. b. What is the probability that a new MBA at Mt. Kinley will become partner? We can perform a similar analysis at the manager level, which indicates that the flow rate there is 10 consultants. In order to have 10 consultants as a flow rate at the manager level, we need to promote 10 associates to manager level (remember, the firm is not recruiting to the higher ranks from the outside). Hence, every year, we dismiss 40 associates and promote 10 associates to the manager level (the odds at that level are 20%) Now, consider the partner level. The flow rate there is 2 consultants per year (obtained via the same calculations as before). Thus, from the 10 manager cases we evaluate every year, 8 are dismissed and 2 are promoted to partner (the odds at that level are thereby also 20%). In order to find the odds of a new hire to become partner, we need to multiply the promotion probabilities: 0.2*0.2=0.04. Thus, a new hire has a 4% chance of making it to partner.

ISOM2700 Practice Questions Set 2

2. (Advanced Problem: Little’s Law) The ISOM Dept at HKUST has professors at three ranks: Assistant Professor, Associate Professor, and Full Professor. Assistant professors are hired from graduate schools. 40% of assistant professors leave in an average of 3 years. The remaining 60% are evaluated for tenure after 6.5 (from the time they start at HKUST) and 35% of them get tenure and promoted to associate professor. Those failing to get tenure leave HKUST. In an average year, the ISOM department hires 2.5 assistant professors. Over the last several years, the ISOM department has had an average of 7 associate professors. In addition to those who started at HKUST, this total includes “external hires” who began their careers at other universities. On average, the ISOM department hires one external hire at the associate level every four years. 70% of associate professors (whether developed internally or hired from the outside) are ultimately promoted to full professor. In addition, the department hires some senior scholars at the full professor level. Over the last several years, the ISOM department has had an average of 12 full professors. The average full professor remains at the department for 20 years from the time they are promoted to full or hired from the outside. The flow of faculty through the ranks is shown below: External Hires

Assistant Prof.

Leaving before Tenure

Tenure Review

Denied Tenure

External Hires

Associate Prof.

Promotio n Review

Full Prof.

Leaving before Promotion

a. On average, how many professors are there in the ISOM department across all ranks? The question is asking for the “inventory” in the department. We have assistant or associate or full professors as “inventory”. The problem says the dept. has an average of 7 associate professors and 12 full professors. So obviously “at any point in time” the inventory of associate and full professors is 7+12=19. It remains to calculate the inventory of assistant professors. We do this as follows: there are two parts to count for the inventory of the assistant professors: Part I: 40% *2.5 *3= 3. These are the assistant professors who join and leave after 3 years. At any point in time their inventory is flow rate * flow time. Flow rate is 40%*2.5 and flow time is 3 years. Part II : 60% *2.5 *6.5 = 9.75. These are the ones who join and stay for 6.5 years. Part I+ Part II = 12.75 Assistant Professors Therefore, there are 12.75+7+12= 31.75 professors across all ranks in the ISOM department.

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ISOM2700 Practice Questions Set 2

b. How many “new professors” are hired by the department in an average year? Every year the department hires 2.5 new assistant professors. Also there is one new (external) hire at the associate professor level every four years. That means 0.25 associate professors a year. So far, we know the department hires 2.5 + 0.25 = 2.75 professors every year at the assistant and associate professor level. It remains to see how many new professors are hired at the full professor level. To get that, we need to have the flow rate to the full professor position. The flow rate to the full professor position is easy to compute. This is because we are given the “inventory” at that level and we also know the flow time! The inventory is 12 and the flow time is 20 years. By Little’s Law the flow rate = 12/20 = 0.45 /Year. This flow rate though has two components: The first is the ones who get promoted from the associate professor rank to the full professor rank. The other is the new hires (if any). We can compute how many are getting promoted in a year from associate level to full level: (0.35*0.6*2.5+0.25)*0.7=0.5425/Year. What is in parentheses is the yearly flow into the associate professor level and 0.7 is the fraction of the associates who actually get promotion to full professorship. Let’s see what we have so far! We said that the flow rate is 0.6/Year at the level of full professors. We also said the flow from associate level to full professor level is 0.5425/Year. This means on average the department hires 0.0575 full professor per year (or 1 full professor about every 17 years). So at the end, we only have 2.5+0.25+0.0575=2.8075 new hires every year. The solution is 2.8075/Year.

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ISOM2700 Practice Questions Set 2

3. (Quality Management: Process Capability Index) HVAC Manufacturing produces 6-inch-diameter round metal ducting, with an acceptable tolerance of ±0.03 inch. Anything produced outside specifications is considered defective. The supervisor for this product has data showing that the actual diameter of finished product is 5.99 inches with a standard deviation of 0.01 inch. a. Calculate the process capability index for this example. ,-./012

𝐶& = 𝑚𝑖𝑛 +

34

,

01 2/.-. 34

07.83/9.::

6 = 𝑚𝑖𝑛 +

8.83

,

9.::/9.:; 8.83

=0.667 6 = min0{1.333,0.667}

b. What does this figure tell you about the process? The process capability is less than 1. The process is producing more defective items with the diameter smaller than the LSL than the defective items with a diameter larger than USL. Further, the process has a less than 3-sigma quality because Cp < 1.

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ISOM2700 Practice Questions Set 2

4. (Quality Management: X-bar Chart) Resistors for electronic circuits are manufactured on a high-speed automated machine. The machine is set up to produce a large run of resistors of 1,000 ohms each. To set up the machine and to create a control chart to be used throughout the run, 15 samples were taken with four resistors in each sample. The complete list of samples and their measured values are as follows:

Readings (In Ohms) Sample Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

X1 1010 995 990 1015 1013 994 989 1001 1006 992 996 1019 981 999 1013

X2

X3

991 996 1003 1020 1019 1001 992 986 989 1007 1006 996 991 993 1002

X4

985 1009 1015 1009 1005 994 982 996 1005 1006 997 991 989 988 1005

986 994 1008 998 993 1005 1020 996 1007 979 989 1011 1003 984 992

Develop an X-bar chart for 99.7% confidence and plot the values. From the charts, what comments can you make about the process?

𝑋E =999.1, 𝑅G = 21.733

Sample 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1 1010 995 990 1015 1013 994 989 1001 1006 992 996 1019 981 999 1013

2 991 996 1003 1020 1019 1001 992 986 989 1007 1006 996 991 993 1002

3 985 1009 1015 1009 1005 994 982 996 1005 1006 997 991 989 988 1005

4 986 994 1008 998 993 1005 1020 996 1007 979 989 1011 1003 984 992

Control limits for X-bar chart: UCL, LCL = 𝑋E ± 𝐴I 𝑅G = 999.1 + 0.73(21.733) = 1014.965, 983.235

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Mean 993.00 998.50 1004.00 1010.50 1007.50 998.50 995.75 994.75 1001.75 996.00 997.00 1004.25 991.00 991.00 1003.00

Range 25 15 25 22 26 11 38 15 18 28 17 28 22 15 21

ISOM2700 Practice Questions Set 2

The process is in statistical control.

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ISOM2700 Practice Questions Set 2

5. (Quality Management: P Chart) The state and local police departments are trying to analyze crime rates so they can shift their patrols from decreasing-rate areas to areas where rates are increasing. The city and county have been geographically segmented into areas containing 5,000 residences. The police recognize that not all crimes and offenses are reported: people do not want to become involved, consider the offenses too small to report, are too embarrassed to make a police report, or do not take the time, among other reasons. Every month, because of this, the police are contacting by phone a random sample of 1,000 of the 5,000 residences for data on crime. (Respondents are guaranteed anonymity.) Here are the data collected for the past 12 months for one area: Month January February March April May June July August September October November December

Crime Incidence 7 9 7 7 7 9 7 10 8 11 10 8

Crime rate 0.007 0.009 0.007 0.007 0.007 0.009 0.007 0.01 0.008 0.011 0.01 0.008

Construct a p-chart for 95 percent confidence (this means we build a 1.96-sigma control chart and not 3-sigma control chart. Each industry has its own standard) and plot each of the months. (Remark: after you construct the control limits using the existing data, if there are any points outside the control limits, we don’t need to reconstruct the control limits again for this case.) If the next three months show crime incidences in this area as January = 10 (out of 1,000 sampled) February = 12 (out of 1,000 sampled) March = 11 (out of 1,000 sampled) What comments can you make regarding the crime rate?

𝑝2 =

(8.88;L8.88:L⋯L8.88N)

= 0.008333

PI

&2(P/&2)

𝜎& = R

S

8.88N333(P/8. 88N333)

=R

P888

= 0.00287

UCL = 𝑝2 + 1.96 𝜎& = 0.008333 + 1.96(0.00287) = 0.01396 LCL = 𝑝20- 1.96 𝜎& = 0.008333 - 1.96(0.00287) = 0.0027

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ISOM2700 Practice Questions Set 2

The process is in control. Therefore, it can be stated that the crime rate has not increased. However, there appears to be a gradual increase in the crime rate.

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ISOM2700 Practice Questions Set 2

6. (Quality Management: X-bar Chart) The manager of Champion Cooling Company has recently implemented a statistical process control method. The accompanying table shows the results of several different samples of walkin coolers that were produced in the previous month.

Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 𝑥2 R

Sample 1 104.0 106.4 101.8 105.6 100.6 103.68 5.8

Sample 2 100.1 104.4 103.6 101.4 100.7 102.04 4.3

Sample 3 101.1 100.4 103.0 101.0 104.9 102.08 4.5

Sample 4 102.0 105.9 104.6 102.1 107.0 104.32 5

For your reference, the A2 value for determining the three-sigma control limits is given below: n 3 4 5 6

A2 1.023 0.729 0.577 0.483

1) What is the value of the center line for the chart that monitors performance? X-double bar = (103.68+102.04+102.08+104.32)/4 = 103.03 2) What is the value of the upper control limit and lower control limit? Sample size n = 5 A2 = 0.577 R-bar = 4.9 LCL = 103.03 - 0.577*4.9 = 100.203 UCL = 103.03 + 0.577*4.9 = 105.857 3) Construct an x-bar chart using the data in the table. Is the process in control? In control

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ISOM2700 Practice Questions Set 2

7. Hoosier Manufacturing operates a production shop that is designed to have the lowest unit production cost at an output rate of 100 units per hour. In the month of July, the company operated the production line for a total of 175 hours and produced 16,900 units of output. What was its capacity utilization rate for the month? 16,900 𝐶𝑎𝑝𝑎𝑐𝑖𝑡𝑦0𝑢𝑡𝑖𝑙𝑖𝑧𝑎𝑡𝑖𝑜𝑛0𝑟𝑎𝑡𝑒 = = 96.57% (175) ∗ (100)

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