Problem 1: Lagoon Corp. is a manufacture of fragrant candles, and faces seasonal demand. Its demand forecast and capacity (in units) for the coming year is as follows.
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Spring
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Summer
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Fall
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Winter
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Demand
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2,400
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1,900
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2,700
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3,100
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Capacity
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Regular time
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2,100
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1,500
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2,300
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2,350
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Overtime
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400
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300
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600
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600
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Subcontract
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600
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600
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600
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600
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The company currently has 200 units in inventory. The costs data is as follows.
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Regular-time production cost
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$80/unit
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Overtime production cost
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$120/unit
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Subcontracting cost
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$140/unit
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Inventory carrying cost
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$6/unit/quarter
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(a) Suggest a demand management method when facing seasonal/cyclical demand fluctuations so that the firm better utilize capacity during low demand period.
(b) Develop a zero-inventory plan (i.e., no inventory is left in each month) that uses minimum amount of overtime and subcontracting capacity. Calculate the cost of the plan. No backorder is allowed.
(c) Develop a plan with inventory, without subcontract that uses minimum amount of overtime capacity. No backorder is allowed.
A template for the production plan:
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Spring
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Summer
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Autumn
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Winter
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Total
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Demand
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2400
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1900
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2700
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3100
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10100
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Beginning Inventory
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200
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Regular time production
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2100
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1500
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2300
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2350
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8250
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Overtime production
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Subcontract
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Ending Inventory
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Regular time production cost
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Overtime production cost
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Subcontracting cost
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Inventory carrying cost
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Total:
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A group of seven jobs must go through Processes I and II in that sequence (Process I first, then Process II). Each of the processes is operated by a dedicated machine, Machine I and Machine II respectively.
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Job
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Process I (hours)
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Process II (hours)
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A
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9
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3
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B
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5
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8
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C
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2
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1
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D
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4
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5
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E
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8
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9
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F
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2
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3
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G
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6
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7
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(a) Use Johnson's rule to determine the optimal sequence to schedule the jobs. Draw the Gantt chart of the schedule. What is the proportion of time that Machine II is idle from the beginning until the end of the schedule?
(b) Suppose that the manager has started to process the jobs according to the schedule found in part (a). At the moment when job B has been processed on Machine I for 2 hours, the manager finds that the process I of job B takes more time than expected, and hence updates the total time requirement of job B to 9 hours (Process I) and 8 hours (Process II). Assume that the in-process jobs can be interrupted and resume arbitrarily. How would you adjust the schedule at that moment? Why?
Problem 2: Dave Commerce is a wholesaler that sources industry IoT sensors from oversea suppliers and sells them to local partners. You are asked to help Dave Commerce to develop an inventory policy for a product with the following demand and cost information. Assume the demand is distributed according a Normal distribution. (Standard Normal distribution table is attached at the end of the exam paper.)
Ordering cost = $900 per order
Inventory holding cost = $0.05 per unit per week
Expected demand = 600 units per week
Standard deviation of weekly demand = 100 units per week
Lead time = 3 weeks
(a) For this type of product, which inventory model will you use? Briefly explain.
(b) Using the inventory model you select in part (a), how much safety stock should be kept in order to achieve no more than 20% risk of stockout during lead time (i.e., 80% service level)? Develop the corresponding inventory policy. Show your detailed calculations.
(c) How much safety stock is needed in order to achieve no less than 99% service level? Compare the safety stock of an 80% service level and that of a 99% service level; comment on the comparison by using Pareto Principle.
(d) If the lead time is extended to 5 weeks due to the trade war and recent chaos in Hong Kong, but the safety stock is still at the level specified in part (b), what is the risk of stockout during the new lead time?
Note: Only the safety stock remains unchanged. The d-L amount is adopted according to the new lead time.
Problem 3:
Kelle Inc. is a construction company that builds highway, bridges, and tunnels. The company bids for a contract from government to build a link bridge connecting a recreation park with residential area. A major criterion for selecting the winning bid besides low cost is the time requested to finish the construction. However, if the company is awarded the contract it will be held strictly to the completion date specified in the bid, and any delays will result in severe financial penalties. In order to determine the project completion time to put in its bid, the company identified the project activities, precedence relationships, and activity times, which are shown in the following table:
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JOB No.
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PREDECESSOR JOB(s)
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TIMES (WEEKS)
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Expected Time
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Activity Variance
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a
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m
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b
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A
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---
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5
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6
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7
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6
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B
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A
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3
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4
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5
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C
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A
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4
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11
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12
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D
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B
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9
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10
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17
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11
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E
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B
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1
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3
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5
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3
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F
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C, E
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5
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6
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7
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6
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G
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D
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1
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2
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3
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2
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H
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D, F
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4
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6
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8
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6
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I
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G, H
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7
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9
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11
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9
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You don't need to fill the table in your answer scripts, but you may need to calculate some missing values (not all) on your draft paper.
(a) Construct the network diagram for this project.
(b) Using the expected times, find the earliest and latest start and finish times, and slacks for all activities.
(c) What is the critical path? What are the expected project duration and its variance?
(d) If Kelle Inc. wants to be 95% certain that it can finish the project without incurring a penalty, what time frame should it specify in the bid?
(e) Suppose that Kelle Inc. wins the bid. The government rewards $5,000 for each week the company finishes the project early before the contracted completion time. Use the expected activity time to do the time-cost trade-off analysis according to the information below. What is the optimal crashing schedule for Kelle Inc.?
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Activity
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Normal Time (Weeks)
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Crash cost / week ($)
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Weeks Allowed to Crash
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A
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6
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10,000
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2
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B
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4,000
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1
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C
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6,000
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2
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D
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11
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3,500
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3
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E
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3
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1,000
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1
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F
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6
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2,000
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2
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G
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2
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10,000
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1
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H
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6
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4,000
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1
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I
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9
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15,000
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3
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Problem 4:
(This question is made up according to a recent event. Please do not directly use news facts as your answers because the question details and numbers below are not real.)
To attract tourists and local expenditure during holiday, the Hong Kong Tourism Board (HKTB) held a New Year Countdown Lucky Draw between 6:00pm and 11:30pm on Dec 31, 2019. Participants can join a real-time lucky draw for small prizes and a final lucky draw for big prizes. In the real-time lucky draw, participants play a game to win the chance for a reward; multiple attempts are allowed.
The bottleneck of the event system lies in a lucky draw system, which consists of one server that verifies identity, processes real-time lucky draw, and records results to a database. Suppose that there is enough buffer space for waiting requests in the lucky draw system. It takes an average of 0.01 second to process a lucky draw request. In different time slots, it is forecast that lucky draw requests arrive at the following rates.
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Time slots
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Request rate
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6:00pm - 6:30pm
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5300 requests per minute
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6:30pm - 7:00pm
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5400 requests per minute
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7:00pm - 8:00pm
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5800 requests per minute
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8:00pm - 9:00pm
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5960 requests per minute
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9:00pm - 10:00pm
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6600 requests per minute
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10:00pm - 10:30pm
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8200 requests per minute
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10:30pm - 11:00pm
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9000 requests per minute
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11:00pm - 11:30pm
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10200 requests per minute
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Assume that all inter-arrival and service times are exponentially distributed, and that the request rate is stable within each time slot.
(a) For the 6:00pm-6:30pm timeslot: What is the average time (Ws) for a lucky draw request to pass the lucky draw system (including the waiting and processing time)? What is the average number of requests in the lucky draw system?
(b) Calculate the system load of the lucky draw server for each timeslot, respectively. In which time slot the system gets overloaded for the first time (let's call this as timeslot X)? For each of the two time slots right before timeslot X, calculate the average numbers of requests in the lucky draw system. What situation would happen in timeslot X?
(c) The director of the event received the above demand forecast in the evening of Dec. 31, and decided to remove the verification code requirement to save time in processing lucky draw request. In this case, the server only needs 0.007 second to process a request. Do you think this measure can solve the problem? Why?
(d) After the event, some HKTB members debate whether they should have used a one-time aggregate lucky draw instead of the real-time individual lucky draw. In a one-time aggregate lucky draw, participants can register in an extended period before the lucky draw; the results are not known upon registration; only one lucky draw will be conducted after the registration period to allocate all rewards. Analyse the difference of the service packages provided by these two lucky draw modes, briefly comment on their Pros and Cons.
(e) To learn from the fiasco, the government decides to estimate the social cost of this event. One important index is the time each participant spent in the event system. Summarized data shows that a total number of 1.63 million people participated in the 5.5 hours period. We observed around 30,000 concurrent access (includes filing form, playing game, waiting) to the entire event system at an average moment. What is the average amount of time each participant spent in accessing the event system?
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