Problem 1 (10 points) An operation of a company relies on a machine which has 3 different bearings with identical distribution of the life time as shown in Table 1. The machine will fail to operate if any one of the bearings fails. In case of machine failure, a mechanic is called to repair the machine and the delay time for the mechanic to arrive is random with a distribution shown on Table 2. The time for a mechanic to repair the machine depends on the total number of bearings that need to be replaced. The mechanic takes 10 minutes to replace 1 bearing but, to replace all 3 bearings, the mechanic needs 20 minutes. Table 3 lists all the costs associated with the maintenance of the machine. Lifetime (hour) probability 1000 0.1 1100 0.13 1200 0.25 Table 1 1300 0.19 1400 0.13 1500 0.12 1600 0.06 1700 0.02 Delay time (minute) 10 probability 0.20 Table 2 15 0.30 20 0.50 Cost of bearing HK$400 per bearing Downtime cost of machine HK$5 per minute Direct on-site cost of mechanic HK$400 per hour Table 3 There are 2 maintenance policies for the management to consider. Policy 1 is to replace only the bearing that fails, and other operable bearings remain unchanged. Policy 2 is to replace all 3 bearings whenever any one of the 3 bearings fails. Monte Carlo simulation models for the 2 policies has been built using Microsoft Excel. The simulation model is in an Excel file named "Q1.xlsm which has been placed to the Moodle system of this course. The model simulates the lifetime of 60 bearings. For policy 1, it is assumed that the bearings are never fail at the same time. Hence, there is no more than 1 bearing that need to be replaced at the same time. The performance measures of the policies are defined as the total cost per 10,000 bearing-hours. (a) Assume that the performance measures for both policies are normally distributed. Modify the simulation model to include experiment of 30 replications for both policies. Estimate the performance measures for both policies and give your estimations together with 95% confidence intervals. (b) If a bearing is cheap, policy 2 can improve the performance by reducing the frequency of repairs without too much additional cost. However, if a bearing is expensive, replacing all 3 bearings at the same time becomes costly. The management is interested to know the decision point of the cost of bearing, i.e., if the cost of bearing is lower than the decision point, the result will favour policy 2. However, if it is higher than the decision point, policy 1 will result a lower cost per 10,000 bearing-hours. Modify the experiment in (a) to 15 experiments with different costs of bearing from HK$400 to HK$1100 with a separation of HK$50. Based on your result, suggest the decision point of the cost of bearing to the management. NB: Use a single Excel file for both questions and clearly name the worksheets for the questions. Problem 1 (10 points) An operation of a company relies on a machine which has 3 different bearings with identical distribution of the life time as shown in Table 1. The machine will fail to operate if any one of the bearings fails. In case of machine failure, a mechanic is called to repair the machine and the delay time for the mechanic to arrive is random with a distribution shown on Table 2. The time for a mechanic to repair the machine depends on the total number of bearings that need to be replaced. The mechanic takes 10 minutes to replace 1 bearing but, to replace all 3 bearings, the mechanic needs 20 minutes. Table 3 lists all the costs associated with the maintenance of the machine. Lifetime (hour) probability 1000 0.1 1100 0.13 1200 0.25 Table 1 1300 0.19 1400 0.13 1500 0.12 1600 0.06 1700 0.02 Delay time (minute) 10 probability 0.20 Table 2 15 0.30 20 0.50 Cost of bearing HK$400 per bearing Downtime cost of machine HK$5 per minute Direct on-site cost of mechanic HK$400 per hour Table 3 There are 2 maintenance policies for the management to consider. Policy 1 is to replace only the bearing that fails, and other operable bearings remain unchanged. Policy 2 is to replace all 3 bearings whenever any one of the 3 bearings fails. Monte Carlo simulation models for the 2 policies has been built using Microsoft Excel. The simulation model is in an Excel file named "Q1.xlsm which has been placed to the Moodle system of this course. The model simulates the lifetime of 60 bearings. For policy 1, it is assumed that the bearings are never fail at the same time. Hence, there is no more than 1 bearing that need to be replaced at the same time. The performance measures of the policies are defined as the total cost per 10,000 bearing-hours. (a) Assume that the performance measures for both policies are normally distributed. Modify the simulation model to include experiment of 30 replications for both policies. Estimate the performance measures for both policies and give your estimations together with 95% confidence intervals. (b) If a bearing is cheap, policy 2 can improve the performance by reducing the frequency of repairs without too much additional cost. However, if a bearing is expensive, replacing all 3 bearings at the same time becomes costly. The management is interested to know the decision point of the cost of bearing, i.e., if the cost of bearing is lower than the decision point, the result will favour policy 2. However, if it is higher than the decision point, policy 1 will result a lower cost per 10,000 bearing-hours. Modify the experiment in (a) to 15 experiments with different costs of bearing from HK$400 to HK$1100 with a separation of HK$50. Based on your result, suggest the decision point of the cost of bearing to the management. NB: Use a single Excel file for both questions and clearly name the worksheets for the questions