R. C. COLEMAN Case Study Case Background: In this case, R.C. Coleman is facing a challenge of completing a warehouse expansion project within a required 40-week time frame. There are uncertainties and variations in the activity times, and management needs to determine the optimal crash cost and duration for each activity to achieve the target completion time. The project manager needs to analyze the project network, calculate the expected completion time, and identify the critical path to make informed decisions on how to shorten the project duration while minimizing the cost impact. R. C. Coleman has decided to automate its warehouse operation by installing a computer-controlled order-picking system, along with a conveyor system. The purpose is to reduce high labor costs and increase productivity. This report presents the activity schedule, expected project completion time for the warehouse expansion project, and a project network. Objective The warehouse expansion project aims to expand the current warehouse space to accommodate the company's growing inventory needs. The goal is to expand the warehouse capacity while minimizing the impact on the ongoing business operations. R.C. Coleman aims to complete the project within 40 weeks and at a minimum cost while meeting quality standards. Methodology The methodology used to conduct the analysis for this case is Critical Path Analysis (CPA), which is a management science tool used to identify the critical path of a project and determine the minimum time required to complete it. CPA involves identifying all the activities required to complete a project, their dependencies, and their estimated durations and then analyzing the network to identify the critical path. The available data for this case includes the estimated durations for each activity, their predecessors, and their associated costs for normal and crashed times. We can use this data to create a project network diagram and calculate the expected completion time of the project. To conduct the analysis, we first create a project network diagram to identify the sequence of activities and their dependencies. Then we calculate the expected duration and variance of each activity using the formula: Expected duration = (optimistic time + 4 x most likely time + pessimistic time) / 6 Variance = ((pessimistic time - optimistic time) / 6)? We then calculate the earliest start time, earliest finish time, latest start time, and latest finish time for each activity using forward and backward pass methods. This helps us to identify the critical path, which is the longest path through the network and has zero slack time. The critical path activities must be completed within their estimated durations to complete the project on time. We also calculate the probability of completing the project within a given time frame using the normal distribution formula: Probability of completing the project on time = (Z - p) / o where Z is the desired time frame, u is the expected completion time, and o is the standard deviation.Based on the analysis, we can make recommendations to the management regarding the required completion time, crashing activities to shorten the duration, and the probability of meeting the desired completion time. Based on the given table, the expected time, slack, variance, standard deviation, and critical activity of each task are provided. The critical activities are those that have zero slack, meaning any delay in completing them would cause a delay in the overall project completion time. Activity A has a duration of (4t4(6)+8)/6 = 10, an expected time of 6, a slack of 3, a variance of 0.44, a standard deviation of 0.66, and is not a critical activity since it has some slack. Activity B has a duration of (6+4(8)+16)/6 = 10, an expected time of 9, no slack, a variance of 2.78, a standard deviation of 1.67, and is a critical activity since it has zero slack. Activity Chas a duration of (2+4(4)+6)/6 = 4, an expected time of 4, no slack, a variance of 0.44, a standard deviation of 0.66, and is a critical activity since it has zero slack. Activity D has a duration of (8+4(10) +24)/6 = 14, an expected time of 12, a slack of 4, a variance of 7.11, a standard deviation of 2.67, and is not a critical activity since it has some slack. Activity E has a duration of (7+4(10)+13)/6 = 11, an expected time of 10, no slack, a variance of 1, a standard deviation of 1, and is a critical activity since it has zero slack. Activity F has a duration of (4+4(6)+8)/6 = 6, an expected time of 6, no slack, a variance of 0.44, a standard deviation of 0.66, and is a critical activity since it has zero slack. Activity G has a duration of (4+4(6)+20)/6 = 10, an expected time of 8, a slack of 8, a variance of 7.11, a standard deviation of 2.67, and is not a critical activity since it has some slack. Activity H has a duration of (4+4(6)+8)/6 = 6, an expected time of 6, no slack, a variance of 0.44, a standard deviation of 0.66, and is a critical activity since it has zero slack. Activity I has a duration of (4+4(6)+14)/6 = 7, an expected time of 7, a slack of 3, a variance of 2.78, a standard deviation of 1.67, and is not a critical activity since it has some slack.Activity J has a duration of (3+4(4)+5)/6 = 4, an expected time of 4, no slack, a variance of 0.11, a standard deviation of 0.33, and is a critical activity since it has zero slack. Activity K has a duration of (2+4(4)+6)/6 = 4, an expected time of 4, no slack, a variance of 0.44, a standard deviation of 0.66, and is a critical activity since it has zero slack. After calculating the data, the critical path has been shown as; B-C-E-F-H-J-K, and a standard deviation of 2.380. With zero slack time it has been determined that it would take 43 weeks to complete the expansion of the warehouse; thus, not meeting the required 40-week completion time line provided. Furthermore, by using the z score of -0.5477 and a normal distribution, it can confidently be decided that the probability of the project being completed within the deadline of 40 weeks is 10.38% (0.1038). To calculate the amount by which the expected project completion time should be shortened to achieve an 80% chance of completion within 40 weeks, we need to use the normal distribution table. From part (1), we found that the project completion time has a mean of 36.7 weeks and a standard deviation of 2.82 weeks. To achieve an 80% chance of completion within 40 weeks, we need to find the z-score associated with the 80th percentile of the normal distribution. We can use the standard normal distribution table or calculator to find this value, which is approximately 0.84. The formula for finding the new expected project completion time, T', is: T' = T - (2 * 0') Where T is the original expected project completion time, z is the z-score associated with the desired probability (0.84 for an 80% chance), and o' is the new standard deviation after shortening the activity times. Since the variance in the project completion time is the same as before, the new standard deviation will also be 2.82 weeks. Therefore: T' = 36.7 - (0.84 * 2.82) T' = 34.4 weeks (rounded to one decimal place) So, the expected project completion time should be shortened by approximately 2.3 weeks (from 36.7 to 34.4 weeks) to achieve an 80% chance of completion within 40 weeks. Recommendation: Based on this analysis, it is unlikely that the project can be completed within the required 40-week time frame with a probability of only 10.38%%. Therefore, it is recommended that R.C. Coleman consider crashing some of the critical path activities to reduce the project completion time. At 40 = T