Case Problem of R.C. Coleman Warehouse - Report Example

Assignment 2: Case Problem R.C. Coleman Your TBS 912 Quantitative Methods for Decision Making Sydney Business School University of Wollongong November 3, 2010 Executive Summary R. C. Coleman distributes a variety of food products that are sold through grocery store and supermarket outlets. At present, warehouse operation is performed manually that incur high labor costs and relatively low productivity. As a result, management decided to automate the warehouse operation. This report analysed present activity schedule and expected project completion time for the warehouse expansion project using PERT/CPM and Linear programming models. The analysis indicated present project completion time of 43 weeks that has only about 10% chance of meeting the 40-week completion time goal. Further, the project completion time should be shortened to38 weeks to achieve an 80% chance of 40-week completion time. Thus, recommendation made that activities B, F, J and K should be reduced (crashed) by 2, 1, 1, and 1 weeks, respectively to achieve planned project completion time of 38-weeks with the total crash cost of $2,250. Table of Contents Introduction (Problem Statement) 4 Assumptions & Approximations Made 5 Solution Approach/Computer Program Used 6 Analysis and Results 6 Project Network 6 Activity’s expected time and variance 6 Activity Schedule and Expected Project Completion Time 6 Chance of Completing the Project in 40-weeks or Less 7 Expected Project Completion Time for 80% Chance of Meeting the 40- week Deadline 8 Activity Crashing Decisions 9 Activity Crash Cost Range for Optimal Solution (Sensitivity Analysis) 14 Revised Activity Schedule 15 Conclusion and Recommendation 16 Introduction (Problem Statement) R. C. Coleman distributes a variety of food products that are sold through grocery store and supermarket outlets. Presently, Warehouse operation is performed manually that incur the high labor costs and relatively low productivity of hand order picking. Because of this, management has decided to automate the warehouse operation by installing a computer-controlled order-picking system, along with a conveyor system for moving goods from storage to the warehouse shipping area. The project manager, after consultation with members of the engineering staff and warehouse management personnel compiled a list of activities associated with the project. The list of activities along with the optimistic, most probable and pessimistic times (in weeks) is shown in table 1. Table 2 shows the crashed activity times along with activity’s normal and crashed costs. Table 1: Activity’s immediate predecessor and optimistic, most probable and pessimistic times Activity Description Immediate Predecessor Optimistic Time Most Probable Time Pessimistic Time A Determine equipment needs - 4 6 8 B Obtain vendor proposals - 6 8 16 C Select vendor A, B 2 4 6 D Order system C 8 10 24 E Design new warehouse layout C 7 10 13 F Design warehouse E 4 6 8 G Design computer interface C 4 6 20 H Interface computer D, F, G 4 6 8 I Install system D, F 4 6 14 J Train system operators H 3 4 5 K Test system I, J 2 4 6 Table 2: Normal and Crash Activity Data Activity Crashed Activity Time (weeks) Normal Cost ($) Crashed Cost ($) A 4 1,000 1,900 B 7 1,000 1,800 C 2 1,500 2,700 D 8 2,000 3,200 E 7 5,000 8,000 F 4 3,000 4,100 G 5 8,000 10,250 H 4 5,000 6,400 I 4 10,000 12,400 J 3 4,000 4,400 K 3 5,000 5,500 This report will present the activity schedule (with project network) and expected project completion time for the warehouse expansion project. Further, this report will consider following issues: 1. Whether a 40-week completion time established by the R. C. Coleman’s top management for the project can be achieved. 2. How much should the expected project completion time be shortened to achieve the goal of an 80 percent chance of completion within 40 weeks? 3. The activity crashing decisions and revised activity schedule for the warehouse expansion project for shortened project completion time. This report will use CPM/PERT and Linear programming models to analyse above issues. Assumptions & Approximations Made The recommendations inside of this report are all based on mathematical models. These models are idealizations of the real life and the resulting recommendations cannot be treated as being absolutely accurate. The type of model used in this report is a stochastic type of mathematical model, as we need to make some assumptions to solve. The mathematical models used are PERT/CPM model and linear programming model. The PERT/CPM can be used to plan, schedule, and control a wide variety of projects by developing PERT/CPM project network. PERT/CPM project network depicts the activities and their precedence relationships. First Assumption made is that the activity times follow a Beta distribution. Because the optimistic, most probable and pessimistic time for activity is given that is the three-time estimate approach. Activity’s mean completion time and variance is given by: and Where, a = the optimistic completion time estimate, b = the pessimistic completion time estimate m = the most likely (probable) completion time estimate Second assumption made is that the project completion time follows a normal probability distribution. Third assumption made is that the activity’s crash cost is uniformly distributed over the difference between normal and crashed time of activity. Solution Approach/Computer Program Used The solution is approached by developing PERT/CPM model and linear programming model, regarding the objective function and the correlating constraints. The calculations and presentations for the output are realized by using The Management Scientist v6.0 software. Analysis and Results Project Network Figure 1 shows the project network for the warehouse expansion project with predeceasing activities. Figure 1: Project network for the warehouse expansion project Activity’s expected time and variance Figure 2 shows the activity’s expected time and variance by using PERT/CPM model in The Management Scientist. Activity Schedule and Expected Project Completion Time Figure 3 shows the activity schedule by using PERT/CPM model in The Management Scientist. The project completion time equals the maximum of activities earliest finish time. The expected project completion time, E(T) is 43 weeks. The critical path of activities is the path with 0 slack times. The critical path activities are B-C-E-F-H-J-K. Figure 2: Expected times and variances for activities Figure 3: Activity schedule Chance of Completing the Project in 40-weeks or Less The variance of the critical path is The z score is {Using Excel Function =NORMSDIST(-1.2603)} There is about a 10.4% chance that a 40-week completion time established by the R. C. Coleman’s top management for the project can be achieved. Therefore, it is recommended that R. C. Coleman should consider crashing project activities so that they can achieve the 40-week completion time for the project. Expected Project Completion Time for 80% Chance of Meeting the 40- week Deadline Figure 4 shows the distribution of planned project completion time for 80% chance of meeting the desired 40-week completion time. Figure 4: Distribution of planned project completion time For 80% chance, Z = 0.8416. Therefore, The expected project completion time should be shortened to 38 weeks to achieve the goal of an 80% chance of completion within 40 weeks. Therefore, it is recommended that R.C. Coleman should crash activities to reduce the expected project completion time to 38 weeks. Activity Crashing Decisions This section will consider expected activity times as normal times and use a linear programming model based on expected times to make the crashing decisions. The maximum reduction in time and crash cost per week for the activity is calculated as below: Table 3 shows the maximum reduction in time and crash cost per week for the activity. Table 3: Maximum reduction in time and crash cost per week for activity Activity Normal Time Crash Time Normal Cost Crash Cost Maximum Reduction in Time Crash Cost per week A 6 4 1,000 1,900 2 450 B 9 7 1,000 1,800 2 400 C 4 2 1,500 2,700 2 600 D 12 8 2,000 3,200 4 300 E 10 7 5,000 8,000 3 1000 F 6 4 3,000 4,100 2 550 G 8 5 8,000 10,250 3 750 H 6 4 5,000 6,400 2 700 I 7 4 10,000 12,400 3 800 J 4 3 4,000 4,400 1 400 K 4 3 5,000 5,500 1 500 When crashing decisions are made, it is important to consider crash cost per week of the activity. It is desirable to crash those activity that have minimum crash cost per week. Linear programming model will be used to decide the activity(s) that should be crashed to achieve planned project completion time of 38-weeks. The decision variables are defined as follows: Let = the finish time for activity = the amount of time activity is crashed The activity should be crashed such that the total crash cost of crashed activity is minimum. Therefore, optimal function will be defined as Minimize: Subject To: 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) All , 0 (non-negativity) There are 22 decision variables and 27 constraints. Using the Management Scientist, the results for the above linear programming model is given in figure 5, 6, 7 and 8. Table 4 shows the optimal crashing decisions for the activity. Table 4: Optimal crashing decisions Crash Activity Weeks Cost B 2 800 F 1 550 J 1 400 K 1 500 Total 2,250 It is recommended that Activities B, F, J and K should be reduced by 2, 1, 1, 1 and weeks respectively to achieve planned project completion time of 38-weeks. The total crash cost is $2,250. Figure 5: Optimal solution (Part 1) Figure 6: Optimal solution (Part 2) Figure 7: Objective coefficient ranges Figure 8: Right hand side ranges Activity Crash Cost Range for Optimal Solution (Sensitivity Analysis) Table 5 shows the crash cost per week ranges (objective coefficient ranges) for the crashed activity for optimal solution. Table 5: Objective coefficient ranges for the crashed activity Activity Lower Limit Current Value Upper Limit B No lower Limit 400 550 F 500 550 600 J No lower Limit 400 550 K No lower Limit 500 550 The optimal solution of crashing of activities will not change for the above ranges (lower and upper limit). For example, for activity F crash cost range of $500 to $600, the solution will be same. Similarly, for Activity B crash cost range of $0 to $550, the solution will remain optimal. Revised Activity Schedule Figure 9 shows the expected time for the activity after crashing activities B, F, J and K. Figure 9: Activity’s expected time after crashing Using the Management Scientist, a revised activity schedule based on crashing decisions is shown in figure 10. Figure 10: Revised Activity Schedule The new project completion time is 38 weeks for the warehouse expansion project that is required to achieve the goal of an 80% chance of completion within 40 weeks. Conclusion and Recommendation In conclusion, the project completion time with given optimistic, most probable and pessimistic times (in weeks) of activity and immediate predecessor is 43 weeks. The critical path of the project is B-C-E-F-H-J-K. There is only about a 10.4% chance that a 40-week completion time established by the R. C. Coleman’s top management for the project can be achieved. The expected project completion time should be shortened to 38 weeks to achieve the goal of an 80% chance of completion within 40 week. The activities that should be crashed to achieve planned project completion time of 38-weeks are B, F, J and K by 2, 1, 1, and 1 weeks respectively. Based on the analysis, the main recommendations to R. C. Coleman’s project team are: R. C. Coleman should consider crashing project activities so that they can achieve the 40-week completion time for the project. R.C. Coleman should crash activities to reduce the expected project completion time to 38 weeks. Activities B, F, J and K should be reduced by 2, 1, 1, and 1 weeks respectively to achieve planned project completion time of 38-weeks with the total crash cost of $2,250. Read More