Project Scheduling by PERT/CPMReference Books: Anderson, Sweeney, and Williams, AN INTRODUCTION TO MANAGEMENT SCIENCE, QUANTITATIVE APPROACHES TO DECISION MAKING, 7th edition, West Publishing Company,1994 Hamdy A. Taha, OPERATIONS RESEARCH, AN INTRODUCTION, 5th edition, Maxwell Macmillan International, 1992 Daellenbach, George, McNickle, INTRODUCTION TO OPERATIONS RESEARCH TECNIQUES, 2nd edition, Allyn and Bacon. Inc, 1983 Lawrence Lapin, QUANTITATIVE METHODS for Business Decisions with Cases, 4th edition Harcourt Brace Jovanovich, Inc., 1988 T. A. Burley and G O’sullivan, OPERATIONAL RESEARCH, MacMillan Education Ltd., 1990 Lecture 1 1. Introduction A project defines a combination of interrelated activities that must be executed in a certain order before the entire task can be completed. An activity in a project is usually viewed as a job requiring time and resources for its completion. Project management has evolved as a field with the development of two analytical techniques for planning, scheduling, and controlling of projects. These are the project evaluation and review technique (PERT) and the critical path method(CPM). These techniques were developed by two groups almost simultaneously. CPM was developed by E. I. Du Pont de Nemours & Company as an application to construction projects and was later extended to a more advanced status by Mauchly Associates. PERT was developed by the U.S. Navy by a consulting firm for scheduling the research and development activities for the Polaris missile program. Although PERT and CPM were developed independently, they are similar in principle. Today, PERT and CPM actually comprise one technique and the differences, if any, are only historical. Consequently, both technique are referred to as “project scheduling” techniques. Project scheduling by PERT-CPM consists of three basic phases: Planning • breaking down the project into distinct activities; • determining the time estimates for these activities; 1 • constructing a network diagram with each arc representing the activity; Scheduling • constructing a time chart showing the start and the finish times for each activity as well as its relationship to other activities in the project; • pinpointing the critical (in view of time) activities that require special attention if the project is to be completed on time. • Showing the amount of slack (or float) times for the non-critical activities; Controlling • Using the network diagram and the time chart for making periodic progress reports; • updating the network. 2. Network Diagram Representations and Network Construction The network diagram represents the interdependencies and precedence relationships among the activities of the project. An arrow is commonly used to represent an activity, with its head indicating the direction of progress in the project. An event represents a point in time that signifies the completion of some activities and the beginning of new ones. The following diagram shows an example, where activities (1, 3) and (2, 3) must be completed before activity (3, 4) can start. Head event 1 3 2 4 Tail Rules for constructing a network diagram: 1. Each activity is represented by one and only one arrow in the network; 2. No two activities can be identified by the same head and tail events (a dummy activity is introduced in such situations); A A B B D In this case, D is the dummy activity. 2 C. Develop a list of sources for financing. Analyse the financial records of Tiny Ltd. Construct the network diagram. Tiny Ltd. To ensure the correct precedence relationship in the network diagram.). The whole procedure involves four activities: A.3. Submit a proposal to a lending institution. Develop a business plan (sales projections. etc. The precedence relationship of these four activities is described as in the Table below. D. B. the following questions must be answered as every activity is added to the network: • What activities must be completed immediately before this activity can start? • What activities must follow this activity? • What activities must occur concurrently with this activity? Example 1: The Galaxy plc is to buy a small business. C 1 B A 3 C D 4 2 3 . Activity A B C D Immediate Predecessor B A. cash flow projections. which is not true according to the specification. which has C as its immediate predecessor. E. The solution is to add a dummy activity between C and node 3 in order to add E correctly. 3 A 1 B C 2 E D 4 5 The first completed 7-activity network is shown as follows: 4 . F Activity A B C D E F G For activities A. the network portion is as follows: 3 A 1 B 2 D 4 C When activity E. This is shown below. is to be added. C. it has to be after A as well in this arrangement. Immediate Predecessor B A. C C C D. B. we come cross a problem because activities A and C both end at node 3. If activity E is to happen after C. and D.Example 2: Construct the network based on the Table of information . 3 A 1 B C 2 E F D G 4 5 6 It is seen that activities E and F share the same head and tail events. A non-critical activity is an activity that has time to spare (known as slack or float time) within the entire project. We will discuss the determination of the critical path through the following example. dummy activities should be introduced. Determination of the Critical Path An activity is said to be critical if a delay in its start will cause a delay in the completion date of the entire project. 3 A E F B C 2 7 D G 1 4 5 6 The above network describes correctly the relationships among the 7 activities. which is in conflict with Rule 2. In such situations. He hopes to add 8 to 10 new business or tenants to the shopping complex. A critical path is a sequence of connected critical activities that leads from the source node to the sink node. The specific activities that make up the 5 . Example The owner of a shopping centre is considering modernising and expanding the current 32-business shopping complex. Lecture 2 3. H Completion Time (weeks) 5 6 4 3 1 4 14 12 2 Total 51 We are now asked to answer the following questions: 1) What is the total completion time of the project? 2) What are the scheduled start and completion time for each activity? 3) Which activities are critical and must be completed exactly as scheduled in order to keep the project on schedule? 4) How long can the non-critical activities to be delayed before they cause a delay in the completion time for the project? To solve the problem. Activity Description Prepare architectural drawings Identify potential new tenants Develop prospectus for tenants Select contractor Prepare building permits Obtain approval for building permits Perform construction Finalise contracts with tenants Tenants move in Activity A B C D E F G H I Immediate Predecessor A A A E D. together with information on immediate predecessor and completion time. C G. F B. D 2 E A 5 F G 3 1 4 4 14 I 5 C 4 6 1 B H 2 7 6 3 12 Completion time for activity H 6 .expansion project. are listed in the following table. we need first construct the network according to the problem specification. Starting at the network’s source node (node 1) we will have to compute the earliest start time and the earliest finish time for each activity in the network. Let’s assume that ES = earliest start time for a particular activity EF = earliest finish time for a particular activity t = expected completion time for the activity The earliest finish time can be calculated by the following expression for a given activity: EF = ES + t For example. Using activity A as an example. the following rule determines the earliest start time for activities. 5] EF 2 1 5 Expected completion time Since activities leaving a node cannot be started until all immediate preceding activities have been completed. Earliest Start Time Rule The earliest start time for an activity leaving a particular node is equal to the largest of the earliest finish time for all activities entering the node. the earliest start and finish times for each activity are written onto the network. we have ES Activity A [0. for activity A ES = 0 and t = 5. which now looks as follows: 7 . We will write ES and EF directly on the network in brackets. thus the earliest finish time for activity A is EF = 0+5 = 5. Using this rule. This process gives the earliest completion time of the entire project.6] 4 [9.26] 5 4 [5.8] [6.10] F 2 [0.9] B [0.21] H 6 1 6 2 7 12 3 As has demonstrated. computing a latest start time and latest finish time for each activity. The PERT/CPM network with both [ES. Latest Finish Time Rule The latest finish time for an activity entering a particular node is equal to the smallest of the latest start times for all activities leaving the node. but we will put them within a pair of round brackets. 8 . we trace back through the network.D [5.6] 4 14 [10. In the case of the shopping centre. Starting at the sink node (node 7)and using a latest finish time of 26 weeks for activity I. Let LS = latest start time for a particular activity LF = latest finish time for a particular activity The latest start time is given by the following expression: LS = LF . which is the earliest finish time for the last activity. proceeding in a forward pass through the network. We now continue the algorithm for finding the critical path by making a backward pass calculation. we can establish the earliest start time and then the earliest finish time for each activity. The following rule determines the latest finish time for any activity in the network.5] A E 5 G 3 1 C [5.t The latest start and finish times are also to be displayed on the network. the total time required for project completion is 26 weeks. LF) for the example is shown below. EF] and (LS.24] I [24. Slack is defined as the length of time an activity can be delayed without affecting the total time required to complete the project.10) [6.10) [10. 2) What are the scheduled start and completion time for each activity? See the above Table.24) I [24.5) E 5 14 G 3 1 [5.9] 4 4 (6.26] (24.6) C 4 (8.6] (6. Activity A B C D E F G H I ES 0 0 5 5 5 6 10 9 24 LS 0 6 8 7 5 6 10 12 14 EF 5 6 9 8 6 10 24 21 26 LF 5 12 12 10 6 10 24 24 26 Slack 0 6 3 2 0 0 0 3 0 Critical Path? Yes Yes Yes Yes Yes We can now answer the questions we were asked before: 1) What is the total completion time of the project? The project can be completed in 26 weeks if the individual activities are completed on schedule.8] (7.12) 12 3 From the above diagram.24] (10.21] H (12.10] F 2 [0. 9 . The amount of slack is computed as follows: Slack = LS .ES = LF .24) 5 6 [9. we find the amount of slack or free time associated with each of the activities.26) 1 B 2 7 6 [0.6] (5.5] A A (0. we arrive at the following table of information (the project schedule) for the shopping centre project.12) [5. According to the finished PERT/CPM network.D [5.EF Activities with zero slack are the critical path activities. 10 . Step 3 Estimate the completion time for each activity. Now. determine the earliest start and finish times for each activity by making a forward pass through the network. Step 6 Using the project completion time identified in Step 5 as the latest finish time for the last activity. and I are the critical path activities. The earliest finish time for the last activity in the project identifies the total time required to complete the project. F. This. Lecture 3 4. Step 5 Using the network and the activity time estimates. the critical activities. make a backward pass through the network to identify the latest start and finish times for each activity. obviously. It is evident that • • it is the critical paths that determine the project completion time. Step 2 Determine the immediate predecessor activities for each activity listed in the project. these are the critical path activities.3) Which activities are critical and must be completed exactly as scheduled in order to keep the project on schedule? A. E. will give the project manager a clear picture for his control over the project. Step 7 Use the difference between the latest start time and the earliest start time for each activity to identify the slack time available for the activity. G. 4) How long can the non-critical activities to be delayed before they cause a delay in the completion time for the project? Table above shows the slack time associated with each activity. Step 9 Use the information from Steps 5&6 to develop the activity schedule for the project. changing time of the non-critical activities within the permissible range will not affect the project completion time. The project schedule is based on the given cost and finish time of the individual activities. Step 1 Develop a list of activities that make up the project. Step 8 Find the activities with zero slack. let us summarise the PERT/CPM critical path procedure. Step 4 Draw a network depicting the activities and immediate predecessors listed in Steps 1&2. Consideration of Time-cost Trade-offs From the shopping centre example. and the slack times of the non-critical activities. it is seen that the PERT/CPM can answer questions such as the total project completion time. but changing time of the critical activities may cause the project completion time to change. sometimes.In practice. Usually.9] D (9. These are the considerations of time-cost trade-offs. The complication which arises is that as the critical activities are sped up more and more. We are asked to: 1) calculate the normal completion of the project. Activity A (1-2) B (1-3) C (1-4) D (2-4) E (2-5) F (3-6) G (4-6) H (5-6) Normal completion time (days) 6 8 5 3 5 12 8 6 Shortest completion time (days) 4 4 3 3 3 8 6 6 Cost of reduction per day (£) 80 90 30 40 200 50 - We first set-up the network according to the description of the project. • the shortest time and the associated cost. and the critical path. 4. The overhead cost of general site activities is £160/day.17] H 6 (14. 3 [0. Example: A project consisting of 8 activities are described in the following table. EF. LF times and the critical activities.12) [0. its cost. to avoid penalty clauses in contracts or. or completing a project in minimum time.20 ) 4 G [9.9) 11 5 . These are shown in the following network and table.12) 3 E [6.6] 2 [6.11] 5 (9.1 Completion of projects at minimum cost By adding more resources.14) F 12 [8. we need to know the possible reduction in time and extra cost for reduction per unit time for each activity. we sometimes demand more than this. To obtain the minimum cost or the minimum time.20) [11.8] B 1 A 6 8 (0.20) 6 (3. we establish the ES. LS. other activities also become critical. We may be interested in completing a project at minimum cost.8) C [0.5] 5 (7.17] 8 (12. 2) calculate and plot on a graph paper the cost/time function for the project and state: • the minimum cost and the associated time. the purpose of speeding up is to save money on project overheads.20] (8. to earn bonuses for early completion. Then using the PERT/CPM scheduling technique discussed earlier. The cost for completion of these 8 activities is £5800 excluding the site overhead. We will discuss the algorithm through the following example. a project may be sped up. it is necessary to reduce time along all the critical paths simultaneously.9] D (6.5] 5 (4. According to the project description. 3 [0.4] B 1 A 5 4 (0. The rule is to speed up firstly the critical activity that cost the least to do so.5) C [0. The cost of completing the project at normal speed is £5800 + 20 × £160 = £9000 Now.5] 5 (3. Obviously.10) (5.17] 8 (9.8] D 3 12 E [5. As indicated in the following diagram.8) [0. This action further reduces the project completion time into 16 days.17 ) G [9. The cheapest way this can be done in this example is to save one day on activities A and B the same time.16] 8 (8.5) . The amount of time to speed up is determined based on (1) the reduction should reduce the project completion time the most.5] B 1 A 6 5 (0.17) [11. This reduces the completion time to 17 days. the activity B can be shortened by 8-4=4 days.10] 5 (5.16) 6 (0.16] H 5 6 (10.17] (5.6] 2 [6. The critical paths remain the same.6) The new cost accordingly is now: £9000 . the activity to speed up is B.11) 5 F 12 [5. all activities except C become critical because of this.17) 6 4 (0.16] (4.8) F 12 [4.4) C [0. 3 [0. Let us speed up 3 days for B. and (2) the reduction should cause as many activities to become critical as possible. which costs £90 for speeding up one day. we wish to speed up the project so that the project will cost the least.5] 2 [5.9) [0.Activity A B C D E F G H ES 0 0 0 6 6 8 9 11 EF 6 8 5 9 11 20 17 17 LS 3 0 7 9 9 8 12 14 LF 9 8 5 12 14 20 20 20 Slack 3 0 7 3 3 0 3 3 Critical? Yes Yes The above shows that the normal completion time is 20 days and the critical activities are B(1-3) and F(3-6).17] H 6 (11.3 × £160 + 3 × £90 = £8790 In order to achieve any further saving.9) 3 E [6.16) [10.16 ) 4 G [8.11] 5 (6. 15] (4.4] B 1 A 4 4 (0.13] H 5 6 (7.7) [0.9) (0.15) 6 4 (4. F.7) [0. to reduce time further on all the critical paths.7) (0.4] B 1 A 4 4 (0.15 ) G [7.4) C [0.4) C [0. we need to consider activities F and A which have 4 days and 1 day. to spare. 3 days.7] D 9 F [4.13] (4.15] H 5 6 (9. respectively. Reduction of two days on these activities makes the total projection time to 13 days. and G (2 days.15] 8 (7.5] 5 (2.4) 13 . 3 [0. and 2 days available respectively).5] 5 (2.7] 3 (4.7] D F 11 [4.The total cost under this circumstance is £8790 .4] 2 [4.1 × £160 + 1 × £80 + 1 × £200 = £8920 The project can still be sped up by reducing time on activities E. Activity C still has 2 days slack time while all the others are critical.7) 3 E [4. We can only reduce one day on both of these and the total completion time is now reduced to 15 days.9] 5 (4.4) The total cost is now £8800 .13) 6 (4. 3 [0.1 × £160 + 1 × £90 + 1 × £80 = £8800 Now.13) [7.7) 3 E [4.4] 2 [4.13] 6 (7.13 ) 4 G [7.15) [9. The total cost in this case is £8920 . we summarise in the following table the completion times and costs of the project. For the purpose of plotting the required cost/time graph. This concludes the example.2 × £160 + 2 × (£200 + £50 + £40) = £9180 13 days is the minimum completion time for the project because no further time reduction is available on the critical path 1-2-5-6.2 Completion of projects in minimum time In some circumstances the primary interest when completing a project is to use the least possible time even if this does not mean the least possible cost. and that the minimum possible completion time is 13 days costing £9180. One example for 14 . Completion time (days) 20 17 16 15 13 Cost (£) 9000 8790 8800 8920 9180 Completion time VS Cost 9200 9100 9000 (£) 8900 8800 8700 8600 8500 20 17 16 Days 15 13 Cost (£) It is evident that the minimum cost for completing the project is £8800 in 17 days. Lecture 4 4. Possible reduction time (days) 1 2 4 1 3 2 1 3 2 2 3 2 Extra cost for reduction (£/day) 300 200 700 400 200 200 400 300 600 300 100 500 Activity A(1-2) B(1-3) C(1-4) D(2-3) E(2-5) F(3-4) G(4-5) H(4-6) I(4-7) J(5-6) K(6-8) L(7-8) Completion time (days) 5 8 15 4 12 6 7 11 10 8 9 10 Cost of activity (£1.this is the situation when the equipment being used for the project is urgently needed for more profitable work else where.e. if it is the minimum time that is of interest. Example: The data shown in the following table relates to a contract being undertaken. i. and the cost. like the method used in the last example. You are required to: (1) calculate and state the time for completion on a normal basis. However. One way of finding the minimum time for completion of a project is to start with the normal completion network and gradually make reductions in critical activities until minimum time is reached. Note: To crash an activity is to use the shortest possible time available for the activity. There are also site costs of £500 per day.000) 6 10 17 5 15 8 9 13 12 14 25 13 Answer: 15 . (2) calculate and state the critical path on this basis. then there is another and more efficient way of proceeding.. (3) calculate and state the cost of completion in the shortest possible time. 1) Crash every activity and look at the resulting network • this will certainly give us minimum completion time • but it may be a highly wasteful way of achieving the minimum time 2) Consider the activities which are not critical and allow the most expensive of these to slow down as much as possible without the duration of the project being increased above the desired minimum. .4] A 1 4 (4.29] 6 (23.7] [4.39] 9 (30. (2) The above network shows that there are two critical paths.5) B 8 [0.500 = £166. i.11] (0.29) [0.23) 7 L [19.17] [7.11) B 6 [11. using the shortest completion time for each activity.25] (19.22) I 10 H J [22.9) 3 (9.17] (10. the minimum completion time of the project is found to be 29 days.11] 4 F G 6 (11..23) 6 K [23.21) H J [17.9] [5.17) 5 [11. we first reconstruct the PERT/CPM network by crashing all the activities. the project will require 39 days to complete under the normal situation. A-D-F-G-J-K.6] (1.15) [15.e.39) 8 [0.35] (29. 2 4 (0.30] 8 (22.7) C 11 D 9 E [4. i.30) [15.39) [0. Money can be saved by allowing the slowing down of those non-critical 16 . and C-G-J-K.9) C 15 D E 12 [5. The cost on the normal basis is Σ£(All costs) + 39 × £500 = £147.17) I 8 [11.5] A 5 (5.(1) According to the above table.22] [9.7) 3 (7.19] 8 (15.4) [0.15) 1 [25.500 (3) To find the minimum completion time.11) 4 By crashing all the activities. the PERT/CPM network can be generated as follows.e. 2 3 (0.23] 6 (17.8] (1.26] 11 (19.19] (13.22) 5 [15.29) 4 As it is indicated.15] 6 F G 7 (15.27] 8 (21.000 + £19.15] (0.13] (8.30) 7 L 10 6 K [30.29) 8 [0. we first let activity I to slow down to its normal completion time of 10 days.29] 6 (23.29) [0.29) [0.7) 3 [11.4] A 1 4 (4.23] 6 (17.11) [21.29) 8 [0.11] (0. It should be mentioned that the slowing down of the non-critical activities should not increase the minimum completion time of the project. Secondly. we let activity H take its full 11 days. which leads to the following diagram.23) 7 L [21.6] (1.17) 5 [11.17] [7.6] (1.4) B 6 [0.29] 8 (21.13] (8. 2 3 (0.21) H J [17.19] 8 (15.13] (8.29] (21.7) C 11 D 9 E [4.21) 4 17 .17) 5 [11.activities which are most expensive to speed up.17) I 10 [11. So.7) C 11 (7. The follow table lists all the non-critical activities and their costs to speed up.11) 4 The change of the completion time of activity I from 8 days to 10 days makes activities I and L critical without increasing the minimum project completion time.17] [7.21] (11.21] (11.7) 3 (7. This will result in the following network diagram.7] [4.7] E [4.11) [11.23) [11.4] A 1 4 (4. 2 3 (0.11] 4 F G 6 (11.23) 6 K [23.23) 7 L 8 6 K [23.23] 6 (17. Non-critical activities B E H I L Cost to speed up (£/day) 200 200 300 600 500 Obviously.29) 8 [0.11] (0.17) I 10 H J [17.11] F G 6 (11.11) 4 D 9 [4. the most expensive non-critical activity to speed up is I.29] 6 (23.4) B 6 [0.22] 11 (12. 17) 5 [11.16] (5.11) [21.17) I 10 H J [17.11] (0.17] [7.23) 6 K [23.7) 3 (7. the least possible cost of completing in 29 days is: Σ£(All costs) + 29 × £500 + 1 × £300 + 1 × £200 + 4 × £700 + 1 × £400 + 2 × £200 + 1 × £400 + 2 × £300 + 3 × £100 18 . Hence.7) C 11 D E 12 [4. 2 3 (0.16] (5.21] (11.11] 4 F G 6 (11. 2 3 (0.23] 6 (17.Thirdly.11) [11.7] [4.22] 11 (12.29) 8 [0. leading to the diagram below.21] (11.7) 3 (7.23) 7 L [21. we allow activity E to take its full 12 days.4] A 4 (4.29] 6 (23.23] 6 (17.29) [0.7] [4.21) 4 No further savings are possible as the three non-critical activities are all now at normal duration.11) 1 B 6 [11.29] (21.29) 8 [0.7) C 11 D E 12 [4.4) B 7 [0.17) 5 [11.4) [0.21) 4 Fourthly.23) [11.29] 6 (23. we allow activity B to take 7 days.17) I 10 H J [17.22] 11 (12.23) 7 L 8 6 K [23.29) [0.11) [11. which makes B critical as indicated in the following diagram.11] (0.29] 8 (21.11] 4 F G 6 (11.6] (1.4] A 1 4 (4.17] [7.7] (0. 400 = £167.500 + 2 × £500 + £6.= £147.000 + £14.900 19 . E Activity H I G 2.Exercise 2 Scheduling with PERT/CPM 1. Assume that the project in problem 2 has the following activity times: 20 . Add the dummy activities that will eliminate the problem that the activities have the same starting and ending nodes. B D. 2 A B 1 C 4 H I 5 G 6 J 7 D E 3 F a. C B. 3. Add dummy activities that will satisfy the following immediate predecessor requirements: Immediate predecessor B. Consider the PERT/CPM network shown below. E The project is completed when activities F and G are both complete. b. Construct a PERT/CPM network for a project having the following activities: Activities A B C D E F G Immediate predecessor A A C. C D. B C. Activities A B C D E F G Time (months) 4 6 2 6 3 3 5 a. C F. Can activity C be delayed without delaying the entire project? If so. A project involving the installation of a computer system consists of eight activities. how many weeks? d. The immediate predecessor and activity times are shown below. 4. Identify the critical path. How long it will take to complete the project? c. What is the schedule for activity E? 5. Consider the following project network (the times shown are in weeks): C 2 7 5 A 5 1 6 D H 8 F 3 G 4 6 B 3 3 10 E 7 a. The project must be completed in 1½ years. b. Do you anticipate difficulty in meeting the deadline? Explain. G Time (weeks) 3 6 2 5 4 3 9 3 a. C D E B. 21 . Activity A B C D E F G H Immediate predecessor A B. b. how many weeks? e. Draw the PERT/CPM network for this project. Can activity D be delayed without delaying the entire project? If so. Find the critical path. D 2 4 A 9 1 6 3 H 6 C E 0 I 6 7 3 Activity time in weeks B 6 3 G 2 F 5 3 a. Does it appear reasonable that construction of the athletic complex could begin 1 year after the decision to begin the project with the site survey and initial design plans? What is the expected completion time for the project? 7. What is the expected project completion time? 6. 22 . G Time (weeks) 6 8 12 4 6 15 12 8 Activity A B C D E F G H a. Develop the activity schedule for the project. Show the activity schedule for this project. Immediate predecessor A. constructing a shelter house. Piccadily College is considering building a new multipurpose athletic complex on campus. The activities that would have to be undertaken before beginning constructing are shown below. classrooms. E E F. purchasing picnic equipment. Identify the critical path. B C C D. Project development activities include cleaning playground and picnic areas. What are the critical path activities? c. Hamilton Country Parks is planning to develop a new park and recreational area on a recently purchased 100-acre tract. scheduling. What is the critical path for this network? b. constructing road. d. Description Survey building site Develop initial design Obtain board approval Select architect Establish budget Finalise design Obtain financing Hire contractor Develop a PERT/CPM network for this project. b. and intramural facilities. The complex would provide a new gymnasium for intercollegiate basketball games. The PERT/CPM network shown below is being used in the planning. expanded office space. and so on. and controlling of this project. c.b. The park commissioner would like to open the park to the public within 6 months from the time the work on the project is started. 8. Does this opening date appear feasible? Explain. Find out the minimum project completion time using the crashing method. What is the minimum cost associated with the crashed project completion time? 23 . What is the total project cost using the normal times? c.c. Find the critical path and the expected project completion time on the normal basis. Consider the project network with activity times shown in days: 2 A 3 1 C 5 E 4 6 6 B 2 3 D 2 5 F 5 G 2 The crash data for this project are as follows: Time (days) Normal Crash 3 2 2 1 5 3 5 3 6 4 2 1 2 1 Total Cost ($) Normal Crash 800 1400 1200 1900 2000 2800 1500 2300 1800 2800 600 1000 500 1000 Activity A B C D E F G a. b. d. 9. What are the critical path activities. has developed a proposal for introducing a new computerised office system that will improve word processing and interoffice communications for a particular company. What is the added project cost to meet the 6-month completion time? 24 . Information about the activities is shown below. c. b. Develop an activity schedule for the project. e. Contained in the proposal is a list of activities that must be accomplished to complete the new office system project. and attempt to make the crashing decisions by inspection. and what is the expected project completion time? d. Immediate predecessor A B A D C. Inc. f. E Time (weeks) Normal Crash 10 8 8 6 10 7 7 6 10 8 3 3 Cost ($000’s) Normal Crash 30 70 120 150 100 160 40 50 50 75 60 60 Activity A B C D E F Description Plan needs Order equipment Install equipment Setup training lab Conduct training Test system a. Develop an activity schedule for the crashed project. What crashing decisions would be recommended to meet the desired completion time at the least possible cost? Work through the network. Show the network for the project.. Office Automation. Assume that the company wishes to complete the project in 6 months or 26 weeks.