Project Management and. Scheduling CHAPTER CONTENTS

Transcription

1 6 Proect Management and Scheduling HAPTER ONTENTS 6.1 Introduction 6.2 Planning the Proect 6.3 Executing the Proect Monitor ontrol losing 6.4 Proect Scheduling 6.5 ritical Path Method PM alculations AON Network omutations 6.6 PERT Methodology 6.7 Time / ost Trade-off Tyes of cost rashing of Proect Time-ost Trade-off Assumtions Additional onsideration

2 ha. 6 / Proect Management and Scheduling Proect management is a set of rinciles, methods, and techniques that eole use to effectively lan and control roect work. It establishes a sound basis for effective lanning, scheduling, resourcing, decision making, controlling, and relanning. The obective of roect management is to ensure that roects meet agreed goals of time, cost, and scoe. Today, modern roect management has emerged as a remier solution in business oerations. Large and small organizations recognize that a structured aroach to lanning and controlling roects is a necessary core cometency for success. 6.1 INTRODUTION Proect work and traditional functional work differ in significant ways. Functional work is routine, ongoing work. A manager is assigned to the secific function and rovides worker training and suervision. In contrast, a roect is a temorary endeavor undertaken to create a unique roduct or service. A roect manager is resonsible for the aroved obectives of a roect, such as budget, schedule, and scoe. The need for roect management is aarent in the world today as seed, quality, and cost control are becoming increasingly imortant. Imlementing a roect management system requires a long-term commitment and management suort. It is imortant to understand how your organization is structured so you can decide how to fit roect management techniques into it. Organizational structures tyically san the sectrum from functional to roect, with a variety of matrix structures in between. A functional organization is a hierarchy in which eole are groued into functional divisions, such as marketing or roduction. Each emloyee has one clear suerior. In a roect organization, roects are centralized in a searate division of skilled roect managers that serves the roect management needs of all divisions of the comany. This is often referred to as a roect office. Matrix organizations are a blend of functional and roect organizations. A weak matrix has many of the characteristics of a functional organization and the roect manager role is more that of a coordinator or exediter with limited authority. A strong matrix organization has many of the characteristics of a roect organization, with a full-time roect manager who has significant authority and a roect administrative staff. In a matrix organization, the roect team has a dual reorting role to a roect manager, coordinator, or exediter (who rovides roect management skills) and a functional manager (who rovides technical and functional skills). In a strong matrix organizational structure, the roect manager has more ower than the functional manager. In a weak matrix structure, the balance of ower leans toward the functional manager. It is imortant to set u a formal lanning and control system that is flexible enough to oerate in the real world, but still rigorous enough to rovide control. A Algorithms for Sequencing & Scheduling 6. 2

3 ha. 6 / Proect Management and Scheduling roect management system must allow for adustments to the lan as needed throughout the roect s life. The system hels you define the roblem or oortunity, establish roect obectives, develo the roect lan, begin roect work, monitor and control the work, and then close the roect. 6.2 PLANNING THE PROJET Some eole ut a minimum of effort into lanning. They argue that since things invariably change during the life of the roect, it is a waste of effort to make extensive u-front lans. The average organization sends only 5 ercent of the total roect effort on lanning. More successful organizations send u to 45 ercent. A good rule of thumb is to send 25 ercent of the roect effort in concet and develoment and 75 ercent in imlementation and termination. Although it is true that factors might be introduced during the life of the roect that necessitate minor or maor adustments to the lan, it is still imortant to have a solid lan in lace. Without one, a roect becomes even more chaotic in the face of change. If lans are made using roect management software, it is easy to make adustments to the lan as needed. In an ideal world, a roect would be lanned and aroved, and then work would start and be comleted according to the lan. In actual ractice, however, you might have to adust the lan throughout the life of the roect. Therefore, any good lanning and control system must be flexible enough to oerate in the real world, and yet be rigorous enough to rovide control. Some roects are managed in ieces. Because of time constraints or other factors, the roect manager might have to develo a lan for only art of the roect, get it aroved, and begin that ortion while other arts of the roect are still in the lanning stage. Often, lanning continues to some extent throughout the life of the roect. Recognizing this reality, the successful roect manager establishes a roect management system that allows for adustments to the lan as needed. Figure 6.1 shows how a roect management system allows a roect to react to changing conditions. The key stes in lanning are as follows: Define the roblem or oortunity that this roect addresses. Establish roect obectives in terms of time, cost, and scoe. Perform roect reviews to ensure the roect is needed, feasible, and ractical. Define the work (activities) that must be done to comlete the roect. Estimate the cost and time needed to accomlish each activity. Algorithms for Sequencing & Scheduling 6. 3

4 ha. 6 / Proect Management and Scheduling 6.3 EXEUTING THE PROJET Sequence the activities into a logical order, considering the deendencies between activities. alculate the critical ath to determine the longest sequence of activities. Schedule the activities by alying calendar dates. Preare resource lans by assigning secific ersonnel and equiment to each activity. Preare budget lans to determine what funds are needed at what times. Plan for risk to be ready to resond to events that may effect the roect for better or worse. Get arovals and comile a formal roect lan. When all lans are in lace, aroved, and communicated to roect ersonnel, roect work can begin Monitor As roect work rogresses, the roect manager gather status information and comare it to the lan to determine variances. Deviations from the lan are then analyzed to determine if corrective action should be taken ontrol When necessary, the roect manager takes corrective action to get the roect back on track. Some deviations might require re-sequencing activities, rescheduling, re-budgeting, or reallocating resources. Larger deviations can necessitate renegotiating the basic roect obectives of cost, time, and scoe. In some cases, the situation might be serious enough to warrant readdressing the roblem or oortunity to determine if it has been identified correctly and if the organization has the resources, exertise, and commitment necessary to handle it. Planning, monitoring, and controlling are not one-time events. They continue throughout the life of the roect to refine and adust to current conditions (see Figure 6.1) losing A good roect management methodology includes formal stes to close the roect. The urose of roect closure is to verify that all work has been accomlished as agreed and that the client or customer accets the final roduct. Algorithms for Sequencing & Scheduling 6. 4

5 ha. 6 / Proect Management and Scheduling 6.4 PROJET SHEDULING Figure 6.1 Proect Management System In terms of classical scheduling theory, roect scheduling is concerned with execution of m obs using infinite number of machines with recedence constraints. However, scoe of roect scheduling is highly exanded and alies to vast variety of activities in real world. It includes every shere of disciline not merely manufacturing. The term roect connotes a synchronized and well coordinated effort by a multitude of inuts (men, machines, money) to achieve a well defined target in a secific amount of time while living within the resources. Building of a hosital, construction of suer highway, installation of etroleum refining lant, laying of a gas Algorithms for Sequencing & Scheduling 6. 5

6 ha. 6 / Proect Management and Scheduling ie line from one continent to another continent and develoment of comuter network on a university camus are some examles of term roect. Tyically every roect is unique as it comrises distinct set of activities (obs) with secific time and resource constraints. The tasks inter-relationshi lays a decisive role in roect scheduling. If resources constraint is not an issue, it is termed as unconstrained roect scheduling. In this scheduling scenario, scheduling decision roblem requires these two basic arameters as inut; Processing times ( durations) of the activities Precedence relationshi of the activities in the roect However, activities do require resources for their execution. Often the same resources are required by different activities at the same time resulting in resource-use conflict situation. Scheduling methodology has to be modified in such scenarios. A class of roblems called Resource-constrained scheduling has emerged over years to tackle these situations. PM and PERT has gained wide sread use to solve unconstrained roect scheduling roblems. In this chater, we resent these two methodologies (PM & PERT) to deal with unconstrained roect scheduling. In next chater, detailed account of Resource-constrained scheduling will be resented. 6.5 RITIAL PATH METHOD (PM) ritical Path Method (PM) is a roect management technique, which has been created out of the need of industrial and military establishments to lan, schedule and control comlex roects. PM was the discovery of M. R. Walker of E.I.Du Pont de Nemours & o. and J. E. Kelly of Remington Rand, circa The comutation was designed for the UNIVA-I comuter. The first test was made in 1958, when PM was alied to the construction of a new chemical lant. In March 1959, the method was alied to maintenance shut-down at the Du Pont works in Louisville, Kentucky. PM heled the comany to reduce unroductive time from 125 to 93 hours. PM rovides an integrated frame work for lanning, scheduling and control of roect management. The scheduling of a roect includes answers to imortant questions, like; How long will the entire roect take to be comleted? What are the risks involved? Which are the critical activities or tasks in the roect which could delay the entire roect if they were not comleted on time? Is the roect on schedule, behind schedule or ahead of schedule? If the roect has to be finished earlier than lanned, what is the best way to do this at the least cost? Algorithms for Sequencing & Scheduling 6. 6

7 ha. 6 / Proect Management and Scheduling A roect is defined by a set of distinguishable, indivisible and distinct set of actions called activities. The activities have deendency relationshi among themselves. Tyically, an activity will have some activities receding it and some activities following it. Then each activity will require time for its execution and incur costs. In addition, an activity would use secific tyes of resources (man ower, equiment) for its comletion. Five useful questions to ask when collecting data about activities are; Is this a Start Activity? Is this a Finish Activity? What Activity Precedes this? What Activity Follows this? What Activity is oncurrent with this? Some activities are serially linked. The second activity can begin only after the first activity is comleted. In certain cases activities are concurrent, because they are indeendent of each other and can start simultaneously. This is esecially the case in organizations which have suervisory resources so that work can be delegated to various deartments which will be resonsible for the activities and their comletion as lanned. The information collected about activities for roect lanning and scheduling is well deicted by constructing recedence network diagram. The network diagram uses nodes and arcs to ortray information about the activities. Two tyes of recedence network diagrams are constructed for resenting roect activities data. Activities-on-arcs (AOA) network diagrams use directed arcs to resent an activity. Nodes reresent start and terminal occurrence of various roect activities. Besides reresenting activities, the arcs also contain information about recedence relationshis and duration of activities. Figure 6.2 Activity-On-Arc Precedence Network Diagram Algorithms for Sequencing & Scheduling 6. 7

8 ha. 6 / Proect Management and Scheduling The recedence network diagram in Figure 6.2 resents 11-activity AOA roect network with activities and their attributes as rovided in Table 6.1 Table 6.1 Proect network data in AOA and AON format Sr. No. Activity Duration Immediate Predecessor 1 (1, 2) (1, 3) (2, 4) 8 (1, 2) 4 (3, 4) 3 (1, 3) 5 (2, 5) 4 (1, 2) 6 (3, 6) 2 (1, 3) 7 (4, 5) 2 (2, 4), (3, 4) 8 (4, 7) 4 (2, 4), (3, 4) 9 (4, 6) 1 (2, 4), (3, 4) 10 (5, 7) 3 (2, 5), (4, 5) 11 (6, 7) 6 (3, 6), (4, 6) Activity-on-Node (AON) recedence network diagram is the second way of resenting roect data. The nodes resent activities. Directed arcs resent recedence relationshi which oin the nodes. The duration of activities is written inside or above the node. The data in Table is converted to Activity-on-node (AON) network format from activity-on-arc (AOA) network format and is shown in Table 6.2. AOA Format Table 6.2 Proect network data in AOA and AON format AON Format Duration Immediate Predecessor For AOA Format Immediate Predecessor For AON Format ( 1,2 ) A ( 1, 3 ) B ( 2, 5 ) 4 ( 1, 2 ) A ( 2, 4) D 8 ( 1, 2 ) A ( 3, 4 ) E 3 ( 1, 3 ) B ( 3, 6 ) F 2 ( 1, 3 ) B ( 4, 5 ) G 2 ( 2, 4 ), ( 3, 4 ) D, E ( 4, 7 ) H 4 ( 2, 4 ), ( 3, 4 ) D, E Algorithms for Sequencing & Scheduling 6. 8

9 ha. 6 / Proect Management and Scheduling AOA Format AON Format Duration Immediate Predecessor For AOA Format Immediate Predecessor For AON Format ( 4, 6 ) I 1 ( 2, 4 ), ( 3, 4 ) D, E ( 5, 7 ) J 3 ( 2, 5 ), ( 4, 5 ), G ( 6, 7 ) K 6 ( 3, 6 ), ( 4, 6 ) F, I The recedence network diagram for the data in Table 6.2 using AON format is shown in Figure 6.2. Note two extra nodes have been added in the diagram. Both start node and end nodes are dummy nodes and, resent starting as well as terminating activities of a roect. Activity Name, 4 Activity Duration A, 7 D, 8 G, 2 J, 3 Start H, 4 End B, 6 E, 3 I, 1 K, 6 Precedence Relationshi F, 2 Figure 6.2 Activity-on-node (AON) recedence network diagram After setting u the network, PM technique finds the longest ath through the activity network. The longest ath comrises a set of activities which are designated as critical activities. These are the activities which must start and finish on exact dates and time, because the entire roect comletion is deendent uon them. PM technique identifies these activities. When execution of roect starts, these activities may be assigned to resonsible ersons. Management resources could be otimally used by concentrating on the few activities which determine the fate of the entire roect. Algorithms for Sequencing & Scheduling 6. 9

10 ha. 6 / Proect Management and Scheduling Non-critical activities can be re-lanned, rescheduled. Resources for these activities can be reallocated flexibly, without affecting the whole roect PM alculations ritical ath method (PM) minimizes the roect comletion time ( max ) of the roect by finding longest ath from source to sink node in the recedence network diagram. Let s define mathematical notations for activity-on-arc (AOA) recedence network as follows; i, = Duration for activity (i, ) E i i i, = Earliest occurrence time for event i = Latest occurrence time for event i S = Earliest start time for activity (i, ) = Earliest comletion time for activity (i, ) i, S = Latest start time for activity (i, ) i, = Latest comletion time for activity (i, ) i, ψ t = Total slack (float) for activity (i, ) ψ f = Free slack for activity (i, ) The comutations are carried out in two stages. First stage comrises forward ass calculations to comute earliest event times. The stes are as follows: 1. Set the start time of the initial event to zero. 2. Start each activity as soon as its redecessor events occur. 3. alculate early event times as the maximum of the earliest comletion time of activities terminating at the event. In mathematical terms; E 1 0 And, E = max { E +,E +,..., E t } i i, i i, i n Where, i 1, i 2,..., i n indicate the receding events of the n activities that terminate at event. Earliest start and comletion time for an activity (i, ) is calculated from the following exressions. S = i, E i + i, = E i i, in, Algorithms for Sequencing & Scheduling 6. 10

11 ha. 6 / Proect Management and Scheduling Proect comletion time, max = E n Second stage comrises backward ass calculations to comute latest occurrence of event. The stes are as follows: 1 Set the latest time for the terminal event equal to the earliest time for that event. 2 Start each activity at the latest time of its successor event less the duration of the activity. 3 Determine event times as the minimum of the latest start times of all activities emanating from the event. Set, L n = E n And, L i = min(l t, L t,..., L t ) i n 1 i, 1 2 i, 2 v i, < v Where; 1, 2,., v indicate the successor events of v activities that emanate from event i. Latest start and comletion time for an activity (i, ) is calculated from the following exressions. S = i, L i, = L i, Total slack time for activity (i, ) is comuted from; ψ = L E t i, i i, Free slack is the amount of time that activity comletion can be delayed without affecting the early start ( S i, ) time for any other activity in the network. ψ f i, = E E i i, ritical ath consists of all activities with zero slack on the network. Algorithms for Sequencing & Scheduling 6. 11

12 ha. 6 / Proect Management and Scheduling Examle 6.1 The data for an 8-activity roect is given in following table with activities, their durations and corresonding recedence relationshis. Activity ( i, ) Duration i, Immediate Predecessor ( 1, 2 ) ( 2, 4 ) 7 ( 1, 2 ) ( 2, 3 ) 8 ( 1, 2 ) ( 2, 5 ) 6 ( 1, 2 ) ( 4, 6 ) 15 ( 2, 3 ), ( 2, 4 ) ( 3, 5 ) 9 ( 2, 3 ) ( 5, 6 ) 12 ( 2, 5 ), ( 3, 5 ) ( 6, 7 ) 8 ( 4, 6 ), ( 5, 6 ) Draw activity-on-arc (AOA) recedence network diagram, find roect comletion time and identify critical ath and find slack times. Solution: An Activity-on-arc (AOA) network uses arcs to reresent activities and nodes to reresent occurrence of events. The recedence network diagram is shown in Figure Figure 6.3 AOA Precedence network diagram Early & late occurrence times calculation of events is shown in Table 6.4. Algorithms for Sequencing & Scheduling 6. 12

13 ha. 6 / Proect Management and Scheduling Table 6.4 Early and late occurrence times of Events Event E L omutations of various arameters ertaining to activities including early start time S, early comletion time, late start time S, late comletion time i, i,, total slack ψ t and i, ψ f are shown in Table 6.5. i, Activity (i,) Table 6.5 alculated Parameters for Examle 6.1 S i, i, S i, ψ i, t ψ f Status (1,2) ritical (2, 4) Slack (2, 3) ritical (2, 5) Slack (4, 6) Slack (3, 5) ritical (5, 6) ritical (6, 7) ritical The schedule generated by PM technique in Table 6.5 is shown on Gantt hart in Figure 6.4. (1,2) (2, 3) (3, 5) (5, 6) (6, 7) (2, 5) (2, 4) (4, 6) Figure 6.4 Gantt chart for Schedule Algorithms for Sequencing & Scheduling 6. 13

14 ha. 6 / Proect Management and Scheduling AON Network omutations Activity-on-node (AON) network calculations are also carried out in two stages. Forward ass calculations are carried out in following stes. 1. Set early start time for source activity (node) equal to zero. Then, early comletion time for source node will be equal to = S + = 0 + = source source source source source 2. For activity i, calculate early comletion time by the exression; i = max{,,..., } v i Where, Activities 1, 2, 3,..., v are the redecessor activities of activity i and v < n as shown in Figure i v Figure 6.5 Set of redecessor activities for activity i 3. Then, roect comletion time; max = sink In second stage, backward ass calculations are carried out to find late start and finish times of activities. 1. Set late comletion time for last (sink) activity equal to early comletion time; n n 2. For any activity k (k < n), find late comletion time from the exression; k = min{ 1 1, 2 2,..., v v } Where, Activities 1, 2, 3,., v are the successor activities of activity k and v < n as shown in Figure 6.6. Algorithms for Sequencing & Scheduling 6. 14

15 ha. 6 / Proect Management and Scheduling 1 2 k 3 v Figure 6.6 Set of successor activities for activity k 3. Activities on the critical ath have zero slack time. Slack time for any activity is found from the exression; ψ t = Examle 6.2 hange the data in Examle 6.1 to reresent activity-on-node (AON) frame work. Draw recedence network diagram and find critical ath. Solution: The AON version of the data of Examle 6.1 is shown in Table 6.6 Activity Name Table 6.6 Activity-on-node (AON) data format Activity ( i, ) Duration i, Immediate Predecessor A ( 1, 2 ) B ( 2, 4 ) 7 ( 1, 2 ) A ( 2, 3 ) 8 ( 1, 2 ) A D ( 2, 5 ) 6 ( 1, 2 ) A E ( 4, 6 ) 15 ( 2, 3 ), ( 2, 4 ) B, F ( 3, 5 ) 9 ( 2, 3 ) G ( 5, 6 ) 12 ( 2, 5 ), ( 3, 5 ) D, F H ( 6, 7 ) 8 ( 4, 6 ), ( 5, 6 ) E, G The recedence network diagram is develoed and shown in Figure 6.7 Algorithms for Sequencing & Scheduling 6. 15

16 ha. 6 / Proect Management and Scheduling Activity Name B, 7 E, 15 Activity Duration A, 4, 8 F, 9 H, 8 D, 6 G, 12 comletion Figure 7.7 Activity-on-node (AON) Precedence network diagram The time attributes of the activities including early start S, early i,, late start S and late comletion are shown in Table 6.7 i, i, i, Activity (i,) Table 6.7 alculated arameters for Examle 6.2 S i, i, S i, i, ψ t ψ f Status A ritical B Slack ritical D Slack E Slack F ritical G ritical H ritical The generated schedule in Table 6.7 is shown on the Gantt chart in Figure 6.8 A F G H D 10 B E Figure 6.8 Gantt chart for Schedule in Table 6.7 Algorithms for Sequencing & Scheduling 6. 16

17 ha. 6 / Proect Management and Scheduling 6.6 PERT METHODOLOGY PERT (Program Evaluation Review Technique) was devised in 1958 for he POLARIS missile rogram by the Program Evaluation Branch of the Secial Proects office of the US Navy, heled by the Lockheed Missile Systems division and the onsultant firm of Booz-Allen & Hamilton. The calculations were so arranged so that they could be carried out on the IBM Naval Ordinance Research omuter (NOR) at Dahlgren, Virginia. PM technique deals with roects, where there is high certainty about the outcomes of activities. When there is learning rocess involved, degree of uncertainty is much higher and duration of activities involve considerable degree of estimation. In such situations, the PERT aroach is useful, because it can accommodate the variation in activities comletion times, based on an exert s or an exert committee s estimates. The activities durations are estimated using three horizons, namely; otimistic, most likely and essimistic. Let s assign notations to activities durations for PERT methodology as follows: = Otimistic (minimum) duration of th activity n = Most likely (normal) duration of th activity = Pessimistic (maximum) duration of th activity P = Exected duration of th activity, and is calculated by following; P = + 4 n + 6 µ = Exected duration of th activity which is on the critical ath (Note that, µ = P ) The PM technique is alied on the roblem data using P values to find critical activities as well as critical ath. Let A is the set of activities on the critical ath. Then, an estimate of the roect comletion time is E( max ) = µ A c Variance of the duration of th activity is given by Algorithms for Sequencing & Scheduling 6. 17

18 ha. 6 / Proect Management and Scheduling σ = 6 2 To obtain an estimate of the variance of the roect comletion time ( max ), consider only the activities on the critical ath in the network. Since these critical activities occur one after the other, add the variance of rocess durations of the critical activities to obtain an estimate of variance of the roect comletion time V( max ) = σ A c 2 The distribution of the roect comletion time ( max ) is assumed to be normal with a mean of E( max ) and variance V( max ). Examle 6.3 onsider a PERT network roblem. The otimistic, most likely and essimistic task durations (in DAYS) are given in Table 6.8 below. Also, the Table contains recedence relationshis. Table 6.8 Problem data for Examle 6.3 Activity Activity Duration Immediate Otimistic Most Likely Pessimistic Predecessor , , , , , ,10 Algorithms for Sequencing & Scheduling 6. 18

19 ha. 6 / Proect Management and Scheduling Activity Activity Duration Immediate Otimistic Most Likely Pessimistic Predecessor , a) What is the exlicit distribution (mean, variance) of the roect comletion time? b) What is the robability of finishing roect no later than time 41 days? c) What is the robability of finishing no earlier than time 38 days? d) Suose tasks 2 and 3 finish at time 10. What is the robability of finishing the roect by time 41 days? Solution: The recedence network diagram is shown in Fig , 6 4, 6 8, 12 11, 6 15,10 Start 2, 8 5, 6 9, 12 12, 9 16, 6 End 3, 9.5 6, 3 10,11 13, 8 17, 7 Task, Duration 7, , 8 Figure 6.9 Precedence network diagram for Examle 6.3 Proect calculations are shown in the following Table 6.9. Algorithms for Sequencing & Scheduling 6. 19

20 ha. 6 / Proect Management and Scheduling Activity () P Table 6.9 alculated arameters for Examle 6.3 S S Status µ SLK SLK *RTL* SLK SLK SLK *RTL* SLK SLK *RTL* SLK SLK SLK *RTL* SLK SLK *RTL* σ The critical ath on the network is shown in Figure , 6 4, 6 8, 12 11, 6 15,10 Start 2, 8 5, 6 9, 12 12, 9 16, 6 End 3, 9.5 6, 3 10,11 13, 8 17, 7 Task, Duration 7, , 8 Figure 6.10 ritical Path on the network (Examle 6.3) Algorithms for Sequencing & Scheduling 6. 20

21 ha. 6 / Proect Management and Scheduling a) Exected roect comletion time: ritical Path = { } E( max ) = µ 3 + µ 7 + µ 10 + µ 14 + µ 17 = 43 days Exected variance of the roect comletion time; V( max ) is found from; V( max ) = A c σ 2 = σ σ σ σ σ 2 17 = 9.06 b) Probability that roect is comleted no later than 41 days is: Probability (X <= 41 days) x µ = P( ) = P(z 0.664) = σ Area under the normal curve from - to is as shown in Figure 6.11 Figure 6.11 Area under normal curve P(z 0.664) c) Probability that roect is comleted no earlier than 38 days is: Probability (X >= 38 days) = 1 P(X <= 38) x µ = 1 - P( ) = 1 P(z 1.66) = = σ Area under the normal curve from - to is as shown in Figure 6.12 Algorithms for Sequencing & Scheduling 6. 21

22 ha. 6 / Proect Management and Scheduling Figure 6.12 Area under normal curve P(z 1.66) d) If Tasks 2 and 3 finishes at time 10, what is the robability that roect will be comleted by 41 days. When Task 2 and Task 3 finish at time 10, the network is revised. The new values of earliest and latest comletion times are shown in the network (Fig 6.13) (6, 10) (12, 16) (28, 28) (34, 34) (44, 44) 1, 6 4, 6 8, 12 11, 6 15,10 (0, 0) (10, 10) (16, 16) (28, 29) (37, 38) (43, 44) (44, 44) Start 2, 10 5, 6 9, 12 12, 9 16, 6 End (10, 13) 3, 10 (13, 17) 6, 3 (28.5, 29) 10,11 (36.5, 38) 13, 8 (43.5, 44) 17, 7 (17.5, 18) (36.5, 37) 7, , 8 Figure 6.13 Revised recedence network with udated calculations i) The new value of max = 44. ii) New critical ath is { }. iii) Exected Duration (µ) = 44 and, Variance = 3(0.44) = 4.11 Probability ( X <= 41 days) x µ = P( ) = P(z 1.48) = σ 2.03 Algorithms for Sequencing & Scheduling 6. 22

23 ha. 6 / Proect Management and Scheduling Hence, there is 7% robability that roect will comlete by 41 days. Area under the normal curve from - to is as shown in Figure 6.14 Figure 6.14 Area under normal curve P(z 1.48) Whereas PERT is used to estimate durations of activities, the method has certain limitations. The activity time estimates are somewhat subective and deend on udgment. For inexerienced lanners, duration of activities may be only a guess. In other cases, if the erson or grou erforming the activity estimates the time there may be bias in the estimate. Even if the activity times are well-estimated, PERT assumes a beta distribution for these time estimates, but the actual distribution may be different. Even if the beta distribution assumtion holds, PERT assumes that the robability distribution of the roect comletion time is the same as that of the critical ath. Because other aths can become the critical ath if their associated activities are delayed, PERT consistently underestimates the exected roect comletion time. The underestimation of the roect comletion time due to alternate aths becoming critical is erhas the most serious of these issues. To overcome this limitation, Monte arlo simulations can be erformed on the network to eliminate this otimistic bias in the exected roect comletion time. Algorithms for Sequencing & Scheduling 6. 23

24 ha. 6 / Proect Management and Scheduling 6.7 TIME / OST TRADE-OFF One of the maor obectives in roect scheduling is minimization of roect comletion time ( max ). There is a relationshi between a roects comletion time ( max ) and its cost. The relationshi deends uon the tye of costs. For some tyes of costs, the relationshi is direct roortion; i.e., as max increases, the total roect cost increases. In order to accelerate the ace of the roect (minimize max ), more resources are acquired increasing the cost of the roect. Because of these two tyes * of costs, there is an otimal roect comletion time ( max ) for minimal cost. By understanding the time-cost relationshi, one is better able to redict the imact of a schedule change on roect cost Tyes of osts The costs associated with a roect can be classified as direct costs or indirect costs. Direct costs are those directly associated with roect activities, such as salaries, travel, and direct roect materials and equiment. If the ace of activities is increased (to decrease duration of activities) thereby decreasing roect comletion time, the direct costs generally increase since more resources must be allocated to accelerate the ace. Indirect costs are those overhead costs that are not directly associated with secific roect activities such as office sace, administrative staff, and taxes. Such costs tend to be relatively steady er unit of time over the life of the roect. As such, the total indirect costs decrease as the roect duration decreases. The roect cost is the sum of the direct and indirect costs rashing of Proect rashing the roect schedule refers to the acceleration of the roect activities in order to comlete the roect earlier than normal time. Since roect comletion time ( max ) is determined by the activities on the critical ath, so to crash a roect schedule one must focus on critical ath activities. A rocedure for determining the otimal roect time is to determine the normal comletion time (duration) for each critical ath activity and a crash time (duration). The crash time (duration) is the shortest time in which an activity can be comleted. The direct costs then are calculated for the normal and crash durations of each activity. Let, n = Normal duration cost of th activity c = rash duration cost of th activity n = Normal duration of th activity Algorithms for Sequencing & Scheduling 6. 24

25 ha. 6 / Proect Management and Scheduling c = rash duration of th activity = cost of reducing duration by one time unit of th activity o = Fixed cost of roect er unit time k = Total roect cost at k th iteration. Sloe of each activity s cost versus time trade-off ( ) can be found by Sloe = (rash ost Normal ost) (Normal Duration rash Duration) In terms of mathematical notations, is determined by; = P c n P n c The grahical determination of sloe is shown in Figure ost n Sloe, c c n (Duration ) Figure 6.15 Sloe calculations ( ) for Time-ost Trade-Off * To obtain an otimal value of roect comletion time ( max ), the activities having the lowest values of should be shortened first. In this way, one can ste through the critical ath activities and create a grah of the total roect cost versus the roect time. The indirect, direct, and total roect costs then can be calculated for different roect durations. The otimal oint is the duration resulting in the minimum roect cost. Attention should be given to the critical ath to make sure that it remains Algorithms for Sequencing & Scheduling 6. 25

26 ha. 6 / Proect Management and Scheduling the critical ath after the activity duration is reduced. If a new critical ath emerges, it must be considered in subsequent time reductions. To minimize the cost, those activities that are not on the critical ath can be extended to minimize their costs without increasing the roect comletion time Time-ost Trade-off Assumtions The time-cost model described above relies on the following assumtions: The normal cost for an activity is lower than the crash cost. There is a linear relationshi between activity time and cost. The resources are available to shorten the activity. A formal methodology is described by taking following stes to reduce roect comletion time yielding minimum total roect cost. i. Use normal durations of all activities and solve roect network roblem by PM/PERT technique. Find max and, identify critical ath/s on the network. ii. For normal duration, total cost of the roect will be equal to 0 = o x max. iii. Set iteration counter k = 1. To reduce roect cost, find among the unmarked activities on the critical ath having minimum value of. iv. Pick minimum cost unmarked activity on the critical ath with least cost, and reduce duration by one unit of time. Mark this activity if its duration c has reached value. Find max and new critical ath/s for revised roect network. Proect cost for the revised network will be equal to; k = 0 max + Where, U is the set of least cost activities in the resent network. K k 1 IF THEN go to next ste, otherwise STOP. v. Set k = k + 1. From the new critical aths found in ste (iv), find if all obs are at their c value. If yes, STOP. There is no further ossibility of reduction in max value. If No, go to ste (ii). The rocedure to reduce max by single unit of time in every iteration may require enormous amount of calculations. It is ossible to seed u the calculations by making large reductions in an activity s duration in one iteration. However, new critical aths may emerge in this aroach. The activities on the critical aths may hit their minimum, making the critical ath irrelevant. Secial algorithms have to be devised for such aroaches. U Algorithms for Sequencing & Scheduling 6. 26

27 ha. 6 / Proect Management and Scheduling Examle 6.4 onsider the recedence grah in Figure S 5 E 3 4 Figure 6.16 Precedence network diagram for Examle 6.4 The time / cost data for the roblem is as under. Table 6.11 Normal and rash Data for Examle 6.4 Activity () max min Take o = 12 and find minimum roect comletion time and corresonding cost using Time-ost Trade-Off. Solution: i) omutation of max is shown in Figure 6.17, when activities are executed at normal duration ( ) value, max (0, 0) S (3, 3) (9, 9) 1, 3 2, 6 (4, 6) (7, 9) 3, 4 4, 3 Activity number Activity duration 5, 2 E (11, 11) (, ) Figure 6.17 omutations of Early and Late comletion times Algorithms for Sequencing & Scheduling 6. 27

28 ha. 6 / Proect Management and Scheduling max = 11, ritical Path: ost of Proect, P = o x max = = 132 ii) The activities on the critical ath along with crash costs are; Activities Since, activity 2 has minimum value, reduce duration of activity 2 by one time unit in the roect network. So, c 2 = 5 The network with c 2 = 5 is shown in recedence network diagram (Figure 6.18) (0, 0) S (3, 3) (8, 8) 1, 3 2, 5 (4, 6) (7, 9) 3, 4 4, 3 5, 2 (10, 10) (, ) Activity number Activity duration E Figure 6.18 omutations of comletion times with activity 2 being erformed at crash duration For this network, max = 10. ritical Path: ost of Proect, 1 P = o x max + 2 = = 126 iii) Unmarked Activities on critical ath have the following features; Activity n c Activity 2 is hitting its minimum value. Hence, there is no further reduction in duration of activity 2. So activity 2 is removed from candidate s list. Out of the remaining two activities, activity 1 has lower value of. So, reduce duration of Algorithms for Sequencing & Scheduling 6. 28

29 ha. 6 / Proect Management and Scheduling activity 1 by one time unit in the roect network. Set, c = 1 2. Re-comute comletion times with c = 1 2 as shown in Figure The max value of this network is 9: (0, 0) S (2, 2) (7, 7) 1, 2 2, 5 (4, 4) (7, 7) 3, 4 4, 3 5, 2 (9, 9) (, ) Activity number Activity duration E Figure 6.19 omutations of comletion times with activity 1 being erformed at crash duration There are two critical aths now; ost of Proect, 2 P = o x max = = 126 There are two critical aths in the above network. The common activity is 5. Activities n c Set c 5 = 1 in the new network. The comutations of max are shown below in Figure Algorithms for Sequencing & Scheduling 6. 29

30 ha. 6 / Proect Management and Scheduling (0, 0) (2, 2) (7, 7) 1, 2 2, 5 Activity number Activity duration S (4, 4) 3, 4 (7, 7) 4, 3 5, 1 (8, 8) (, ) E Figure 6.20 omutations of comletion times with activity 5 being erformed at crash duration max = 8 Two critical aths are; ost of Proect, 3 P = o x max = = 130 The calculations are summarized in table 6.11 Table 6.11 Summary of alculations Iteration (k) max Total ost x 11 = x = x = x = 130 The otimal value of roect cost is 126. There are two alternative critical aths with max values of 9 and Additional onsiderations There are other considerations besides roect cost. For examle, when the roect is art of the develoment of a new roduct, time-to-market may be extremely imortant and it may be beneficial to accelerate the roect to a oint where its cost is much greater than the minimum cost. In contract work, there may be incentive ayments associated with early comletion or enalties associated with late comletion. A time-cost model can be adated to take such incentives and enalties into account by modeling them as indirect costs. Algorithms for Sequencing & Scheduling 6. 30

31 ha. 6 / Proect Management and Scheduling EXERISES 6.1 Assume we are to construct a um station. The following table lists the activities to construct a um station. Activity No. Activity Descrition Duration in Days 1 Start 0 Preceding Activity 2 Mobilize Survey Grade site Trench footings 5 3, 4 6 Form and our concrete 5 5, 8 7 ure concrete oncrete and material design Sec refab metal building Plumbing materials, um Electrical materials, lights, anel Install um 7 7, 9, Erect structural steel 4 7, 9, Install roofing and siding Install lights and anels 3 11, Test um Paint End 0 16, 17 a. Draw the network diagram for construction of a um station and erform the required comutations. When constructing the network diagram use the following format for each node: Activity Name Activity Number Activity Duration Start Date Finish Date Early start Time Early finish Time Late start Time Late finish Time Algorithms for Sequencing & Scheduling 6. 31

32 ha. 6 / Proect Management and Scheduling b. Determine critical ath. c. Based on the early start and early finish times for each activity and assuming that the roect will start on March 1 st, 2007, construct calendar dates for construction of a um station. (i.e., assign a calendar date to the beginning of the first activity and convert the time durations on each activity to calendar date.). d. Draw the Gantt chart for construction of um station. 6.2 The following table gives the data for small roect. Proect Phase Preceding Phase Normal Time rash Time Normal ost rash ost A ,000 12,000 B A ,000 15,300 A ,500 10,000 D A ,000 22,000 E, D ,000 15,000 F B, E ,000 7,000 a. Draw the network diagram and erform the required comutations. b. Determine critical ath. c. Draw the Gantt chart. d. There is a enalty of 3000 SR er day beyond the normal PM duration. Perform crashing analysis to comare total normal cost to total crash cost. First crash only the critical activities and then crash all activities. e. Assume that the variances of the durations of the roect hases are as given below: Phase A B D E F Duration Variance Find the robability that total lateness enalty will be less than 3000 SR. f. Reeat art e above for robability that total lateness enalty will be less than 9000 SR. Algorithms for Sequencing & Scheduling 6. 32

33 ha. 6 / Proect Management and Scheduling 6.3 For a secific roect to be accomlished, you are given the following time and redecessor for each activity in the roect: Activity a m b Immediate Predecessor A B A D B&E E A a. Draw Network Diagram. b. Perform forward schedule. c. Perform backward schedule. d. In table format, determine the ES, EF, LS, LF, and slack for all activities. e. Draw Gantt chart for the Early Start Schedule for the roect (5 oints). f. Find the ritical Path. g. What is the variance in comletion time for the critical ath found? 6.4 Develoment of a new deluxe version of a articular software roduct is being considered. The activities necessary for the comletion of this roect are listed in the table below. Activity Normal Time (week) rash Time (week) Normal ost rash ost Immediate Predecessor A 4 3 2,000 2,600 - B 2 1 2,200 2, D 8 4 2,300 2,600 A E ,200 B F 3 2 3,000 4,200 G 4 2 1,400 2,000 D, E a. What is the roect exected comletion date? b. What is the total cost required for comleting this roect on normal time? Algorithms for Sequencing & Scheduling 6. 33

34 ha. 6 / Proect Management and Scheduling c. If you wish to reduce the time required comleting this roect by 1 week, which activity should be crashed, and how much will this increase the total cost? 6.5 A roect has an exected comletion time of 40 weeks and a standard deviation of 5 weeks. It is assumed that the roect comletion time is normally distributed. a. What is the robability of finishing the roect in 50 weeks or less? b. What is the robability of finishing the roect in 38 weeks or less? c. The due date for the roect is set so that there is a 90% chance that the roect will be finished by this date. What is the due date? 6.6 onsider the following recedence grah shown below: A D Source Sink B E The time/cost data for the roblem is shown in the following table: Activity () A B D E max min Given that O is 15 do the following: a. Find roect comletion time ( max ) as well as roect cost if all activities max are erformed at b. Find roect comletion time ( max ) as well as roect cost if all activities min are erformed at c. Use time-cost trade-off method, and find Minimum ost roect Schedule. Algorithms for Sequencing & Scheduling 6. 34

35 ha. 6 / Proect Management and Scheduling 6.7 Precedence grah of a small roect comrising 11 obs/tasks is shown in Fig.1. The time/cost trade-off data including maximum rocess time ( ), minimum rocess min time ( ) and cost of the obs/tasks for reducing rocess time from maximum to minimum value er day ( ) is shown in Table. Take fixed cost of the roect er time unit ( 0 ) as equal to 12. max Job () A B D E F G H I J K max min Find the otimal roect comletion time to rovide minimum total costs. Also show the critical ath/s for this schedule. A F G J D B E H I K Precedence Grah 6.8 For a secific roect to be comleted, you are given the following activities and their redecessor, durations, and cost in the table below: Activity Immediate Predecessor Otimistic time Most likely time Pessimistic time A ost B A D E D F A Algorithms for Sequencing & Scheduling 6. 35

36 ha. 6 / Proect Management and Scheduling Activity Immediate Predecessor Otimistic time Most likely time Pessimistic time ost G E & F H B I H J F & I K F & I L K M J & K N M O A P D & O Q P R G & Q Then, erform the following: a. Draw network diagram and do the forward and the backward schedule for the roect. b. Determine the exected comletion time and the critical ath of this roect. c. In table format, determine the early start, early finish, late start, late finish, and slack for all activities, d. Draw Gantt chart for the early start schedule for the roect and show the critical ath. e. What is the robability of finishing the roect in 120 days? 6.9 Assume that the status of the roect given in the question 6.8 on the 58 days in the roect time is as follows: Activity Percent comleted Activity Percent comleted Activity Percent comleted A 100 G 0 M 0 B 100 H 100 N I 100 O 50 D 100 J 75 P 0 E 50 K 25 Q 0 F 100 L 0 R 0 The total exenditures to date are Then, erform the following: Algorithms for Sequencing & Scheduling 6. 36

37 ha. 6 / Proect Management and Scheduling a. Analyze the rogress of the roect from both a budget and time oint of view. b. If the roect is not comleted on the exected comletion time, then, the event of celebrating the end of the roect will be cancelled and this will cost Thus, there are two suggestions to reduce the times of the remaining activities as follows: Reduce the most likely and essimistic times for activity L by 50% and this will cost Reduce activity N by three days and this will cost Then, which suggestion or combination of suggestion should you consider adoting Perform the PM analysis for the given roect using the following data in the table below: Activity Predecessor Duration A - 2 B A 4 B 6 D A 3 E B 2 F E 1 G,D 2 H G 3 I H 4 When erforming the analysis assume the following: a. The roect cannot start until 3 days from time zero (i.e. ES at start node = 3). b. There is deadline of 30 days (i.e. L at finish node = 30). Draw the PM network and erform the required comutations. For every activity, show in a table ES, LS, E, L, and TS. Also, in the same table for every activity show whether the activity is critical or non-critical. Algorithms for Sequencing & Scheduling 6. 37

38 ha. 6 / Proect Management and Scheduling 6.11 Perform the PERT analysis for the roect using the following data in the table below: Activity No a m b Preceding activity , For every activity, comute the average activity duration and the standard deviation of activity duration. Then, draw the PERT network and erform the required comutations. For every activity, list in a table format the following: ES, LS, E, L, and TS. Also, for every activity show in the same table whether the activity is critical or non-critical. Then, find the robabilities of finishing the roect in 15 unit of time. Also, find the robabilities of finishing the roect in 21 unit of time. Algorithms for Sequencing & Scheduling 6. 38