OR PERT CPM AND JOB SCHEDULING

By: Jignesh Kariya, NJSMTI VERAVAL 1

2640003 - Operations Research (OR) Unit - 5


Contents

Project Management and Scheduling1. Project Management (CPM & PERT)• Network concepts• Components• Rules for network construction• Critical path method (CPM) • Project evaluation and Review Techniques (PERT)2. Production scheduling (job sequencing)• Introduction• Johnson’s algorithm for n jobs 2 machines• Johnson’s algorithm for N jobs m machines• 2 jobs m machines using graphical method.


Project Management and Scheduling


Introduction• A project is an endeavor to create a unique product or service. It is

specific, timely, usually multi-disciplinary and always conflict-ridden. Projects are a part of an overall programme and are broken down into well-defined set of tasks (Jobs), subtasks and further if desired, all of which must be completed within a specified time, along with minimum cost.

• Examples of projects include, construction of a bridge, highway, power plant, repair and maintenance of an oil refinery or an air plane; design. development and marketing of a new product, research and development work, etc. Such projects involve a large number of interrelated activities (or tasks) that must be completed in a specified time, in a specified sequence (or order) and require resources such as personnel, money, materials, facilities and/or space.

• The main objective before starting any project is to schedule the required activities in an efficient manner so as to complete it on or before a specified time limit at a minimum cost of its completion


Introduction• Hence, before starting any project, it is necessary to prepare a plan for

scheduling and controlling the various activities (or tasks) involved in the given project. The techniques of operations research used for planning, scheduling and controlling large and complex projects are often referred to as network analysis, network planning or network planning and scheduling techniques.

• All these techniques are based on the representation of the project as a network of activities. A network is a graphical plan consisting of a certain configuration of arrows and nodes for showing the logical sequence of various activities to be performed to achieve project objectives.


Introduction• PERT and CPM are two methods where (Programme Evaluation and

Review Technique) was developed in 1956-58 by a research team to help in the planning and scheduling of the US Navy's Polaris Nuclear Submarine Missile project, which involved thousands of activities. The objective of the team was to efficiently plan and develop the Polaris missile system. Since 1958, this technique has proved to be useful for all jobs or projects that have an element of uncertainty in the estimation of duration, as is the case with new types of projects the likes of which have never been taken up before.

• CPM (Critical Path Method) was developed by E.L. DuPont company along with Remington Rand Corporation. The aim behind its development was to provide a technique for the control of the maintenance of company's chemical plants. In course of time, the use of CPM was extended to the field of cost and resource allocation.


Basic Difference Between PERT & CPMBasis PERT CPM

Stands for PERT stands for Programme Evaluation and Review Technique

CPM stands for Critical Path Method.

Model It is a probabilistic model under which the result estimated in a manner of probability.

It is a deterministic model under which the result is ascertained in a manner of certainty.

Time IT deals with the activities of uncertain time.

It deals with the activities of precise well known time.

Jobs It is used for onetime projects that involve activities of non-repetitive nature

It is used for completing of projects that involve activities of repetitive nature.

Orientation It is activity oriented in as much as its result is calculated on the basis of the activities.

It is even oriented, in as much as its results are calculated on the basis of the events.

Dummy Activities

It does not make use of dummy activities. It makes use of dummy activities to represent the proper sequencing of the activities.

Cost It has nothing to do with cost of a project. It deals with the cost of a project schedules and their minimization.

Estimation It finds out expected time of each activity on the basis of three types of estimates.

Its calculation is based on one type of time estimation that is precisely known.

Time PERT is restricted to time variable. CPM includes time-cost trade off.


Basic Difference Between PERT & CPM• There are no essential differences between PERT and CPM as such. Both of

them share in common the determination of a critical path and are based on the network representation of activities and their scheduling that determines the most critical activities to be controlled so as to meet the completion date of a project. However. The following are the some of their other major differences.

PERT : Project Evolution And Review Technique1. In PERT, three estimates are used to form a weighted average of the expected

completion time of each activity, based on the probability distribution of completion times.

2. It is basically a tool for planning and control of time.3. To perform PERT analysis on a project, the emphasis is on the completion of a

task rather than the activities required to be performed to reach a particular event or task. Thus, it is also called an event oriented technique.

4. It is used for onetime projects that involve activities of non-repetitive nature (i.e. activities that may never have been performed before), in which time estimates are uncertain, such as redesigning an assembly line or installing a new information system.


Basic Difference Between PERT & CPM 5. It helps in identifying critical areas in a project so that necessary

adjustments can be made to meet the scheduled completion date of the project.

CPM : Critical Path Method6. In CPM, there is only one estimate of completion time of each activity. This

technique was developed in connection with a construction and maintenance project in which the duration of each activity was known with certainty.

7. It allows an explicit estimate of costs in addition to time. Thus, it is suitable for establishing a trade off for optimum balancing between schedule time and cost of the project.

8. It is used for completing of projects that involve activities of repetitive nature.


Basic Difference Between PERT & CPM Significance of Using PERT / CPM1. A network diagram helps to translate highly complex project into a set of

simple and logical arranged activities and therefore, • Helps in the clarity of thoughts and actions. • Helps in clear and unambiguous communication developing from top to bottom and vice versa among the people responsible for executing the project.

2. Detailed analysis of a network helps project incharge to peep into the future because

• Difficulties and problems that can be reasonably expected to crop up during the course of execution can be foreseen well ahead of its actual execution. • Delays and holdups during course of execution arc minimized. Corrective action can also be taken well in time..


Basic Difference Between PERT & CPM

3. Isolates activities that control the project completion and therefore, results in expeditious completion of the project.

4. Helps in the division of responsibilities and therefore, enhance effective coordination among different departments/agencies involved.

5. Helps in timely allocation of resources to various activities in order to achieve optimal utilization of resources.


Phases of Project ManagementIn general, project management consists of three phases.1. Project planning phase2. Scheduling phase3. Project control phase

4. Project Planning Phase :• In order to visualize the sequencing or precedence requirements of the

activities in a project, it is helpful to draw a network diagram.

For this the following tips are adopted:(a) Identify various activities (task or work elements) to be performed in

the project, that is, develop a breakdown structure (WBS).

(b) Determine the requirement of resources such as men, materials, machines, money, etc., for carrying out activities listed above.


Phases of Project Management(c) Assign responsibility for each work package. The work packages corresponds to the smallest work efforts defined in a project and forms the set of elemental tasks that are the basis for planning, scheduling and controlling the project.

(d) Allocate resources to work packages.

(e) Estimate cost and time at various levels of project completion.

(I) Develop work performance criteria.

(g) Establish control channels for project personnel.


Phases of Project Management2. Scheduling Phase :Once all work packages (i.e. tasks) have been identified and given unique names or identifiers, scheduling the project, i.e. when each of the activities are required to be performed, is taken up. It prepares an analysis/estimate of the likelihood of the project to be completed on or before the specified time. The various steps involved during this phase are listed below:

(a) Identify all people who will be responsible for each task.

(b) Estimate the expected duration(s) of each activity, taking into consideration the resources required for their execution in the most economic manner.

(c) Specify the interrelationship (i.e. precedence relationship) among various activities.


Phases of Project Management2. Scheduling Phase cont..(d) Develop a network diagram, showing the sequential interrelationship between various activities. For this, tips such as; what is required to be done; why it must be done, can it be dispensed with how to carry out the job; what must precede it; what has to follow; what can be done concurrently, may be followed.(e) Based on these time estimates, calculate the total project duration, identify critical path; calculate floats; carry out resources smoothing (or levelling) exercise for critical (or search) resources, taking into account the resource constraints (if any).

3. Project Control PhaseProject control refers to the evaluation of the actual progress (status) against the plan. If significant differences are observed, then the scheduling and resource allocation decisions are changed in order to update and revise the uncompleted pan of the project. In other words, remedial (modifying planning) or reallocation of resources (cost minimization) measures are adopted in such cases.


Phases of Project ManagementRelationship between phases of Project Management


PERT/CPM Network Components & Precedence Relationships• PERT/CPM network consists of two major components.

1. Events2. Activities

Events • Events in the network diagram represent project milestones, such

as the start or the completion of an activity (task) or activities, and occur at a particular instant of time at which some specific part of the project has been or is to be achieved. Events are commonly represented by circles (nodes) in the network diagram.

• The events can be further classified into the following two categories:

1. Merge Events2. Burst Events


PERT/CPM Network Components & Precedence Relationships• Merge Event : An event which represents the joint completion of more than one activity is known as a merge event.

• Burst Event :An event that has represents the initiation (beginning) of more than one activity is known as burst event.

Events

Events


PERT/CPM Network Components & Precedence Relationships• Activities • Activities in network diagram represent project operations or tasks

to be conducted. As such each activity except dummy activates requires resources and takes a certain amount of time for completion. An arrow is commonly used to represent an activity with its head indicating the directions of progress in the project.

• Activities are identified by the numbers of their starting (tail or initial) event and ending (head or terminal) event.

Activities can be further classified into the following three categories. 1. Predecessor Activity2. Successor Activity3. Dummy Activity


PERT/CPM Network Components & Precedence Relationships1. Predecessor Activity :

An activity which must be completed before one or more other activities start is known as predecessor activity.

2. Successor activity An activity which starts immediately after one or more of other activities

are completed is known as successor activity. 3. Dummy activity :

An activity which does not consume either any resources and/or time is known as dummy activity.A dummy activity in the network id added only to establish the given precedence relationship among activities of the project. It is needed when

> Two or more parallel activates in a project have same head and tail events.> Two or more activities have some (but not all) of their immediate

predecessor activities in common.


PERT/CPM Network Components & Precedence Relationships


PERT/CPM Network Components & Precedence RelationshipsNetwork Shows the sequential relationships among activities using nodes and arrows. Network models use the following two types of precedence network to show precedence requirements of the activities in the project.AON (Activity–on-Node) Network : In this type of precedence network each node (or circle) represents a specific task while the arcs represent the ordering between tasks. AON network diagrams place the activities within the nodes, and the arrows are used to indicate sequencing requirements. Generally these diagrams have no particular starting and ending nodes for the whole project . The lack of dummy activities in these diagrams always make them easier to draw and interpret.

AOA (Activity–on-Arrow) Network : In this type of precedence network at each end of the activity arrow is a node. These nodes represent points in times or instants, when an activity is starting or ending.


PERT/CPM Network Components & Precedence RelationshipsAOA Network Cont…The arrows itself represents the passage of time required for that activity to be performed.

These diagrams have a single beginning node from which all activities with no predecessors may start. The diagram then works its way from left to right, ending with a single ending node, where all activities with no followers come together.

Three important advantages of using AOA are as follows :1. Many computer programs are based on AOA network.2. AOA diagrams can be superimposed on a time scale with the arrows drawn, the correct length to indicate the time requirement.3. AOA diagrams give a better sense of the flow of time throughout a project.


PERT/CPM Network Components & Precedence RelationshipsRules for AOA Network Construction


PERT/CPM Network Components & Precedence RelationshipsRules for AOA Network Construction cont..1. In network diagram, arrows represent activities and circles the events.

The length of an arrow is of no significance.

2. Each activity should be represented only by one arrow and must start and end in a circle called event. The tail of an activity represents the start, and head the completion of work.

3. The event numbered 1 denotes the start of the project and is called “initial event”. All activities emerging (or taking off)from event 1 should not be preceded by any other activity or activities . An event carrying the highest number denotes the completion event. A network should have only one initial event and only one terminal event.

4. The general rule for numbering the event is that the head event should always be numbered such that for each activity (i,j), i < j.


PERT/CPM Network Components & Precedence RelationshipsRules for AOA Network Construction cont..5. An activity must be uniquely identified by its starting and completion

event, which implies that: (a) An event number should not get repeated or duplicated.(b) Two activities should not be identified by the same completion event.(c) Activities must be represented either by their symbols or by the corresponding ordered pair of Starting - completion events.

6. The logical sequence (or interrelationship) between activities must follow these rules:

(a) An event cannot occur until all its incoming activities have been completed.

(b) An activity cannot start unless all the preceding activities, on which it depends, have been completed.

(c) Though a dummy activity does not consume either any resource of time, even then it has to follow the rules 6(a) and (b).


PERT/CPM Network Components & Precedence RelationshipsErrors and Dummies in Network :Looping and Dangling :• Looping• Looping (cycling) and dangling are considered as faults in a network.

Therefore, these must be avoided.• A case of endless loop in a network diagram, which is also known as

looping, where activities A, Band C form a cycle.• Due to precedence relationships, it appears from Fig, that every activity in looping (or cycle) is a predecessor of itself. In this case it is difficult to number three Events associated with activity A, B and C So as to satisfy rule 6 of constructing the network.


PERT/CPM Network Components & Precedence RelationshipsErrors and Dummies in Network cont..• Dangling• A case of disconnect activity before the completion of all activities, which

is also known as dangling, is shown in Fig. • In this case, activity C does not give any result as per the rules of the

network The dangling may be avoided by adopting rule S of constructing the network.


PERT/CPM Network Components & Precedence RelationshipsDummy (or Redundant) Activity:• The following are the two cases in which the use of dummy activity may

help in drawing the network correctly, as per the various rules.(i) When two or more parallel activities in a project have the same head and tail events, i.e. two events are connected with more than one arrow. (ii) When two chains of activities have a common event, yet are wholly or partly independent of each other, as shown in Fig. (a). A dummy which is used in such a case, to establish proper logical relationships, is also known as logic dummy activity.



In fig (a) : If head event of C and D do not depend on the completion of activities A and B then the network can be redrawn, as in fig (b), otherwise fig (a) must be followed.


ExamplesEx : 1


ExamplesSolution : 1

ACTIVITY Predecessor Activity

A -

B A

C A

D B

E B,C

F E

G D,F

H G


ExamplesEx : 2


ExamplesSolution : 2


A -

B A

C B

D B

E B

F C

G C

H F, G

I D, E ,H

J I


ExamplesEx : 3


Examples

Solution : 3


A -

B A

C A

D C

E B

F C

G D,E

H G

I H

J F

K I,J

L K

M L

N J

O M,N




Examples


Critical Path Analysis Critical path is the longest path through the project network, the activities on the

path are the critical activities therefore any delay in their completion must be avoided to prevent delay in project completion.

The objective of critical path analysis is to estimate the total project duration and to assign starting and finishing times to all activities involved in the project. This helps to check the actual progress against the scheduled duration of the project. The duration of individual activities may be uniquely determined (in case of CPM) or may involve the three time estimates (in case of PERT), out of which the expected duration of an activity is computed.

Having done this, the following factors should be known in order to prepare the

project scheduling.(i) Total completion time of the project.(ii) Earlier and latest start time of each activity.(iii) Critical activities and critical path.(iv) Float for each activity, i.e. the amount of time by which the completion of a non-critical activity can be delaying without delaying the total project completion time.


Critical Path Analysis


Critical Path Analysis For calculating the earliest occurrence and latest allowable times of events, the following two methods are there

1. Forward pass method We use forward and backward pas in CPM / PERT to find Earliest and Latest occurrence times of events.

• Forward pass calculations are used to find Earliest occurrence times of events (Ei). Forward pass calculations are from Left to Right. Start value is equal to zero. (Earliest occurrence time of 1st event = 0).

• When two or more activities merge in an event, the maximum value is taken as the Earliest occurrence time for that event.

• Forward pass time= Earliest time of Tail event + Activity time


Critical Path Analysis 2. Backward pass method

Backward pass calculations are from Right to Left. Backward pass calculations are used to find Latest occurrence times of events (Li). For the last event, Latest time = Earliest time.

If there is more than one activity coming back in an event, in backward pass we take minimum value.

Backward pass time= Latest time of Head event – Activity time


Critical Path Analysis 1. Forward Pass Method (For Earliest Event Time)


Critical Path Analysis 2. Backward Pass Method (For Latest Allowable Event Time)


Critical Path Analysis Float (slack) of an activity and Event


Critical Path Analysis Float (slack) of an activity and Event






Example :1


Example :1 conti…


Solution : 11. Forward Pass method


Solution : 1 cont…2. Network Diagram


Solution : 1 cont..3. Backward Pass Method


Solution : 1 cont..4. Total And Free Float


Example :2


Solution : 2


Solution : 2 cont...


Solution : 2 cont...


Project Scheduling With Uncertain Activity Times

PERT was developed to handle projects where the time duration for each activity is not known with certainty but is a random variable that is characterized by β Beta distribution. To estimate the parameters : mean and variance, of the β – distribution three time estimates for each activity are required to calculate its expected completion time.

The three time estimates that required are as under :

Optimistic time (t0 or a) The shortest possible time (duration) in which an activity, can be performed assuming that everything goes well.

Pessimistic time (tp or b) The longest possible time required to perform an activity under extremely bad conditions. However, such conditions do not include natural calamities like earthquakes, flood, etc.

Most likely time ( tm or m) The time that would occur most often to complete an activity, if the activity was repeated under exactly the same conditions many times. Obviously, it is the completion time that would occur most frequently (I.e. model value).


Project Scheduling With Uncertain Activity TimesThe β – distribution is not necessarily symmetric, the degree of skewness depends on the location of tm to t0 and tp.

The range of optimistic time t0 and pessimistic time tp is assumed to enclose every possible duration of the activity. The most likely completion time for an activity may not be equal to the midpoint (t0 + tp )/2 and may occur to its left or to its right.


Project Scheduling With Uncertain Activity TimesIn Beta Distribution the midpoint (t0 + tp )/2 is given half way weightage than that of most likely point tm . Thus, the expected or mean ( te or µ ) time of an activity, that is also the weighted average of three estimates, is computed as arithmetic mean of ( t0 + tp ) /2 and 2tm .

If the duration of activities associated with project is uncertain, then variance describes the dispersion (variation) in the activity time values.

The calculations are based on the concept of normal distribution where 99% of area under normal curve falls within + 3σ from the mean or fall within the range approximately 6 standard deviation in length.

Therefore, the interval (t0, tp ) or range (tp - t0) is assumed to enclose about 6 standard deviations of a symmetric distribution.

Thus if σ1 is the standard deviation of the duration of activity i





Estimation of Project Completion Time


Project Scheduling With Uncertain Activity TimesExample : 1


Project Scheduling With Uncertain Activity TimesSolution:





E values and L values are shown in below figure :

c :The critical path is : 1 -4 -7 and expected completion time for the project is 42.8 weeks.



D


Extra SumExample - 1


Extra SumSolution - 1


Production Scheduling (Job Sequencing)


Introduction• The optimal order (sequence) shows the minimum time in which jobs,

equipment, people, materials, facilities and all other resources are arranged to support the production schedules to give low costs and high utilization.

• Other objective of calculating optimal production schedule are minimizing customers waiting time for a product or service, meeting promised delivery dates, keeping stock levels low providing preferred working pattern and so on.

• The general scheduling or sequencing problem may be described as: Let there be n jobs to be performed, one at a time, on each of m machines. The sequence (order) of the machines in which each job should be performed is given. The actual or expected time required by the jobs on each of the machines is also given. The general sequencing problem, therefore, is to find the sequence out of (n !)m possible sequences, which minimize the total elapsed time between the start of the job in the first machine and the completion of the last job on the last machine.


Introduction• In particular, if there are n=3 jobs to be performed and m=3 machines are

to be used, then the total number of possible sequences will be (3 !)3 =216. Theoretically, it may be possible to find the optimum sequence but this would require a lot of computational time. Thus, one should adopt the sequencing technique.

• To find the optimum sequence, we first need to calculate the total elapsed time for each of the possible sequences. As stated earlier, even if the values of m and n are very small, it is difficult to get the desired sequence with the total minimum elapsed time. However, due to certain rules designed by Johnson, the task of determining an optimum sequence has become quite easy.


Notations, Terminology and Assumptions• Notations :

tij = Processing time (time required) for job i on machine j.T = Total elapsed time for processing all the jobs. This includes idle time, if any.Iij= Idle time on machine j from the end of job (i - 1) to the start of job i.

• Terminology:Number of Machines: The number of machines refer to the number o f service facilities through which a job must pass before it is assumed to be completed.

Processing Time : This is the time required by a job on each machine.

Processing Order : This refers to the order(sequence) in which machines are required for completing the job.


Notations, Terminology and AssumptionsIdle Time on a Machine : This is the time during which a machine does not have a job to process, i.e. idle time from the end of job (i - 1) to the start of job i.

Total Elapsed Time : This is the time interval between starting the first job and completing the last job, including the idle time (if any), in a particular order by the given set of machines.

No Passing Rule :This refers to the rule of maintaining the order in which jobs are to be processed on given machines. for example, if n jobs are to be processed on two machines, M1 and M2 in the order M1M2, then each job should go first to machine M1 and then to M2.


Notations, Terminology and AssumptionsAssumptions:1. The processing time on different machines are exactly known and are

independent of the order of the jobs in which they are to be processed.

2. The time taken by the job in moving from one machine to another is negligible.

3. Once a job has begun on a machine, it must be completed before another job can begin on the same machine.

4. All jobs are known and are ready for processing before the period under consideration begins.

5. Only one job can be processed on a given machine at a time.

6. Machines to be used are of different types.

7. The order of completion of jobs are independent of the sequence of jobs.


Processing n Jobs Through Two Machines Example -1

Solution -1


Processing n Jobs Through Two Machines



Solution -2


Processing n Jobs Through Two Machines



Solution -3


Processing n Jobs Through Three Machines

Example -4

Processing n Jobs Through Three Machines


Processing n Jobs Through Three Machines Solution -4


Processing n Jobs Through m Machines

Example -5

Processing n Jobs Through m Machines


Processing n Jobs Through m Machines Solution - 5


Processing n Jobs Through m Machines Solution – 5 cont..


Processing n Jobs Through m Machines Example -6

Solution - 6


Processing n Jobs Through m Machines Solution – 6 cont..


Processing n Jobs Through m Machines Example - 7


Processing n Jobs Through m Machines Solution – 7


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OR PERT CPM AND JOB SCHEDULING