sk-mb0048 (set-1)

SMU - MBA - MB0048 – Operation Research Semester: 2 - Assignment Set: 1

MBA SEMESTER IIMB0048 –Operation Research- 4 Credits

(Book ID: B1137)Assignment Set- 1 (60 Marks)

Q1: Outline the broad features of the Judgment phase and Research phase of the scientific method in OR. Discuss in detail any of these phases.

Important features of OR are:

i. It is System oriented: OR studies the problem from over all points of view of organizations or

situations since optimum result of one part of the system may not be optimum for some other

part.

ii. It imbibes Inter – disciplinary team approach. Since no single individual can have a

thorough knowledge of all fast developing scientific knowhow, personalities from different

scientific and managerial cadre form a team to solve the problem.

iii. It makes use of Scientific methods to solve problems.

iv. OR increases the effectiveness of a management Decision making ability.

v. It makes use of computer to solve large and complex problems.

vi. It gives Quantitative solution.

vii. It considers the human factors also.

The scientific method in OR study generally involves the following three phases:

i) Judgment Phase: This phase consists of

a) Determination of the operation.

b) Establishment of the objectives and values related to the operation.

c) Determination of the suitable measures of effectiveness and

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d) Formulation of the problems relative to the objectives.

ii) Research Phase: This phase utilizes

a) Operations and data collection for a better understanding of the problems.

b) Formulation of hypothesis and model.

c) Observation and experimentation to test the hypothesis on the basis of additional data.

d) Analysis of the available information and verification of the hypothesis using pre-established

measure of effectiveness.

e) Prediction of various results and consideration of alternative methods.

iii) Action Phase:

It consists of making recommendations for the decision process by those

who first posed the problem for consideration or by anyone in a position to make a decision,

influencing the operation in which the problem is occurred.

Q2: Operation Research is an aid for the executive in making his decisions by providing him the needed quantitative information, based on scientific method analysis. Discuss.

Operation Research is a scientific method of providing executive departments with a

quantitative basis for decisions regarding the operations under their control. Morse &

Kimball Operations research is a scientific approach to problem solving for executive

management. – H.M. Wagner Operations research is an aid for the executive in making

these decisions by providing him with the needed quantitative information based on the

scientific method of analysis. The mission of Operations Research is to serve the entire

Operations Research (OR) community, including practitioners, researchers, educators, and

students. Operations Research, as the flagship journal of our profession, strives to publish

results that are truly insightful. Each issue of Operations Research attempts to provide a

balance of well-written articles that span the wide array of creative activities in OR. Thus,

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the major criteria for acceptance of a paper in Operations Research are that the paper is

important to more than a small subset of the OR community, contains important insights,

and makes a substantial contribution to the field that will stand the test of time. Operational

research, also known as operations research, is an interdisciplinary branch of applied

mathematics and formal science that uses advanced analytical methods such as

mathematical modeling, statistical analysis, and mathematical optimization to arrive at

optimal or near-optimal solutions to complex decision-making problems.

It is often concerned with determining the maximum (of profit, performance, or

yield) or minimum (of loss, risk, or cost) of some real-world objective. Originating in

military efforts before World War II, its techniques have grown to concern problems in a

variety of industries. Operational research, also known as OR, is an interdisciplinary branch

of applied mathematics and formal science that uses advanced analytical methods such as

mathematical modeling, statistical analysis, and mathematical optimization to arrive at

optimal or near-optimal solutions to complex decision-making problems. It is often

concerned with determining the maximum (of profit, performance, or yield) or minimum (of

loss, risk, or cost) of some real world objective.

Originating in military efforts before World War II, its techniques have grown to

concern problems in a variety of industries. Operational research encompasses a wide range

of problem-solving techniques and methods applied in the pursuit of improved decision-

making and efficiency. Some of the tools used by operational researchers are statistics,

optimization, probability theory, queuing theory, game theory, graph theory, decision

analysis, mathematical modeling and simulation.

Because of the computational nature of these fields, OR also has strong ties to

computer science. Operational researchers faced with a new problem must determine which

of these techniques are most appropriate given the nature of the system, the goals for

improvement, and constraints on time and computing power. Work in operational research

and management science may be characterized as one of three categories:

Fundamental or foundational work takes place in three mathematical disciplines:

probability, optimization, and dynamical systems theory. Modeling work is concerned with

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the construction of models, analyzing them mathematically, implementing them on

computers, solving them using software tools, and assessing their effectiveness with data.

This level is mainly instrumental, and driven mainly by statistics and econometrics.

Application work in operational research, like other engineering and economics' disciplines,

attempts to use models to make a practical impact on real-world problems.

The major sub disciplines in modern operational research, as identified

by the journal Operations Research, are:

• Computing and information technologies

• Decision analysis

• Environment, energy, and natural resources

• Financial engineering

• Manufacturing, service sciences, and supply chain management

• Policy modeling and public sector work

• Revenue management

• Simulation

• Stochastic models

• Transportation

Q3: A furniture manufacturer makes two products: chairs and tables. Processing of these products is done on two machines A and B. A chair requires 2 hours on machine A and 6 hours on machine B. A table requires 5 hours on machine A and no time on machine B. There are 16 hours per day available on machine A and 30 hours on machine B. Profit gained by the manufacturer from a chair and a table is Rs 2 and Rs 10, respectively. What should be the daily production of each of the two products?

Let X1 be the number of chairs to be manufactured

And X2 be the number of tables to be manufactured.

Let Z=Profit earned (objective function which is to be maximized).

Z is to be maximized under the following constraints.

Z=2 X1+10 X2

2 X1+5 X2 ≤ 16 (Availability constraint of machine A).

6 X1 ≤30 (Availability constraint of machine B).

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X1 ≥ 0.

X2 ≥ 0.

Solving graphically,

The first constraint 2 X1+5 X2≤16, written in a form of equation

2 X1+5 X2=16

Put X1=0, then X2=16/5=3.2

Put X2=0, than X1=8

The coordinates are (0,3.2) and (8,0)

The second constraint 6 X1+0 X2≤30, written in a form of equation

6 X1=30 – > X2=5

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The corner points of fesible regions are A, B and C. So the coordinates for the corner points are A (0,3.2)

B(5,1.2) (Solve the two equations 2 X1+5 X2=16 and X1=5 to get the coordinates) C= (5,0)

We know that Max Z=2 X1+10 X2

At A (0,3.2)

Z-2(0)+10(3.2)=32

At B (5,1.2)

Z=2(5)+10(1.2)=22

At C (5,0)

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Z=2(5)+10(0)=10

we get :

Maximum Z=32, X1=0, X2=3.2.

The manufacturer should produce approximately 3 tables and no chair to get the maximum profit.

Q4: Given a general linear programming problem, explain how you would test whether a basic feasible solution is an optimal solution or not. How would you proceed to change the basic feasible solution in case it is not optimal?

Answer:Linear programming (LP) is a mathematical method for determining a way to achieve the

best outcome (such as maximum profit or lowest cost) in a given mathematical model for some list of requirements represented as linear relationships.

More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear inequality constraints. Given a polytope and a real-valued affine function defined on this polytope, a linear programming method will find a point on the polytope where this function has the smallest (or largest) value if such point exists, by searching through the polytope vertices.

Linear programming is a considerable field of optimization for several reasons. Many practical problems in operations research can be expressed as linear programming problems. Certain

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special cases of linear programming such as network flow problems and multicommodity flow problems are considered important enough to have generated much research on specialized algorithms for their solution LP problems as sub-problems. Historically, ideas from linear programming have inspired many of the central concepts of optimization theory, such as duality, decomposition, and the importance of convexity and its generalizations. Likewise, linear programming is heavily used in microeconomics and company management, such as planning, production, transportation, technology and other issues. Although the modern management issues are ever-changing, most companies would like to maximize profits or minimize costs with limited resources. Therefore, many issues can be characterized as linear programming problems.

With this introduction, we now give a fairly formal definition of the class of problem we are going to study.

Definition Suppose that one is given a linear (strictly an affine) function of n real variables

z = ƒ (x1, x2, … xn) = c1x1 + c2x2 + …. + cnxn + dand a set of linear inequalities and/or equations, called constraints

a11x1+a12x2+ …. + a1nxn ≤ or = or ≥ b1,a21x1+a22x2+ …. + a2nxn ≤ or = or ≥ b2, (1.1)

am1x1+am2x2+ …. + amnxn ≤ or = or ≥ bmn,

Where in each line either, ≤ or ≥ occurs. The problem of finding x in Rn, where x = (x1, x2,…,

xn), that satisfies the constraints (1.1) and makes z a maximum (or minimum) is called a Linear Programming Problem. We saw in Section 1.1 that, if it is convenient, we can always restrict attention to just having “≤” signs, at the expense of having more constraitns.

We shall assume that every Linear Programming Problem has included in its constraints, the non-negativity restrictions

Xj ≥ 0 for j = 1,2, …. n.And these will be written separately from the other constraints. We will see, in Section 4.5 that this does not in fact limit the class of problems that can be handled by the methods to be discussed. These non-negativity constraints are sometimes known as reality constraints. In examples they typically represent quantities that for physical reasons are non-negative.

Any x satisfying the constraints (1.1) and inequalities (1.2) is called a feasible solution. The set of feasible solutions is the feasible region. Somewhat perversely, any x satisfying the constraints (1.1) but not (1.2) is called a non-feasible solution.

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The function ƒ is called the objective function and z the objective variable. If x is a feasible

solution that makes ƒ (x1,….,xn) a maximum (or minimum) then x is an optimal solution and the corresponding value z is the output value.

Finally a convention; an element x in Rn will be treated as a row vector or as a colum vector

according to the context.

Q5. State and discuss the methods for solving an assignment problem. How is Hungarian method better than other methods for solving an assignment problem?

Answer: In the world of trade Business Organization are confronting the conflicting need for

optimal utilization of their limited resources among competing activities. When the information available on resources and relationship between variables is known we can use LP very reliably. The course of action chosen will invariably lead to optimal or nearly optimal results. The problems which gained much importance under LP are: Transportation problems

Assignment problems

The assignment problem is a special case of transportation problem in which the objective is to assign a number of origins to the equal number of destinations at the minimum cost (or maximum profit). Assignment problem is one of the special cases of the transportation problem. It involves assignment of people to projects, jobs to machines, workers to jobs and teachers to classes etc., while minimizing the total assignment costs. One of the important characteristics of assignment problem is that only one job (or worker) is assigned to one machine (or project). Hence the number of sources are equal the number of destinations and each requirement and capacity values is exactly one unit.

Although assignment problem can be solved using the techniques of Linear Programming or the transportation method, the assignment method is much faster and efficient. This method was developed by D.Konig, a Hungarian mathematician and is therefore known as the Hungarian method of assignment problem. In order to use this method, one needs to know only the cost of making all the possible assignments. Each assignment problem has a matrix (table) associated with it. Normally, the objects (or people) one wishes to assign are expressed in rows, whereas the columns represent the tasks (or things) assigned to them. The number in the table would then be the costs associated with each particular assignment.

It may be noted that the assignment problem is a variation of transportation problem with two characteristics

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i) The cost matrix is a square matrix, and ii) The optimum solution for the problem would be such that there would be only one

assignment in a row or column of the cost matrix.

Mathematical Statement of Problem

An assignment problem is a special type of linear programming problem where the objective is to minimize the cost or time of completing a number of jobs by a number of persons. Furthermore, the structure of an assignment problem is identical to that of a transportation problem.

Application Areas of Assignment Problem –Though assignment problem finds applicability in various diverse business situations, we discuss some of its main application areas –

i) In assigning machines to factory ordersii) In assigning sales/marketing people to sales territories.iii) In assigning contracts to bidders by systematic bid evaluationiv) In assigning teachers to classesv) In assigning accountants to accounts of the clients

Hungarian method

Algorithm for Solving

There are various ways to solve assignment problems. Certainly it can be formulated as a linear program (as we saw above), and the simplex method can be used to solve it. In addition, since it can be formulated as a network problem, the network simplex method may solve it quickly. However, sometimes the simplex method is inefficient for assignment problems (particularly problems with a high degree of degeneracy). The Hungarian Algorithm developed by Kuhn has been used with a good deal of success on these problems and is summarized as follows –

1) Determine the cost table from the given problema. If the no. of sources is equal to no. of destinations, go to Step 3.b. If the no. of sources is not equal to the no. of destination, go to step 2.

2) Add a dummy source or dummy destination, so that the cost table becomes a square matrix. The cost entries of the dummy source/destinations are always zero.

3) Locate the smallest element in each row of the given cost matrix and then subtract the same from each element of the row.

4) In the reduced matrix obtained in the step 3, locate the smallest element of each column and then subtract the same from each element of that column. Each column and row no have at least one zero.

5) In the modified matrix obtained in the step 4, search for the optimal assignment as follows:

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a. Examine the rows successively until a row with single zero is found. Enrectangle this row ( ) and cross off (X) all other zeros in its column. Continue in this manner until all the rows have been taken care of.

b. Repeat the procedure for each column of the reduced matrix.c. If a row and/or column has two or more zeros and one cannot be chosen by

inspection then assign arbitrary any one of the zeros and cross off all other zeros of that row/column.

d. Repeat (a) through (c) above successively until the chain of assigning ( ) or cross (X) ends.

6) If the number of assignment ( ) is equal to n (the order of the cost matrix), and optimum solution is reached. If the number of assignment is less than n (the order of the matrix), go to the next step.

7) Draw the minimum number of horizontal and/or vertical lines to cover all zeros of the reduced matrix.

8) Develop the new revised cost matrix as follows:a. Find the smallest element of the reduced matrix not covered by any of the lines.b. Subtract this element from all uncovered elements and add the same to all the

elements laying at the intersection of any two lines.9) Go to step 6 and repeat the procedure until and optimum solution is attained.

Q6: Compare and contrast CPM and PERT. Under what conditions would you recommend scheduling by PERT? Justify your answer with reasons.

Program Evaluation and Review Technique (PERT) and Critical Path Method (CPM) are tools widely used in project scheduling. Both are based on network diagrams applicable for both the planning and control aspects of production. Visual display of the network enhances the communication and highlights the interdependency of the various activities required for project completion. Perhaps the greatest contribution of these tools is the identification of sequentially time-critical activities that require the closest monitoring.

BACKGROUND

In the early 1900s the Gantt chart was widely hailed as the reason that ships were built in record time. Developed by an engineer named Henry Gantt, this horizontal bar chart shows the

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scheduled times for individual jobs to be accomplished by specific resources. However, this tool is static in nature, and requires frequent manual updating, especially when activities are sequentially dependent.

In Figure 1, the Gantt chart shows the prospective times for five activities in a project, but does not show an underlying dependency of Activity D on the completion of Activity B.

In the 1950s, two groups independently developed what has become known as the PERT/CPM method of project scheduling. Each of these techniques improved on the Gantt chart by building into the tool the explicit sequencing of activities.

PERT was developed by the U.S. Navy, the Lockheed Corporation, and the consulting firm of Booz, Allen and Hamilton to facilitate the Polaris missile project. As time was a primary issue, this technique used statistical techniques to assess the probability of finishing the project within a given period of time.

By contrast, CPM was created in the environment of industrial projects, where costs were a major factor. In addition to the identification of the time-critical path of activities, representatives from the Du Pont Company and Sperry-Rand Corporation also developed a time-cost tradeoff analysis mechanism called crashing.

These two tools differ in the network diagram display. PERT historically uses the activity-on-arrow (AOA) convention, while CPM uses activity-on-node (AON). For most purposes, these two conventions are interchangeable; however some propriety software requires the logic of a specific convention. Both forms of network diagrams use arrows (lines implying direction) and nodes (circles or rectangles) to define the set of project activities or tasks. The flow of logic is from left to right. To simplify the diagram, letters are frequently used to represent individual activities. Figures 2 and 3 illustrate the differences for the same simple project.

Figure 2 illustrates the AOA convention, in which arrows depict activity requiring time and resources. The node represents an event, which requires neither time nor resources; this event is actually recognition that prior tasks are completed and the following tasks can begin. While the length of the arrow is not necessarily related to the duration of the task, there may be a tendency on the part of the analyst to sketch longer arrows for longer activities. To maintain the integrity of the network, there may be need for a dummy activity, as it is not acceptable to have two tasks that share the same beginning and ending nodes.

In Figure 3, the AON uses nodes to represent activities. The arrows have no implication of time, used only to indicate sequential flow. Since the AOA convention requires the use of dummy activities, the simpler AON convention will be used here to illustrate an example.

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USING CPM TO SCHEDULE AND CONTROL A PROJECT

Scheduling is an important part of the planning of any project. However, it is first necessary to develop a list of all the activities required, as listed in the work breakdown structure. Activities require both time and the use of resources. Typically, the list of activities is compiled with duration estimates and immediate predecessors.

To illustrate the use of CPM, we can imagine a simple cookie-baking project: the recipe provides the complete statement of work, from which the work breakdown structure can be developed. The resources available for this project are two cooks and one oven with limited capacity; the raw materials are the ingredients to be used in preparing the cookie dough. As listed in Table 1, the activities take a total of 80 minutes of resource time. Because some activities can run parallel, the cooks should complete the project in less than 80 minutes.

Table 1 displays some of the planning that will save time in the project. For example, once the oven is turned on, it heats itself, freeing the cooks to perform other activities. After the dough is mixed, both batches of cookies can be shaped; the shaping of the second batch does not have to wait until the first batch is complete. If both cooks are available, they can divide the dough in half and each cook can shape one batch in the same four-minute period. However, if the second cook is not available at this time, the project is not delayed because shaping of the second batch need not be completed until the first batch exits the oven.

Description of Activity Duration (minutes) Immediate Predecessor(s)

A. Preheat oven 15 minutes —

B. Assemble, measure ingredients 8 minutes —

C. Mix dough 2 minutes B

D. Shape first batch 4 minutes C

E. Bake first batch 12 minutes A, D

F. Cool first batch 10 minutes E

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Description of Activity Duration (minutes) Immediate Predecessor(s)

G. Shape second batch 4 minutes C

H. Bake second batch 12 minutes E, G

I. Cool second batch 10 minutes H

J. Store cookies 3 minutes F, I

Total time 80 minutes

Some expertise is required in the planning stage, as inexperienced cooks may not recognize the independence of the oven in heating or the divisibility of the dough for shaping. The concept of concurrent engineering makes the planning stage even more important, as enhanced expertise is needed to address which stages of the project can overlap, and how far this overlap can extend.

After beginning the project at 8:00 A.M., the first batch of dough is ready to go into the oven at 8:14, but the project cannot proceed until the oven is fully heated—at 8:15. The cooks actually have a one-minute cushion, called slack time. If measuring, mixing, or shaping actually takes one additional minute, this will not delay the completion time of the overall project.

Figure 4 illustrates the network diagram associated with the cookie-baking project. The set of paths through the system traces every possible route from each beginning activity to each ending activity. In this simple project, one can explicitly define all the paths through the system in minutes as follows:

A-E-F-J = 15 + 12 + 10 + 3 = 40A-E-H-I-J = 15 + 12 + 12 + 10 + 3 = 52B-C-D-E-F-J = 8 + 2 + 4 + 12 + 10 + 3 = 39B-C-D-E-H-I-J = 8 + 2 + 4 + 12 + 12 + 10 + 3 = 51B-C-G-H-I-J = 8 + 2 + 4 + 12 + 10 + 3 = 39

The critical path is the longest path through the system, defining the minimum completion time for the overall project. The critical path in this project is A-E-H-I-J, determining

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that the project can be completed in 52 minutes (less than the 80-minute total of resource-usage time). These five activities must be done in sequence, and there is apparently no way to shorten these times. Note that this critical path is not dependent on the number of activities, but is rather dependent on the total time for a specific sequence of activities.

The managerial importance of this critical path is that any delay to the activities on this path will delay the project completion time, currently anticipated as 8:52 A.M. It is important to monitor this critical set of activities to prevent the missed due-date of the project. If the oven takes 16 minutes to heat (instead of the predicted 15 minutes), the project manager needs to anticipate how to get the project back on schedule. One suggestion is to bring in a fan (another resource) to speed the cooling process of the second batch of cookies; another is to split the storage process into first- and second-batch components.

Other paths tend to require less monitoring, as these sets of activities have slack, or a cushion, in which activities may be accelerated or delayed without penalty. Total slack for a given path is defined as the difference in the critical path time and the time for the given path. For example, the total slack for B-C-G-H-I-J is 13 minutes (52–39 minutes). And the slack for B-C-D-E-H-I-J is only one minute (52–51), making this path near critical. Since these paths share some of the critical path activities, it is obvious that the manager should look at the slack available to individual activities.

Table 2 illustrates the calculation of slack for individual activities. For projects more complex than the simplistic cookie project, this is the method used to identify the critical path, as those activities with zero slack time are critical path activities. The determination of early-start and early-finish times use a forward pass through the system to investigate how early in the project each activity could start and end, given the dependency on other activities.

Activity Early Start Early Finish Late Start Late Finish Slack

A 8:00 8:15 8:00 8:15 0

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Activity Early Start Early Finish Late Start Late Finish Slack

B 8:00 8:08 8:01 8:09 1

C 8:08 8:10 8:09 8:11 1

D 8:10 8:14 8:11 8:15 1

E 8:15 8:27 8:15 8:27 0

F 8:27 8:37 8:39 8:49 12

G 8:10 8:14 8:23 8:27 13

H 8:27 8:39 8:27 8:39 0

I 8:39 8:49 8:39 8:49 0

J 8:49 8:52 8:49 8:52 0

The late-time calculations use the finish time from the forward pass (8:52 A.M.) and employ a backward pass to determine at what time each activity must start to provide each subsequent activity with sufficient time to stay on track.

Slack for the individual activities is calculated by taking the difference between the late-start and early-start times (or, alternatively, between the late-finish and early-finish times) for each activity. If the difference is zero, then there is no slack; the activity is totally defined as to its time-position in the project and must therefore be a critical path activity. For other activities, the slack defines the flexibility in start times, but only assuming that no other activity on the path is delayed.

CPM was designed to address time-cost trade-offs, such as the use of the fan to speed the cooling process. Such crashing of a project requires that the project manager perform contingency planning early in the project to identify potential problems and solutions and the costs associated with employing extra resources. Cost-benefit analysis should be used to compare the missed due-date penalty, the availability and cost of the fan, and the effect of the fan on the required quality of the cookies.

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This project ends with the successful delivery of the cookies to storage, which brings two questions to mind: First, should the oven be turned off? The answer to this depends on the scheduling of the oven resource at the end of this project. It might be impractical to cool the oven at this point if a following project is depending on the heating process to have been maintained. Second, who cleans up the kitchen? Project due dates are often frustrated by failure to take the closeout stages into account.

USING PERT TO SCHEDULE AND CONTROL A PROJECT

In repetitive projects, or in projects employing well-known processes, the duration of a given activity may be estimated with relative confidence. In less familiar territory, however, it may be more appropriate to forecast a range of possible times for activity duration. Using the same cookie-baking project example, Figure 4 still accurately represents the sequencing of activities.

Table 3 illustrates the project with three time estimates for each activity. While m represents the most likely time for the activity, a suggests the optimistic estimate and b is the pessimistic estimate. The estimated time and or standard deviation for each activity (E) are calculated from the formula for the flexible beta distribution. With a reasonably large number of activities, summing the means tends to approximate a normal distribution, and statistical estimates of probability can be applied.The mean is calculated as [(a + 4m + b) ÷6], an average heavily weighted toward the most likely time, m. The standard deviation for an activity is [(b − a) ÷6], or one-sixth of the range. Managers with a basic understanding of statistics may relate this to the concept of the standard deviation in the normal distribution. Since ±3 standard deviations comprise almost the entire area under the normal curve, and then there is an intuitive comparison between a beta standard deviation and the normal standard deviation.

Using these new estimates for activity duration, the activity paths through the system have not changed, but the estimates of total time (T) are as follows:

A-E-F-J = 40.66 minutesA-E-H-I-J = 53 minutesB-C-D-E-F-J = 40.66 minutesB-C-D-E-H-I-J = 53 minutesB-C-G-H-I-J = 40.66 minutes

There are two factors that should be considered coincidental to the comparison of PERT and CPM in the example. First, there are two critical paths of T = 53 minutes each in the PERT

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analysis. Second, all the other paths have the same duration of T = 40.66 minutes. These concepts are neither more nor less likely to happen under PERT as opposed to CPM; they are strictly a function of the numbers in the estimates. However, the serendipity of two critical paths allows us to address the issue of which would be considered the more important of the two.In Table 4, each of the critical paths is considered. Relevant to this analysis is the sum of the variances on the critical path; note that summing variances

Duration (minutes)

Description of Activity a m b Et Vt St

A. Preheat oven 12 15 18 15.00 1 1

B. Assemble, measure ingredients 6 8 12 8.33 1 1

C. Mix dough 2 2 2 2.00 0 0

D. Shape first batch 3 4 9 4.67 1 1

E. Bake first batch 10 12 16 12.33 1 1

F. Cool first batch 5 10 11 9.33 1 1

G. Shape second batch 3 4 9 4.67 1 1

H. Bake second batch 10 12 16 12.33 1 1

I. Cool second batch 5 10 11 9.33 1 1

J. Store cookies 2 3 10 4.00 1.78 1.33

Total times 58 90 114

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Duration (minutes)

Path = A–E–H–I–JDescription of Activity

a m b Et Vt St

A. Preheat oven 12 15 18 15.00 1 1




J. Store cookies 2 3 10 4.00 1.78 1.33

Total variance 5.78

Standard deviation 2.40

Duration (minutes)

Path = B–C–D–E–H–I–JDescription of Activity

a m b Et Vt St

B. Assemble, measure ingredients

6 8 12 8.33 1 1

C. Mix dough 2 2 2 2.00 0 0

D. Shape first batch 3 4 9 4.67 1 1



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Duration (minutes)

Path = A–E–H–I–JDescription of Activity

a m b Et Vt St


J. Store cookies 2 3 10 4.00 1.78 1.33

Total variance 6.78

Standard deviation 2.60

is mathematically valid, while summing standard deviations is not. Path A-E-H-I-J has a total variance of 5.78 minutes, while path B-C-D-E-H-I-J has a variance of 6.78. Thus, path B-C-D-E-H-I-J, with the larger variance, is considered the riskier of the two paths and should be the primary concern of the project manager. We assign the entire project a variance of 6.78 minutes, and the standard deviation (the square root of the project variance) is 2.60 minutes.

Armed with this project standard deviation, the next step is to estimate the probability of finishing the project within a defined period. Applying the critical path time of 53 minutes to the normal distribution, the probability of finishing in exactly T = 53 minutes is 50/50. The relevant formula for calculating the number of standard normal distributions is as follows:

where T = total time of the critical path (T = 53)S = standard deviation of the project (S = 2.60)C = arbitrary time for end of project

If C = 9:00 a.m., then Z = [(9:00 − 8:53) ÷2.60] = 7 ÷2.60 = 2.69 standard normal deviations. Referring to a cumulative standard normal table, we find that Z = 0.99632, or a 99.632 percent chance of finishing by 9:00 A.M.

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If C = 8:50 A.M., then Z = [(8:50 − 8:53) ÷2.60] =−3 ÷2.60 = −1.15. In this case, we use (1 − table value) for the probability = 1 − 0.87493 = 0.1251, or a 12.51 percent chance of finishing 3 minutes earlier than predicted.

From a managerial viewpoint, it should be reiterated that there is only a 50/50 chance of completing the project within the sum of the activity-time estimates on the critical path (T). This perspective is not emphasized in the CPM analysis, but is likely relevant in that context also. Adding a buffer to the promised due date (where C > T) enhances the probability that the project will be completed as promised.

There may be competitive advantages to bidding a project on the basis of a nearer-term completion date (where C < T), but managers can assess the risks involved using PERT analysis. In the cookie example, there may be a promised delivery time riding on this project estimate, or the resources (cooks and oven) may be promised to other projects. By using PERT, managers can allocate the resources on a more informed basis.Both PERT and CPM rely heavily on time estimates, as derived from local experts, to determine the overall project time. While the estimating process may intimidate local managers, this may suffice to produce an estimate that becomes a fait accompli, as managers strive to meet the goal rather than explain why they failed to do so.

These two project management tools, frequently used together, can assist the project manager in establishing contract dates for project completion, in estimating the risks and costs of contingencies, and in monitoring project progress. Many commercial software packages exist to support the project manager in tracking both costs and time incurred to date through-out the project duration.

The Framework for PERT and CPM

Essentially, there are six steps which are common to both the techniques. The procedure is listed below:

I. Define the Project and all of its significant activities or tasks. The Project (made up of several tasks) should have only a single start activity and a single finish activity.

II. Develop the relationships among the activities. Decide which activities must precede and which must follow others.

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III. Draw the "Network" connecting all the activities. Each Activity should have unique event numbers. Dummy arrows are used where required to avoid giving the same numbering to two activities.

IV. Assign time and/or cost estimates to each activity

V. Compute the longest time path through the network. This is called the critical path.

I. Use the Network to help plan, schedule, monitor and control the project.

The Key Concept used by CPM/PERT is that a small set of activities, which make up the longest path through the activity network control the entire project. If these "critical" activities could be identified and assigned to responsible persons, management resources could be optimally used by concentrating on the few activities which determine the fate of the entire project.

Non-critical activities can be replanned, rescheduled and resources for them can be reallocated flexibly, without affecting the whole project.

Five useful questions to ask when preparing an activity network are:

Is this a Start Activity? Is this a Finish Activity?

What Activity Precedes this?

What Activity Follows this?

What Activity is Concurrent with this?

Drawing the CPM/PERT Network

Each activity (or sub-project) in a PERT/CPM Network is represented by an arrow symbol. Each activity is preceded and succeeded by an event, represented as a circle and numbered.

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.

At Event 3, we have to evaluate two predecessor activities - Activity 1-3 and Activity 2-3, both of which are predecessor activities. Activity 1-3 gives us an Earliest Start of 3 weeks at Event 3. However, Activity 2-3 also has to be completed before Event 3 can begin. Along this route, the Earliest Start would be 4+0=4. The rule is to take the longer (bigger) of the two Earliest Starts. So the Earliest Start at event 3 is 4.

Similarly, at Event 4, we find we have to evaluate two predecessor activities - Activity 2-4 and Activity 3-4. Along Activity 2-4, the Earliest Start at Event 4 would be 10 wks, but along Activity 3-4, the Earliest Start at Event 4 would be 11 wks. Since 11 wks is larger than 10 wks, we select it as the Earliest Start at Event 4. We have now found the longest path through the network. It will take 11 weeks along activities 1-2, 2-3 and 3-4. This is the Critical Path.

The PERT (Probabilistic) Approach

So far we have talked about projects, where there is high certainty about the outcomes of activities. In other words, the cause-effect logic is well known. This is particularly the case in engineering projects.

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However, in Research & Development projects, or in Social Projects which are defined as "Process Projects", where learning is an important outcome, the cause-effect relationship is not so well established.

In such situations, the PERT approach is useful, because it can accommodate the variation in event completion times, based on an expert’s or an expert committee’s estimates.

For each activity, three time estimates are taken

The Most Optimistic The Most Likely

The Most Pessimistic

The Duration of an activity is calculated using the following formula:

Where te is the Expected time, to is the Optimistic time, tm is the most probable activity time and tp is the Pessimistic time.

It is not necessary to go into the theory behind the formula. It is enough to know that the weights are based on an approximation of the Beta distribution.

The Standard Deviation, which is a good measure of the variability of each activity is calculated by the rather simplified formula:

The Variance is the Square of the Standard Deviation.

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