Chapter#10 Techniques for making better engineering management decisions
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CHAPTER#10TECHNIQUES FOR MAKING BETTER ENGINEERING MANAGEMENT DECISIONS
Lecture No. 08 Course: Engineering Management
MED DEPARTMENT, U.E.T TAXILA
COURSE INSTRUCTOR : PROF. DR. SHAHAB KHUSHNOOD
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INTRODUCTION
• During middle of 1940s, a significant growth of quantitative management techniques occurred.
• The increase in the use of computers and the modern problem complexities have enhanced the importance of many of these techniques.
For Example:• Linear Programming.• Non-Linear Programming.
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Today, the mostly management techniques which are in practice: Linear ProgrammingDecision TreesExponential SmoothingDiscounted Cash Flow Analysis
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Optimization Techniques
There are two widely known optimization techniques known as:1. Lagrangian Multiplier Technique
(Technique of undetermined Multipliers)2. Linear Programming Technique
(Advanced form of simplex method technique)
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1-LAGRANGIAN MULTIPLIER TECHNIQUE
• This technique is demonstrated for a two variable function.
• However on similar lines it can be extended for n variables.
• Assume that we have defined function f(y1,y2)
Subject to the constraint function k(y1,y2)=0
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1-LAGRANGIAN MULTIPLIER TECHNIQUE
• With the aid of these two functions the Lagrangian function , L(y1,y2,λ) formulated as follows:
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1-LAGRANGIAN MULTIPLIER TECHNIQUE
• Subject to the following necessary conditions for estimating a relative maximum or minimum value:
• Expressions for y1,y2 and λ can be obtained by solving the simultaneous equations.
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EXAMPLE-1
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Solution:The Lagrange function is formulated as follows
Taking the partial derivative with respect to y1,y2 and λ results in
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2y1+2λ = 0
2y2- λ=0
2y1-y2=0
SoY1=8/5
Y2=-8/10
λ =-8/5Thus the critical point of f, subjected to specified condition, is (8/5,-8/10) .
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2-LINEAR PROGRAMMING
• In real life situations the objective and constraint functions will be rather complex.
• However the simplest form of linear programming problem formulation may be expressed as follows
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This can be subject to
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Where
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Example 2For above equations following values for symbols are defined
Write down the resulting equations by assuming that the objective function is to be maximized.
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Example 2 (cont.…)
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EXAMPLE-3
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Solution:
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(1)
(2)
(3)
(4)
(5)
(6)
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Plot Equation (1) to (6) as:
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From Plot of equations 1-6:Company profit is optimum at point F.
(F point satisfies all the constraints)F point shows maximum profit.The optimum values of Z1 and Z2 at point F are 6.5 and 2. substitute these values in equation.1.
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Discounted Cash Flow Analysis
• In various engineering investment decisions, the time value of money plays an important role.
• Therefore it is necessary for the engineers to have some knowledge of engineering economics.
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Discounted cash flow analysis
The basics of engineering economics are
1. Simple Interest• This is the interest which is calculated on the original sum of money,
called the original principal, for the period in which the lent or borrowed sum is being utilized.
• The simple interest, St ,is given by:
(1)WhereM is the principal amount lent or borrowedi is the interest rate per period ( this is normally a year)k is the interest periods (these are usually years)
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Simple Interest (Cont.…)
The total amount of money, Mt, after the specified lent or borrowed period is given by
(2)
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Example: 4
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Solution:(1)
(2)
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2. Compound Interest
• In this case at the end of each equal specified period, the earned interest is added to the original principal or amount lent or borrowed at the beginning of that period
• Thus, this new principal, or amount, acts as a principal for the next period and the process continues.
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• To calculate compound amount, Mck, the resulting formula is developed as follows:
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Example:5
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Solution:
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3.Present Worth• The present worth of a single payment is given
by:
• In simple terms, this formula is used to obtained the present worth, M, of money, Mck, after k periods, discounted at the periodic interest rate of i.
• Sometime above equation can be written as
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Example # 6
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Solution:
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Formula For Uniform Periodic Payments• For using formula it is assumed that at the end of
each of the K period or years, the depositor adds D amount of money. The money is invested at the interest rate i, compounded annually or periodically.
• Thus the total amount of money(1)
where
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• It should be noted in this equation that the D amount of money is first time deposited at the end of first year or period.
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Example # 07
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Solution:
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By substituting k=3, D=2000 and i= 0.15 in equation (1):
From above equation the total amount of money after three year period is:
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Example # 8
Given Data: (From Example 7)
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Solution:
Substitute the data in given equation:
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PRESENT VALUE OF UNIFORM PERIODIC PAYMENTS In present value of uniform periodic payment we wish to
find the present value of uniform periodic payments after K periods or years instead of total amount.
Thus the present worth, PW, of uniform periodic payments is given by
Multiply both side with (1+i)-1,The resulting present worth formula
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Example # 9
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Solution:In this example following values are given:
Substitute these values in equation:
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Example 9 Cont.
Thus decision to purchase the machine will be profitable investment.
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Depreciation Techniques• The term “Depreciation” means a decline in
value.• In order to take into consideration the change
in the value of the product, the depreciation charges are made during the useful life of the engineering products.
• The three depreciation techniques are:1- Declining-Balance Depreciation Method
2- Straight-line Depreciation Method
3- Sum Of Digits Depreciation Method
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Depreciation Techniques (Cont..)1. Declining-Balance Depreciation Method• This method dictates the accelerated write-off of
the product worth in its early productive years and corresponding lower write-off near the end of useful life years.
• The depreciation rate αd is given by
where
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The declining balance techniques requires a positive value of s. The product book value, Vbv (M), at the end of year M is given by:
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• The annual depreciation charge, DC(M), at the end of year M is given by
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Example # 10
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Solution:From example statement
Substitute in equation
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Depreciation Techniques (Cont..)
2. Straight-line Depreciation Method• This is the most widely used technique and is very
easy to apply.• In this method it is assumed that the annual
depreciation is constant during the productive life of the product.
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Straight-line Depreciation Method (Cont..)
• Thus the annual depreciation charge is given by
• The book value of the product at the end of year M is
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Example # 11
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Solution:
Substituting the values in equation:
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Depreciation Techniques (Cont..)3. Sum Of Digits Depreciation Method• The name of the technique is obtained from the calculation
approach.• For this technique, in the initial years the depreciation
charge is larger than in the final years of the product useful life.
• The annual depreciation charge,DCa,is given by
(1)
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3. Sum Of Digits Depreciation Method (cont…)
The sum of denominator of equation is given by:
(2)
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3. Sum Of Digits Depreciation Method (cont…)
Then substitute equation (2) in equation (1)
The product book value at the end of year M is:
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Example # 12
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Solution:Substituting values in equation:
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Business Operations Analysis• Business operations analysis is concerned
with finding the:Point of maximum profit.Point of maximum investment rate. Point of maximum economic production
when the cost of production is described by the parabolic equation.
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Business Operations Analysis (Cont.…)• We assume that the cost of production is defined by
WhereC is the cost of production Z is the number of units producedΑ,β,θ are the constants
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Business Operations Analysis (Cont.…)Differentiating above equation with respect to Z yields:
(1)
The unit cost, Cu , is given by:
(2)From this equation the quantity of units to be produced to maximize profit is given by:
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Business Operations Analysis (Cont.…)
• By Equating equations (1) and (2)we get
Solving for Z we get
The investment rate ,Ir ,is defined by
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Business Operations Analysis (Cont.…)To find the maximum economic production, we have ( from REF [7])
Where r denote the lowest acceptable rate of return.Since
Solving for Z, the maximum economic production quantity is given by
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Example # 13
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Solution:
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Forecasting• An engineering manager makes many decisions
which deal with some point in the future.• Therefore in order to make such decisions
intelligently, projection into the future is necessary.• For example, when planning for the development of
new product, it would not be wise to plan without having a knowledge of the new product demand in the future.
• Forecasting is the process of estimating a future event using the past data.
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Forecasting (Cont..)When using forecasting techniques care must be given to the fact that these forecasts will be valid only if they are used according to specified conditions or the conditions under which the forecasts are made.
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Forecasting Techniques• There are various techniques which are utilized to
make product demand and other forecasts.• The forecasting technique selection may be subject
to any one or more of the following factorsData accuracyForecast development costPrediction interval lengthExpected accuracy from the forecasted resultsPast data availabilityComplexity of factors affecting operations in time to
comeAnalysis time
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Forecasting Techniques (Cont..)Forecasting techniques can be categorized into the following two groups
1) Empirical techniques2) Analytical techniques
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Forecasting Techniques (Cont..)
1. Empirical Techniques
These techniques use input data coming from subjective judgments.
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Empirical Techniques (Cont.…)a) Jury of executive opinion approach• This approach utilizes forecasts made by the
various executives of a company in question.• The consensus reached from this data is usually
reviewed by a group of senior executives or by the company president.
• In addition this approach is normally used if the cost is low as a first step in forecasting procedure.
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Empirical Techniques (Cont.…)b) Sales Force Opinion Approach• This technique is very similar to the jury of
executive opinion approach.• However the only major difference is that in
this approach the opinions of sales force personnel are used instead of those of executive personnel.
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Empirical Techniques (Cont.…)c) Customer’s Expectation Technique• The input data for this technique is obtained from
the users of the company product or service.• This information is used to develop the forecast.• This technique is very practical when
The number of users is small.A small number of big companies dominates the
demand for the service or goods.The company needing forecasting information is
small in size and the resources required to use other forecasting approaches are not within reach.
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2. Analytical TechniquesThese are those techniques which make use of
mathematics to obtained desired results.
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Analytical Techniques(MODELS)
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2. Analytical Techniques (Cont.…)
Model 1: Simple Average• In this method the demands of all earlier periods
are given equal weight.• Thus the simple average, As, of past data is given
below
Wheredi denotes the demand for the ith past period
k denotes the total number of all the past demand periods
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2. Analytical Techniques (Cont.…)
• Model 1: Simple Average (Cont..)
The advantage of this model is that the effects of randomness are minimized because all the past periods data are used in the calculation.
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Example # 14
Solution:
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Model 2:Simple Moving Average
• In this model the data from several recent past periods are averaged and used as a forecast for the next period.
• The number of recent past-periods data to be used in computing the moving average has to be decided by the forecaster and then held constant throughout.
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Model 2:Simple Moving Average (Cont.…)
• In this situation, the average “moves” over time because the oldest period demand data is discarded and the newest period demand data is included in the calculation of forecast for the next period.
• This process is shown in figure for the fixed past periods.
• Figure shows that the data for only four recent past periods are used to compute moving average
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Four Period Moving Average
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Model 2:Simple Moving Average (Cont.…)
• Similarly the data for period 2,3,4 and forecast period 1 are used to calculate moving average results for the forecast period 2.
• In this situation the period 1 data is discarded and the forecast period 1 data added.
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Model 2:Simple Moving Average (Cont.…)
• Thus the g-period moving average Am is given by
• Where di denotes the ith period demand; for i=1 means the oldest period demand in the g-period, for i=g means the most recent period demand in the g-period average
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Example # 15
![Page 86: Chapter#10 Techniques for making better engineering management decisions](https://reader031.fdocuments.net/reader031/viewer/2022020319/56813588550346895d9cea95/html5/thumbnails/86.jpg)
Solution:Substitute given values in equation:
![Page 87: Chapter#10 Techniques for making better engineering management decisions](https://reader031.fdocuments.net/reader031/viewer/2022020319/56813588550346895d9cea95/html5/thumbnails/87.jpg)
Model 3: Weighted Moving Average• This model is similar to model 2.• However in this model the forecaster assigns a
weight to each demand of past periods in question.
• This way the forecaster can assign weights to past g-period demands according to his desire instead of having equal weight for all the past period demands.
![Page 88: Chapter#10 Techniques for making better engineering management decisions](https://reader031.fdocuments.net/reader031/viewer/2022020319/56813588550346895d9cea95/html5/thumbnails/88.jpg)
Model 3:Weighted Moving Average• The Weighted moving average, Awm is given by
Where• Wi denotes the weight associated with ith period
demand di; for i=1 means the weight associated with the oldest period demand, for i=g means the weight associated with the most recent period demand.
![Page 89: Chapter#10 Techniques for making better engineering management decisions](https://reader031.fdocuments.net/reader031/viewer/2022020319/56813588550346895d9cea95/html5/thumbnails/89.jpg)
Example # 16
![Page 90: Chapter#10 Techniques for making better engineering management decisions](https://reader031.fdocuments.net/reader031/viewer/2022020319/56813588550346895d9cea95/html5/thumbnails/90.jpg)
Solution:
![Page 91: Chapter#10 Techniques for making better engineering management decisions](https://reader031.fdocuments.net/reader031/viewer/2022020319/56813588550346895d9cea95/html5/thumbnails/91.jpg)
Model 4: Exponential Smoothing
• This method is widely known and is frequently utilized in operations management.
• The reasons for its wide usage may be as followsAvailability in the standard computer software
packages.The requirements for data storage and
computation facilities are relatively low.• Basically exponential smoothing is an averaging
method and is useful for forecasting one period ahead.
![Page 92: Chapter#10 Techniques for making better engineering management decisions](https://reader031.fdocuments.net/reader031/viewer/2022020319/56813588550346895d9cea95/html5/thumbnails/92.jpg)
Model 4:Exponential Smoothing (Cont.…)
• In this approach, the most recent past period demands is weighted most heavily.
• For this approach the forecast,ft, for demand one period ahead is given by
(1)
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Model 4:Exponential Smoothing (Cont.…)
Wheret is the time periodft is the forecast for demand one period ahead
dt-1 is the actual demand for the most recent past periodft-1 is the demand forecast for the most recent past periodθ is the weighting factor or the smoothing constant
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Model 4:Exponential Smoothing (Cont.…)
• Figure shows various equal time periods
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Model 4:Exponential Smoothing (Cont.…)• From above equation the forecast for the period
just ending is given by
(2)
Where ft-2 is the forecast for the time period (t-2)dt-2 is the actual demand for the time period (t-2)
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Model 4:Exponential Smoothing (Cont.…)
Similarly, the forecast for the period (t-2) is given by
(3)Wheredt-3 is the actual demand for the period (t-3)
ft-3 is the forecast for the time (t-3)
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Model 4:Exponential Smoothing (Cont.…)
Substitute equation (3) into (2)
(4)Substitute (4) into (1)
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Model 4:Exponential Smoothing (Cont.…)
• The above equation leads to the following generalized equation
(5)
Where k is the number of past periods.
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Model 4:Exponential Smoothing (Cont.…)
• It can be seen from this equation that the weight associated with each observation of this equation are not equal but rather the successively older observations weights decrease by factor (1- θ)
• In other words the successive terms
decrease exponentially.• The frequently used values for θ are between 0.01 and
0.3.• At θ=1 in above equation, the consumption or demand
of the last period is the forecast for the period ahead.
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Example # 17
![Page 101: Chapter#10 Techniques for making better engineering management decisions](https://reader031.fdocuments.net/reader031/viewer/2022020319/56813588550346895d9cea95/html5/thumbnails/101.jpg)
Solution:(1)
(1)
![Page 102: Chapter#10 Techniques for making better engineering management decisions](https://reader031.fdocuments.net/reader031/viewer/2022020319/56813588550346895d9cea95/html5/thumbnails/102.jpg)
Solution (cont…)
![Page 103: Chapter#10 Techniques for making better engineering management decisions](https://reader031.fdocuments.net/reader031/viewer/2022020319/56813588550346895d9cea95/html5/thumbnails/103.jpg)
Solution(cont…)(5)
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Decision Trees• The decision tree analysis is used to deal with
sequential problems.• A decision tree may be simply described as a
schematic diagram of a sequence of alternative decisions as well as the conclusions of those decisions.
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Decision Trees (Cont.…)Some of the advantages of decision tree areIt can take into consideration the actions of
more than one decision makerIt simplifies the expected value calculationsIt is a visibility tool to represent the sequential
decision process
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Decision Trees (Cont.…)• To perform the decision tree analysis there are
basically three steps involved.• These are diagramming, estimation and
evaluation and selection.Identification and sequence of decisions and
their alternatives Identification of chance eventsConstruction of the tree diagram depicting
decisions and chance events sequence
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Decision Trees (Cont.…)• The estimation aspect is concerned with
estimating the chance event occurrence probability and financial consequences of all possible outcomes.
• Finally, the evaluation and selection are concerned with computing the actions expected values and choosing the actions with the best expected value.
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Example # 18
![Page 109: Chapter#10 Techniques for making better engineering management decisions](https://reader031.fdocuments.net/reader031/viewer/2022020319/56813588550346895d9cea95/html5/thumbnails/109.jpg)
Solution:Decision Tree for the specified data:
![Page 110: Chapter#10 Techniques for making better engineering management decisions](https://reader031.fdocuments.net/reader031/viewer/2022020319/56813588550346895d9cea95/html5/thumbnails/110.jpg)
Solution (cont…)
From decision tree the paths 1 to 6 probabilities are
![Page 111: Chapter#10 Techniques for making better engineering management decisions](https://reader031.fdocuments.net/reader031/viewer/2022020319/56813588550346895d9cea95/html5/thumbnails/111.jpg)
Solution(cont…)
![Page 112: Chapter#10 Techniques for making better engineering management decisions](https://reader031.fdocuments.net/reader031/viewer/2022020319/56813588550346895d9cea95/html5/thumbnails/112.jpg)
Solution(cont…)
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Fault trees• This is a technique widely used in industry to
evaluate reliability of complex engineering systems.
• This technique is schematically resembles and name similarity to the decision tree concept.
• The fault trees analysis starts by identifying an undesirable event, called the top event associated with a system.
• Events which could cause the occurrence of a top event are generated and connected by logic gates known as AND,OR and so on.
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Fault Tree Symbology1. Circle• This is used to represent the failure of an
elementary component.• The element component failure parameters such
as constant failure and repair rates, probability and unavailability and calculated from the field data or other sources.
• This is shown as
![Page 115: Chapter#10 Techniques for making better engineering management decisions](https://reader031.fdocuments.net/reader031/viewer/2022020319/56813588550346895d9cea95/html5/thumbnails/115.jpg)
Fault Tree Symbology (Cont.…)2. Diamond• This symbol is used to represent a fault event
whose causes will not be faulty developed due to lack of interest or data.
• This is shown as
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Fault Tree Symbology (Cont.…)3. Rectangle• This symbol is used to represent an event
which occurs due to combination of fault events through the input of an gate.
• The rectangle symbol is given as
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Fault Tree Symbology (Cont.…)4. And Gate• This gate is used to represent a situation such
that an output fault event will occur only if all the input fault events occur.
• The and gate symbol is shown as
![Page 118: Chapter#10 Techniques for making better engineering management decisions](https://reader031.fdocuments.net/reader031/viewer/2022020319/56813588550346895d9cea95/html5/thumbnails/118.jpg)
Fault Tree Symbology (Cont.…)5. Or Gate• This is opposite of an AND gate.• OR GATE signifies that an output fault event occurs if
one or more of the input fault events occurs.• The OR GATE symbol is shown as
![Page 119: Chapter#10 Techniques for making better engineering management decisions](https://reader031.fdocuments.net/reader031/viewer/2022020319/56813588550346895d9cea95/html5/thumbnails/119.jpg)
Fault Tree Construction• There are various advantages of the fault tree
construction Identifies system weakness in a visible formActs as a visual tool in communicationEvaluates the system design adequacyActs as a visual tool to perform trade of
studies
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Example # 19
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Solution: (Fault Tree)
![Page 122: Chapter#10 Techniques for making better engineering management decisions](https://reader031.fdocuments.net/reader031/viewer/2022020319/56813588550346895d9cea95/html5/thumbnails/122.jpg)
Solution(cont…)
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Quantitative Analysis Of Fault Trees
• Quantitative analysis of fault trees concerned with the evaluation of probability of occurrence of top fault event.
• Basic probability theory is used to compute the probabilities of OR and AND gates top event occurrence.
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Quantitative Analysis Of Fault Trees (Cont.…)1. OR Gate:• A “K” independent input fault event OR gate is shown.
• The occurrence probability of event ET is given by
(1)
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Quantitative Analysis Of Fault Trees (Cont.…)
P(ET) is the occurrence probability of event ETK is the number of OR gate input eventsPj is the occurrence probability of jth input fault event; for j=1,2,3,……..k
• However, if the input event probabilities are very small, then above equation will be
(2)
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Example # 20
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Solution:From equation (1)
From equation (2)
For both equations the result is almost same. The probability of OR gate top event occurrence is approximately 0.07.
![Page 128: Chapter#10 Techniques for making better engineering management decisions](https://reader031.fdocuments.net/reader031/viewer/2022020319/56813588550346895d9cea95/html5/thumbnails/128.jpg)
Quantitative Analysis Of Fault Trees (Cont.…)
2. AND GATE• An “m” independent input fault event AND gate is
shown as
![Page 129: Chapter#10 Techniques for making better engineering management decisions](https://reader031.fdocuments.net/reader031/viewer/2022020319/56813588550346895d9cea95/html5/thumbnails/129.jpg)
2. AND GATE (cont…)• Probability of occurrence of top event, ET, is given
by
WhereP(ET) is the occurrence probability of top event ETm is the number of AND gate input eventsPj is the occurrence probability of jth input fault event; for j=1,2,3,……..K
![Page 130: Chapter#10 Techniques for making better engineering management decisions](https://reader031.fdocuments.net/reader031/viewer/2022020319/56813588550346895d9cea95/html5/thumbnails/130.jpg)
Example # 21
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Solution:
![Page 132: Chapter#10 Techniques for making better engineering management decisions](https://reader031.fdocuments.net/reader031/viewer/2022020319/56813588550346895d9cea95/html5/thumbnails/132.jpg)
Fault Tree Evaluation• Fault tree evaluation is concerned with
calculating the probability of occurrence of a fault tree top event
• The AND and OR gate probability evaluation equations are used to evaluate the occurrence probability of the fault top tree event.
• The fault tree top event probability of occurrence evaluation is demonstrated in the following example
![Page 133: Chapter#10 Techniques for making better engineering management decisions](https://reader031.fdocuments.net/reader031/viewer/2022020319/56813588550346895d9cea95/html5/thumbnails/133.jpg)
Example # 22
![Page 134: Chapter#10 Techniques for making better engineering management decisions](https://reader031.fdocuments.net/reader031/viewer/2022020319/56813588550346895d9cea95/html5/thumbnails/134.jpg)
Solution:
![Page 135: Chapter#10 Techniques for making better engineering management decisions](https://reader031.fdocuments.net/reader031/viewer/2022020319/56813588550346895d9cea95/html5/thumbnails/135.jpg)
Solution(cont…) :
![Page 136: Chapter#10 Techniques for making better engineering management decisions](https://reader031.fdocuments.net/reader031/viewer/2022020319/56813588550346895d9cea95/html5/thumbnails/136.jpg)
Solution(cont…):
![Page 137: Chapter#10 Techniques for making better engineering management decisions](https://reader031.fdocuments.net/reader031/viewer/2022020319/56813588550346895d9cea95/html5/thumbnails/137.jpg)
Solution(cont…)
![Page 138: Chapter#10 Techniques for making better engineering management decisions](https://reader031.fdocuments.net/reader031/viewer/2022020319/56813588550346895d9cea95/html5/thumbnails/138.jpg)
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