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

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.

Today, the mostly management techniques which are in practice: Linear ProgrammingDecision TreesExponential SmoothingDiscounted Cash Flow Analysis

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)

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

1-LAGRANGIAN MULTIPLIER TECHNIQUE

• With the aid of these two functions the Lagrangian function , L(y1,y2,λ) formulated as follows:

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.

EXAMPLE-1

Solution:The Lagrange function is formulated as follows

Taking the partial derivative with respect to y1,y2 and λ results in

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) .

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

This can be subject to

Where

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.

Example 2 (cont.…)

EXAMPLE-3

Solution:

(1)

(2)

(3)

(4)

(5)

(6)

Plot Equation (1) to (6) as:

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.

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.

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)

Simple Interest (Cont.…)

The total amount of money, Mt, after the specified lent or borrowed period is given by

(2)

Example: 4

Solution:(1)

(2)

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.

• To calculate compound amount, Mck, the resulting formula is developed as follows:

Example:5

Solution:

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

Example # 6

Solution:

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

• 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.

Example # 07

Solution:

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:

Example # 8

Given Data: (From Example 7)

Solution:

Substitute the data in given equation:

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

Example # 9

Solution:In this example following values are given:

Substitute these values in equation:

Example 9 Cont.

Thus decision to purchase the machine will be profitable investment.

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

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

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:

• The annual depreciation charge, DC(M), at the end of year M is given by

Example # 10

Solution:From example statement

Substitute in equation

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.

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

Example # 11

Solution:

Substituting the values in equation:

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)

3. Sum Of Digits Depreciation Method (cont…)

The sum of denominator of equation is given by:

(2)

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:

Example # 12

Solution:Substituting values in equation:

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.

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

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:

Business Operations Analysis (Cont.…)

• By Equating equations (1) and (2)we get

Solving for Z we get

The investment rate ,Ir ,is defined by

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

Example # 13

Solution:

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.

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.

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

Forecasting Techniques (Cont..)Forecasting techniques can be categorized into the following two groups

1) Empirical techniques2) Analytical techniques

Forecasting Techniques (Cont..)

1. Empirical Techniques

These techniques use input data coming from subjective judgments.

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.

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.

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.

2. Analytical TechniquesThese are those techniques which make use of

mathematics to obtained desired results.

Analytical Techniques(MODELS)

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

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.

Example # 14

Solution:

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.

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

Four Period Moving Average

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.

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

Example # 15

Solution:Substitute given values in equation:

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.

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.

Example # 16

Solution:

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.

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)

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

Model 4:Exponential Smoothing (Cont.…)

• Figure shows various equal time periods

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)

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)

Model 4:Exponential Smoothing (Cont.…)

Substitute equation (3) into (2)

(4)Substitute (4) into (1)

Model 4:Exponential Smoothing (Cont.…)

• The above equation leads to the following generalized equation

(5)

Where k is the number of past periods.

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.

Example # 17

Solution:(1)

(1)

Solution (cont…)

Solution(cont…)(5)

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.

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

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

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.

Example # 18

Solution:Decision Tree for the specified data:

Solution (cont…)

From decision tree the paths 1 to 6 probabilities are

Solution(cont…)

Solution(cont…)

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.

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

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

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

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

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

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

Example # 19

Solution: (Fault Tree)

Solution(cont…)

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.

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)

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)

Example # 20

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.

Quantitative Analysis Of Fault Trees (Cont.…)

2. AND GATE• An “m” independent input fault event AND gate is

shown as

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

Example # 21

Solution:

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

Example # 22

Solution:

Solution(cont…) :

Solution(cont…):

Solution(cont…)

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